La
367
34
UC-NRLF
SEM
L. P. SHIDY
FORMULAS AND TABLES
ARCHITECTS AND ENGINEERS
CALCULATING THE STRAINS AND CAPACITY
OP
STRUCTUKES IN IRON AND WOOD,
BY
F. SCHUMANN, C. E.
ILLUSTRATED WITH MORE THAN THREE HUNDRED DIAGRAMS, DESiGSEL .uSi>
ENGRAVED ESPECIALLY FOR THIS WORK BY J. C. LYONS.
WASHINGTON CITY:
WARREN CHOATE & CO.
1873.
Entered according to Act of Congress, in the year 1873, by
F. SCHUMANN,
In the Office of the Librarian of Congress, at Washington.
IN ME?/!Of?IAM
KL
STEREOTYPED BY
tt OILL A WITHEROW,
WASHINGTON, D. C.
THIS VOLUME
is
RESPECTFULLY DEDICATED
TO
A. B. MULLETT,
gUPERVISING ARCHITECT OF THE U. 8. TREASURY DEPARTMENT,
BY THE AUTHOR.
(iii)
CONTENTS.
EEEATA.
On page 4, 10th line from bottom, read - - instead of 30.
On page 4, 10th line from bottom, read 10.0036 instead of
10.036.
a
On page 4, 14th, 15th, and 16th lines from bottom, read -
instead of a.
On page 32, Fig. 70, insert I = distance between supports.
On page 34, Fiq. 72, insert I = distance between supports.
On page 34, Fig. 74, insert I = length of beam.
On pages 38 and 39 w = total weight of beam between supports.
On page 39, 5th line from top, read 1099000 instead of 1000000.
On page 39, 5th line from top, read 1754 instead of 1757.
On pages 144 and 145, in formulas for H n , change places of last
minus sign with foregoing plus sign. (See 13th line from top.)
Page 145, lines 1 to 7 from bottom, ~| Change places of Cand T
Page 146, lines 1 to 3 from top, > under strains in Figs.
Page 146, lines 13 to 22 from top, J 225, 226, 227, and 228.
On page 149, 1st line from bottom, read -- instead
//,
of-
N
On page 197, 7th line from bottom, read 3.14159 instead of
1.14159.
On page 204, 1st line from bottom, read A-\- A, instead ofAA,.
Static and moving loads on bridges of wrought iron.,,... 192, 193
(vii)
CONTENTS.
PAGES.
Summary of definitions and general principles 1-5
^Moments of inertia and resistance of various sections.... 5-25
Strength of materials, &c 26-29
Resistance to cross-breaking and shearing 29
Capacity and strength of beams 29-99
Pressure on supports 100-102
Compressive strains and pressure on supports 102
Sloping beams, rafters, &c 102-103
Resistance to crushing 103
Strength of columns, pillars, and struts 103-110
Parallelogram of forces Ill
Strains in frames 112-114
Strains in boom derricks 114-115
Strains in trusses.... 115-121
Strains in trussed beams 122-125
Strains in trusses with parallel booms 126-146
Strains in parabolic curved trusses 147
"Bow-string girders" 147-153
Capacity and strength of parabolic arched beams or ribs
originally curved 153, 154
Strains in a polygonal frame 154, 155
Strains in roof trusses 156-178
Pressure of wind on roofs 178, 179
Pressure of snow on roofs 180
Tie rods and bars 181, 183
Joints or connections in iron constructions 184-186
Dimensions of bolts and nuts 187, 188
Compound strain in horizontal and sloping beams 188-190
Weight of moving loads 191
Static and moving loads on bridges of wrought iron 192, 193
(vii)
Vlll CONTENTS.
MISCELLANEOUS.
PAGES.
Geometry 197-201
Center of gravity of planes 202-204
Trigonometrical formulas 205
Trigonometrical functions 206-217
Circumference, area, and cubic contents of circles 218-223
Specific gravities of materials 224-226
Weight of a superficial inch of wrought and cast iron... 227
Weight per square foot of metals 228
Weight of a lineal foot of flat and square bar iron 229-233
Weight of a lineal foot of rolled round iron 234
Weight of bolts, nuts, and heads 235-237
Weight of materials used in building 238 1
Divisions of a foot expressed in equivalent decimals 239
Table for comparing measures and weights of different
countries 240, 241
To cut the strongest and stiffest beam from a log 242
FORMULAS AND TABLES
FOR
ARCHITECTS AND ENGINEERS.
Summary of Definitions and General Principles.
EXTERNAL FORCES are those forces (loads, &c.) which cause or
tend to cause the rupture of a structure.
INTERNAL FORCES are those forces which resist the external
forces; when one balances the other, the structure is said to pos
sess Stability
EXTERNAL FOKCKS. INTERNAL FORCES.
Compressive strain. Resistance to Compression.
Tensional strain. Resistance to Tension.
Shearing strain. Resistance to Shearing.
Cross-breaking strain. Resistance to Cross-breaking.
COMPRESSION causes the material to fail by crushing or buck
ling, or both.
RESISTANCE to direct Crushing: In case pillars, blocks, struts,
or rods, along which the strains act, are not so long in propor
tion to their diameter as to have a tendency to give way by
bending sideways. This includes
Stone and brick pillars and blocks, of ordinary proportions;
Pillars, struts, and rods of cast iron, in which the length is
not more than five times the diameter, approximately;
Pillars, struts, and rods of wrought iron, in which the length
is not more than ten times the diameter, approximately ;
Pillars, struts, and rods of dry timber, in which the length is
not more than twenty times the diameter.
Let W == Crushing load in Ibs.
C= Ultimate resistance of material to crushing in
Ibs. per square inch.
A = Sectional area of pillar, &c., in square inches.
Then will TP = A X C; and A = -^-
C
TENSION, causes the material to be torn asunder.
(1)
AND GENERAL PRINCIPLES.
Resistance of bars, &<?., to teaiing: the ultimate strength of a
bir (co :te.fcring) is : whssa
T= Ultimate resistance of the material to tearing, in
Ibs. per square inch.
W= Tearing load in Ibs.
A = Sectional area of bar, in square inches.
W
Then will TF= A X T; and A =
SHEARING causes the fibres of the material to be shorn by each
other ; when a bolt pulls out of its nut, the threads of the screw
are said to be sheared.
Resistance of bars, bolts, &c., when sheared at one place, is:
when
S = Ultimate resistance, of material to shearing, in
Ibs. per square inch.
W= Shearing load in Ibs.
A = Sectional area of bar, &c., in square inches.
W
Then will W= A X 8; and A = -~
o
CROSS-BREAKING a beam, &c., supported at one or both ends,
with a force at right angles to its length, sufficient to cause rup
ture, is said to be broken across.
Resistance to cross-breaking is the resistance of the material
to compression, tension, and shearing combined ; .
The flanges or booms, in beams or trusses, resist the bending
moment, or moment of rupture.
The web or braces, in beams or trusses, resist the shearing
forces.
CAPACITY means the load or pressure a structure is capable of
sustaining with safety.
DEFLECTION is the displacement of a beam from a horizontal,
caused by its own weight or the applied load, or both.
CAMBER is given a beam to counter balance the deflection, so
that the beam may be horizontal when loaded ; the camber should
be equal to the computed deflection.
To find the effect of combining several loads on one beam, whose
separate actions are known: for each cross section, the shearing
force is the sum of the shearing forces, and the bending moment
the sum of the bending moments, which the loads would produce
separately.
When a load on a structure is partly static and partly moving,
multiply each part of the load by its proper factor of safety, and
DEFINITIONS AND GENERAL PRINCIPLES.
add together the products : the sum will be the load to which the
structure is to be adapted.
For a Bridge with two platforms, one carrying a road and the
other a railway, those two loads are to be combined.
Of Iron Ties, Struts, and Beams.
In designing ordinary structures of wrought iron, it saves time
and expense to use iron bars of such forms of cross section as are
usually to be met with in the market. An engineer should
avoid introducing new sections for bars into his designs, except
when, by so doing, some important purpose is to be served, or
some decided advantage to be gained. The most common forms
of rolled bars is shown by the following enumerated figures :
No. of
figure.
Name of Form.
Applicable for
4
Square iron
Ties.
13
Round iron
Ties, bolts, and rivets.
2
Flat iron
Ties.
29
I or double T-iron
Beams rafter* and struts
30
Channel iron .
Rafters and struts.
37
T-iron
Rafters and struts
47
L or angle iron
Various purposes.
1
Deck Beam
Beams and rafters
Avoid the use of cast iron for ties, also trussed cast-iron beams.
When a member of a structure acts alternately as a strut and
as a tie, it must have sufficient total sectional area, and sufficient
stiffness, to resist the greatest compressive strain that can act, and
sufficient effective sectional area to resist the greatest tensional
strain which can act along it.
Let all pins, bolts, rivets, &c., exposed to a shearing strain,
fit tight in its hole or socket.
Roof trusses, the divisions of a rafter, and also the struts, may
be considered as hinged at the ends.
In members under a compound strain, as for instance a trussed
beam with a distributed load, be careful to take into account the
additional compression, caused by the bending moment.
The best distribution of the material in a section of a cast-iron
T s Q
beam, for cross-breaking, is that = ; or = -
s s / s T
When T= Ultimate resistance of the material to tension.
C= Ultimate resistance of the material to compression.
s Distance from neutral axis to most extended fibres.
s, = Distance from neutral axis to most compressed fibres.
That is, the fibres most in tension should be nearest the neutral
axis of beam.
DEFINITIONS AND GENERAL PRINCIPLES.
In wrought-iron beams, the section may be made alike above
and below the neutral axis.
THE MODULUS OF RUPTURE should be applied to beams with
full section, or beams with continuous web only ; for all open web
beams, or beams with very thin web, the resistance of the mate
rial to crushing or tearing, respectively, must be used instead.
EXPANSION AND CONTRACTION of long beams, which arise from
the changes of atmospheric temperature, are usually provided for
by supporting one end of each beam on rollers of steel or hard
ened cast iron. The following table shows the proportions in
which the length of a bar of certain materials is increased by an
elevation of temperature from the melting point of ice (32 Fahr.,
or Centigrade) to the boiling point of water under the mean
atmospheric pressure, (212 Fahr., or 100 Cent.;) that is, by an
elevation of 180 Fahr., or 100 Cent,:
METALS.
Brass 0.00216
Bronze 0.00181
Copper 0.00184
Cast iron 0.00111
Wrought iron 0.00120
Tin 0.00225
Zino 0.00294
Lead 0.00290
EARTHY MATERIALS.
Brick, common 0.00355
Brick, fire 0.00050
Cement 0.00140
Glass, average 0.00090
Granite 0.00085
Marble 0.00087
Sandstone 0.00105
Slate 0.00104
Reference.
Let u Value given in above table, for a certain material.
I Length of a bar at Centigrade,
and Z x its length at a given number of degrees Centigrade.
a Given number of degrees, at which I, is required.
A = Superficial area of a plate ;
and A, its area at a given number of C.
B = Cubic contents of a body,
and .#,= its contents at a given number of C.
Then will I, = I (1 +au);
A, = A(l + 2au) ,
B / = B (1 -j- 3 a u).
Example : A bar of wrought iron 2 inches square, is 10 feet
long at a temperature of Centigrade ; what is its length at an
increased temperature of 30 ?
Ans : I, = 10 (1 + 30 X 0.00120) = 10.036 feet.
THE NEUTRAL Axis, in a cross section of a beam, is that layer of
fibres which are neither in compression or tension, by the action
of a load on the beam ; it always passes through the centre of
gravity of the section : provided the limits of elasticity of the
material is not exceeded. A beam supported at both ends, with
a load acting perpendicular between the supports, will cause the
fibres above the neutral axis to be compressed, and those below,
extended: the farther from the fibres to the neutral axis, the
greater the stress.
MOMENTS OF INERTIA AND RESISTANCE. 5
Under MOMENT OF INERTIA of a cross section, may be under
stood : an internal force at rest ; a static force resisting an exter
nal force; it means the sum of all the area elements, multiplied
by the square of their perpendicular heights from the neutral
axis of the section. The moment of inertia, which we have
denoted with I, depends on the form and dimensions of the cross
section, and is expressed in square inches.
MOMENT OF RESISTANCE of a cross section is that static force
resisting an external force of compression or tension ; it is equal to
the moment of Inertia divided by the distance of the most ex
tended or compressed fibres, respectively, from the neutral axis.
MOMENTS OF INERTIA AND RESISTANCE OF VARIOUS
SECTIONS.
To find the moment of inertia of any given cross section
FIRST. Divide the section into as many simple figures as possi
ble. (See diagram, fig. 1.)
SECOND. Find the moment of inertia of each of the simple figures
about its own neutral axis, and insert the value in the following
formula :
Reference.
Letters A, A /t A //t = area of simple figure, respectively; and
d, d /t d //t = its distance from its centre of gravity
to that of the whole section.
i Vi V/ = moment of inertia of simple figures, re
spectively.
For neutral axis see centre of gravity.
Fig. 1. B y
Formula.
1= (i + dU) + (v + d,*A,) +
(i // -{- djfAji) + <fcc., = moment
of inertia of whole section.
MOMENTS OF INERTIA I AND MOMENTS OF RESISTANCE
I
Reference.
m n = neutral axis of section.
r = radius.
s = distance from neutral axis to most compressed or
extended fibres.
6, h, &c. = dimensions.
A = area.
MOMENTS OF INERTIA AND RESISTANCE.
No. of Figure.
Form of Section.
2 and 3
It
Tfo
LJL7I
JHfr
m
n
771
MOMENTS OF INEETIA AND EESISTANCE.
Moment of Inertia 7.
Moment of Resistance-
= T v Ah*
bh*
h*
6
12
h* -
Qh
12
1/2
MOMENTS OF INERTIA AND RESISTANCE.
No. of Section.
VI.
No. of Figure,
Form of Section.
VII.
VIII.
10
IX.
11
12
MOMENTS OF INERTIA AND RESISTANCE.
Moment of Inertia /.
Moment of Resistance -
A
A
A 1
A
jly 6^3 =
10
MOMENTS OF INERTIA AND RESISTANCE.
No. of Section. No. of Figure.
XI.
13
Form of Section.
XII.
14
XIII.
XIV.
15
16
XV.
17 and 18
MOMENTS OF INERTIA AND RESISTANCE.
11
Moment of Inertia /.
Moment of Resistance JL
} TT r = J Ar
J
i*
s = 0.576/fc = (1 A_ ) h
12
MOMENTS OF INERTIA AND RESISTANCE.
No. of Section. No. of Figure.
XVI.
19
Form of Section.
XVII.
20
XVIII.
21
XIX.
XX.
22
23
TTU
- -
MOMENTS OF INERTIA AND RESISTANCE. 13
Moment of Inertia /.
Moment of Resistance
- bh* = & Alt
15 - 10
MOMENTS OF INERTIA AND RESISTANCE.
No. of Section.
No. of Figure.
Form of Section.
XXI.
24
XXII.
XXIII.
25
26
? /*
/
XXIV.
27
XXV.
28, 29, and 30
MOMENTS OF INERTIA AND RESISTANCE. 15
Moment of Inertia /.
Moment of Resistance -
I A [i V cos*u + Jf A 2 sin*v]
A [A 2 cos 2 ?; + V siri*v]
12
I
*/,
16
MOMENTS OF INERTIA AND RESISTANCE.
No. of Section.
No. of Figure.
Form of Section.
XXVI.
31
XXVII.
XVIII.
32
33
fel kM
XXIX.
34
XXX.
35
MOMENTS OF INERTIA AND RESISTANCE. 17
Moment of Inertia /.
12
Moment of Resistance .
b (h* - A/)
~~~~
1
~o/r
18-
MOMENTS OF INERTIA AND RESISTANCE.
No. of Section.
XXXI.
XXXII.
No. of Figure.
36 and 37
38
Form of Section.
FF^li
. V \ . A .
XXXIII.
XXXIV.
XXXV.
39
41
-^~"~^j3i
MOMENTS OF INERTIA AND RESISTANCE. 19
Moment of Inertia /.
Moment of Resistance -
A ( bh * + W)
6/1
A (*6 + &,*)
hb* +
66
A l(**S 3625 A
A t^/ 4
6 s &/
(A Z)) 4 s ] 0.049 Id*
IT
20
MOMENTS OF INERTIA AND RESISTANCE.
No. of Section.
Nj. of Figure.
Form of Section.
XXXVI.
42
XXXVII.
XXXVIII.
XXXIX.
XL.
44
45
46, 47, and 48
MOMENTS OF INERTIA AND RESISTANCE.
21
Moment of Inertia I.
Moment of Resistance -
s
- V)
A
__
1*7
12(M" 6,
22
MOMENTS OF INERTIA AND RESISTANCE.
No. of Section
No. of Figure.
Form of Section.
XLI.
49, 50, and 51
XL1I.
XLIII.
52
53
-i
XLIV.
XLV.
55
MOMENTS OF INERTIA AND KESISTANCE.
23
Moment of Inertia /.
(bh* - 6 A 2 ) 2 - *&/*&/
12
^ r* \X3~0.5413r 4
Moment of Resistance
(bh*- b^/) *- 4bhb / h / (h-h / ) >
= O.G381
= 0.5413 (r* r/)
= 6381 (?* r/)
MOMENTS OF INERTIA AND RESISTANCE.
No. of Section. No. of Figure.
XLVI.
XLVII.
XLVIII.
56
57
58
Form of Section.
^
T I
i
""7
- A
XLIX.
59
\ I
L.
60
MOMENTS OF INERTIA AND RESISTANCE.
25
Moment of Inertia J.
Moment of Resistance I .
s
n / = number of sides.
^j w/r 4 sin . v (2 -f- cos . v)
A n / r3 s ^ ?l v (^ 4~ cos v )
n / = number of sides.
b = length of a side.
TV ^ (3/& 2 -}~ J O 2 )
^1
i _^ (3^2 ^_ 1 12)
h
oJfi b/hf -j- b/h/fi
w _ w ; w
12
6A
2b^hh // " "
/
/
7
//
26
STRENGTH OF MATERIALS.
STRENGTH OF MATERIALS, &o. f
In H>s., avoirdupois, per square inch of cross-section.
Materials.
rt ^j
o|
"5
Ultimate Resistance to
Modulus of
elasticity.
Tearing.
Crushing.
Shearing.
Oross-br k
Modulus o
Rupture.
METALS.
Brass, ca^t, average
505.7
533
524
537
540
18000
40000
30000
10000
30000
3;ooo
00000
105(10
13400
to
20000
10300
9170000
14230000
9900000
17000000
17000000
14000000
to
22900000
29000000
25300000
15000000
29000000
to
42000000
720000
4000000
13000000
1000000
1350000
" wire.
Bronze or gun metal, (cop
per 8, tin 1)
Copper cast
117000
" she^t
" bolt**
" \viro
Iron ca^t avcnvo
445
4)54
to
450
112000
80000
to
115000
27700
" various
" beams, average..
28800
17000
33000
to
43500
38000
" solid rect. bars,
various quailities.
Iron, wrought, average
481
05000
30000
to
40000
50000
plates
joints, d ble
riveted.
Iron, wrought, joints, single
riveted.
Iron, wrought, bars and
bolts.
hoop, best best
wire
51000
35700
28000
GdOOO
to
70000
04000
70000
to
100000
00000
wire ropes....
Steel, average
490
80000
" bars
100000
to
130000
80000
3:500
4(300
7000
to
8oOO
17000
0300
11500
12000
7730
15500
15000
" plates
Lead, sheet
712
402
430
47
43
Tin, cast
Zinc
TIMBER, (well seasoned and
dry.)
Ash
9000
9300
1400
12000
to
14000
9000
to
20000
Bamboo
Beech
STRENGTH OF MATERIALS.
27
Materials.
Weight of a
cubic foot.
Ultimate resistance to
If
Tearing.
Crushing.
Shearing
Cross-br k
Modulus o
Rupture
||
TIMBER Continued.
Birch
44
80
33.4
34
74.5
37
37
33
52
47
52.5
44
62
35
49
52.5
47.4
15000
200O(
100O(
to
1300<
140iK
120CX
to
14000
12401
9QOC
to
10* -00
25000
20000
23400
1COOO
ll.XOO
8000
to
21800
10600
10000
o
19800
6400
10300
5300
10300
19-)00
5375
to
6200
5900
5570
11000
7300
9000
9900
8200
6500
10000
7700
6100
6000
5300
5400
12000
12000
11000
6500
4000
550
to
800
1100
1700
417
to
612
11701
1086
6004
to
2700
7 UK
to
9541
990,
to
1 JjOt
50. <
to
10001
1735!
1 1 ..00
1200
10000
10000
to
13(00
87 0<
10601)
9600
12000
to
17460
6600
1645000
1140000
700000
to
1340000
146 000
to
1900000
UOO 00
to
1800000
900000
to
1 .".60000
10-tUOOO
1255000
1200000
to
1750000
21 50000
2400000
Box.... ....
Chestnut
Elm
1400
Ebony, West Indian....
Fir Red Pine ..
500
to
8(iO
600
970
to
1700
" Spruce
" Larch
Hickory
Hornbeam
Lance wood
Locust
Lignum vitse
Mahogany
Maple......
2300
Oak, British
" Dantzic
" American white....
42
54
346
29
37
48
62.5
18000
10250
11500
15000
13000
15000
red
Pine, American, white
" yellow
Sycamore
Teak, Indian
Water gum
Walnut
40
25
50
125
135
37.5
loo
8000
14000
8000
280
to
300
Willow, various
Yew
STOXES, (natural and arti
ficial.)
Brick, weak red
" strong red
" fire
" work
Cement
89
280
to
300
STRENGTH OF MATERIALS.
Materials.
Weight of a
cubic foot.
Ultimate resistance to
Modulus of
elasticity.
Tearing
Crushing.
Shearing.
>oss-br k.
Jodukis ot
Rupture.
STONES Continued.
Chalk
145.5
173
168
118
9400
330
8000000
13000000
to
16000000
Glas=!
Granite
5500
2360
1100
5000
Limestone marble
172
to
11000
5500
4000
to
4500
About
4-10 cut
stone.
5500
3300
to 4400
" granular
197
100
to
170
50
109
116
Rubble masonry
Sandstone, strong ")
" ordinary (
144
" weak )
Slate
178
9600
to
12800
25000
14000
6300
4200
5200
7700
MISCELLANEOUS.
Flaxen yarn
Hempen ropes...
Hide, ox undressed ...
Leather ox
Silk fibre
Whalebone
MODULUS OF RUPTURE R.
According to Professor Rankine, the modulus of rupture is
eighteen times the weight that is required to break a bar of a
given material one inch square (section) and one foot between
supports, the weight concentrated at the middle.
MODULUS OF ELASTICITY E
Is that power (in Ibs. generally) through which a prismatic body
of a given material, of section = 1, is assumed to be extended
double its length, or compressed to 0.
Let A = Sectional area of a rod of the material.
W= Weight or power in Ibs., which causes the extension
or compression of the rod.
I = Length in inches of rod before W is applied.
Y = The extension or compression of the rod in inches,
caused by W.
Wl W
RESISTANCE TO CROSS -BREAKING AND SHEARING. 29
FACTORS OF SAFETY k.
The ultimate resistance of material should be divided by
A Wrolgift t I 1 e on nd For Proof stren S th - Foi> Working Stress.
Steady load 2
Moving load 4 to 6
Cast Iron.
Steady load 2 to 3 3 to 4
Moving load 6 to 8
Timber.
Average 3 8 to 10
RESISTANCE TO CROSS-BREAKING AND SHEARING.
CAPACITY AND STRENGTH OF BEAMS.
Reference.
A = Area of cross-section of beam.
D Deflection of beam from a horizontal.
E = Modulus of elasticity.
J= Moment of inertia of cross section.
M Maximum moment of rupture, or bending moment.
R = Modulus of rupture.
S = Vertical shearing force.
V = Pressure on supports.
W= Capacity or weight of load,
c, d, I == Dimensions in units of length.
k = Factor of safety.
w = Weight of load per unit of length.
= Moment of resistance of cross-section.
I
R I
For the stability of a beam : M=. K = . .
k s
The web of a metal beam must have sufficient area to resist the
shearing force 8; that is, A = -rrr-: : :
Ultimate resistance to shearing.
The weight of the beam must be added to W, except in small
beams, under 60 Ibs. per lineal foot, when it may be disregarded.
[NOTE. Always use the same units of dimensions or weight.]
30 RESISTANCE TO CROSS BRKAKING AND SHEARING.
No. of Figure.
Manner of loading and fastening beams.
61
. -P " " " V x
62
63
64
.
If
o
p
xim
ent
ure
W.I
-
W.
W.
5.333
ity
sec
K
I
4 K
~
5.333-
K- ------ - ------- -
.65
r "V
RESISTANCE TO CROSS BREAKING AND SHEARING.
31
Maximum deflec
tion D.
Distance from
A to point of
maximum/).
Shearing force S.
Pressure on sup
ports V.
W I 3
.EM 3
J
At any point.
W
W
W I s
E I 8
I
At any point.
w .d
W
W p
- lO ls"
z
2
At any point.
IF
~Y
V - V - W
1/1 - ^2--^
IF Z 3
"J 0.00931
J-j. J.
0.553J
1 TT.-L
".-".-?-
& W P
2
"T
At any point,
d<d / ;
(i-0
fi-F._i
8 E.I 48
32 RESISTANCE TO CROSS-BREAKING AND SHEARING:
66
68
69
Manner of loading and fastening beams.
Maximum mo
ment of rup
ture M.
\
/MM
A
^ <rl-
A
w|< \ \ 2- -
^
\-L
r*-f * i
^ -------- _ ------ ^
-
IF.
12
W .1 +
W
tion.
Capaci
any
8.
12 "T-
-.K
RESISTANCE TO CROSS BREAKING AND SHEARING.
|CQ
^"S
Maximum deflec
tion D.
11
p r
Shearing force 5.
Pressure on sup
ports F.
"DQTJH C
P
W Z 3
j
ir
W
E.I 4.48
O
^
W Z 3
00 r )4
Q. 572.1
n tt 1 " ,T\
v v w
E.I
( 8 1
W P
l
*<**
E.I 8.48
o
-(!-)
^
(Irr) +
At any point be
/ W l ^ 3 >
i
tween loads.
( y - ~ ) ~t~
^
S= W .S l =
II 1 \\ l \ n 2
/.J. )
~m~ _i_ ~nr ^
(\
W 4- w l + W z
E.I ~3")
At any point and
under any load.
_ }y 1 2
V -?- W
W /3 72/2
1 fj 6j
I
1
E.I 3 I 2 I 2
Constant bet, A & IF
J
s w ll
V 2 = 1- W
Constantbet. Z?& IF
3-1 RESISTANCE TO CROSS-BREAKING AND SHEARING.
i Maximum moment of
Manner of loading and fastening beam?, j rupture M.
A
x:.:v ^ /C>v rrK I
1 m ^ l ^
T^ !
IF./,
W.I,
When ^>^
[(4-) - . ]
When Z < Zi \/8 ;
RESISTANCE TO CROSS-BREAKING AND SHEARING.
35
^"c
Capacity W of any
section.
Maximum
deflection D.
sll
I s !
Shearing
force 8.
Pressure
on sup
ports V.
5
W
E.I
K
h
a
2
W
w 2
h
i 2
w l *
F 1= =
Kl
~~i~w
^( 1 -^)
1 1
2 Ti
1
i w
W 1 2 t l
- 8 E.I
K
Wl^
w
F! = F 2
h
D,- l
W
\ 2 3
2 (1 + ^i) g
(4) - .-
w .l or
w.-JL-
W
The greater
value to be
2
taken.
2(1+21,) K
"
36 RESISTANCE TO CROSS- BREAKING AND SHEARING.
Manner of loading and fastening beams.
Maximum moment of
rupture M.
When c? >(Z c) ;
w-
W-
78
1^ (!+,)]
RESISTANCE TO CROSS-BREAKING AND SHEARING.
37
Capacity W of any
section.
MAximum
deflection D.
S oq
lag
a|
1-1
|^l
Shearing
force 8.
Pressure
on sup
ports F".
1 K
e (l__d) A
2K
(l-r-d)*
213 E
W
-^
3 E.I
IS(1IJ
1^(31-1^(1-1^)
I 2
P K
Vff-y"
38 EESISTANCE TO CROSS-BREAKING AND SHEARING.
EXAMPLE. Capacity of wrought-iron l-shaped beams; top and
bottom flange alike ; load equally distributed ; ends not fixed.
Dimensions of Cross-section.
h = Height = 10 inches.
b = Width of flange 4 inches.
t = Thickness of flange = 0.8 inches.
t / = Thickness of web = 0.5 inches.
7t /= = h 2t; b, = b t,.
Distance between supports = 20 feet = 240 inches. Factor of
safety = 3.
MOMENT OF RESISTANCE.
_Z_ ft/* 3 M/ 3 4 x 1Q 3 3.5 X 8.4* _
V = 6* ~ 6 X 10
Capacity W.
w = (4 X 0.8 X 2 + 8.4 x 0.5) x 240 X 0.28 = 712.32 Ibs.
K=*. ^J*. 32.09 = 406473.33.
. k s 3
W=8^--w^8.~^ -- 712.32 = 12836.72 Ibs.
li ^41)
EXAMPLE. Capacity of cast-iron i-shaped beams; load equally
distributed; ends not fixed; flange down.
Dimensions of Cross- section.
Let h = Height = 18 inches.
b = Width of flange = 9 inches.
t = Thickness of flange = 1.25 inches.
t / = Thickness of web = 1 inch.
7i / = h t; by = b t/.
Area = 28 square inches. Distance between supports = 20
feet = 240 inches. Factor of safety k = 4.
MOMENT OF RESISTANCE.
sh, (h h,Y
* L bh 2 2b / hh / + b,h/
-*[-
2b / hh / +
(9 x 18 2 8 x 16.752)2
9 X 18 2 2 X 8 X 18 X 16.75 + 8 X 16.75 2
4X9X18X8X 16.75 (18 16.75) 2
9 X 18 2 2 X 8 X 18 X 16.75+8 X 16.
5) 2 n
6.75 2 J
RESISTANCE TO CROSS-BREAKING AND SHEARING. 39
_ rj452256.25_ _135675.00_-] = 15-
* L 336.5 336 5 J .
Capacity W.
w = 28 x 240 X 0.261 = 1754. lbs.
JT^;* J- = -^-. 157 = 1099000.
k s 4
JF= 8 A - = 8 . -- ~S" - 1757 = 34879 lbs -
For light beams no attention need be paid to weight of beam w.
CAPACITY If OF ROLLED I -SHAPED BEAMS.
Load equally distributed.
The calculations are based upon (lie patterns or section* used
by the Phcenixville Iron Company. Practically this applies to
all similar beams rolled in the United States, the difference in the
profile of section being slight.
In the following table the factor of safety k = 2.53:
Reference.
W= Load in tons of 2,000 lbs., equally distributed.
w = Weight of beam in tons of 2,000 lbs.
L = Distance between supports in feet.
I = Distance between supports in inches.
iu / = Weight per square foot of floor.
W,= Capacity of coupled or trebled beams in tons of 2,000 lbs.
D = Deflection in inches at centre, between supports.
d = Distance between centres of beams, when spacing for
floors, in feet.
W W, r> W+w J?
7.0 tons, d -- , or d= - -, D f . .
L.to, L.w / &.L 4b
K 1 = Constant, computed by formulas. (See under examples.)
40
RESISTANCE TO CROSS -BREAKING AND SHEARING.
The rivets for coupled or trebled beams should be about inch
in diameter, and 8 inches apart.
Trebled Beams.
Coupled Beams.
W,= WX 5.33.
:JQ ; = 17.2 tons. This is also found at the intersection of
Fig. 79.
Examples explanatory of the following Table.
EXAMPLE.- What is the capacity of a 15-inch light beam, load
equally distributed, distance between supports = 20 feet?
7-1 /f 8 1 TT7 Kl K 1
A = r^ , and W ; for 15-inch light beam ----- =
-Li J_j
345. 19
20 ~
20 feet and column under capacity W.
EXAMPLE. What distance apart should 9-inch medium beams
be placed, the distance between supports being 20 feet, and to
carry a total load of 140 Ibs. per square foot of floor surface?
Ans. 4.4 feet; being found at the intersection of the horizontal
line from 20 feet and the vertical column under 140 Ibs.
EXAMPLE.- What is the capacity of 12- inch light beams trebled .
load equally distributed, distance between supports = 25 feet?
Ans. W for 12-inch light beam == 9.19 and W, = W X 5.33 =
9.19 X 5.33 = 48.98 tons.
RESISTANCE TO CROSS-BREAKING AND SHEARING. 41
CAPACITY OF ROLLED BEAMS.
Explanation of Tables for I Beams.
The first column gives the distance between supports in feet.
The second column gives the capacity in tons of 2,000 Ibs.,
equally distributed.
The third column gives the deflection in inches at centre of
beam.
The fourth column gives the weight of beam in Ibs. for length
between supports.
The fifth to fifteenth column (inclusive) gives the distance in
feet that the beams should be spaced from centre to centre, for
weight in Ibs., per sq. ft. of surface for floors.
Pounds in decimals of a ton.
Ibs. tons.
GO == 0.03
70 = 0.035
80 = 0.0-1
90 = 0.045
100 = 0.05
140 = 0.07
160 = 0.03
180 = 0.085
200 = 1
250 = 0.125
300 == 0.15
In using these beams for floors, with brick arching, the ends
resting on supports should have a bearing of about 8 inches,
resting on a cast-iron plate, 8 X 12 in. sq,, by 1 in. thick.
Tie rods should be used where floors are subject to heavy con
centrated moving loads, (as trucks with merchandise, &c.;) these
rods should be about 8 times the depth of beam apart, fastened
about -J from the bottom of beam.
When beams are used to support walls, or as girders to carry
floor beams, and put side by side (II,) they should be fastened to
gether with cast-iron blocks, fitting between the flanges, so as to
securely combine the two beams. The blocks may be put about
the same distance apart as the tie-rods.
42
RESISTANCE TO CROSS BREAKING AND SHEARING.
Fig. 81.
15" "Heavy " Beam. Weight per If. = 66.66 Ibs.
Sectional area = 20.0"
Moment of inertia / = 652.42
Constant # =434.95
K
W .
L
s "S
5
a
fl
05
Deflec. in inches.
Weight in Ibs.
1
Distance d bet. centres of beams in feet, for
weight in Ibs. per sq. foot of
1
8
i
.0
i
o
|
GO
.D
g
<n
.0
j
B
1
8
i
8
i
i
1
20.1
17.6
14.7
12.8
io!2
8.9
8.0
7.2
6.7
5.9
5.5
50
4.7
4.2
3.9
3.6
3.4
3.2
3.0
2.8
2.6
2.5
2.3
2.2
2.1
2.0
1.9
1.8
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
72.49
62.13
54.35
4832
43.48
39.54
36.24
33.45
31.05
28.99
27.18
2558
24.16
22.89
21.73
20.71
19.58
18.91
18.12
17.39
16.72
16.10
15.53
14.99
14.49
14.03
13.59
13.17
12.79
12.42
12.08
11.75
11.43
11.15
10.87
0.037
0.050
0.065
0.084
0.104
0.126
0.150
0.177
0.205
0.236
0.270
0305
0.342
0.383
0.426
0471
0.515
0.569
0.623
0.677
0.735
0.795
0860
0.925
0.994
1.067
1.141
1.219
1.304
1.384
1.473
1.564
1.656
1.754
1.854
400.0
466.6
5333
600.0
666.6
733.3
800.0
866.6
933.3
1000.0
1066.6
1133.3
1200.0
1266.6
1333.3
1400.0
1466.6
1533.3
1600.0
1666.6
1733.3
1800.0
1866.6
1933.3
2000.0
2066.6
2133.3
2200.0
2266.6
2333.3
2400.0
2466.6
2533.3
2600.0
2666.6
20.9
17.7
15.5
13.5
12,0
10.7
9.6
8.6
7.9
7.1
6.7
6.0
5.6
5.2
4.7
4.4
4.1
3.8
3.6
3.3
3.1
3.0
2.8
2.6
2.5
2.4
2.2
2.1
22.1
19 >
79 3
21.2
19.6
16.7
15.2
13.5
12.5
ll.l
10.5
9.4
8.6
8.0
7.4
6.9
6.8
6.0
5.6
5.3
4.9
4.7
4.4
4.1
39
3.7
3.5
3.3
18.8
17.0
14.9
13.4
12.0
11.5
9.8
9.4
8.3
7.7
7.2
6.7
6.2
5.7
5.3
5.0
4.7
4.4
4.1
3.9
37
3.5
3.3
3.1
2.9
LG.9
15.0
13.4
12.0
10.8
9.8
8.9
8.2
7.5
6.9
6.4
5.9
5.5
5.1
4.8
4.5
4.2
3.9
3.7
3.5
3.3
3.1
3.0
2.8
2.7
21.4
19.1
17.6
15.5
14.7
127
11.8
10.7
10.2
9.2
8.5
7.9
7.4
6.9
6.4
6.0
5.7
53
5.0
4.7
4.4
4.2
4.0
3.8
2-i o
19 7
21.7
19.7
17.8
17.1
15.1
14.4
12.8
11.9
11.0
10.7
9.8
9.0
8.4
7.9
7.5
7.1
6.6
e.3
6.0
5.7
5.4
21.0
18.8
17.3
16.7
14.9
13.8
12.9
12.0
11.3
10.0
9.9
9.4
8.8
8.4
7.9
7.5
7.1
6.7
18.9
16.7
15.5
15.2
13.4
12.3
11.5
10.7
10.0
9.4
8.8
8.2
7.9
7.4
7.0
6.6
6.3
6.0
21.4
19.8
18.2
17.2
16.1
15.0
14.0
13.3
12.5
11.8
11.1
10.8
10.0
9.5
9.0
21.5
19.9
18.3
17.1
15.8
14.8
13.8
12.9
12.0
11.4
10.7
10.1
9.5
9.1
8.5
8.1
7.7
RESISTANCE TO CROSS BREAKING AND SHEARING.
43
Fig. 82.
15" "Light" Beam. Weight per If. = 51.66 Ibs.
Sectional area = 15.5"
Moment of inertia / 517.78
Constant K =345.19
K
W =-.
L
3
00
<D
Distance d bet. centres of beams in feet, for
o
a
c
o
00
weight in Ibs. per sq. foot of
d
CO CJ
^
a
3
|.s
|
i
lip
|
.0
Ul
.C
Vl
.0
?]
|
&
JS
j5
09
X3
72
&
ft
1
^c
<S
9
8
o
Srj
o
&
1
>g
1
CM
1
6
57.52
0037
310.0
7
49.31
0.050
361.6
8
43.13
0.065
413.3
9
38.35
0.084
4650
10
34.50
0.103
51 6.6
23.0
11
31 .38
0.124
567.3
22.9
19.0
12
28.76
0.150
620.0
19 T15.9
13
26.55
0.176
671. 6
22.7
^0 4
164 13.6
14
24. G5
0.205
793 ^
22.0
19.5
17.6 14.0 11.7
15
23.01
0.236
775.0
21.9
19.1
17.0
15.3 12.3 10.2
16
21.57
0.269
806.6
19.2
16.8
14.9
13.4 10.71 8.9
17
20.30
304
858.3
17 9
14.9
13.2
11.9 95! 7.Q
18
19.16
OJ341
830.0
21.3
15^2
13.3
11.8
10.6
8.5 j 7.1
19
18.15
0.381
981.6
21.3
19.1
13.6
11.9
10.6
9.5
7.6
6.3
20
17.24
0.424
1033.3
21.5
19.1
17.2
12.3
10.7
9.5
8.6
6.9
5-7
21
16.43
0.469
1085.0
19.5
17.4
15.6
11.1
9.8
8.7
7.8
6.2
5.2
22
15.68
0.515
1136.6
20.3
17.8
15.8
14.2
10.1
8.9
7.9
7.1
5.7
4.7
23
15.00
0.565
1187.3
21.7
18.7
16.3
14.5
13.0
9.3
8.1
7.2
6.5
5.2
4.3
24
14.38
0.620
1240.0
19.9
17.1
14.9
13.3
11.9
8.5
7.4
6.6
59
4.8
3.9
25
13.80
0.674
1291.6
18.4
15.8
13.8
12.3
11.0
7.8
6.9
6.1
5.5
4.4
3.6
2(5
13.27
0.732
1343.3
17.0
14.5
12.8
11.3
10.2
7.2
6.3 5.6
5.1
4.0
3.4
27
12.78
0.791
1395.0
15.7
13.6
11.8
10.5
9.4
6.7
5.9 5.2
4.7
3.7
3.1
28
12.32
0.855
1446.6
14.6
12.5
11.0
9.7
8.8
6.2
5.5
4.8
4.4
3.5
2.9
29
11.93
0.921
1498.3
13.7
11.8
10.2
9.1
8.2
5.8
5.1
4.5
4.1
3.2
2.7
30
11.50
0.989
1550.0
12.7
10.9
9.5
8.5
7.6
5.4
4.7
4.2
3.8 3.0
2.5
31
11.13
1.060
1601.6
11.9
10.3
8.9
8.0
7.1
5.1
4.4
3.9
3.5 2.8
2.3
32
10.78
1.133
1653.3
11.2
9.(
8.4
7.4
6.7
4.8
4.2
3.7
3.3 26
2.2
33
10.46
1.211
1705.0
10.5
9.0
7.9
7.0
6.3
4.5
3.9
3.5
3.1
2.5
2.1
34
10.14
1.292
1750.6
9.9
8.5
7.4
6.6
5.9
4.2
3.7
3.3
2.9
2.3
1.9
35
9.86
1.375
1808.3
9.3
8.(
7.0
6.2
5.6
4.0
3.5
3.1
2.8
2.2
1.8
36
9.58
1.463
1860.0
8.8
7.6
6.6
5.9
5.3
3.8
3.3
2.9
2.6
2.1
1.7
37
9.32
1.553
1911.6
8.3
7.2
6.2
5.6
5.0
3.5
3.1
2.7
2.5
2.0
1.6
38
9.08
1.645
1963.3
7.
6.8
5.9
5.3
4.7
3.4
2.9
2.6
2.3
1.9
1.5
39
8.85
1.742
2015.0
7!5
6.5
5.6
5.0
4.5
3.2
2.8
2.5
2.2
1.8
1.4
40
8.62
1.841
2066.6
7.1
6.1
5.3
4.7
4.3
3.0
2.6
2.3
2.1
1.7
1.4
RESISTANCE TO CROSS- BREAKING AND SHEARING.
Fig. 83.
12" "Heavy" Beam. Weight per If. = 56.66 Ibs.
Sectional area = 17.0"
Moment o l inertia / = 373.53
Constant K = 311.28
K
W = .
g
d
00
Distance d bet. centres of beams in feet, for
o
ft .
G
x>
weight in Ibs. per sq. foot of
ft-g
d
d
"-I
2
u
d
.2
I S
o
Gj
6
o
I
00
32
_Q
OQ
&
i
A
J
|
n
ft
C3
o
ft
o
o
o
o
o
o
o
g
1
6
51.88
0.046
340.0
7
44.54
0.063
396.6
8
38.70
0.082
453.3
<)
34.58
0.105
510.0
10
31.12
0.131
566.6
20.7
28.29
0.158
623 ?
"
17.1
12
25.9^
0.188
680.0
21.6
17.2
14.4
-iq
23.94
0.222
736.6
23.0
20.4
18.4
14.7
12.2
lo
14
22/22
258
793.3
22.(
19.8
17.6
15.8
12.6
10*.5
15
20.75
0.297
850.0
10 7
17.2
15.3
138
11.0
9.2
16
19*.5(
0.339
906.6
17.4
15.2
13.5 12.1
9.7
8.1
17
0.383
963*3
21.5
15.3
13.4
11.9 10.7
8.6
7.1
18
17/29
0.431
1020.0
21. P
19.2
13.7
12.0
10.6 9.6
7.6
6.4
19
16.38
0.481
1076.6
21.5
19.1
17.2
12.3
10.7
9.5 8.6
6.8
5.7
20
15.61
0.538
1133.3
19.5
17.: J
15. (
11.1
9.7
8.6
7.8
6.2
5.2
21
14.82
0.592
1190.0
20.1
17. (
15.(
14.1
10.0
8.8
7.8
7.0
5.6
4.7
22
14.1-1
0.652
1246.6
21.4
18.3
16.(
142
12.8
9.1
8.0
7.1
6.4
5.1
4.2
23
13.53
0.717
1303.3
19.0
16.8
14."~
130
11.7
8.4
7.3
6.5
5.8
4.7
3.9
24
12.9"
0.786
1360.0
18.0
15.4
13.5
12.(
10-8
7.7
6.7
6.0
5.4
4.3
3.6
25
12.4
0.855
1416.6
16.6
14.2
12.4
11.0
9.r
7.1
6.2
5.5
4.9
3.9
3.3
26
11.9
0.927
1473.3
15.3
13.1
11.5
10.1
9.1
6.5
5.7
5.1
4.G
3.6
3.0
27
11.5
1.003
1530.0
14.2
12.1
10.0
9.4
8-5
6.C
5.3
4.7
4.2
3.4
2.8
28
11.1
1.084
1586.6
13.2
11 .f
9.9
8.8
7-i
5.C
4.9
4.4
3.9
3.1
2.6
29
10.7
1.170
1643.3
12.3
10.5
9.2
8.2
7.4
5.2
4.0
4.1
3.7
2.9
2.4
30
10.3
1.257
1700.0
11.5
9.8
8.0
7.G
6.9
4.!
4.3
as
3.4
2.7
2.3
31
10.0
L350
1756.6
10.
9.2
8.0
7.1
6.4
4.C
4.0
3.6
3.2
2.5
2.1
32
9.7
1.443
1813.3
10.1
8.6
7.5
67
6.0
4.3
3.7
3.4
3.0
2.4
2.0
33
9.4
1.546
1870.0
9.5
8.2
7.1
6.P
5.7
4.(
3.5
3.1
2.8
2.2
1.9
34
9.1
1.650
1926.6
8.9
7.6
6.7
5.9
5.3
3 .8
3.8
2.9
2.6
2.1
1.7
35
8.8
1.758
1983.3
8.4
7.2
6.;
5.(
5-0
3.f
3.1
2.8
2.5
2.0
1.6
36
8.6
1.871
2040.0
8.0
6.8
6.0
5.3
4.8
3.4
3.0
2.6
2^4
1.9
1.6
37
8.4
1.987
2096.6
7.5
6.4
5.6
5.0
4.5
3.L
2.8
2.5
2.2
I .s
1.5
38
8.1
2.104
2153.3
7.1
6.1
5.3
47
4.3
3.(
2.6
2.3
2.1
1.7
1.4
39
7.9
2.234
2210.0
6.8
5.8
5.1
4.5
4.0
2.9
2.5
2.2
2.0
l.C
1.3
40
7.7
2.336
2266.6
6.4
5.5
4.8
4.3
3.8
&u
2.*4
2.1
1.9
1.5
1.2
RESISTANCE TO CROSS BREAKING AND SHEARING.
45
Fig. 84.
12" "Light" Beam. Weight per If. =41.66 Ibs.
Sectional area = 12.5"
Moment of inertia 1= 275.92
Constant K = 229.94
K
1
CB
d
Distance d bet. centres of beams in feet, for
ft^5
3
d
.2
1
weight in Ibs. per sq. foot of
CO O
d
6
d
GO
GO
^
o!
CD
P
CO
CO*
GO
.0
ja
J3
.0
JO
,0
ft
1
1
.Q
i
1
i
g
8
o
1
|
o
I
1
6
39.31
0.047
250.0
7
32.84
0.063
291.6
8
28.74
0.083
333.3
24.0
9
25.54
0.105
375
23.0
18.9
10
22.98
0.131
416.6
22.0
18.3
15.3
11
20.90
0.158
458.3
23.0
21.0
19.0
15.2
12.6
12
19.16
0.189
500.0
22.0
19.9
17.7
15.9
12.7
10.6
13
17.68
0.222
541.6
10 4
17.0
15.1
13.6
10.9
9.0
14
16.42
0.258
583.3
16.7
14.6
13.0
9.3
7.8
15
15.32
0.297
625.0
22.0
20.0
14.5
12.7
11.3
iol2
8.1
6^7
16
14.37
0.339
666.6
22.0
19.9
17.9
12.8 11.2
9.9
8.9
7.1
5.9
17
13.52
0.383
708 3
19.9
17.7
15.9
11.3 9 9
8.8
7.9
6.3
5.3
18
12.77
0.431
750.0
20.0
17.7
15.7
14.1
10.1
8.8
7.8
7.1
5.6
4 .7
19
12.10
0.481
791.6
21.0
18.3
15.9
14.2
12.7
9.1
7.9
7.0
6.3
5.1
4.2
20
11.48
0.538
833.3
19.1
16.4
14.3
12.7
11.4
82
7.1
6.3
5.7
4.5
3.8
21
10.94
0.592
875.0
17.3
15.0
13
ll.(
10.4
7.4
6 5
5 7
5 2
4 1
3.4
22
10.44
0.652
916.6
15.8
13^5
111 8
me
9^5
G!?
5.9
5.2
4.7
3.7
3.1
23
9.99
0.717
958.3
14.4
12.5
10.8
9.7
8.6
6.2
5.4
4.8
4..
3.4
2.8
24
9.58
0.786
1000.0
13.3
11.4
9.9
8.8
7.9
5.7
4.9
4.4
3.9
3.1
2.6
25
9.19
0.855
1041.6
12.2
10.5
9.1
8.2
7.3
5.2
4.5
4.0
3 (
2.9
2.4
26
8.84
0.927
1083.3
11.3
9.7
8.5
7.5
6.8
4.8
4.2
3.7
3.4
2.7
2.2
27
8.51
1.003
1125.0
10.6
9.0
7.8
7.0
6.3
4.5
3.9
3.5
3.1
2.5
2.1
28
8.21
1.084
1166.6
9.7
8.3
6.5
5.8
4.1
3-.C
3.2
2.9
2.3
1.9
29
7.92
1.170
1208.3
9.1
7.8
6.8
6.1
5.4
3.8
3.4
3.(
2 .7
2.1
1.8
30
7.66
1.257
1250.0
8.5
7.2
6.3
5 (
5.1
3.6
3.1
2.8
2.f
2.0
1.7
31
7.41
1.350
1291.6
7.9
6.8
5.9
5^3
4.8
3.4
2.9
2,(
2.3
1.9
1.5
32
7.18
1.443
1333.3
7.4
6.4
5.6
4.9
4.4
3.2
2.8
2.4
2.2
1.7
1.4
33
6.96
1.542
1375.0
7.C
6.0
5.2
4.7
4.2
3.0
2.6
2.3
2.1
i.r
1.4
34
6.75
1.645
1416.6
6.6
5.6
4.9
4.4
3.9
2.8
2.4
2.2
2.0
1.5
1.3
35
6.57
1.754
1458.3
6.2
5.3
4.7
4.1
3.7
2.6
2.3
2.0
1.8
1.5
1.2
36
6.38
1.871
1500.0
5.9
5.0
4.4
3.9
3.5
2.5
2.2
1.9
1.7
1.4
1.1
37
6.21
1.987
1541.6
5.5
4.8
4.2
3.7
3.3
2.3
2.0
1.8
1.6
1.3
1.1
38
6.05
2.109
1583.3
5.3
4.5
3.9
3.5
3.1
2.2
1.9
1.7
l.f
1.2
1.0
39
5.89
2.229
1625.0
5.0
4.J
3.7
3.3
3.0
2.1
1.8
1.6
1.4
1.1
1.0
40
5.74
2.366
1666.6
4.7
4.1
3.5
3.1
2.8
2.0
1.7
1.5
1.3
1.0
0.9
46
RESISTANCE TO CROSS BREAKING AND SHEARING.
10.5
Fig. 85.
10.5" Seam. Weight per If. = 35 Ibs.
Sectional area = 10.5"
Moment of inertia I = 179.44
Constant K =170.903
K
-2
c
Distance d bet. centres of beams in feet, for
g; ,
c
d
weight in Ibs. per sq. foot of
E"
c
"7
2
c5
1
c
c5
o
09
1
K
A
.c
<R
J
|
|
|
1
|
0*
&
3
Q
s
o
Q
I
I
o
00
I
I
I
I
1
I
1
6
2U8
0.053
2100
2L41
0.072
2450
23.2
8
21.3(5
0.095
280.0
21.3
17.8
9
18 OS
0.120
315.0
23.4
21.1
17.0
14.0
10
17.09
0.149
350.0
21 3
18 9
17.0
13.6
11.4
11
15.53
0.181
385.0
20.1
17.6
15.6
14.1
11.3
9.3
12
14.21
0.216
420.0
1(5.9
14.8
13.1
11.8
9.4
7.9
13
13 14
0.254
455.0
20,6
202 144
12.6
11.1
10.1
8.1
6 7
14
12.2(1
0.295
490.0
21.7
19.2
17.4
12.4
10.9
9.6
8.7
6.9
5*.8
15
11.38
0.340
525.0
21.9
18.9
17.0
15.1
10.8
9.4
8.4
7.5
6.0
5.0
16
1068
0.389
560.0
22.2
19.0
16.6
14.9
13.3
9.5
8.3
7.4
6.6
5.3
4.4
17
10.05
0.439
595.0
19.7
17.0
14.7
13.2
11.8
8.4
7.3
6.5
5.9
4.7
3.9
18
9.49
0.494
630.0
17.5
15.0
131
11.7
10.5
7.6
6.5
5.8
5.2
4.2
3.5
19
8.99
0.553
6650
15.7
13.0
11.7
10.5
9.4
6.7
5.9
5.2
4.7
3.7
3.1
2)
8.54
0.614
700.0
14.2
12.2
10.6
9.4
85
6.1
5.3
4.7
4.2
3.4
2.8
21
8.13
0.681
735.0
12.9
11.1
9.6
8.6
7.7
5.5
4.8
4.3
3.8
3.1
2.5
22
7.75
0.752
770.0
11.7
10.0
9.1
7.8
7.0
5.0
4.4
3.9
3.5
2.8
2.3
2>
7.43
0.823
805.0
10.7
9.2
8.0
7.2
6.4
4.6
4.0
3.5
3.2
2.5
2.1
24
7.12
0.903
840.0
9.8
8.4
7.4
6.5
5.9
4.2
3.7
3.2
2.9
2.3
1.9
25
6.83
0.980
875.0
9.1
7.8
6.8
6.0
5.4
3.9
3.4
3.0
2.7
2.1
1.8
2 ,
657
1.067
910.0
8.4
7.2
6.3
5.6
5.0
3.6
3.1
2.8
2.5
20
1.6
27
G.32
1.154
945.0
7.8
6.7
5.8
5.2
4.6
3.3
2.9
2.6
2.3
1.8
1.5
28
6.10
1.251
980.0
7.2
6.2
5.4
4.8
4.3
3.1
2.7
2.4
2.1
1.7
1.4
2!)
5.89
1.346
1015.0
6.7
5.8
5.0
4.5
4.0
2.9
2.5
2.2
2.0
1.6
1.3
30
5.69
1.450
1050.0
6.3
5.4
4.7
4.2
3.7
2.7
2.3
2.1
1.8
.5
1.2
31
5.51
1.556
1085.0
5.9
5.1
4.4
3.9
3.5
2.5
2.2
1.9
1.7
A
1.1
32
5.31
1.672
1120.0
5.5
4.7
4.1
3.7
3.3
2.3
2.0
1.8
1.6
.3
1.1
33
5.17
1.783
1 155.0
5.2
4.4
3.9
3.4
3.1
2.2
1.9
1.7
1.5
.2
1.0
34
5.02
1.906
1190.0
4.8
4.2
3.6
3.2
2.9
2.1
1.8
1.6
1.4
!T
35
4.88
2.033
1225.0
4.6
4.0
3.4
3.1
2.7
1.9
1.7
1.5
1.3
.1
36
4.69
2.143
1200.0
4.3
3.7
3.2
2.8
2.6
1.8
1.6
1.4
1.3
1.0
37
4.61
2.297
1295.0
4.1
3.5
3.1
2.7
2.4
1.7
1.5
1.3
1.2
38
4.50
2.444
1330.0
3.9
3.3
2.9
2.6
2.3
1.6
1.4
1.3
1.1
39
4.38
2589
1365.0
3.6
3.2
2.8
2.5
2.2
1.6
1.4
1.2
1.1
40
4.20
2.711
1400.0
3.5
3.0
2.6
2.3
2.1
1.5
1.3
1.1
1.0
RESISTANCE TO CROSS- BREAKING AND SHEARING.
47
Fig. 80.
9 /x " Heavy" Beam. Weight per If. = 50 Ibs.
Sectional area = 15.0"
Moment of inertia / = 188.55
Constant /f 209.50
K
03
03
Distance d bet. centres of beams in feet, for
N
o
fl
C
03
weight in Ibs. per sq. foot of
3 o
O3 O
U
"3
J.S
6
6
o
il
03
a
03
g
03
OB
w
g
JC
-A
,0
.2
Q
ft
CS
o
o
Q
I
p
,Q
O
So
i
8
O
1
1
8
1
6
36.91
0.065
300.0
7
29.92
0.084
3500
8
26.1 8
0.111
400.0
21.8
9
23.27
0.141
450.0
20. (i
17.2
10
20.95
0.174
500.0
23.2
20.9
16.7
13.9
11
19.01
0.211
550.0
21.6
19.1
17.3
13.7
11.5
12
17.45
0.253
600.0
20.7
18.1
16.1
14.5
11.6
9.6
13
16.11
0297
650.0
17.C
15.4
13.8
12.3
9.9
8.2
14
14.96
0.345
700.0
21.3
15.2
13.3
11.8
10.C
8.5
7.1
15
13.96
0.398
750.0
20.6
18.0
13.2
11.6
10.3
9.3
7.5
6.2
16
13.09
0.454
800.0
20.4
19.5
16.3
ll.(j
10.2
9.0
8.1
6.5
5.4
17
12.32
0.515
850.0
20.7
18.1
16.1
14.5
10.3
9.0
8.0
7.2
5.7
4.8
18
11.63
0.580
900.0
21.5
18.4
16.1
14.3
12.9
9.2
8.0
7.1
6.4
5.1
4.3
19
11.02
0.648
950.0
19.3
16.5
14.5
12.8
11.6
8.2
7.2
6.4
5.8
4.6
3.8
20
10.47
0.722
1000.0
17.4
14.9
13.0
11.4
10.4
7.4
6.5
5.8
5.2
4.1
3.4
21
9.97
0.799
1050.0
15.8
13.5
11.8
10.5
9.4
6.8
5.9
5.2
4.7
3.4
3.1
22
9.52
0.883
11000
14.4
12.3
10.3
9.6
8.2
6.1
5.4
4.8
4.3
3.4
2.8
23
9.10
0.968
1150.0
13.1
11.3
9.8
8.7
7.9
5.6
4.9
4.3
3.9
3.1
2.6
24
8.72
1.002
1200.0
12.1
10.3
9.0
8.0
7.2
5.1
4.5
4.0
3.6
2.9
2.4
25
8.34
1.152
1250.0
11.1
9.5
8.3
7.4
6.6
4.7
4.1
3.7
3.3
2.G
2.2
26
8.05
1.258
1300.0
10.3
8.8
7.7
6.8
6.1
4.4
3". 8
3.4
3.0
2.4
2.0
27
7.75
1.363
1:550.0
9.5
8.0
7.1
6.3
5.7
4.1
3.5
3.1
2.8
2.2
1.9
28
7.48
1.477
1400.0
8.9
7.6
6.6
5.9
5.3
3.8
3.3
2.9
2.6
2.1
1.7
29
7.22
1.593
1450.0
8.3
7.2
6.2
5.5
4.9
3.5
3.1
2.7
2.5
1.9
1.6
30
6.98
1.718
1500.0
7.7
6.6
5.8
5.1
4.6
3.3
2.9
2.5
2.3
1.8
1.5
31
6.75
1.846
1550.0
7.3
6.2
5.4
4.8
4.3
3.1
2.7
2.4
2.1
1.7
1.4
32
6.54
1.982
1GOO.O
6.7
5.8
5.1
4.4
4.0
2.9
2.5
2.2
2.0
1.6
1.3
33
6.34
2.119
1650.0
6.4
5.6
4.8
4.2
3.8
2*.7
2.4
2.1
1.9
1.5
1.2
34
6.16
2.265
1700.0
6.0
5.1
4.5
4.0
3.6
2.5
2.2
2.0
1.8
1.4
1.2
35
5.98
2.416
1750.0
5.6
4.8
4.2
3.7
3.4
2.4
2.1
1.8
1.7
1.3
1.1
36
5.81
2.577
1800.0
5.3
4.6
4.0
3.5
3.3
2^3
2^0
1.7
1.6
1.2
1.0
37
5.66
2.742
1850.0
5.0
4.3
3.8
3.3
3.0
2.1
1.9
1.6
1.5
1.2
38
5.51
2.918
1900.0
4.8
4.1
3.6
3.2
2.9
2.0
1.8
1.6
1.4
1.1
39
5.37
3.098
1950.0
4.5
3.9
3.4
3.0
2.7
1.9
1.7
15
1.3
1.1
40
5.27
3.289
2000.0
4.3
3.7
3.2
2.9
2.6
1.8
1.6
1.4
1.3
l.C
43
RESISTANCE TO CROSS BREAKING AND SHEARING.
9" " Medium" Beam. Weight per If. = 30 Ibs.
Sectional aren = 9.0"
Moment of inertia I = 111.32
Constant K = 123.09
K f
oc
p
<D
Distance d bet. centres of beams in feet, for
o
1
weight in Ibs. per sq. foot of
<n
i
ft
.2
~j
G5 S
.0
O
9
Itf)
J
d
g
"
J3
|
00
JO
s
JO
1
n
.0
s
6
O
ft
9
p
o
s
8
8
1
1
1
6
20.00
0.062
1800
22
7
17.07
0.085
210.0
25.0
20. (
16.0
8
15.40
0.111
240.0
21.0
21.0
in n
15.0
12^0
9
13.74
0.141
270.0
21.0
19.0
100 15.0
11.0
10.0
10
12.36
0.174
300.0
17.0
15.0
13.0 12.0
9.8
8^2
11
11.24
0.211
330.0
22.0
20.0
14.0
12.0
11.01 10.0
8.1
6.8
12
10.30
0.252
300.0
21.0
19.0
17.0
12.0
10.0
9.5 1 8.5
6.8
5.7
13
9.51
0.297
390.0
2J.O
18.0
10.0
14.0
10.0
9.1
8.1! 7.3
5.8
4.8
14
8.83
0.345
420.0
21.0
18.0
15.0
14.0
12.0
9.0
7.8
7.0i 6.3
5.0
4.2
15
8.24
0.398
450.0
18.0
15.0
13.0
12.0
10.0
7.8
6.8
6.11 5.4
4.3
3.6
16
7.73
0.455
480.0
10.0
13.0
12.0
10.0
9.6
6.9
6.0
5.3; 4.8
3.8
3.2
17
7.21
0.511
510.0
14.0
12.0
10.0
9.4
8.4
6.0
5.3
4.7i 4.2
3.2
2.8
18
6.87
0.580
540-0
12.0
10.0
9.5
8.4
7.6
5.4
4.7
4.2^ 3.8
3.0
2.5
19
6.51
0.650
570.0
11.0
9.7
8.5
7.0
6.8
4.8
4.2
3.8 3.4
2.7
2.2
20
6.18
0.722
000.0
10.0
8.8
7.7
6.8
6.1
4.4
3.8
3.4
3.0
2.4
2.0
21
5.88
0799
630.0
9.3
8.0
7.0
6.2
5.6
4.0
3.5
3.1
2.8
2.2
1.8
22
5.02
0.884
660.0
8.5
7.2
6.3
5.0
5.1
3.6
3.1
2.8
2.5
2.0
1.7
23
5.37
0.969
6DO.O
7.7
6.6
5.8
5.1
4.6
3.3
2.9
2.5
2.3
1.8
1.5
24
5.15
1.065
720.0
7.1
6.1
5.3
4.7
4.2
3.0
. 2.0
2.3
2.1
1.7
1.4
25
4.94
1.157
750.0
6.5
5.6
4.9
4 3
3.9
2.8
2.5
2.1
1.9
1.5
1.3
26
4.83
1.277
780.0
6.1
5.3
4.6
4J
3.7
2.6
2.3
2.0
1.8
1.4
1.2
27
4.58
1.365
810.0
5.6
4.8
4.2
3.7
3.3
2.4
2.1
1.8
1.6
1.3
1.1
28
4.41
1.476
840.0
5.2
4.5
3.9
3.5
3.1
2.2
1.9
1.7
1.5
1.2
1.0
29
4.20
1.593
870.0
4.8
4.1
3.6
3.2
2.9
2.0
1.8
1.6
1.4
1.1
30
4.12
1.718
900.0
4.5
3.9
3.4
3.0
2.7
1.9
1.7
1.5
1.3
1.0
31
3.99
1.846
930.0
4.2 3.0
3.2
2.8
2.5
1.8
1.6
1.4
1.2
1.0
32
3.80
1.982
900.0
4.0 3.4
3.0
2.6
2.4
1.7
1.5
1.3
1.2
33
3.74
2.119
9900
3.7
3.2
2.8
2.5
2.2
1.0
1.4
1.2
1.1
34
3.03
2.235
1020.0
3.5
3.0
2.6
2.3
2.1
1.5
1.3
1.1
1.0
35
3.53
2.416
1050.0
3.3
2.8
2.5
2.2
2.0
1.4
1.2
1.1
1.0
36
3.43
2.577
1080.0
3.1
2.7
2.3
2.1
1.9
1.3
1.1
1.0
37
3.34
2.742
1110.0
3.0
2.5
2.2
2.0
1.8
1.2
1.1
1.0
38
3.25
2.918
1140.0
28
2.4
2.1
1.9
1.7
1.2
1.0
39
3.17
3.098
1170.0
2.7
2.3
2.0
1.7
1.6
1.1
1.0
40
3.09
3.289
1200.0
2.5
2.2
1.9
1.6
1.5
1.1
RESISTANCE TO CROSS BREAKING AND SHEARING.
Fig. 88.
9" "UgU" Beam. Weight per If. = 23.33 Ibs.
Sectional area = 7.0"
Moment of inertia /== 91.00
Constant K r =101.2
K
3
CQ
e
CO
o
Distance d bet. centres of beams in feet, for
1
3
.5
JG
o
c
rc
,Q
weight in pounds per sq. foot of
CO 0)
G
fl
J.S
o
d
HP
a!
i
-;
02
1
.0
|
,0
1
.a
JD
01
ft
6
ft
8
i
8
5
g
1
g
1
i
1
g
16.86
0.002
140.0
22.0
18.7
7
14.45
035
103.3
i;> 5
79 ()
20.0
.0.0
13.7
12.05
0.111
180.0
25.0
19.7
17.5
15.8
[2.0
10.5
9
11.24
0.141
21o o
17.8
15.6
13.8
10.1
8*.6
10
10.12
0.175
233.3
22.0
20.0
14.4
12.0
11.2
iai
8.0
6.7
11
9.20
0.212
250.0
21.0
18.7
10.7
11.9
10.4
9.2
8.3
0.7
5.5
12
8.43
0.253
280.0
23.0
20.0
17.5
15.0
14.0
10.0
8.7
7.8
7.0
5.6
4.6
13
7.78
0.297
303.3
19.9
17.9
14.9
13.4
11.9
8.5
7.4
0.0
5.9
4.8
3.0
14
7.22
0.345
320.0
17.1
14.7
12.8
11.4
10.3
7.3
0.4
5.7
5.1
4.1
3.4
15
6.74
0.398
350.0
14.9
12.9
11.2
10.0
8.9
6.4
5.0
5.0
4.5
3.0
2.$
16
6.31
0.453
373.3
13.1
11.2
9.8
8.7
7.8
5.0
4.9
4.3
3.9
3.1
2.6
17
5.95
0.510
390.0
11.6
10.0
8.7
7.8
7.0
5.0
4.3
3.8
3.5
2.8
2.3
18
5.02
0.579
420.0
10.4
8.9
7.8
0.9
0.2
4.4
3.9
3.4
3.1
2.5
2.0
19
5.32
0.048
443.3
9.3
8.0
7.0
0.2
5.0
4.0
3.5
3.1
2.8
2.2
1.8
20
5.00
0.721
466.6
8.4
7.2
03
5.0
5.0
3.0
3.1
2.8
2.5
2.0
1.0
21
4.81
0.797
490.0
7.6
G.5
5.7
5.1
4.5
3.2
2.8
2.5
2.2
1.8
1.5
22
4.59
0.879
513.3
0.9
5.9
5.2
4.0
4.1
2.9
2.0
2.3
2.1
1.7
1.3
23
4.40
0.908
536.6
0.3
5.5
4.7
4.2
3.8
2.7
2.3
2.1
1.9
1.5
1.2
24
4.21
1.000
500.0
5.8
5.0
4.3
3.8
3.5
2.5
2.1
1.9
1.7
1.4
1.1
25
4.04
1.151
583.3
5.3
4-G
4.0
3.5
3.2
2.3
2.0
1.7
1.0
1.2
1.0
26
3.89
1.254
GOO.G
4.9
4.2
3.7
3.3
2.9
2.1
1.8
1.0
1.4
1.1
27
3.74
1.359
030.0
4.0
3.9
3.4
3.0
2.7
1.9
1.7
1.5
1.3
10
28
3.00
1.400
653.3
4.2
3.0
3.2
2.8
2.5
1.8
1.0
.4
1.2
1.0
29
3.48
1.582
676.6
4.0
3.4
3.0
2.0
2.4
1.7
1.5
.3
1.1
30
3.37
1.711
700.0
3.7
3.2
2.8
2.5
2 2
1.0
1.4
1.2
1.1
31
3.20
1.837
723.3
3.5
3.0
2.0
23
2.1
1.5
1.3
.1
1.0
32
3.10
1.968
746.6
3.2
2.8
2.4
2.1
1.9
1.4
12
l.o
33
3.00
2.104
770.0
3.0
2.0
2.3
2.0
1.8
1.3
1.1
34
2.97
2.250
793.3
2.9
2.4
2.1
1.9
1.7
1.2
l.o
35
2.89
2.399
810.G
2.7
2.3
2.C
1.8
1.0
1.1
36
2.81
2.505
840.0
2.0
2.2
1.9
1.7
1.5
1.0
37
2.73
2.723
803.3
2.4
2.1
1.8
1.0
1.4
38
2.06
2.898
880.6
2.3
2.0
1.7
1.5
1.4
39
2.59
3.070
910.0
9 9
1
i r
1 4
1 .}
40
2.52
3.250
2.1
1.8
1.5
1.4
1.2
50
RESISTANCE TO CROSS-BREAKING AND SHEARING.
8" Beam. Weight per If. -= 21.66 Ibs.
Soctionnl area = 6.5"
Moment of inertia I = 05.09
Constant, K = 82.49
K
-2
"tf
=Q
C
o>
Distance d bet. centres of beams in feet, for
p.
P.*-
c c
3
"3
c
ri
JB
weight in pounds per pq. foot of
"c c
ffi
c;
c
d
09
32
e
2
tC
1
1
1
Ja
J
|
5
B
_C
i
1
w
X>
cr
s
6
&
3
i
55
i
1
g
i
o
o
i
1
G
13.74
0.070
130.0
oo 9
18 3
Ti
7
11.78
0.095
151.6
21.
18. ">
10.8
13.-;
11.2
g
10.30
0.124
173.3
>- Y
18.3
1 0.0
14. 3
12.8
10.3
8.5
9
9.16
0.158
195.0
. ().:>
1 l.fi
12.7
11.3
10.1
8.1
6.7
10
8.23
0.198
216.6
J0.5
1X.3
10.4
11.7
10.2
9.1
8.2
6.5
5.4
11
7.49
0.238
238.3
22.C
19.4
17.0
15.1
l:j.G
9.7
8.5
7.5
6.8
5.4
4.5
12
6.87
0.284
260.0
19.0
163
L4.8
12.7
11.4
8.1
7.1
6.3
5.7
4.5
3.8
13
634
0.335
281.6
16.2
14.0
12.1
10.9
7
0.9
6.0
5.4
4.8
3.9
3.2
14
5.89
0.390
303.3
14.0
12.0
10.5
9.:}
8.4
0.0
5.2
4.0
4.2
3.3
2.8
15
6.49
0447
325.0
12,2
10.5
9.1
8,1
7.3
5.2
4.5
4.0
3.6
2.9
2.4
16
5.15
0.511
346.6
10.7
9.1
8.0
7,1
6.4
4.5
4.0
3.6
3.2
2.5
2.1
17
4.85
0.580
368.3
9.5
.0
7.1
6.3
5.7
4.0
3.7
3.2
2.8
2.3
1.9
18
4.58
0.653
390.0
8.4
7.2
6.3
5.6
5.1
3.8
3.1
2.7
2.5
2.0
1.7
19
4.34
0.731
411.6
7.6
65
5.7
5.1
4.5
3.3
29
B5
22
1.8
1.5
20
4.11
0.810
433.3
6.8
5.8
5.1
4.5
4.1
29
2.5
2.2
2.0
1.6
1.3
21
3.92
0.898
455.0
0.2
5.3
4.6
4.1
3.7
2.6
2.3
2.0
1.8
1.4
1.2
22
3.73
0.989
476.6
5.6
4.8
4.2
3.7
3.4
2.4
2.1
1.8
1.6
1.3
1.1
23
358
1.090
498.3
5.1
4.4
3.8
3.4
3.1
2.2
1.9
1.7
1.5
1.2
1.0
24
3.42
1.192
520.0
4.7
4.0
3.5
3.1
2.8
2.0
1.7
1.5
.4
1.1
25
3.29
1.300
541.6
4.3
3.7
3.2
2.9
2.6
1.8
1.5
1.4
1.3
1.0
26
3.17
1.417
563.3
4.0
3.4
3.0
2.7
2.4
1.7
1.4
1.3
1.2
27
3.05
1.536
585.0
3.7
3.2
2,8
2.5
2.2
1.6
1.4
1.2
1.1
28
2.94
1.602
606.6
3.5
3.0
2.6
2.3
2.1
1.5
1.3
1.1
1.0
29
2.84
1.795
628.3
3.2
2.8
2.4
2.1
1.9
1.4
1.2
1.0
30
2.73
1.923
650.0
3.0
2.6
2.2
2.0
1.8
1.3
1.1
|
31
2.66
2.080
671.6
2.8
2.4
2.1
1.9
1.7
1.2
1.0
2.56
2.219
693.3
2.6
2.2
2.0
1.7
1.0
1.1
33
2.49
2.383
715.0
2.5
2.1
1.8
1.6
1.4
1.0
34
2.42
2.550
736.6
2.3
2.0
1.7
1.5
1.4
35
2.35
2.722
758.3
2.2
1.9
1.6
1.4
1.3
36
2.29
2.907
780.0
2.1
1.8
1.5
1.4
1.2
37
2.22
3.084
801.6
2.0
1.7
1.5
1.3
12
38
2.17
3.290
823.3
1.9
1.6
1.4
1.2
1.1
39
2.11
3.484
845.0
1.8
1.5
1.3
1.2
1.0
40
2.06
3.702
866.6
1.7
1.4
1.2
1.1
1.0
]
BESISTANCE TO CEOSS-BBEAZING AND SHEABING.
51
Fig. 90.
7" Seam. Weight per If. = 18.33 Ibs.
Sectional area == 55"
Moment of inertia / = 44.84
Constant # =64.06
K f
e
jj
02
Distance d bet. centres of beams in feet, for
O
p.
1
A
o
m
.0
weight in Ibs. per sq. foot of
13 O
2
.s
"""
OD CD
5
c5
d
JbK)
03
so
,
|
t
|
.JO
i
i
X
s
OS
O
1
i
|
i
O
05
3
1
1
1
6
10.67
0.080
110.0
25.4
22 2
H).7
17.7
14.2
11 8
7
9.15
0.109
18.6
16.3
14/>
13,0
10.5
8 .7
8
8.00
0.143
14&6
25. 6
22.2
20. (!
14.2
12.5
11.1
10.0
8.0
6.6
9
7.11
0.181
165.0
2*2.9
19.7
17.5
16.8
11.2
9.8
8.7
7.9
6.3
5.2
10
6.40
0.224
183.3
2L3
18.2
16.0
14.2
12.8
9.1
8.0
7.1
6.4
5.1
4.2
11
5.82
0.272
201.6
17.6
15.3
13.2
11.7
10.5
7.5
6.6
5.8
5.2
4.2
3.5
12
5.33
0.325
220.0
14.8
12.6
11.1
9.8
8.8
6.3
5.5
4.9
4.4
3.5
2.9
13
4.92
0.382
238.3
12.6
10.9
9.4
8.3
7.5
5.4
4.7
4.1
3.7
3.0
2.5
14
4.56
0.444
256.6
10.8
9.3
8.1
7.2
6.5
4.6
4.0
3.6
3.2
2.6
2.1
15
4.27
0.513
275.0
9.4
8.2
7.1
6.3
5.7
4.0
35
3.1
2.8
2.2
1.8
16
3.99
0.585
293.3
8.3
7.1
6.2
5.5
4.9
3.5
3.1
2.7
2.4
1.9
1.6
17
3.76
0.665
311.6
7.3
6.5
5.5
4.9
4.4
3.1
2.7
2.3
2.1
1.7
1.4
18
3.55
0.749
330.0
6.5
5.6
4.9
4.3
3.9
2.8
24
2.1
1.9
1.5
1.3
19
3.37
0.840
348.3
5.9
5.1
4.4
3.9
3.5
2.5
2.2
1.9
1.7
1.4
1.1
20
3.20
0.936
366.6
5.3
4.5
4.0
3.5
3.2
2.2
2.0
1.7
1.6
1.2
1.0
21
3.05
1.038
385.0
4.8
4.1
3.6
3.2
2.9
2.0
1.8
1.6
1.4
1.1
22
2.91
1.146
403.3
4.4
3.7
3.3
2.9
2.6
1.8
1.6
1.4
1.3
1.0
23
2.78
1.257
421.6
4.0
3.4
3.0
2.7
2.4
1.7
1.5
1.3
1.2
24
2.66
1.381
440.0
3.6
3.1
2.7
2.4
2.2
1.6
1.3
1.2
1.1
25
2.56
1.504
458.3
3.4
2.9
2.5
2.2
2.0
1.5
1.2
1.1
1.0
26
2.45
1.630
476.6
3.1
2.6
2.3
2.0
18
1.4
1.1
27
2.37
1.775
495.0
2.9
2.5
2.1
1.9
1.7
1.3
1.0
28
2.27
1.871
513.3
2.7
2.3
2.0
1.8
1.6
1.2
29
2.20
2.075
531.6
2.5
2.1
1.8
1.7
1.5
11
30
2.12
2.229
550.0
2.3
2.0
1.7
1.5
1.4
1.0
52
RESISTANCE TO CROSS- BREAKING AND SHEARING.
6" Seam. Weight per If. = 13.33 Ibs.
I 0.28"
^pfc
Sectional area =. 4.0"
Moment of inertia I = 22.5
Constant K = 37.64
K
W .
-
K
or:
0)
Distance d bet. centres of beams in feet, for
a .
1
00*
weight in Ibs. per sq. foot of
*
d
.5
s
?c
d
d
".S
05
6
o
a
n
J
j
03
jQ
>
JO
2
J
J
s
Bi
o
o
Q
1
i
i
i
o
Ci
8
o
o
CD
1
1
1
6
6.27
094
80.0
23.2
20.9
14.9
13.0
11.6
10.4
8.3
6.9
7
5.37
o!l28
93.3
19.1
17.3
L5.3
lo .o
9.5
8.4
7.6
6.1
5.1
8
.168
106^6
19.5
16.8
14.6
13.0
11.7
8.5
6.5
5.8
4.7
3.9
9
4J8
0.213
120.0
15.4
13.4
11.6
10.4
9.2
6.6
sis
5.1
4.G
3.7
3.1
10
3.75
0.263
133.3
12.5
10.7
9.3
8.3
7.5
5.3
4.7
4.1
3.7
3.0
2.5
11
3.42
0.320
146.6
10.3
9.0
7.7
6.9
6.2
4.4
3.8
3.4
3.1
2.4
2.0
12
3.13
0.382
160.0
8.6
7.0
6.5
5.7
5.2
3.7
3.2
2.9
2.6
2.0
1.7
13
2.89
0.450
173.3
7.4
6.4
5.5
4.9
4.4
3.1
2.7
2.4
2.2
1.7
1.4
14
2.68
0.524
186.6
6.3
5.4
4.7
4.2
3.8
2.7
2.3
2.1
1.9
1.5
1.2
15
2.51
0.607
200.0
5.5
4.8
4.2
3.7
3.3
2.3
2.1
1.8
1.6
1.3
1.1
16
2.34
0.689
213.3
4.8
41
3.6
3.2
2.9
2.0
1.8
1.6
1A
1.1
17
2.21
0.786
226.6
4.3
3.7
3.2
2.8
2.5
1.8
1.6
1.4
1.8
18
2.09
0.888
240.0
3.8
3.3
2.9
2.5
2.3
1.6
1.4
1.2
1.1
19
1.98
0.995
253.3
3.4
3.0
2.6
2 3
2.1
1.4
1.3
1.1
20
1.88
1.110
266.6
3.1
2.7
2.3
2.1
1.8
1.3
1.1
21
1.79
1.231
280.0
2.8
2.4
2.1
1.9
1.7
1.2
1.0
22
1.70
1.350
293.3
2.5
2.2
1.9
1.7
1.5
1.1
23
1.63
1.493
306.6
2.3
2.0
1.7
1.5
A
1.0
24
1.56
1.641
320.0
2.1
1.8
1.6
1.4
.3
25
1.50
1.787
333.3
2.0
1.7
1.5
1.3
.2
26
1.44
1.950
346.6
1.8
1.5
1.3
1.2
.1
27
1.39
2.129
360.0
1.7
1.4
1.2
1.1
28
1.33
2.286
373.3
1.5
1.3
1.1
29
1.29
2.489
386.6
1.4
1.2
1.0
30
1.25
2.698
400.0
1.3
1.1
RESISTANCE TO CROSS-BREAKING AND SHEARING.
53
CAST-IRON BEAMS.
Factor of rupture for cast-iron beams of various sections.
The factor C is based on practical experiments by Hodgkinson-
Its value alters with the different proportions of the cross-sections
of beam.
Beam supported at the ends ; load concentrated at the center.
Reference.
= Factor of rupture.
W= Breaking weight in Ibs.
A = Sectional area of beam in square inches.
I = Distance between supports in inches.
h = Height of beam in inches.
Dimensions in inches. 6 = Thickness of web at center is the
unit.
Fig. 92. 0.32 = 0.726
5.125 = 11.646
0.44=6
0.47 = 1.066
10.52 = 1.186
Fig. 93.
2.27 = 5.156
. = 3.20 (7=27292
1.74 = 5.86
JJT
5.125 = 17.086
.0.26 = 0.866
0.30 = 6
. 55 = 1.736
l. 78~="5?936
= 2.73 (7=28513
BESISTANCE TO CEOSS-BEEAKING AND SHEAEING.
Fig. 2^. 1.07 = 3.346
=0.946
5.125 = 166
2.10 = 6.566
= 2.88 (7=30330
. 95.
1.6 = 4.216
"
5.125 = 13.486
. 315 = 0.826
0.38 = 6
0.53 = 1.396
Fig. 96.
4.16 = 10.946
4.33 C= 35262
2.33 = 8.756
5.125=19.266
0.31 = 1.166
= 6
= 2.486
6.67 = 25.076
6.23 (7=44176
RESISTANCE TO CROSS-BREAKING AND SHEARING.
55
10
r^
CD
CO
a
(M
JO O
56
EESISTANCE TO CROSS-BREAKING AND SHEARING.
-5 ^
o"*
C -<3j
rn ~
&
S
jo O
8
RESISTANCE TO CROSS-BREAKING AND SHEARING.
57
o
CD
r-i W) ^ T3
2 >> fl
Cf- -FH -ua o O>
00
^D
S 1
f 1 IS!
3
-s
a
CX .2 cj a K
^
T 1
2 H H g^
O
rH
a g
5 .2
CN O -
1-1 -Is
I S r& O
CD CD CQ
T-\ "* "
^
-j
e8
|
a
3
1 S
S
o
CD
O M
58 RESISTANCE TO CROSS- BREAKING AND SHEARING.
Load concentrated at center: W= r-, or K l L W.
t
Beam fixed at one end; principal flange at top.
Load equally distributed: TF= -^-p or JT 1 = 2 .1. W.
. V
K l
Load concentrated at free end: W -7-7-, or K l = 4 .1. W.
. 4 .1
[NOTE. The more the sectional area is contained in coefficient ITi, the
more is the section economical.]
EXAMPLE. Section No. 34. Load equally distributed; beam
supported at both ends; thickness of web = 1 inch: thickness
of flange = 1} inch; height = 10 inches; width of flange = 5.9
inches. Distance between supports = 20 feet = 240 inches.
K l 658
W = -jy- = ^- = 5.48 tons capacity.
T Tfi
The moment of resistance of cross-section = - r-
RESISTANCE TO CROSS-BREAKING AND SHEARING.
Number of
section.
^3
2P-S
|IB
Sectional
area in
square
incnes.
Coefficient
K.I
Fig. 102.
1
6
5.0
10.0
238
2
6}
5.2
10.7
280
^ ft
^ i
3
7
5.5
11.5
322
^ j
JT
4
7}
5.7
12.2
364
1
5
8
6.0
13
420
N**.
_Z"]lpl^iP^ _.$.,
6
gi
6.2
13.7
476
Fig. 103.
7
9
6.5
14.5
532
\
lj.
8
9}
6.7
15.2
602
9
10
6.9
15.9
672
4c
10
10}
7.1
16.6
742
11
11
7.4
17.4
812
>
^^^i _ N /___
12
11}
7.6
18 1
882
K""*" -4
13
12
79
18.9
966
Fig. 104.
t27
14
12}
8.1
19.6
1050
!
*
15
13
8.4
20.4
1134
^
i
16
13}
8.6
21.1
1232
%
\ . ~
17
14
8.8
21.8
1316
%
i
1 Q
1/11
5^^^4^~. 1 "
lo
14} ;
9.0
22.5
1428
k-"--B -i"
19
15
9.3
23.3
1526
Fig. 105.
20
15}
9.5
24.0
1624
: * I F
21
16
9.8
24.8
1750
.: 1 I
/ ! \ -\
22
16}
10.0
25.5
1848
\k
I !
23
17
10.3
26.3
1960
1 !
24
17}
10.5
27.0
2086
_ .. J. " ;/;:,;% *
[
jg ->;_
25
18
10.8
27.8
2212
60
RESISTANCE TO CROSS-BREAKING AND SHEARING.
Number of
section.
.2
Width B
of lower
flange in
inches.
Sectional
area in
square
inches.
Coefficient
K.i
Fig. 106.
26
6
4.5
10.4
224
i- -->
27
6}
4.6
11.1
266
t i
28
7
4.8
11.8
322
y-
29
7j
5.0
12.5
364
- \
P
30
8
5.2
13.2
420
31
gl
5.4
13.9
476
Fig. 107.
32
2
9
5.6
14.7
532
\J"/
33
9}
5.7
15.4
588
P
-- -y
34
10
5.9
16.2
658
"
35
10$
6.1
16.9
728
36
11
6.3
17.6
798
|
i
1
37
11}
6.5
18.3
882
,- k >^- >;
38
12
6.7
19.1
952
Fig. 108.
39
12}
6.9
19.8
1036
^
ir
40
13
7.1
20.6
1134
|
i
41
13}
7.3
21.3
1218
rr
,
42
14
7.5
22.1
1316
I
^
1 !
43
14}
7.7
22.8
1414
E]l^
!<
:. -#- >! ?*
44
15
7.9
23.6
1512
Fig. 109.
45
15J
8.0
24.3
1610
>i \/gi
i IT
46
47
16
16J
8.2
8.4
25.1
25.8
1722
1834
i
48
17
8.6
26.5
1946
1 *
^ 1
1
49
1VJ
8.8
27.2
2072
sk
KA
I Q
9.0
28.0
2198
,_..-S- H-
O\J
J.O
RESISTANCE TO CKOSS-BREAKING AND SHEAEING.
61
Number of
section.
Height H
in inches.
*!t
Sectional
area in
square
inches.
Coefficient
Ji
I
ig. 110.
51
6
4.2
10.5
224
\
2 !
1" T
52
6}
4.3
11.4
266
1 \
% i
53
7
4.5
12.3
308
" f
54
7}
4.6
12.9
364
^ I
55
8
4.7
13.6
406
z^flltll
^
1 A O
K
B >
56
8 i
4.8
14.3
462
I
fy. 111.
57
9
5.0
15.0
532
V
7
58
91
5.1
15.7
588
m
%
1
\
59
10
5.3
16.5
658
1
J
60
10}
5.4
17.2
728
1
\
\
r^r,
61
11
5.6
17.9
798
62
11}
5.7
18.6
868
*
^-- -
63
12
5.9
19.4
952
f
7 ^. 112.
64
12}
6.0
20.1
1036
p"~*
^
65
13
6.3
20.9
1120
3T
66
13}
6.4
21.6
1204
M
w
!
67
14
6.6
22.4
1302
W
1 ^
\ -**
i A f\r\
HHf
wSm^.
68
14}
6.7
23.1
14UU
<_ J
J- >|
69
15
6.9
23.8
1498
J
V 113.
70
15}
7.0
24.5
1610
\jj
_^
i f
hf Q
oc q
i >7ns
1
lo
/ .4
ZiO . O
1 <UO
1
\
72
16}
7.3
26.0
1820
i
73
17
7.5
26.8
1932
i
}
1
74
17}
7.7
27.5
2058
4^
fer-L
B- !
75
18
7.9
28.3
2184
62 RESISTANCE TO CROSS -BREAKING AND SHEARING;
o .
^g
c^c
-so
tt
s
^ o
31&I
SM 1
I SgJ
o^
*
a>
W.S
1 * %-
o
o
Fig. 114.
I frr?
76
6
4.0
12.0
224
i
77
7
4.1
13.1
308
i
*
78
8
4.2
14.4
406
-F?>. 115.
79
9
4.4
15.7
518
VT
~A~
1 i
80
10
4.6
17.1
644
*
| I
81
11
4.8
18.6
784
"|P|| ~~] I
i *- -H
82
12
5.0
20.0
938
%, 116.
1" ^ ""
i
83
13
5.2
21.4
1106
aJ,
^
84
14
5.5
22.9
1288
1
TS2P 7
<L jB >|
85
15
5.7
24.4
1484
Fig. 117.
^ ^T-
86
16
5.9
25.8
1694
1 1 1
i 1L
87
17
6.2
27.3
1918
1 1 i
i_
88
18
6.4
28.8
2156
^---^""^
RESISTANCE TO CROSS-BREAKING AND SHEARING.
63
Number of
section.
51
II
"o
gj
(33 ^ fl
5 -^ rf^
Sectional
area in
square
incnes.
Coefficient
JZX
Fig. 118.
89
6
5.6
12.9
294
W. ^
90
6}
5.8
13,8
336
I U I I
91
7
6.0
14.7
392
B
_
92
ff
6.2
15.5
448
v
93
8
6.4
16.4
518
j0gH|BHB-i* -.
94
ii
6.6
17.3
588
s \e. .ft _>|
2
Fig.V&l
95
9
6.9
18.3
658
. //
96
9}
7.1
19.2
742
T
97
10
7.4
20.2
826
&
98
10}
7.6
21.1
910
99
11
7.9
22.1
1008
i
100
Hi
8.1
23.0
1106
.
101
12
8.4
23.9
1204
Fig. 120.
102
8.6
24.8
1302
w~ \~
103
13
8.9
25.8
1414
^ 7T
m JL
104
13}
9.1
26.7
1526
1 t - ^-" :
I !
105
1 HA
14
9.4
9r>
27.7
1652
1 7/2/1
i&"
lUb
*?J
. b
28.5
1754:
E- -J5- r^-
107
15
9.8
29.4
1890
Fig. 121.
\% " \%?
108
15}
10.0
30.3
2030
ITT
109
16
10.3
31.3
2156
% \
| i
110
16}
10.5
32.2
2296
tj *i
111
17
10.8
33.2
2436
1 !
112
171
11.0
34.1
2590
-i
113
18
11.3
35.0
2730
<____JB V;
64
RESISTANCE TO CROSS-BREAKING AND SHEARING.
I
\
ig. 122.
2%r
Number of
section.
1
X3 O bJD _e
T3~ G ^
Sectional
a r e a i n
square
inches.
Coefficient
jffl.
114
115
116
6
6}
7
5.3
5.4
5.6
13.6
14.4
15.3
280
336
392
I
J5
r
117
7}
5.7
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118
119
120
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17.8
18.7
518
588
658
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Fig. 123.
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6.4
19.6
742
1
122
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6.6
20.5
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; 124
11
7.0
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994
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125
126
127
128
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12
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13
7.2
7.4
7.6
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26.1
1092
1190
1288
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Fig. 124.
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129
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8.0
27.0
1512
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130
131
132
14
14}
15
8.2
8.4
8.6
27.9
28.8
29.8
1624
1750
1876
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Fig. 125.
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133
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8.8
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137
138
17
18
9.4
9.6
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34.4
35.3
2422
2562
2716
B-
EESISTANCE TO CROSS-BREAKING AND SHEARING. 65
Jf
I
Fig. 127.
W
Fig 128.
. 129.
o .
II
C o
I s
139
140
141
142
143
144
145
146
147
148
149
150
151
10
11
12
13
14
15
16
17
18
5.0
5.1
5.3
5.5
5.7
6.0
6.3
6.5
6.8
7.4
7.7
8.0
15.0
16.4
18.0
19 7
21.4
23.2
25.0
26.8
28.6
30.5
32.3
34.2
36.0
66
RESISTANCE TO CROSS -BREAKING AND SHEARING.
VH
N
(
^>
2 d
2
2*00
^ .5 2 cr
d
2
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Fig. 130.
152
6
6.3
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153
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6.5
17.2
406
1
154
7
6.7
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155
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6.9
19.3
546
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156
8
7.1
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157
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7.3
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700
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</. 131.
158
9
7.5
22.6
784
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159
91
7.7
23.6
882
:
A
:
JB:
160
161
10
8.0
8.2
24.7
25.8
980
1078
;
162
11
8.4
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1190
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166
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167
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168
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9.8
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10.0
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2100
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170
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10.3
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10.5
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172
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174
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11.3
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1
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175
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11.5
41.3
3080
1 1 i_
176
18
11.8
42.5
3262
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RESISTANCE TO CEOSS-BEEAKING AND SHEARING. 67
Number of
section.
b/Ofl
w.s
*!
Sectional
area in
square
inches.
Coefficient
AI.
Fig. 134.
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177
6
6.0
18.0
336
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178
7
6.1
19.7
402
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179
8
6.3
21.6
602
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Fig. 135.
180
9
6.6
23.6
770
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181
10
6.9
25.7
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182
11
7.2
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Fig. 136.
183
12
7.5
30.0
1400
1
f
184
13
7.8
32.2
1652
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ft
185
14
8.2
34.4
1932
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186
15
8.5
36.7
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Fig. 137.
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187
16
8.9
38.8
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188
17
9.2
41.0
2370
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to
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189
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9.6
43.2
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68
RESISTANCE TO CROSS-BREAKING AND SHEARING.
Number of
section.
ta5
^ o>
Xi^C SJC^
Sectional
area in
square
inches.
Coefficient
Kl.
Fig. 138.
190
6
7.0
21.0
392
H -A-
^^ *i
191
7
7.1
23.0
532
2
i
192
8
7.4
25.2
714
Fig. 139.
193
9
7.7
27.6
896
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194
10
8.0
30.0
1120
2"
i ^
195
11
8.4
32.5
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196
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8.8
35.0
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197
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9.1
37.5
1932
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198
14
9.6
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199
15
10.0
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2590
Fig. 141.
^
^"/ M/e"/
\ IT
1 -^f
200
201
16
17
10.4
10.8
45.2
47.8
2954
3346
v
202
18
11.2
50.4
3766
22* >i
RESISTANCE TO CROSS-BREAKING AND SHEARING.
Number of
section.
^ 03
ill
Sectional
area in
square
incnes.
Coefficient
JO.
^. 142.
L5$
203
6
8.0
24.0
448
/</ : . A
^8
i
204
7
8.1
26.2
616
3T
r /.,,/
yjjl \
205
S
8.4
28.8
812
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, J
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206
9
8.8
31.5
1036
1
2",
P !
207
10
9.1
34.3
1274
pP
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ft i
208
11
9.6
37.1
1554
I
i
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209
12
10.0
40.0
1862
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2"
m
A
210
13
10.4
42.9
2198
fc
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211
14
10.9
45.8
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212
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11.4
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213
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11.8
51.7
3374
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214
17
12.3
54.6
3822
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1 1
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215
18
12.8
57.6
4298
,--
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70
EESISTANCE TO CROSS- BREAKING AND SHEARING.
Number of
section.
^ 03
-C O
^ oqs.rt
2 1||
Sectional
area in
square
inches.
Coefficient
JT1.
Fig. 146.
1
6
6
1.4
11.4
294
r
2
6
7
1.9
12.9
336
1
3
6
8
2.3
14.3
392
1/7 1 ^\
4
6
9
2.7
15.7
448
^
5
6
10
3.1
17.1
504
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6
6
11
36
18.6
560
Fig. 147.
7
6
12
4.0
20.0
602
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8
6
13
4.4
21.4
658
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9
6
14
4.9
22.9
714
r\ #
10
6
15
5.3
24.3
770
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11
6
16
5.7
25.7
826
1 9
&
1 7
69
97 9
QGQ
Fig. 148.
l^j
13
o
6
1 <
18
. /a
6.6
At.Zi
28.6
ODO
924
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14
7
6
1.2
12 2
350
1 !
15
7
7
1.7
13.7
420
Jr
2-1
16
7
8
2.1
15.1
490
%
I i ^
17
H
9
2.6
16.6
560
^i r _Z"
18
7
10
3.0
18.0
616
[^ JFJ. ^j ^>.
Fig. 149.
19
7
11
3.4
19.4
686
k^^l i^.^^,
20
7
12
3.9
20.9
756
i^S ^wiffi
21
7
13
4.3
22.3
826
1 ff"
22
23
7
7
14
15
4.8
5.2
23.8
25.2
896
966
L-LJ
24
7
16
5.7
26.7
1022
^/" ^ \ /./ :
25
7
17
6.1
28.1
1092
RESISTANCE TO CROSS-BREAKING AND SHEARING.
Number of
section.
^ o5
gc.
it ll
5 2*
CS 2 JQ
Coefficient
*> 1
Fig. 146.
26
7
18
6.5
29.5
1162
.rfeipi A"
27
8
6
1.0
13.0
434
28
8
7
1.5
14.5
504
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29
8
8
1.9
15.9
588
1
30
1
8
9
i n
2.4
20
17.4
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672
k Br- >i
Fig. 147.
ol
32
8
1U
11
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3.3
lo .0
20.3
826
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33
8
12
3.7
21.7
910
j
i i
34
8
13
4.2
23.2
994
35
8
14
4.6
24.6
1078
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36
8
15
5.1
26.1
1148
j[" \% , \~fy
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lo
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38
8
17
6.0
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1316
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39
8
18
6.4
30.4
1386
z;-j
40
9
7
1.3
15.3
588
77"
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41
9
8
1.7
16.7
686
42
9
9
2.2
18.2
784
l^^iliil[i$ltl F 7"
43
9
10
2.6
19.6
868
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44
9
11
3.1
21.1
966
k^ * K ^ y
45
9
12
3.5
22.5
1064
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46
9
13
4.1
24.1
1162
f^\m i ^j ^*.
// /^/; % //
47
9
14
4.5
25.5
1246
^| -f" |^
48
9
15
4.9
26.9
1344
1 1 1
49
9
16
5.4
28.4
1442
:;;:/ ;: ^/^tff^
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50
9
17
5.8
29.8
1526
RESISTANCE TO CROSS-BREAKING AND SHEARING.
.
tH fl
^ S
^.2 .fS-S .
C <D
"3
O) O
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5 -rH
iol.s
r % .
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s
55
W.^H
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Fig. 146.
51
9
18
6.3
31.3
1624
v
^-b->\
J
A
52
10
7
1.1
16.1
672
i
i
1
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53
10
8
1.5
17.5
784
1"
1
ZT
54
10
9
2.0
19.0
896
1
i
i
55
10
10
2.4
20.4
1008
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.
1
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56
10
11
2.9
21.9
1106
K -X>-, H
Fig. 147.
57
10
12
3.3
23.3
1218
J?
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58
10
13
3 8
24.8
1330
i
i
59
10
14
4.3
26.3
1428
r" -
i
TJ
60
10
15
4.7
27.7
1540
r
^
y //y////y
~
i
61
10
16
5.2
29.2
1652
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Fia. 148
;
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63
10
10
18
5.7
6.1
30.7
32.1
1750
1862
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wmi"*"
64
11
8
1.3
18.3
896
^
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65
11
9
1.7
19.7
1008
"
j
r
JT
66
11
10
2.2
21.2
1134
?
i
67
11
11
2.7
22.7
1246
^
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68
11
12
3.1
24.7
1372
k jj >i
.%. 149.
69
11
13
3.6
25.6
1498
Kjh !<<>!
70
11
14
4.1
27.1
1610
i tir
"~A~^
~
JZS53TTH
2y/tdJL
71
11
15
4.5 28.5
1736
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-
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72
11
16
5.0
30.0
1862
%
T
|
\
73
11
17
5.5
31.5
1974
i
74
11
18
5.9
32.9
2100
./__
. \
75
12
8
1.1
19.1
994
*
^^^
-3>
EESISTANGE TO CROSS-BREAKING AND SHEARING.
73
Number of
section.
o
^ oc.
*|s
Sectional
area in
square
inches.
Coefficient
Ki.
Fig. 146.
76
12
9
1.5
20.5
1120
S^ U ~b ;
"A"
77
12
10
2.0
22.0
1260
1
78
12
11
2.5
23.5
1400
^1
i
79
12
12
2.9
24.9
1526
1
i
80
12
13
3.4
26.4
1666
1
i
19
3.9
97 Q
1 ROfi
JSflr. 147.
82
.LvH
12
15
4.3
at j
29.3
J.OUO
1932
^JjJ"*1_.-
83
12
16
4.8
30.8
2072
./.jp
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84
12
17
5.3
32.3
2198
f\
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85
12
18
5.7
33.7
2338
v
i
^^
t
86
13
9
1.3
21.3
1232
V 1
IL ^^^MM^M
[i
07
1 3
1 A
i 8
99 ft
1 QQA
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10
iv
loob
jPia. 148.
88
13
11
2.2
24.2
1540
-->!
89
13
12
2.7
25.7
1680
rPl
90
91
13
13
13
14
3.2
3.7
27.2
28.7
1834
1988
1 j
92
13
15
4.1
30.1
2128
r
93
13
16
4.6
31.6
2282
P . II s
j%. 149.
94
13
17
5.1
33.1
2422
k3-* i/
95
13
18
5.5
34.5
2576
i lBl ~*~1P
w%r
96
14
9
1.1
22.1
1358
"J if
/!t J5t ?|
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97
98
14
14
10
11
1.5
2.0
23.5
25.0
1512
1680
i ! i
1 ! t.
99
14
12
2.5
26.5
1834
nnnnn^i
<. _g. 5,1
100
14
13
3.0
28.0
2002
74
RESISTANCE TO CROSS-BREAKING AND SHEARING.
Number of
section.
tB a
^ CD
-C O
B.S
11
^ ol.S
Sectional
area in
square
inches.
Coefficient
J2X
Fig. 146.
101
14
14
3.4
29.4
2170
-A"
102
14
15
3.9
30.9
2324
103
14
16
4.4
32.4
2492
^| 2 r
104
14
17
4.8
33.8
2660
|
105
14
18
5.3
35.3
2814
7/ 7 ill HI v
106
15
10
1.3
24.3
1638
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Fig. 147.
107
15
11
1.8
25.8
1820
>J<--5->!
108
15
12
2.3
27.3
2002
/" |p """ ^""
m
109
15
13
2.7
28.7
2170
p !
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110
15
14
3.2
30.2
2352
^ 1
P
111
15
15
3.7
31.7
2520
^P
112
15
16
4.2
33.2
2702
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113
15
17
4.6
34.6
2884
|-5^j
114
15
18
5.1
36.1
3052
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115
16
10
1.1
25.1
1764
rl k
116
16
11
1.6
26.6
1960
I
1
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117
16
12
2.0
28.0
2156
7"
118
16"
13
2.5
29.5
2338
j< _^, ;>i v >-
^. 149.
119
16
14
3.0
31.0
2534
i<3>- K^^-
120
16
15
3.5
32.5
2730
i ^ """"
121
16
16
3.9
33.9
2912
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122
16
17
4.4
35.4
3108
1 p
123
16
18
4.9
36.9
3290
124
17
11
1.3
27.3
2100
J" V:M$s.
K- --S- H
125
17
12
1.8
28.8
2310
EESISTANCE TO CROSS-BREAKING AND SHEARING.
75
Number of
section.
S
W.2
^*0?B ^
s|||
5 _, ~f c5
J J_ CT p
X! * K ~ <
Coefficient
jsn.
Fig. 146.
126
17
13
2.3
30.3
2506
127
17
14
2.8
31.8
2716
r| jr
I
v ^
128
17
15
3.2
33.2
2926
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Fig. 147.
129
17
16
3.7
34.7
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,jn ^
Jll f" -TT
L 5p ^n
130
17
17
4.2
36.2
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y//, \
?3LmmA
131
17
18
4.7
37.7
3542
">-- -=# ->s
132
18
11
1.1
28.1
2240
^. 148.
jlilz-*-
133
18
12
1.6
29.6
2464
1 * s
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134
18
13
2.0
31.0
2688
1 \
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1 A
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lo
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fc 33 >j "
Fig. 149.
136
18
15
3.0
34.0
3122
&--^ j s^
137
18
16
3.5
35.5
3346
*\ ^
138
18
17
4.0
37.0
3556
139
18
18
4.4
38.4
3780
fcr& >\
76
RESISTANCE TO CROSS-BREAKING AND SHEARING.
o
^i
^s.s .
o *-> a
g33 .
-^ a
a
o
c o
JJH -i-i i-i y
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Number of
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RESISTANCE TO CROSS-BREAKING AND SHEARING.
85
Number of
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Coefficient
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RESISTANCE TO CROSS-BREAKING AND SHEARING.
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10
2.7
29.4
574
!
573
6
11
3.0
32.0
630
. .
2
-2"
574
6
12
3.3
34.6
700
-1 1
575
6
13
3.7
37.4
756
1
1 ^
A C\ C\
OO/j
Fig. 163.
576
577
6
15
4 .0
4.3
40. U
42.6
bZb
882
"F
P:
*
_..
...
~
578
6
16
4.7
45.5
952
i
4
ir
i
579
6
17
5.0
48.0
1008
2
p
&
580
6
18
5.3
50.6
1064
-
JW
^v"
~~~
1
i
i
581
7
9
2.3
28.6
686
Ej^
~i^
582
7
1
2.7
31.4
770
-%. 164.
583
/
7
11
3.0
34.0
854
U--Z
-,
584
7
12
3.4
36.8
938
f \
_?-.
t
585
7
13
3.8
39.6
1036
f/
586
7
14
4.1
42.2
1120
2
s
^
T
587
7
15
4.5
45.0
1204
1
588
7
16
4.9
47.8
1288
M
m
i
1
2"
589
7
17
5.2
50.4
1372
-%. 165.
590
i
7
18
5.6
53.2
1456
, ^ ,
5
591
8
9
2.2
30.4
868
?L
^~^r
592
8
10
2.6
33.2
980
IIP
P !
593
8
11
2.9
35.8
1092
. .-
<<
7
|
!
594
8
12
3.3
38.6
1204
i
i
595
8
13
3.7
41.4
1302
|
i
596
8
14
4.1
44.2
1414
-
s
?
..
597
8
15
4.5
47.0
1526
fr
-
^..
--
-*_
RESISTANCE TO CROSS BREAKING AND SHEARING.
95
mberof
action.
5j
-H*
5 1||
||||
a
J
si.S
^ c.=
$ Cq5
s-5
o
jz
?. 162.
598
8
16
4.9
49.8
1638
-g->_
2
" /7 r
~~^"
599
8
17
5.3
52.6
1750
_!_,
if
!
600
8
18
5.7
55.4
1848
2"
m
JZ"
601
9
9
2.1
32.2
1064
9
i
3F
i
602
9
10
2.5
35.0
1204
"if
~
|^-
603
9
11
2.9
37.8
1330
2^
g. 163.
604
9
12
3.3
40.6
1470
"6
>\
605
9
13
3.7
43.4
I^Qft
1
1
w\
!
606
9
14
4.1
46.2
1736
2
p
-^
607
9
15
4.5
49.0
1876
~/T(^
1
%%%%^
608
9
16
4.9
51.8
2002
11
i
609
Q
n
r q
KA_ f>
i<
j
"
U
O . o
O . O
2142
F*
1. 164.
610
9
18
5.7
57.4
2282
!<--Z
,._,,,
611
10
10
2.4
36.8
1414
1
IT
612
10
11
2.8
39.6
1582
"H
i
613
10
12
3.2
42.4
1736
2 1
"F
614
10
13
3.6
45.2
1904
Hf
i
615
in
i d.
4 A
AQ r\
_
1
M
/ <MftMWf/t
2"
IU
1
.0
4o . U
2058
616
10
i ^
A A
50 H
o o o/~
7
j >
1 U
T: . TC
^c
^
. 165.
617
10
16
4.8
53.6
2380
J
JJ
i
M
!
618
619
10
10
17
18
5.2
5.7
56.4
59.4
2595
2702
1
//I
620
11
10
2.2
38.4
1638
7 "i
7 |
JZ"
621
11
11
2-6
41.2
1820
I
i
622
11
12
3.0
44.0
2016
m
8
i
623
11
13
3.5
47.0
2198
j<-
i
3 >
624
11
14
3.9
49.8
2380
96
RESISTANCE TO CROSS-BREAKING AND SHEARING.
Number of
section.
^ S
*|
.?5I|
Sectional
area in
square
inches.
Coefficient
X
Fig. 162.
625
11
15
4.3
52.6
2576
g
/.
w
3
"T
626
11
16
4.7
55.4
2758
t
627
11
17
5.1
58.2
2954
t
2
^
JL
628
11
18
5.6
61.2
3136
,.
11.
629
630
12
12
11
12
2.4
2.9
42.8
45.8
2086
2296
Fig. 163.
631
12
13
3.3
48.1
2506
~?W
o
~7/ /
%jjr
_._
.
-A
632
12
14
3.7
51.4
2716
J
2
1
633
12
15
4.1
54.2
2940
s
634
12
16
4.6
57.2
3150
1
\ %
1
635
12
17
5.0
60.0
3360
JmMM^m^
j
m
%%;
-y-
636
12
18
5.4
62.8
3570
!< jg
>
Fig. 164.
637
13
11
2.2
44.4
2338
,-N|
638
13
12
2.7
47.4
2576
<
. 1
ii
~2
T
639
13
13
3.1
50.2
2814
|
fa,
i
640
641
13
13
14
15
3.5
4.0
53.0
56.0
3052
3290
w.
y
i
i
642
13
16
4.4
58.8
3528
/
t\
2"
m
&
j
643
13
17
4.9
61.8
3780
Fig. 165.
644
13
18
5.3
64.6
4018
i J-i
5 1
645
14
11
2.0
46.0
2604
646
14
12
2.5
49.0
2870
:
i~r
647
14
13
2.9
51.8
3136
,1
n
i
1
i
648
14
14
3.4
54.8
3402
::
i
649
14
15
3.8
57.6
3668
i
650
14
16
4.2
60.4
3934
^
&
: -^
<~
; v
.y _
651
14
17
4.7
63.4
4208
<-
B-
->
1
RESISTANCE TO CROSS-BREAKING AND SHEARING.
Number of
section.
j="o
.Sf.S
5 ^
-Ifs
~ G
T3 ^ 5="^
c rt * J
Coefficient
Jp.
Fig. 162.
652
14
18
5.1
66.2
4152
f
653
15
12
2.3
50.6
3164
l^M^I 1
W# :
654
15
13
2.7
51.4
3444
<2 ||p -uL
655
15
14
3.2
56.4
3738
%ss 1
656
15
15
3.6
59.2
4032
yOC .
Fig. 163.
657
658
15
15
16
17
4.1
4.5
62.2
65.0
4296
4606
}<- $--->!
659
15
18
4.9
67.8
4900
T
660
16
13
2.5
55.0
3742
sm &
661
16
14
3.0
58.0
4074
\WM \
j%^ \
662
16
15
3.4
60.8
4396
663
16
Q Q
CO
4.71 Q
Fig. 164.
664
16
17
O . <J
4.3
DO . O
66.6
tfc / io
5026
i
^^~ - K-
665
16
18
4.8
69.6
5348
m^\ j :
666
17
13
2.3
56.6
4060
w
2
m &
667
17
14
2.8
59.6
4410
fm
m \
668
17
15
3.2
62.4
4760
mm%^^%^\ / ^~ 3 it
tm^mM/2/m-
669
17
16
3.7
65.4
5110
Fig. 165.
670
17
17
4.1
68.2
5460
I \ \b \
671
17
18
4.6
71.2
5810
EJ { ^~\
672
18
13
2.1
58.2
43^2
tr
v- % > . \>j^""^~"
.H
^ !
673
18
14
2.5
61.0
4746
//( *
^ 41
7 I TT
^ IT
674
675
18
18
15
16
3.0
3.4
64.0
66.8
5124
5502
^ ^
^illlil^ltllll;
676
18
17
3.9
69.8
5080
..
>--B H
677
18
18
4.4
72.8
6258
RESISTANCE TO CROSS BREAKING AND SHEARING.
STRENGTH OF WOODEN BEAMS.
Capacity W in Ibs. of American white and yellow pine beams, joists,
&c., from 1" x 1" to 15 x 15 in.
The modulus of rupture is taken at-
- = 1250 Ibs.. or 8 times safety,
K = tabulated coefficient, to be divided by
I = distance between supports in inches, or length of beams in inches
from support to free end of beam.
a
Coefficient
11
Height in
"o c
1
2
3
4
5
6
7
1
1666
6666
15000
26666
41666
60000
81666
1_1^
2500
10000
22500
39999
62499
90000
122499
2
3333
13333
30000
53333
83333
120000
163333
2*^
4166
16666
37500
66666
104166
150000
204166
3
5000
19999
45000
80000
124999
180000
244999
31^
5833
23333
52700
93333
145833
210000
285833
4 *
6666
26666
60000
106666
166666
240000
326(566
41^
7499
29999
67500
119999
187499
270000
367499
5
8333
33333
75000
133333
208333
300000
408333
5%
9166
36666
82500
146666
229166
330000
449166
6
10000
39999
90000
159999
249999
360000
489999
6%
10833
43333
. 97500
173333
270833
390000
530833
7
11666
46666
105000
186666
291666
420000
571666
71^
12500
49999
112600
199999
312499
450000
612499
8
13333
53333
120000
213333
333333
480000
653333
8 14
14166
566(56
127500
22(5666
354166
510000
694166
9 "
14998
59999
135000
239999
374999
540000
734999
9%
15831
63333
142500
253333
395833
570000
775833
10
16666
66(566
150000
266666
416666
600000
816666
10/4
17500
69999
157500
279999
437499
630000
857599
11
18333
73333
1(55000
293333
458333
660000
898533
11*^
19166
76666
172500
306666
479166
690000
939366
12 *
20000
79999
180000
319999
499999
720000
979999
12^4
20833
83333
187500
333333
520833
750000
1020833
13 "
21666
86066
195000
346666
541666
780000
1061666
1314
22500
89)99
202500
359999
562499
810000
1102499
14
23333
93333
210000
373333
583333
840000
1143333
14K
24166
96(566
217500
386666
604166
870900
1184166
15 *
25000
99099
225000
399999
624999
900000
1224999
RESISTANCE TO CEOSS- BREAKING AND SHEARING.
BEAMS SUPPORTED AT THE ENDS.
K
Load equally distributed, W = or K f = I W. I
K
Load concentrated at centre, W== or K 21W. 2
21
BEAMS FIXED AT ONE END.
K
Load equally distributed, W = or Kf UW. 3
K
Load concentrated at free end, W = or K = SI W. 4
inches.
8
9
10
11
12
13
14
15
106666
135000
166666
201757
240000
281666
326666
375000
159999
202500
249999
302636
360000
422499
489999
562500
213333
270000
333333
403515
480000
563333
653333
750000
266666
337500
416666
504393
600000
704166
816666
937500
319999
405000
499999
605272
720000
844999
979999
1125000
373333
472500
583333
706151
840000
985833
1143333
1312500
426666
540000
666666
807030
960000
1126666
1306666
1500000
479999
607500
749999
907908
1080000
1267499
1469999
1687500
533333
675000
833333
1008787
1200000
1408333
1633333
1875000
586666
742500
916666
1109666
1320000
1549166
1796666
2062500
639999
810000
999999
1210545
1440000
1689999
1959999
2250000
693333
877500
1083333
1311423
1560000
1830833
2123333
2437500
746666
945000
1166666
1412302
1680000
1971666
2286666
2625000
799999
1012500
1249999
1513181
1800000
2112499
2449999
2812500
853333
1080000
1333333
1614060
1920000
2253333
2613333
3000000
906666
1147500
1416666
1714938
2040000
2394166
2776666
3187500
959999
1215000
1499999
1815817
2160000
2534999
2939999
3375000
1013333
1232500
1583333
1916696
2280000
2675833
3103333
3562500
1066666
1350000
1666666
2017575
2400000
2816666
3266666
3750000
1119999
1417500
1749999
2118453
2520000
2957499
3429999
3937500
1173333
1485000
1833333
2219332
2640000
3098333
3593333
4125000
1226666
1552500
1910666
2320211
2760000
3239166
3756660
4312500
1279999
1620000
1999999
2421090
2880000
3379999
3919999
4500000
1333333
1687500
2083333
2521968
3000000
3520833
4083333
4(587500
1386666
1755000
216(5666
2622847
3120000
3661666
4246666
4875000
1439999
1822500
2249999
2723726
3240000
3802499
4409999
5062500
1493333
1890000
2333333
2824605
3360000
3943333
4573333
5250000
1546666
1957500
2416666
2925483
3480000
4084166
4736(566
5437500
1599999
2025000
2499999
3026362
3600000
4224999
4899999
5625000
100
PEESSURE OF SUPPORTS.
PRESSURE ON SUPPORTS.
REACTION OF SUPPORTS.
For a continuous beam, horizontal or inclined. Load W,
equally distributed, and supports equal distance apart. Appli
cable to trussed beams, rafters, or beams supported by three or
more supports.
Reference. (Fig. 166.)
W/ = Weight of load per unit of length in Ibs.
L = Distance between supports in units of length.
P t Pi, P-2, = Pressure on supports in Ibs., counting frorn end
support to center of beam.
M t M lt M 2 = Moments of rupture over supports.
ra, m 11 m 2 = Moments of rupture between supports.
I, l l} 1 2 = The distance from a support to section where
moments m, m lt m z occur.
By this table the pressure upon any support, from 3 to 9 in
number, can be ascertained; also the moments of rupture. The
table is used in calculating the strains in roof trusses, &c.
Fig. 166.
Reactions
or pressure.
Number of Supports.
3
4
5
7
9
P
I
0.375 W t L
1.25 W,L
0.4 W,L
1.1 W,L
0.3929TF,
1.1429 W,L
0.9286 W,L
0.3942 W,L
1.1346 W t L
0.9615 W,L
1.0192 W,L
0.3^43 W,L
1.13401^^
0.9G29 W t L
U)103W,L
0.9948 W t L
M l
M 2
M 3
Ml
0.125 W,L2
0.1 17^2
0.1071 W,L2
0.0714 W t L 2
0.1058 WL 2
0.0769 W t L 2
0.0865 W t L 2
0.1057 W t L 2
0.0773 W t L 2
0.0850 IF, L 2
0.0824^1/2
PRESS flUE ON SUPPORTS.
101
Reactions
or pressure.
Number of Supports.
3
4
5
7
9
m
mi
W*2
m s
0.0703 W t L 2
0.08 W t L i
0.025 W,Li
0.0772^1,2
0.0364^,1,2
0.0777 IF, 2
0.0340^1,2
0.0434:^1/2
0.0777 W 4 L 2
0.0339 ir,L 2
0.04381^1-2
0.0412 W t L 2
I
1
0.375 L
0.4 L
0.5 Z,
0.3928
0.535 L
0.3942 ^
0.5288 L
0.4903
0.3943 I/
0.5283 L
0.4922
0.5025 L
Reference. (Figs. 167, 168, and 169.)
W, TTj, PT 2 = Load in Ibs.
^, ^ 1} 1 2 = Dimensions in units of length.
P, P 1} P 2 = Pressure on supports in Ibs.
Three supports,
unequal distances
apart.
Fig. 168.
Load equally distributed:
One support, and
fixed at one end.
102 COMPRESSIVE STRAIN AND PRESSURE ON SUPPORTS.
ig. 169.
Load concentrated at free end:
One snpport,
and fixed at
one end.
COMPRESSIVE STRAIN AND PRESSURE ON SUPPORTS.
SLOPING BEAMS, RAFTERS, &c.
Load W equally distributed.
For the cross- breaking strain, the rafter, &c., is to be treated
as a horizontal beam of the length I. (See Compound Strains in
Beam, &c.)
Reference.
C= Compression in direction of beam.
jy= Horizontal strain acting on support.
V= Pressure on supports.
Lower end supported vertically and horizontally ; upper end
resting on inclined support :
Fig. 170.
(7 = - sin . v
W
H= - sin.-y cos.-y
2
W
x =-^- (cos.-y) 2
RESISTANCE TO CRUSE ING.
Upper end fized; lower end supported horizontally :
Fig. 171.
103
w
~2~
Upper end resting against a vertical surface ; lower end sup-
>orted vertically and horizontally :
port
rtically and horizontally
Fig. 172.
2 sin. u
W
H=- cotg v
RESISTANCE TO CRUSHING.
STRENGTH OF COLUMNS, PILLARS, AND STRUTS.
Reference.
A = Area of cross-section in inches.
C== Coefficient, depending on the material.
I = Least moment of inertia of cross-section.
W = Capacity of column, pillar, or strut in Ibs.
a = Coefficient, depending on the material in respect to flexure.
c = Coefficient, depending on the material.
h = The least dimension across the section in inches.
k = Factor of safety.
I = Length of column, &c., in inches.
r = Least radius of gyration.
104
RESISTANCE TO CRUSHING.
To find the square of the radius of gyration (r 2 ) of a plane
about a given axis, divide the least moment of inertia by the
sectional area of the plane ; that is, r 2 = .
Values of For Malleable Iron. For Cast Iron.
C= 36,000 Ibs. 80,000 Ibs.
c= 36,000 " 3,200 "
a== 0.000333 0.0025
For Dry Timber.
7,200 Ibs.
3,000 "
0.004
The factor of safety k should be, for wrought iron = 6; for cast
iron = 8; for timber = 10. This applies to moving loads.
Case 1.
Rounded or hinged at both ends, as per
Fig. 173.
For square, rectangular, or circular cross-section :
ir=i._ c ^_
For any other cross-section:
= ~l 7^"
i+-5r
Case 2.
Fixed, or having a flat base at one end, and rounded or hinged
at the other, as per
Fig. 174.
For square, rectangular, or circular cross-section :
W = A
k p
For any other cross-section:
1 CA
1 +
16 J 2
9.c.r 2
RESISTANCE TO CRUSHING.
105
Case 3.
Fixed, or having flat bases at both ends, as per
Fig. 175.
For square, rectangular, or circular cross-section :
1 CA
l+aJ ^-
For any other cross-section:
1 OA
1 +
EXAMPLES.
Case 1.
Rounded at both ends:
What is the capacity of a u rought-iron strut of the annexed
figure and dimensions?
I = 10 feet = 120 inches.
A = 4.68 inches.
r __ 0.9 X 3.58+5.1x0.38
36000 X 4.68
1 +
120 2
36000 X 0.689
12
168480
57600
~^~ 124804"
= 3.227
"37322"
The same as above, in Case 3, fixed at both ends:
36000 x 4.69 168480
1 +
168480
36000 x 0.689
1 +
14400
24804
106
RESISTANCE TO CRUSHING.
For the annexed figure and dimensions ; otherwise, same as
above :
A = 7 inches.
Case 1.
\2: T- 1 X 43+3 x I 3
Rounded at both ends :
Fig. 177.
1 =
12
= 5.(
36000 x 7 , 252000
-T*-1W~ = * 8 = 21 00 lbs
. ^ A. ^ v O
X _
^ ^ 36000 X 0.8
Same as above, in Case 3, fixed at both ends :
36000 x 7 252000
i_i
36000 X 0.8
- = 1- - ~-r = 42 > 000 lbs -
1 . o
Fixed ends :
Case 3.
What is the capacity of a cast-iro7i pillar of the annexed figure
and dimensions?
1= 10 feet= 120 inches.
A = 11 inches.
i X 4 3 7 X 3 3
= 26.9
80000 X 11 880000
~T90~ * S~25" =
1+0.0025 r-f-
RESISTANCE TO CRUSHING.
107
For the annexed figure and dimensions; otherwise, same as
above.
Fig. 179.
V
TP=J-
A 28 inches.
80000 X 28
.
1 + 0.0025 J-
_2240000_ = 179)200U)3 .
1.5625
For the annexed figure and dimensions; otherwise, same as
above.
Fig. 180.
A = 22 inches.
80000 X 22
120 2
1 + 0.0025-
1760000
1.5625
= 140,800 Ibs.
To find the capacity of a Column, Pillar, or Strut of any
cross-section by the following Table :
Find how many times the least dimension h across the section
is contained in the length I of column, &c. that is, then
multiply the corresponding number on the same horizontal line,
under K" , by the sectional area of cross-section. This gives the
capacity in tons of 2,000 Ibs.
Let I = Length of column, &c.
h = Least dimension of cross-section.
K" = Capacity in tons of one square inch of cross-section, to
be multiplied by sectional area of desired cross-
section.
Various sections for which this table is applicable:
Fig. 181. Fig. 182.
108 RESISTANCE TO CEUSHING.
Fig. 183. Fig. 184.
Fig. 186.
Fig. 185.
Fig. 187.
Fig. 188.
[NOTE. This table is strictly correct, only for columns, Ac., with circular
or rectangular cross-section. As the error is small, it may be used for
any cross-section.]
Example explanatory of the following table.
What is the capacity of a cast-iron column 10 feet = 120 inches
long, fixed at both ends, and of the annexed cross-section and
dimensions?
Fig. 189.
7 19()
-f- = -=- = 40 K" for 40=1.000 tons.
h 3
Area= 6 inches.
W=. 6 X 1 = 6 tons, 8 times safety.
BESISTANCE TO CRUSHING.
Column, &c., fixed at both ends.
109
Cast Iron eight times safety.
Wrought Iron six times safety.
I
h
K"
I
h
K"
I
h
K"
I
h
K"
I
~h
K"
I
K"
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
1
4.987
25
1.951
49
0.714
1
2.999
25
2.487
49
1.674
o
4.950
26
1.858
50
0.689
2
2.996
26
2.452
50
1.644
3
4.890
27
1.771
51
0.666
3
2.991
27
2.418
51
1.615
4
4.807
28
1.689
52
0.644
4
2.984
28
2.383
52
1.585
5
4.705
29
1.611
53
0.623
5
2.975
29
2.348
53
1.557
6 | 4.587
30
1.538
54
0.603
6
2.964
30
2.313
54
1.529
7 | 4.450
31
1.469
55
0.584
7
2.953
31
2.277
55
1.501
8 I 4.310
32
1.404
56
0.565
8
2.938
32
2.242
56
1.474
9
10
4.158
4.000
33
34
1.343
1.285
57
58
0.548
0.531
9
10
2.921
2.905
33
34
2.206
2.172
57
58
1.4-18
1.422
11
3.838
35
1.230
59
0.515
11
2.885
35
2.136
59
1 396
12
3.676
36
1.179
60
0.500
12
2.863
36
2.101
60
1.371
13 ! 3.514
37
1.130
61
0.485
13
2.841
37
2.067
01
1.347
14 i 3.355
38
1.084
02
0.471
14
2.817
38
2.032
62
1.323
15 I 3.200
39
1.041
63
0.457
15
2.792
39
1.998
63
1.299
If. 3.048
40
1.000
64
0.445
16
2.766
40
1.963
64
1.276
17
2.902
41
0.961
65
0.432
17
2.738
41
1.930
65
1.253
18
2.762
42
0.924
66
0.420
18
2.711
42
1.896
66
1.228
19
2.628
43
0.889
67
0.409
19
2.680
43
1.863
67
1.209
20
2.500
44
0.856
68
0.398
20
2.650
44
1.831
68
1.187
21
2378
45
0.824
69
0.387
21
2.619
45
1.798
69
1.167
22
2.252
46
0.794
70
0.377
22
2.586
46
1.767
70
1.146
23
2.152
47
0.766
71
0.367
23
2.554
47
1.735
71
1.126
24
2.049
48
0.739
72
0.358
24
2.520
48
1.704
72
1.107
110
KESISTANCE TO CRUSHING.
Strength of Columns, Pillars, or Struts, of seasoned wood, round or
square section.
Fixed at both ends. All dimensions in inches.
Find how many times the least dimension across the section is
TT
contained in the length or height of column, &c.; that is, - ;
then multiply the corresponding figures on the same horizontal
line under K" by the sectional area of cross-section. This gives
the capacity of column, &c., in tons of 2,000 Ibs., 10 times safety.
Reference.
H= Length of column, &c.
D = Least dimension of cross-section.
K" = Capacity in tons of one square inch of cross-section, to
be multiplied by sectional area of desired cross-section.
The coefficient C for white and yellow pine in the following
table is taken at - h -f {} = 600 Ibs. for safety :
For oak at S -J{J-- = 800 Ibs. per square inch for safety.
EXAMPLE. What is the capacity of a pillar of oak, section
4x6 inches, length = 12 feet = 144 inches ?
for 36 _. o.064 x 4 X 6 = 1.536 tons.
Capacity K f/ of one square inch in tons of 2,000 Ibs.
White and Yellow Pine.
Oak.
H
~D "
j\"
H
~D =
K"
H
~D ~
E
H
~D ~
K"
1
0.299
26
0.081
1
0.399
26
0.108
2
0.2:)5
27
0.076
2
0.394
27
0.102
3
0.289
28
0.072
3
0.386
23
0.096
4
0.282
29
0.068
4
0.376
29
0.091
5
0.272
30
0.065
5
0.363
30
0.086
6
0.262
31
0.061
6
0.349
31
0.082
7
0.251
32
0.058
7
0.334
32
0.078
8
0239
33
0.056
8
0,319
33
0.074
9
0.226
34
0.053
9
0.302
34
0.071
10
0.214
35
0.050
10
0.285
35
0.067
11
0.202
36
0.048
11
0239
36
0.064
12
0.190
37
0.046
12
0.254
37
0.061
13
0.179
38
0.044
13
0.238
38
0.059
14
0.168
39
0.042
14
0.224
39
0.056
15
0.158
40
0.040
15
0.210
40
0.054
16
0.148
41
0.038
16
0.197
41
0.051
17
0.139
42
0.037
17
0.185
42
0.049
18
0.130
43
0.035
18
0.174
43
0.047
19
0.123
44
0.034
19
0.163
44
0.045
20
0.115
45
0.033
20
0.154
45
0.044
21
0.108
46
0.031
21
0144
46
0.042
22
0.102
47
0.030
22
0.136
47
0040
23
0.096
48
0.029
23
0.123
48
0.039
21
0.030
49
0.088
24
0.121
49
0.037
25
0.085
50
0.027
25
0.114
50
0.036
PARALLELOGRAM OF FORCES.
Ill
PARALLELOGRAM OF FORCES.
COMPOSITION AND RESOLUTION OF FORCES.
Reference.
A, B, C = Forces, cr strains, acting on a single point,
y, i/, = angles.
Fig. 190.
A =
(7sin. v,
sin. (v + v,)
_ Csin.v .
B = - ; -, when v = v /y A = B
sin. (v + ^/)
sec. v;
whenv+^<90 C =
when t> -f u y > 90 /?_
cos. (v + v,)
= 90
= C cos. v
= C sin. v
(7 =
112
STRAINS IN FRAMES.
STRAINS IN FRAMES.
Reference.
C = Compressive strain in units of weight,
T= Tensile
F= Vertical
H= Horizontal "
W= Load in units of weight.
I = Dimensions in units of length.
v = Angle between horizontal and inclined member.
For 01 oss-breaking strain, see "Resistance to cross-breaking.
Fig. 193.
W
2 sin. v
W
y x = cotg. v = H
Fig. 194.
C / = H = ij W cotg.v = cross-breaking strain at H.
H / = -j- H = j^. j- TP r cotg. v = tension in H/.
I I
H H, = \\. ( -- ) W cotg. v = compression in
F= U W. \ I J
STRAINS IN" FRAMES.
113
Fig. 195.
T7 TT -
V = H / tang, y = - tang, y
= compression.
W.I
cos. y 1^, , cos. y
nressiou.
//= W.I.
When I > ^ 3 the portion t // is in tension = V W =
W Ltang.y-l)
\ Ijj I
When I < Z 3 the portion l y/ is in compression = IF F =
F/ = . TF= tension,
v
/* /. 107.
Ends of beams built into wall or fixed-
F= -L W
V, = V- W = (,--} W, = T, (tension) = C, (.
V V
pression.)
Ill STRAINS IN BOOM DERRICKS.
C= ( J = (compression) = T (tension.)
V 21, J sin. v
II f~ \ TFcotg. v = (tension) = H, (compression.)
Ends of beams not built into wall or fixed:
T/ /r= v W= ( / ) W = C / (compression) = T,
(tension.)
0= = = T (tension.)
sin. v I, sin. v
H= Fcotg. v = TFcotg. v = (tension) = H / (compression.)
STRAINS IN BOOM DERRICKS.
Reference.
C = Compression in boom.
C / = Compression in mast.
T= Tension in tackling.
T,= Tension in guy.
t = Tension in runner from mast head to weight.
t / = Tension in runner from boom head to weight.
W = Weight or load.
H = Horizontal strain.
V = Vertical strain.
v, v lt v 2 = Angles. (See Figure.)
Fifj. 198.
STEAINS IN TRUSSES.
115
TTsin. v 1
sin. (v + Vj
F"== / cosin. v 1
(7= Fcosec. v 2
T= Fcosec. v
TFsin. v
sin. (v + v x
*,=
<?,= PF
2^= Fcotg. -y 3 sec. v 4
STRAINS IN TRUSSES.
Zoac? equally distributed.
Reference.
W= Load equally distributed in Ibs.
I = Distance between abutments.
v = Angle between horizontal and diagonal.
0= Compression in Ibs., (denoted by thick lines.)
T= Tension in Ibs., (denoted by thin lines.)
2 Bays = 4-
Fig. 199.
T-
-* ~ T6 -
3 Bays =
. 200.
116
STEAINS IN TRUSSES.
4 Bays
Fig. 201.
C- W
C 2
3(7 2
cotg.
Fig. 202.
C = T 6C 2 cotg. v
(?!= T 1 = 2(7 2 cotg. t?
W
2 rT 2 = ^ cosec. v
m
STRAINS IN TRUSSES.
I
0= T =
90
2
8C,
2
5 a
cotg. v
cotg. v
a
2
5(7 3
65 " "~T~
T 3 = j cosec.
^4 = 3 ^3
TABLE OF CONSTANTS, BASED ON FOREGOING FORMULA.
Load equally distributed.
Table of constants for strains in respective member of trusses,
from 2 to 6 bays, with diagonals inclined from 5 to 45 :
Reference.
W = Load in Ibs., equally distributed over whole length of
truss, to be multiplied by constant for strain in re-
pective member.
v = Angle between horizontal and diagonal.
0= Compression in Ibs. in respective member.
T= Tension in Ibs. in respective member.
EXAMPLE. Required, the strain in the various members of a
truss of 4 bays. Length = 40 feet ; load W = 80,000 Ibs. ; angle
v = 20.
Members. Constants. W. Strains.
C = T = 1.372 x 80,000 = 109,760 Ibs.
c i= TI= 1.029 X 80,000 = 82,320
<7 2 =0.25 x 80,000= 20,000
<7 3 = 0.375 x 80,000 = 30,000
T 2 = 0.365 x 80,000= 29,200
T 3 = 1.095 x 80,000 = 87,600
[NOTE. When the trusses are inverted, the strains change in kind, but
not in amount]
118
STRAINS IK TRUSSES.
2 Bays = -
Fig. 204.
3 Bays =
Fig. 205.
V
O
C l
T
(7= T
Ci
Tl
5
3.572
0.625
3.584
3.810
0.333
3.820
6
2.972
"
2987
3.170
"
3.186
7
2.544
<
2.562
2.713
2.733
8
2.225
"
2.244
2.370
M
2.393
9
1.972
"
1.997
2.103
"
2.130
10
1.772
(C
1800
1.890
M
1.920
11
1.610
"
1.640
l.MO
"
1.747
12
1.469
"
1.500
1.570
M
1.603
13
1.353
"
1.390
1.444
"
1.483
14
1.253
<
1.290
1.333
1.376
15
1.166
1
1.210
1.243
"
1.286
16
1.087
"
1.134
1.160
(
1.210
17
1.022
"
1.070
1.090
M
1.140
18
0.959
"
1.013
l.<23
"
1.080
19
0.906
H
0.959
0.970
"
1.023
20
0.859
"
0.912
0.917
"
0.973
21
0.813
"
0.872
0.866
"
0.930
22
0.778
"
0.834
0.823
0.890
23
0.734
0.790
0.783
:
0.853
24
0.703
(
0.765
0.750
0.810
25
0.668
"
0.738
0.713
"
0.786
26
0.641
0.712
0.685
0.760
27
0.613
"
0.687
0.653
0.730
28
0.587
(
0.666
0.626
"
0.701
29
0.562
<C
0.644
0.600
.(
0.686
30
0.541
((
0.625
0.643
M
0.666
31
0.519
0.606
0.555
"
0.646
32
0.500
M
0.591
0.533
M
0.630
33
0.481
it
0.575
0.513
0.613
34
0.463
0.559
0.493
0.596
35
0.447
0.544
0.476
"
0.580
36
0.431
0.531
0.460
0.566
37
0.416
"
0.519
0.444
"
0.553
38
0.400
"
0.506
0.426
M
0.540
39
0.384
<
0.497
0.410
0.530
40
0.372
0.487
0.396
"
0.520
41
0.359
0.475
0.385
"
0.506
42
0.347
H
0.466
0.370
0.496
43
0.334
0.456
0.357
"
0.486
44
0.322
<
0.450
0.343
0.480
45
0.312
"
0.444
0.333
"
0.473
STRAINS IN TRUSSES.
119
4 Bays =
Fig. 20(>.
V
C = T
CV-5T!
C 2
Cs
T 2
T 3
5
5.720
4290
0.250
0.375
1.434
4.032
6
4.700
3.570
"
11
1.200
3.600
7
4.008
3.051
"
"
1.025
3.075
8
3.500
2.070
"
0.897
2591
9
3.1G4
2.373
*
;;
0.799
2.397
10
2.8-^2
2.124
0.720
2.160
11
2.508
1.926
;
0.655
1.965
1 2
2.388
1.791
0.601
1.803
13
2.164
1623
0.556
1.608
14
2.000
1.500
0.516
1.548
15
1.804
1.398
0.482
1.446
1C
1.7-10
1.305
0.454
1.362
17
i.632
1.224
0.428
1.284
18
1.532
1.149
0.405
1.215
19
1.448
1.086
0.384
1.152
20
1.372
1.029
"
0.365
1.095
21
1.300
0.975
"
0.349
1.047
22
1.23G
0.927
;
0.334
1.002
23
1.172
0.879
"
0.32)
0.960
24
1.124
0.843
;
"
0.306
0.918
25
1.008
0801
0.295
0.885
26
1.024
0.708
*
0.285
0.855
27
0.080
0.735
0.275
0.825
2-i
0.940
0.705
"
0.266
0.798
2y
0.900
0.675
1
0.258
0.774
30
0.804
0.048
0.250
0.750
31
0.823
0.621
0.243
0.729
32
0.800
O.COO
236
0.708
H,i
0.708
0.576
0.230
0.690
34
0.740
0.655
*
0.224
0.672
35
0.720
0.540
"
*
0.218
0.654
36
0.088
0.516
0.212
0.636
37
O.G04
0.498
0.207
0.621
38
0.640
0.480
"
0.203
0.009
39
0.016
0.462
0.199
0.597
40
0.000
0.450
;
0.195
0.585
41
0.576
0.432
0.190
0.570
42
0.500
0.420
0.186
0.558
43
0.536
0.402
0.183
0.549
44
0.520
390
0.180
0.540
45
0.500
0.375
^
"
0.177
0.531
STRAINS IN TRUSSES.
V
e-jr
C i = Ti
C 2
^3
Tz
n
5
6.858
4.572
0.200
0.400
2.294
4.588
6
5.706
3.804
"
1.912
3.824
7
4.884
3.256
1.640
3.280
8
4.272
2.848
"
1.436
2.872
9
3.786
2.524
.
1.278
2.556
10
3.402
2.268
i
1.152
2.304
11
3.084
2.056
1.048
2.0 )6
12
2.820
1.880
"
0.962
1.01:4
13
2.598
1.732
"
"
0.890
1.780
14
2.406
1.604
M
0.826
1.C.52
15
2.238
1.492
"
0.772
1.544
16
2.088
1392
"
0.726
1.452
17
1.962
1.308
"
"
0.684
1.3(58
18
1.842
1.228
"
0.648
1.296
19
1.740
1.160
M
"
0.614
1.228
23
1.650
1.100
"
"
0.584
1.168
21
1.560
1.040
0.558
1.116
22
1.482
0.988
"
0.534
1.068
23
1.410
0.940
1
0.512
1.024
24
1.350
0.900
41
0.490
0.980
25
1.284
0.856
"
0.472
0.944
26
1.230
0.820
0.456
0.912
27
1.176
0.784
"
0.440
0.880
28
1.128
0.752
"
0.426
0.852
29
1.080
0.720
0.412
0.824
30
1.038
0.692
0.400
0.800
31
0.996
0.664
"
0.388
0.776
32
0.960
0.640
0378
0.756
33
0.924
0.616
"
0.368
0.736
34
0.888
0.592
"
0.358
0.716
35
0.858
0.572
"
0.348
0.696
36
0.828
0.552
"
i
0.340
0.680
37
0.798
0.532
"
0.332
0.664
38
0.768
0.512
;
0.324
0.648
39
0.738
0.492
"
0.318
0.636
40
0.714
0.476
M
0.312
O.C24
41
0.690
0.460
"
0.304
0.608
42
0.666
0.444
0.298
0.596
43
0.642
0.428
"
0.292
0.584
44
0.618
0.412
"
0.288
0.576
45
0.600
0.400
"
0.284
0.568
STRAINS IN TRUSSE.S
121
V
t7- T
Pi -21
Cj 2*2
%
<7 4
e 5
1
T.
T 4
T-,
5
8.5G8
7.G16
4.760
0.1 66
0.250
0.416
0.952
2.856
4.760
G
7
7.123
G.102
6.336
5.424
3.960
3.390
n
u
0.680
2.379
2.041
3.965
3.402
8
5.337
4.744
2.965
;
M
"
0.596
1.788
2.980
g
4.023
4.200
2.G25
"
"
0.530
1.590
2.650
10
4.218
3.776
2.360
"
"
M
0.478
1.434
2.390
11
3.852
3.424
2.140
"
"
0.435
1.305
2.175
12
3.519
3.128
1.955
"
0.399
1.197
1.995
13
3.240
2.880
1.800
"
0.369
1.107
1.845
11
3.006
2.672
1.670
"
"
0.343
1.029
1.715
15
2.799
2.488
1.555
0.320
0.960
1.600
1C
2.610
2.320
1.450
< ;
"
0.301
0.903
1.505
17
2.448
2.176
1.360
"
1
"
0.284
0.852
1.420
18
2.304
2.048
1.280
"
0.269
0.807
1.345
if)
2.1G9
1.928
1.205
"
"
, 0.255
0.765
1.275
ft)
2.001
1.832
1.145
"
"
0.242
0.726
1.210
21
1.944
1.728
1080
"
"
0.231
0.693
1.155
22
1.854
1.G48
1.030
"
0.221
0.663
1.105
2:>
1.7G4
1.568
0.980
"
0.212
0.636
1.060
21
1.G83
1.496
0.935
"
"
0.203
0.609
1.015
23
1.G02
1.424
0.890
"
1
"
0.196
0.588
0.980
2;
1.539
1.368
0.855
"
"
0.189
0.567
0.945
27
1.4G7
1.304
0.815
"
"
0.182
0.546
0.910
28
1.404
1.248
0.780
t
0.177
0.531
0.885
29
1.350
1.200
0.750
"
"
0.171
0.513
0.855
30
1.2)6
1.152
0.720
"
"
0.166
0.498
C.830
:>L
1.242
1.104
0.690
M
0.161
0.483
0.805
32
1.197
1.0G4
0.665
"
"
0.156
0.468
0.780
33
1.152
1.024
0.640
M
0.152
0.45G
0.760
34
1.107
0.984
0.615
"
< ;
0.148
0.444
0.740
35
1.071
0.952
0.595
M
0.144
0.432
0.720
30
1.035
0.920
0.575
M
"
0.141
0.423
0.705
37
0.999
0.888
0.555
M
0.138
0.414
0.690
88
0.954
0.848
0.530
M
0.134
0.402
0.670
39
0.918
0.816
0.510
H
0.132
0.396
0.660
40
0.891
0.792
0.495
i-
t
0.129
0.387
0.645
41
0.8G4
0.768
0.480
0.126
0.378
O.C30
42
0.823
0.736
0.460
M
a
0.123
0.369
0.615
4:5
0.801
0.712
0.445
"
0.121
0.363
0.605
44
0.774
0.688
0.4 JO
"
M
0.119
0.357
0.595
45
0.747
0.664
0.415
"
"
"
0.118
0.354
0.590
122 STRAINS IN TRUSSED BEAMS.
STRAINS IN TRUSSED BEAMS.
When a beam supported at the ends, is required to carry a
greater load than its given capacity, and trussing is resorted to,
it may become necessary to find what portion of the load is borne
by the different members of the trussed beam.
Reference.
Let IF = Load acting on truss at a supported point. (See figure.)
W l = That portion of IFacting on diagonals.
W. 2 = That portion of IF" acting on beam.
A i = Sectional area of diagonal.
A 2 = Sectional area of beam.
E l = Modulus of elasticity of material in diagonals.
E 2 = Modulus of elasticity of material in beam.
a = Length of diagonal.
b = Distance between center of beam and point of support.
c = Distance between abutment and point of support.
{ = Depth of beam.
= Depth of truss.
/ = Distance between center of beam and abutment.
[NOTE. Use the same unit of length and weight.]
No. 1.
Fig. 209
a 3 / 2 A 2 E 2
. -/I. A.. JIM
Wj. a 3 / 2 <4 2 ^ 2
STRAINS IN TEUSSED BEAMS.
- . TF
When load is equally distributed W becomes | TF.
No. 2.
Fig. 210.
211.
W<2 2 a
123
2 a 3 * / 2 A,
>m a3 / 2 ^ 2
A,
2 Wi a 3 / 2
a 3 * / 2
TFl=
"n7 ""
PP,
+ 1
When load is equally distributed W becomes f TF.
124
STRAINS IN TRUSSED BEAMS.
No, 3.
Fig. 212.
A l
W 2
a (a* X
_
h* (Vb^c E l
w l
w l
*. w
+1
When load is equally distributed IF becomes f PP".
STRAINS IN TRUSSED BEAMS.
125
No. 4.
Figs. 213 and 214.
h* (I 2 - 6 2 ) c A l E l
2/2 a(a 2 +6c)
^ _
"
A
A,
/ 2
~~
h* (Pb*)c
2W L / 2 a(a 2 +6c)
W l
. W
W
-+1
When load is equally distributed TF becomes f W.
126 STRAINS IN TRUSSES WITH PARALLEL BOOMS.
STRAINS IN TRUSSES, WITH PARALLEL BOOMS.
(Caused by Static and Moving Loads)
The strain in the upper boom is always compressive.
* The strain in the lower boom is always tensile.
All braces inclined down from the nearest abutment are in
tension.
All braces inclined up from the nearest abutment are in com
pression.
The strains in the verticals and diagonals increase from the
center of truss to abutment.
The strains in the booms decrease from the center of truss to
abutment.
A moving load, advancing over a truss, &c., causes the maxi
mum moment of rupture (which under an equally distributed
load is at the center of truss) to shift to one side of the center,
thereby changing the nature and amount of strain in web only.
This requires either the enlargement of those members consti
tuting the web or the addition of so-called counters, (braces,
struts, or ties.)
To find the point from center of truss to where the addition ol
counters must commence, the following formula is used :
Let d = Distance from center of truss to point where,
maximum moment of rupture occurs, and where
counter bracing must commence.
d / = Distance from nearest abutment to ditto.
Anad /= J d = -^-
2 W /
These results will be found to agree with formulas for " Counter
Strains" when V m becomes negative.
Reference.
N = Total number of bays in a truss.
HK = Horizontal strains in booms.
F n = Strains in verticals.
y n = Strains in diagonals.
V m = Vertical strains acting on counters Y m .
Y m = Strains in counters, opposite in kind to Y n .
STRAINS IN TRUSSES WITH PARALLEL BOOMS. 127
IF = Weight of static load, equally distributed over whole
length of truss.
W,= "Weight of moving load, equally distributed over whole
length of truss.
h = Height or depth of truss between the center of gravity of
booms.
I = Span or length of truss from abutment to abutment.
n Number of member, counting from abutment A.
in = Number of member, between center and abutment B.
r = Half the length of a panel or bay.
s = Length of a panel or bay.
w = Weight of static load per unit of length I.
iv / = Weight of moving load per unit of length I.
v = Angle between horizontal and diagonal.
For other designations, see diagrams and examples.
The angle v for Howe Truss is generally 45.
The angle v for Whipple Truss is generally 45.
The angle v for Lattice Truss is generally 45.
The angle v for Warren Truss is generally 60.
The proportion of height h to span I is from 4 to -^ gener
ally T v
128
STRAINS IN TRUSSES WITH PARALLEL BOOMS.
STRAINS IN" TRUSSES WITH PARALLEL BOOMS. 129
HOWE TRUSS. (Figs. 215, 216, 217, and 218.)
Additional Reference.
,T n Distance from abutment A to center of bay.
y n Distance from abutment A to apex of bay.
Static or Permanent Load, equally distributed over whole length of
Truss,
Strains in Booms.
w w
Strains in Verticals.
w w
F.= - --- -*.
Strains in Diagonals.
^n= V n cosec. v.
Moving and Static Load, each equally distributed per unit of
length.
~
Strains in Booms.
w+w,
~2h~ 2hl~
Strains in Verticals.
Strains in Diagonals.
Y n = F" n cosec. v.
Strains in Counters.
130 STRAINS IN TRUSSES WITH PARALLEL BOOMS.
EXAMPLE. (Figs. 215, 216, 217, and 218.)
Moving Load, (as railway train passing over bridge.)
We will assume W = 50,000 Ibs.
Wt= 100,000 Ibs.
I = 100 feet.
h = 10 feet.
v = 45, (cosec. = 1.414.)
Horizontal Strains in Booms, (compression inupper, tcnsionin lower.)
W-\- Wi JF + TFi 2 __ 50UOO+ 100000
//a = 2/i y " ~ 2hl 2/n = ~20
50000 + 100000_ 2 __ _ ^ 5 2
yn ~~ 2000 ~~ y& ~~ J/a 0-2/n
//! = 7500 . 10 75 . 100 = 67,500 Ibs.
II 2 = 7000.20 75.400 = 120,000 Ibs.
11. 3 = 7000.30 75.900 = 157,500 Ibs.
7/ t = 7500.40 75.1600 = 180,000 Ibs.
J1 5 = 7500.50 75.2500 = 187,500 Ibs.
Strains in Verticals.
W W Wl ,\2__ 500 22__ 5000
n i> r ** "*" W ^ ". ^ 2 100
.(^ .rj 2 = 25000 500. .T n H
Strains in Figs. 215 210 217 218
F 1 = 25000 500.5 +5. 05*= 67625 Ten. Ten. Corn. Com.
F 2 = 25000 500. 15 4-5. 85 2 = 53625 "
F, = 25000 500. 25+5. 7r> a = 40625 "
F 4 = i ) 5000 500. 35 + 5. ()5 2 = 28625 "
Vl = 25000 500. 45 +o.55 2 = 17625 "
Counter Strains (V m ) for Strains in Counters.
V 6 = 20000 500.55 + 5.45* = 7625.
F 7 = 25000 500.65 + 5.85 = 5625.
Strains in Diagonals.
Y n = F n cosec v.
Strains in Figs. 215 210 217 218
y\ = 67625 . 1.414 = 1)5,620 Ibs. Com. Com. Ten. Ten.
Y 2 == 53625 . 1.414 = 75,826 Ibs.
F 3 = 40625 .1 414 = 57,44 i Ibs.
F 4 = 28625 . 1.4 14 = 40,476 Ibs. "
F 5 = 17625 . 1.414 = 24,922 Ibs.
Strains in Counters, (dotted lines, Fig. 215, for example.)
Yn = F m cosec. v.
Strains in Figs. 215 210 2L7 218
F G = 7625 . 1.414 ^r 10,762 Ibs. Com. Com. Tun. Ten.
F 7 =5625 . 1.414= 7,954 Ibs.
STRAINS IN TRUSSES WITH PARALLEL BOOMS.
131
Pig. 219. LATTICE TRUSS WITH VERTICAL NUMBERS.
sssvs.
Fig. 219. Load on either Boom.
To compute the strains in this truss, the
easiest method is to find the values of /f n , F n ,
F m , F n , and F m for a Howe Truss, (Figs. 215,
( 216, 217, and 218 ) loaded in the same man-
| ner, (upper or lower boom.) These values in
the following formulas for the above truss will
i give the required strains:
Strains in Booms. (8.)
<?!=-
4*2~T" jC *8 /^ n r/ -"n l~
: , generally o n =: f
Strains in Verticals. (U.)
Upper boom loaded compression.
Lower boom loaded tension.
W l
constant.
Strains in End Post ( U .)
Upper boom loaded.
U = U -f- ^1= compression.
Lower boom loaded.
U = Si^= compression.
Strains in Diagonals. (D.)
D *=^~
I?_
2
Y*
Generally D n =
i
Strains in Counters.
Generally Z> m =
132 STRAINS IN TRUSSES WITH PARALLEL BOOMS.
Fig. 220. WARREN TRUSS.
Fig. 220. Lower Boom Loaded.
Additional Reference.
#n = Distance from abutment A to center
of diagonal.
3/n = Distance from abutment A to apex
of bay of upper boom.
2n = Distance from abutment A to apex
of bay of lower boom.
Static or Permanent Load, equally distributed
over whole length of Truss.
Strains in Booms.
Upper.
77 W W 2
^ = -ir-2Ar* 2
Lower.
W W
Strains in Verticals.
K = --- a? n ( F n acts at the end
- I
of ar n .)
Strains in Diagonals.
Y^ ==. F n cosec. v.
Moving and Static Load, each equally dis
tributed per unit of length.
Strains in Booms.
Upper.
Lower.
Strains in Verticals.
--?-? +5 "
Strains in Diagonals.
Y* = F n cosec v.
STRAINS IX TRUSSES WITH PARALLEL BOOMS. 133
Strains in Counters.
EXAMPLE. (Fig. 220.)
Moving Load (as railway train passing over bridge) on lower Bocm.
We will assume W = 50,000 Ibs.
Wi= 100,000 Ibs.
I = 100 feet.
A = 10 feet.
?; = 63 20 , (cosec. = 1.12.)
_
Horizontal Strains in Upper Boom. (Compression.)
W+Wi ^ 2 __ 50000 + 100000
2h 2hl 2.10
50000 + 100000 2 _ 150000
Zn ~~ 2.10 . 100 ** ~ ~~20 2n ~~
HI = 7500 . 10 75 . 100 = 67,500 Ibs.
#2 = 7500.20 75.400 = 120,000 Ibs.
H 3 = 7500.30 75.900 = 157,500 Ibs.
H 4 = 7500.40 75. 1600 = 180,000 Ibs.
H 6 = 7500.50 75.2500 = 187,500 Ibs.
Horizontal Strains in Lower Boom. (Tension.)
i JF-f Jh 50000 + 100000
"
2A 2hl 2.10
50000 -f- 100000 f 2 _ 150000 150000
2.10 .~100~ -2/n ~ 20 ^ n 2000~
75.25 = 37500 1875= 35,625 Ibs.
#2 = 7500.15 75.225 =112500 16875= 95,6251bs.
H 3 = 7500 . 25 75 . 625 = 1 87500 46875 = 140,625 Ibs.
Jff 4 = 7500.35 75. 1225 = 262500 91875 = 170,625 Ibs.
H 6 = 7500.45 75.2025 = 337500 151875 = 185,623 Ibs.
134
STRAINS IN TRUSSES WITH PARALLEL BOOMS.
Strains in Verticals.
YU = F n cosec. v.
W W W, , 50000
Fn = .rr n + -nT-(J *) = o
2 2 2
50000
"Too"
r n + ^^T-( 100 *u) 2 = 25 000 500,r n + 5. (100 x u )*
v,
ss
25000
500
. 2.5 +
5 .
, 9506.25
=
71281.25.
r.
5=
25000
500
. 7.5 +
5 ,
, 8556.25
=
64031.25.
F~
S=9
25000
500
. 12.5 +
5 .
7656.25
=
57031.25.
P
as
25000
500
. 17.5 +
5 ,
, 6806.25
=
50281.25.
F 8
25000
500
. 22.5 +
5 ,
. 6006.25
=
43781.25.
^
=
25000
500
. 27.5 +
5
, 5256.25
37531.25.
[ J _
25000
500
. 32.5 +
5 .
. 4556.25
=
31531.25.
P-
_-r^=
25000
500
. 37.5 +
5
. 3906.25
25781.25.
p*
25000
500
. 42.5 +
5 ,
. 3306.25
SBB
20281.25.
l n
( ==
25000
500
. 47.5 +
5 ,
. 2756.25
=
14031.25.
F! j= 25000 -
Fj 2 = 25000 -
F! 3 = 25000-
= 71281.25 ,
= 64031.25
= 57031.25
= 50281.25 ,
= 43781.25
= 37531.25 ,
= 31531.25
= 25781.25
= 20281.25
n = 14031.25 ,
Counter Strains. ( F m .)
500 . 52.5 4- 5 . 2256.25 =
500 . 57.5 + 5 . 1806.25 =
500 . 62.5 + 5 . 1406.25 =
Strains in Diagonals.
Y" n = F" n cosec. v.
10031.25.
528125.
781.25.
1.12 = 79,835 Ibs.
1. 12 = 71, 715 Ibs.
1.12 = 63,875 Ibs.
1.12 = 56,315 Ibs.
1.12 = 49,035 Ibs.
1.12 = 42,035 Ibs.
1.12 = 35,315 Ibs.
1.12 = 28,875 Ibs.
1.12 = 22.715 Ibs.
1.12 = 15,715 Ibs.
Compression in Y l and F 2 ,
Tension in Y 2 and Y 19 .
Compression in Y 3 arid Y li
Tension in Y and ]T 17 .
Compression in Y 5 and Y 1
Tension in Y 6 and Y 15 .
Compression in F" 7 and y r
Tension in Y s and ]T 13 .
Compression in Y g and Y l
Tension in Y lo and Y ll .
Counter Strains.
= T/ m COS6C -
FH= 10031.25 . 1.12 = 11,235 Ibs. Compression in
F 12 = 5281.25.1.12= 5,915 Ibs. Tension in Y 9 i
F 13 = 781.25 . 1.12= 875 Ibs. Compression in
i and .
lif j
STRAINS IN TRUSSES WITH PARALLEL BOOMS. 135
Fiy. 221. WARREN TRUSS.
Fig. 221. Upper Boom Loaded.
Additional Reference.
= Distance from abutment A to center
of bay of upper boom.
2/n = Distance from abutment A to apex of
bay of upper boom.
2 n = Distance from abutment A to apex of
bay of lower boom.
Static or Permanent Load, equally distributed
over whole length of Truss.
Strains in Booms.
Upper.
/ W
Wr*
Lower.
w
2h *""
Strains in Verticals.
_ W W
T a J #n
i I/
Strains in Diagonals.
Y n = V a cosec. v.
p Moving and Static Load, each equally dis
tributed per unit of length.
Strains in Booms.
Upper.
w+Wl ( w + w,
^ 2n, a " ( 2hl * H
2hl
Lower.
TT+TF,
136 STRAINS IN TRUSSES WITH PARALLEL BOOMS.
Strains in Verticals.
-
Strains in Diagonals.
T n = F n cosec. v.
Strains in Counters.
w w w
--- *-^"" (l ~*
= 7 m cosec.
EXAMPLE. (Fig. 221.)
Moving Load (as railway train passing over bridge) on Upper Boom.
We will assume W = 50,000 Ibs.
W l = 100,000 Ibs.
I = 100 feet.
h = 10 feet.
v = 63 20 , r = 5 feet.
Horizontal Strains in Upper Boom. (Compression.)
jT+JFi rjF+TPi 2 ,(W+W } )r*-}_
2/i n L 2M <2a H 2AZ J ""
lupOOO_ ^ 2 150000. 5* 1 _
~200(r~ Zn ^ ^000 J ~
J50000_ rluOOO
20 Za
7500. 2 n [75.2 n 2+ 1875]
jff 1= =7500.5
H 2 = 7500.15
,= 7500.25
75.25 -f 1875"
75.225 + 1875 :
75.625 -f 1875
H= 7500.35 [75.1225 -j- 1875
H 5 = 7500.45 [75.2025 + 1875;
= 33,750 Ibs.
= 93,750 Ibs.
= 138,750 Ibs.
= 168,750 Ibs.
= 183,750 Ibs.
Horizontal Strains in Lower Boom (Tension.)
- .
^==7500.10 75.100 = 67,500 Ibs.
H,= 7500.20 75.400 = 120,000 Ibs.
Hl= 7500.30 75.900 = 157,500 Ibs.
J/ 4 == 7500.40 75.1600 = 180,000 Ibs.
JI 5 = 7500.50 75.2500 = 187,500 Ibs.
STRAINS ITS TRUSSES WITH PARALLEL BOOMS. 137
Strains in Verticals.
F n = -J---^.* + _|L(Z_a; n ) 2 = 25000-500.* +
5.(Z z n ) 2
F 1= = 25000 500.5 + 5.95 2 = 67,625 Ibs.
F 2 = 25000 500.15 + 5.85* = 53,625 Ibs.
F 3 == 25000 500.25 + 5.75 2 = 40,625 Ibs.
F 4 = 25000 500.35 + 5.65 2 = 28,625 Ibs.
F 5 = 25000 500.45 + 5.55 2 == 17,625 Ibs.
Counter Strains.
F G = 25000 500.55 + 5.45* = 7,625 Ibs.
Strains in Diagonals.
F n = Fn cosec.
Y 1 = 67625 . 1.12 = 75,740 Ibs. Tension in Y l and F 10 ;
compression in F a and Y A .
Y 2 = 53625 . 1.12 = 60,060 Ibs. Tension in Y 2 and F 9 ;
compression in F b and F b .
F 3 = 40625 . 1.12 = 45,500 Ibs. Tension in F 3 and Y 6 -
compression in Y c and Y .
F 4 = 28625 . 1.12 = 32,060 Ibs. Tension in F 4 and 7 7 ;
compression in F d and F d .
F 5 = 17625 . 1.12 = 19,740 Ibs. Tension in F 5 and F 6 ;
compression in F e and F e .
Counter Strains.
F m = F m cosec. v.
F 6 = 7625 . 1.12 = 8,540 Ibs. Compression in F 5 and F 6 ;
tension in F fl and F.
133
STRAINS IN TUUSSES \VIT11 PARALLEL BJOMS.
8TEAINS IN TRUSSES WITH PARALLEL BOOMS. 139
LATTICE TRUSS. (Figs. 222, 223, and 224.)
Lower Boom Loaded.
Additional Reference.
r= Half the length of a bay of simple truss. (Figs. 222
and 223.)
x n = Distance from abutment A to center of bay of lower boom.
2/ n Distance from abutment A to apex of bay of upper boom.
z n = Distance from abutment A to apex of bay of lower boom,
The formulas are for the strains in the simple trusses, (Figs.
222 and 223.) Fig. 224 shows the simple trusses combined, con
stituting the Lattice Truss. .
When the upper boom is loaded, treat the strains as acting up
ward and the truss inverted: the strains will be of the same
amount in each member, but different in kind.
Static or Permanent Load, equally distributed over whole length of
Truss.
Strains in Booms.
Upper.
H W ( 4- T \ - W (-I- T V-U Wr2
u ~ "2A r n+ TV ~~ 2/iI V B ~*" ~2~) + W
Lower.
_ W f r \ W f r \2 STFr 2
n ~~^" V^ u 2 / 2/iI \^ m 2 / ~ 8/ti
Strains in Verticals.
Fn = "4 2T * n
Strains in Diagonals.
Moving and Static Load, each equally distributed per unit of
length.
Strains in Booms.
Upper.
n _ W +W(.i r \ W+Wi ( , r \2 (TF+TFi)r 2
Lower.
// _TF-f-IF L / __L\___^ I ^i / _ r V- 3(ir-h)rTF*
140 STRAINS IN TRUSSES WITH PARALLEL BOOMS,
Strains in Verticals.
W W Wi
Strains in Diagonals.
Y n = Vr, cosec v.
Strains in Counters.
W W W l
= Fm cosec< v -
[NOTE. The strains in Fa.b.c, .... are equal in amount, but different
in kind to the strains in H,2, 3, ....
EXAMPLE. (Figs. 222, 223, and 224.)
Moving Load (as railway train passing over bridge) on Lower Boom.
We will assume W = 50,000 Ibs.
Wi= 100,000 Ibs.
I = 100 feet.
h = 10 feet.
v = 63 20 , (cosec. = 1. 12.) r = 5 feet..
Horizontal Straws in Upper Boom. (Compression. Fig. 224.)
= 7500(z n + 2.5) -75(2 n + 2.5) 2 + 468.75
H Q = 7500 . ( + 2.5) 75 . ( + 2.5) 2 + 468.75 = 18,750 Ibs.
l= 7500 . ( 5 + 2.5) 75 . ( 5 + 2.5) 2 + 468.75 = 52,500 Ibs.
//= 7500 . (10+ 2.6) 75 . (10 + 2.5) 2 + 468.75 = 82,500 Ibs.
/= 7500 . (15 + 2.5) 75 (15 + 2.5) 2 + 468.75 = 108,750 Ibs.
JI 4 = 7500 . (20 + 2.5) 75 . (20 + 2.5) 2 + 468.75 = 131,250 Ibs.
//.= 7500 . (25 + 2.5) 75 . (25 + 2.5) 2 + 468.75 = 150,000 Ibs.
l/ ( .= 7500 . (30 + 2.5) 75 . (30 + 2.5) 2 + 468.75 = 165,000 Ibs.
H 7 = 7500 . (35 + 2.5) 75 . (35 + 2.5) 2 + 468.75 = 176,250 Ibs.
/7 8 ==7500 . (40+2.5) 75 . (40+ 2.5) 2 + 468.75^ 183,750 Ibs.
Hf= 7500 . (45 + 2.5) 75 . (45 + 2.5) 2 + 458.75 == 187,500 Ibs.
STRAINS IN TRUSSES WITH PARALLEL BOOMS.
141
Horizontal Strains in Lower Boom. (Tension. Fig. 224.)
_
-n a = -
2h V n I
1 1 2hl V yu "
3(17+ i
7\t
V 75QO (y
, 2.5) 75 . (y n 2.1
Shi
fli = 7500
.( 5_25) 75.
( 5 2.5)21406.25 =
H 2 = 7500 ,
,(10 2.5) 75.
(10 2.5)21406.25 =
#, = 7500
H\ = 7500
. (15 2.5) 75.
.(20 2.5) 75.
(15 2.5) 2 1406.25 =
(202.5)21406.25 =
H 5 = 7500 ,
.(25 2.5) 75.
(252.5)21406.25 =
H 6 = 7500 ,
, (30 2.5) 75.
(302.5)21406.25 =
HI = 7500
.(35 2.5) 75.
(352.5)21406.25 =
HS = 7500
.(40 2.5) 75.
(402.5)21406.25 =
T 9 = 7500
# 10 = 7500
.(45 2.5) 75.
.(50 2.5) 75.
(45_ 2.5)2 1406.25 =
(502.5)21406.25 =
2.5) 2 1406.25
16,875 Ibs.
50,625 Ibs.
80,625 Ibs.
106,875 Ibs.
129,375 Ibs.
148.125 Ibs.
163,1 25 Ibs.
174,375 Ibs.
181,875 Ibs.
185,625 Ibs.
SIMPLE TRUSS. (Fig. 222.)
Strains in Verticals. (F n .)
"T
IT
- ( l ~ **) 2 = 125 -
2.5 .(Z-* n )2
II II II II II
tTuV^V
12500
12500
12500
12500
12500
250 ,
250
250 ,
250 .
250 .
+
.10 +
, 20 +
30 +
40 +
2.5 .
2.5 .
2.5 .
2.5 .
2.8 .
100 2
90 2
80 2
70 2
60 2
=
37,250 Ibs.
30,250 Ibs.
22,500 Ibs.
17,250 Ibs.
11, 500 Ibs.
Com. in U.
Counter Strains. (F m .)
F 6 = 12500 250 . 50 + 2.5 . 50 2 = 6,250 Ibs.
F 7 = 12500 250 . 60 + 2.5 . 40 2 = 1,500 Ibs.
Y 1
Strains in Diagonals.
Y n= V n C0sec -
Tension in Y 1 and F 10 ;
37250 . 1.12 = 41,720 Ibs.
compression in F a and Y
Y 2 = 30250 . 1.12 = 33,880 Ibs.
compression in Y* and Y^.
Tension in Y 2 and F 9
142 STRAINS IN TRUSSES WITH PARALLEL BOOMS.
compression in Y e and Y .
F 4 = 17250 . 1.12 = 19,320 Ibs. Tension in F 4 and F 7 ;
compression in F d and F d .
F 5 = 11500 . 1.12 = 12,880 Ibs. Tension in Y 5 and F 6
compression in F e and F e .
Counter Strains.
Y m= V m cosec. v.
F 6 = 6250 . 1.12= 7,000 Ibs. Compression in F 5 and F 6 ;
tension in F e and Y e .
= 1,680 Ibs. Compression in F 4 and F 7 ;
SIMPLE TRUSS. (Fig. 223.)
Strains in Verticals. ( F n .)
l\ 12500 250 . 5 + 2.5 . 95 2 = 33812 .5.
Vf= 12500 250 . 15 + 2.5 . 85 2 = 26812.5.
T 7 12500 250 . 25 + 2,5 . 75 2 = 20312.5.
F 4 = 12500 250 . 35 + 2.5 . 65 2 == 14312.5.
F 5 = 12500 250 . 45 + 2.5 . 55 2 = 8812.5.
Counter Strains. (V m .)
T 7 G = 12500 250 . 55 + 2.5 . 45 2 = 3812.
Strains in Diagonals.
F n = V n cosec. v.
F 1= 33812.5 . 1.12 = 37,870 Ibs. Compression in Y } and F IO:
tension in F a and F a .
Y,= 26812.5 . 1.12 = 30 ; 030 Ibs. Compression in F, and F {) ;
tension in F b and F b .
Y, 20312 5 . 1.12 = 22,750 Ibs. Compression in F, and F s ;
tension in F c and F c .
F 4 r= 14312.5 . 1.12 = 16,030 Ibs. Compression in F 4 and F 7 :
tension in F d and F d .
F 5 = 8812 5 . 1.12 = 9,870 Ibs. Compression in F 5 and F fi ;
tension in F e and F e .
Counter Strains.
Y m = V m cosec - v -
Y^-= 3812.5 . 1.12 = 4,270 Ibs. Tension in F 5 and F 6 ; com
pression in F e and F e .
STRAINS IN TRUSSES WITH PARALLEL BOOMS.
143
Fly. 225.
Lower boom loaded.
t
144 STRAINS IN TRUSSES WITH PARALLEL BOOMS.
WHIPPLE TRUSS. (Figs. 225, 226, 227, and 228.)
Additional Reference.
# n , y n = Distance from abutment A to end of bay.
Static or Permanent Load, equally distributed over whole length oj
Truss.
Strains in Booms.
W W sW sW
a = "2JT yB ~ W y + ~2hT **- ~4T
Strains in Verticals.
T7- W W
Kn ~~l 2T
Strains in Diagonals.
Y n = V n cosec. v.
Moving and Static Load, each equally distributed per unit of
length.
Strains in Booms.
2hl~ * 2hl
s(W+W 1 )
Strains in Verticals.
TF TF
Strains in Diagonals.
F n = F n cosec. v.
Strains in Counters.
STRAINS IN TRUSSES WITH PARALLEL BOOMS.
145
EXAMPLE. (Figs. 225, 226, 227, and 228.)
(With 20 Bays.)
Moving Load, (as railway train passing over bridge.)
Let W = 50,000 Ibs.
W l = 100,000 Ibs.
I = 100 feet.
h = 10 feet, s = 5 feet.
v 45. (End diagonals v = 26 30 .)
Horizontal Strains in Booms. (Compression in upper, tension in
lower.)
27.
S (W +
"l) h-crnn .. nr o
OfTK a, 1
i ft
T
11 =
7500 .
75. O 2
375
yn ~
o +
y\i \
18750 =
- JLo
18
7C
,7.
RY=
7500 .
5
75,
. 5 2
375
. 5 +
18750 =
52
, r >
JL=
7500 .
10
75,
. 10 2
375 ,
, 10 +
18750 =
82
,51
| 3 =
7500 .
15
75
. 15 2
375
. 15 +
18750 =
108
,7.
7500 .
20
75 .
20 2
375 .
20 +
18750 =
131
,&
// 5 =
7500 .
35
75 .
25 2
375 ,
, 25 +
18750 =
150
,CM
H { ~
7500 .
30
75 ,
. 30 2
375 ,
, 30 +
18750 =
165
, !)(
H 1 =
7500 .
35
75
. 35 2
375 ,
, 35 +
18750 =
176
),
HX
7500 .
40
75 ,
, 40 2
375 ,
, 40 +
18750 =
183
i <
J/ 9 =
7500 .
45
75
. 45 2
375 ,
. 45 +
18750 =
187
ft
_ /
jfefe^A-ii
8*
r"
Strains in Verticals.
~
= 75,000 Ibs.
Fi=
F 2 =
F 3 =
V=
F 5 =
F 6 =
F 7 =
Strains in Figs. 2:
12500 250 . + 2.5 . 100 2 = 37,500 Ibs. C
12500 250. 5+25. 95 2 = 338121bs
12500250.10+2.5. 90 2 = 30,250 Ibs.
12500 250.15+2.5. 85 2 =r 26,812 Ibs.
12500 250 . 20 + 2.5 . 80 2 = 23,500 Ibs.
19^nn o^n of\ i o o ^7 ~9 OA 01 o i u
5 223 227 22
. C. T. T
zouu zou . zo + A.A . 7o 2 = 20,312 IDS.
1 9 : >r>n 9^0 QH 1 9 Pi *7A2 1 n Of^A 11 (
izouu z,o(j . 6(J + Z.o . 70 2 = 17,250 Ibs.
10
146
STRAINS IN TRUSSES WITH PARALLEL BOOMS.
Strains in Figs. 225 225 227 228
V s = 12500 250 . S5 + 2.5 . 65 2 = 14,312 Ibs. C. C. T. T.
V g = 12500 250 . 40+ 2.5 . 60 2 = 11,500 Ibs. "
F 10 = 12500 250 . 45+ 2.5 . 55 2 = 8, 812 Ibs. " " <l "
F n = 12500 250
F r ,= 12500 250 ,
F w = 12500 250 .
V m Acting on Counters.
, 50+2.5 . 50 2 = 6,250 Ibs.
55+2.5 . 45 2 = 3, 812 Ibs.
60+25 . 40 2 = 1,500 Ibs.
Strains in Diagonals.
J n K n cosec. v.
Strains in Figs. 225
Y l = 37500 . 1.117 = 41,887 Ibs. Ten.
F 2 = 33812 . 1.414 = 47,810 Ibs.
F 8 = 30250 . 1.414 = 42,773 Ibs.
F 4 = 26812 . 1.414 = 37,913 Ibs.
F 5 = 23500 . 1.414 = 33,229 Ibs.
F 6 = 20312 . 1.414 = 28,722 Ibs.
7, = 17250 . 1.414 = 24,391 Ibs.
F 8 = 14312 . 1.414 = 20,238 Ibs.
F 9 = 11500 . 1.414 = 16,261 Ibs.
* 10 = 8812 . 1.414 = 12,461 Ibs.
Strains in Counters.
F u = 6250 . 1.414 = 8,837 Ibs.
F 19 = 3812 . 1.414 = 5,391 Ibs.
F,;= 1500 . 1.414 = 2,121 Ibs.
223
Ten.
227
Com.
223
Com.
[NOTE. If counter braces are not inserted, Vn, F"i2,and T r i3,andl r 8>
Yg, and Y\Q will have an additional strain, opposite in kind and equal
to V\ i, V\ 2, and V\ 3, and Y\\, ^12? and Y\ 3 ; but if counters are used,
the strain V\\4 F"i2and Vis will n t occur in the structure, but will
be necessary to determine the strain in FH, Fi 2 ,and F\ 3 only. FH,
FI 2 , and FI 3 will then be inclined in the same direction as the diago
nals from abutment A to center of truss, the character of strain being the
same. (See also "Howe Truss")
Keep in mind that each half truss, as to the character and amount of
strain in the respective members, is alike.]
STRAINS IN PARABOLIC CURVED TRUSSES. 147
STRAINS IN PARABOLIC CURVED TRUSSES " BOW
STRING GIRDERS."
(Figs. 229, 230, 231, 232, 233, and 234.)
The strains in the lower boom (when horizontal) are the greatest,
and equal in every bay, when the load is equally distributed over
the whole length.
The strains in the arch or upper boom are also greatest when
the load is equally distributed over the whole length; the strains
gradually increasing from the middle to the supports.
The strains in the diagonals, whether single or double, in a
bay are, when the load is equally distributed, everywhere null.
When the load is unequally distributed, and one diagonal to each
bay is used, they will be either in compression or tension. The
character of the maximum of strains will be as follows: Assume
the left half of truss to be loaded. All diagonals inclined up from
left to right abutment are in tension; if inclined down, in com
pression. The character of strains will be vice versa when the
right half only is loaded.
The strains in verticals are either compression, tension, or null.
The maximum of compressive strain occurs when the diagonals
in connection are under the greatest strain; that is, under an
unequally distributed load. For other explanation, see diagram
under variously-disposed loads.
In the following formulas and examples the diagonals (for a
moving load) resist a tensional strain only, and the verticals a
compressive. This would not be the case if one diagonal to each
bay were used. In the latter case the diagonals and verticals
would have to resist an alternate compressive and tensional strain.
When the trusses are inverted, the strains are different in kind,
but not in amount.
Reference.
A, B = Reaction of support.
(7= Compression in arch or upper boom.
T Tension in lower boom.
D and H = Rise of arch.
F and / = Vertical forces.
W= Weight of moving and static load per unit of span
or length.
V= Strain in verticals.
N = Total number of bays.
a = Length of a bay.
c = Length of a diagonal.
d and h = Ordi nates to parabola.
I = Distance between supports or span.
k = Total number of verticals = N 1.
m = Number of bays between support and F n .
148 STRAINS IN PAEABOLIC CURVED TRUSSES.
n = Number of a member, counting from support to
middle of truss.
t = Tension in diagonal.
v and z = Angle between horizontal and member of polygon.
w = Weight of static load per unit of span or length.
w/= Weight of moving load, equally distributed per
unit of span or length.
u, x, y = Abscissas.
In the following diagrams, one-half of truss only is shown, the
strains being alike in the respective members of each half:
Fig. 229.
Lower Boom Horizontal.
To find the ordinates h when H is given :
The value of T given, to find h:
Fig. 230.
Lower Boom Curved.
To find the ordinates h or d when H or D is given:
12 d = ~ %
STRAINS IN PAEABOLIC CURVED TRUSSES.
14ST
The value of T given, to find h:
W(la)x n
w
~m~
Load equally distributed Static Load. (Figs. 231 and 232.)
W = The weight of construction and applied load.
Fig, 231.
IF/ 2
Wl *
Lower Boom Loaded.
, Wl
wl
H
= (7 V= - = tension
Loaded.
F=null.
Fig. 232.
Upper Boom Loaded. (C=T.)
Wl 2 Wl 2
V = = tension.
150
STRAINS ITS PARABOLIC CURVED TRUSSES.
Load unequally distributed Moving Load. (Figs. 233 and 234.)
(Strains in Booms, same as for Static Load.)
Fig. 233.
w/l
-
Lower Boom Loaded.
Fn= ^n A= compression.
Boom Loaded.
TFZ
T7
V n = -- = compression.
. 234.
n = r = compression
D)
STRAINS Iff PARABOLIC CURVED TRUSSES. 151
EXAMPLE. (Fig. 233.)
Moving Load on Lower Boom.
Reference.
= 64 feet. c 1= 8.7 i eet. w = 125 Ibs.
H= 8 feet, c 2 = c 3 = 10.0 feet. 10,= 625 Ibs.
a = 8 feet, c 4 = c-= 10.9 feet. W=w + w,= 750 Ibs.
2V = 8, & 7. c 6 = 11.3 feet.
4 v 8 v 3(64 8^ ^i 8.0 8 = Ofeet.
A 1= = X X --4^~ -== 3.5 feet, M 2 = 19.2 16= 3.2 feet.
W3= 40.0 24 = 16.0 feet,
^= 128.0 -32 = 96.0 feet,
4X8X16(64-16)
2== " 2 - =
.
A 4 = IT =8.0 feet.
Tang. t 1= ^ = = 23
Tang. v,= h ~^ = ^-^ = 3 34 30".
y 1= 3,5 x 2.28 = 8.0 feet. y s = 7.5 X 5.37 = 40.0 feet.
2/2= 6.0 X 3.20 = 19.2 feet. y 4 = 8.0 X 16.00 = 128.0 feet.
. = 48,000 Ibs.
(7 n = C sec. v n .
Q= 48000 Xl090=52,3201bs. C 3 = 48000 x 1.017 =48,816 Ibs.
(72=48000x1-047=50,256^8. C 4 = 48000 X 1.0019 = 48,091 Ibs.
*=*%$%-* 8.7 = 5437.5 lb, ,,-
tf= t s = ^-^- X 10.0 = 6250.0 Ibs.
152 STRAINS IN PARABOLIC CURVED TRUSSES.
625 x 64
**= ** = o^o X 10.9 == 6802.5 Ibs.
X o
625 x 64
^ --- X 11-3 = 7062.5 Ibs.
= 2625
2x 8
I = 1875
= 1250
j= 8(125+ 625)-- = 750
2 = 8 (125 + 625) [-^yj-] = 2250
,= 8(125 -J 625) [J^t^L] = 45 00
4 = 8 (125 + 625) I" JL-t^^_"j = 7500
PAKABOLIC ARCHED BEAMS OR BIBS. 153
/ 4 = 750 ( -- )= 562.5
V 96 -f 4 X 8 /
F 1= 6000 =6,000 Ibs. F 3 = 9000 500 =8,500 Ibs.
F 2 = 7812.5 312.5= 7,500 Ibs. F 4 = 9375 562.5 = 8,812.5 Ibs.
CAPACITY AND STRENGTH OF PARABOLIC ARCHED
BEAMS OR RIBS ORIGINALLY CURVED.
Reference. (All dimensions in inches.)
A = Sectional area of beam.
C = Compressive strain in direction of arch.
E = Modulus of elasticity.
H = Horizontal thrust at abutment, or tension on tie rod.
/= Moment of inertia of cross-section of beam.
R = Resistance of material to crushing, (to be divided by factor
of safety.)
W = Concentrated load at crown of arch.
a = Vertical deflection at crown.
b = Horizontal deflection at abutments.
h = Rise of arch.
21 = Distance between abutments = span.
s = Distance between neutral axis and farthest edge of section.
w = Load per unit of length, equally distributed horizontally.
x = Vertical distance from crown to point of arch, intersected
by y, say at on diagram.
y = Horizontal distance .from middle of arch to section where
the amount of strain is desired.
v = Angle between horizontal and tangent to curve.
Horizontal Thrust, (resisted either by abutments or tie rod.)
Fig. 235. (All dimensions to line of pressure.)
154 STKAINS IN A POLYGONAL FRAME.
To determine the curve or line of pressure:
x 7/ 2 y
=|r -f
2x 2<//^T~
Tang, v at any point = - = ^
y i
2h
Tang, v at abutment = ^
t
. Load concentrated at crown or middle of arch:
,_25Z_ A fty 25%2
" V 64/i 5(5Z n ^ 32Z 3 /
__ 25^ IF 81 PF/s
** "cTT A T
1600J
25Z X 1600J
64A( 16007
Load equally distributed:
_ + _ / .
27* p ;^ l-
STRAINS IN A POLYGONAL FRAME IN EQUILIBRIUM.
Load equally distributed over members of Frame.
Reference.
H = Horizontal strain in units of weight at foot.
F n = Vertical strain in units of weight at foot.
(7 n = Compressive strain in units of weight in direction of
member.
TF n = Load in units of weight, equally distributed over a mem
ber of the polygon.
fl n = Angle between horizontal and member.
PARABOLIC ARCHED BEAMS OR RIBS. 155
Fig. 236.
H J TFcotg. v n (7 n = F n cosec. v a
2 2
IT.+
, 3 _ , . .
1 s ^ + 2 z 2 2
_TFi ,
"
2
W.
2
For the equilibrium, v l being given :
Tang. v 4 = -^r = tang. ^
.a
2 + TF 3 ) +
H
The above can be used to compute the strains in ribs for dome
construction.
156 STRAINS IN ROOF TRUSSES.
STRAINS IN ROOF TRUSSES.
Reference. (Figs. 237 to 255.)
C Weight of construction. }
TF = < Pressure of wind. > Load in units of
( Pressure of snow. J
weight, equally distributed over one rafter.
(See Fig. 238.)
C= Compression of member in units of weight.
T Tension of member in units of weight.
L = Total span, or distance between abutments in
units of length.
d, h, I, and S = Dimensions in units of length. (See Figures.)
v, y Angles. (See Figures.)
The diagrams show only one-half of truss, (except Fig. 238.)
the thick lines indicating compression, and the thin ones tension.
(See "Reaction of Supports " for pressure on joints ; also " Compound
Strains in Trussed Beams")
Compression in Rafters. (Trusses Nos. 1, 3, and 4.)
The compressive strain in the rafter gradually increases from
ridge to abutments. Let x = Horizontal distance from abutment
to point where the strain is desired, and I half the span = - .
tg. v
C for Truss No. 1 = IF sin. v (l ) + -HL
Cfor Truss No. 3 = IF sin. v(l } + -^
1/2 tg. (v -|- i\)
C for Truss No. 4 = JFsin. v(l } + -
V I n 2 ig.(v~ Vl )
In the following examples the maximum of C is given :
Truss No. 1.
Fig. 237.
W cos. v
C=W&m.
2 tg.t;
W
T= cotg.v
STEAINS Itf ROOF TRUSSES. 157
EXAMPLE.
Let W 8,000 Ibs.
v = 26 30 .
C= 8000 X 0.44619 + -^ S5T = 10 666 lbs Com
7 8000
When a; = ~ then will C = - 7^7777^ = 8,968 Ibs. Com.
A 2. X v^.TCTrOiy
j_ 80Q 2.00 = 8,000 Ibs. Tension.
Truss No. 2.
Fig. 238.
W W
T= sin. v cos. v = sin. 2v
2 4
EXAMPLE.
Let TT= 8,000 Ibs.
v = 26 30 .
8000
C= - X 0.4462 = 1,785 Ibs. Compression.
a
C 1 = 8000 X 0.895 2 = 6,568 Ibs. Compression.
8000
T= -^ X 0.7986 = 1,597 Ibs. Tension.
[NOTE. When the rafters are fastened together at the ridge, they are
under a cross-breaking strain only. Consequently there is no horizontal
thrust at the abutments ; that is, T=0, and the compression in <?i == W.]
158
Fig. 239.
STRAINS IN ROOF TRUSSES.
Truss No. 3.
n TT7 . W cos.
C = IF sin. v -\
2 t.^-
W cos.
2 sin. (v + Vj)
fy. 240.
Truss No. 4.
0= Ws m.v
W cos. v
2
cos. v
2 sin.(v Vj)
T,= TF-^-~ ^~
sm.(v vj
Let TF= 8,000 Ibs.
v = 26 30 X .
v 1= = 5 O x .
C = 8000 X 0.44619 +
EXAMPLE.
= 12,653 Ibs. Coin.
?1 = 9,920 Ibs. Tension.
0.087
O.ooo
Bl|7201bB> Tension .
STRAINS IN HOOF TRUSSES.
159
Truss No. 5.
Fig. 241.
= ifTFcosec. <y Ci= J TP cotg. v T= jl + -^-- TPcotg. v
When there is no tie T, (7 3 is under a tensile strain == n~,
4/&
7i being the height from C l to ridge.
EXAMPLE.
Let TT= 8,000 Ibs.
? = 22.36 feet.
/ L== 11.18 feet.
v = 26 3CK.
C= -}-|8000 X 2.241 14,566 Ibs. Compression.
Q % 8000 X 2. = 8,OQO Ibs. Compression.
T= } (l + ) 8000 X 2. 9 12,000 Ibs. Tension.
\ ^ii.oo /
Truss No. 6.
Fig. 242.
160
STEAINS IN ROOF TRUSSES.
,== 2(Wi\ W) --
EXAMPLE.
Let TF= 8,000 Ibs.
Z = 20 feet.
/ 1= = 20.6 feet.
h = 10 feet.
A!= 5 feet.
== 22.36.
8000 x 500 1 500 (500 10 X 5)
5 X 2236 - -^^ = 29,264 Ibs. Com.
Q= 0.625 x SOOOfg = 10,000 Ibs. Compression.
1500) JL = 26,780 Ibs. Tension.
_ K
Tj= 2(8000 1500)
= 13,000 Ibs. Tension.
Truss No. 7.
Fig. 243.
= W ~9J- -
M sin. v
(?!= if- IF cosec. v
Let W= 8,000 Ibs.
Z = 20 feet.
sin. 2v
EXAMPLE.
h = 10 feet, v = 26 30 .
Z x = 11.18 feet. vi== 26 30 7 .
STRAINS IN EOOF TRUSSES.
161
n gQOO
9Q
-
2 X 20 X 0.44619
= 8,964 Ibs. Compression.
08125 X 8000 X 2.2411 = 14,567 Ibs. Compression.
2== 0.625 X 8000 X 1-12 = 5,600 Ibs. Compression.
= 0.625 X 8000 = 5,000 Ibs. Tension.
(7
G = :
T= . , .
2i= 0.8125 X 8000 X 2.0 = 13,000 Ibs.
Truss No. 8.
Fig. 244.
c -= . = w
2sin.v 2^sin
s m.(v
= | T7 2
h
sin - ^ i
cos, vain. fa i
"1 _
J ~
Let TF^8,000 Ibs.
v = 26 30 / .
t;!= 9 20 .
up= 19 .
11
EXAMPLE.
162 STRAINS IN ROOF TRUSSES.
9000 + 0.375 x 8000
C= - - = 13,452 Ibs. Compression.
0.892
Ci 0.812 X 8000 ~r = 21,710 Ibs. Compression.
u.zyo
0. 2 = 0.625 X 8000 A 8 ^ = 7,535 Ibs. Compression.
T = 812 X 8000 -~- = 19,702 Ibs. Tension.
(j.2t Jo
Truss No. 9,
5i<7. 245.
C= if fF_J: f IF sin. v
sin. v
1
(;, j| FF--^ = -^fTFcosec. v
r j\= |f 17 cotg. v T V IF cotg. v = -}- IF cotg. v
T,= i IWcotg.v
EXAMPLE.
Let TF^ 8,000 Ibs.
v = 26 30 7 .
C = 0.812x 8000 x 2.241 0.625 X 8000 X 0.4 16 = 12,336 Ibs.
Compression.
Ci= 0.812 X 8000 X 2.241 = 14,56G Ibs. Compression.
C 2 0.625 X 8000 X 0.895 4,475 Ibs. Compression.
T 0.312 X 8000 X 2. = 4,992 Ibs. Tension.
2\= 0.812 X 8000 X 2 0.312X 8000X 2. = 8,0001bs. Tension.
T 2 = 0.812 X 8000 X 2. = 12,992 Ibs. Tension.
STRAINS IN ROOF TRUSSES. 163
Truss No. 1O.
Fig. 246.
~~N I IF sin. v
<7 2 =f JFcos.v.
T _ COS. V COS. Vi
2\= if W^^--~ Tcos. (2v vi) -I Wain. cos. v
W I
EXAMPLE.
Let TF= 8,000 Ibs. Vl = 9 20^ A = 10 feet
v = 2630 / . 1 = 20 feet. A,= 2 feet.
(7=0.8125 X 8000 -- 0.625 x 8000 X 0.446 = 19,517 Ibs.
Compression.
987
(7i= 0.8125 X S 000 ^ 1 ^- == 21.747 Ibs. Compression.
OF= 0.625 x 8000 x 0.895 = 4,475 Ibs. Compression
o, 2 5 X 3000 x
0.895 2 ] = 7,163 Ibs. Tension.
164 STRAINS IN ROOF TRUSSES.
Tl = _ X -j- = 10,000 Ibs. Tension.
T= 0.8125 X 8000 = 19,720 Ibs. Tension.
Truss No. 11.
Fig. 247.
(7=13 IF . C S - Vl -- - f TF sin. *
cos. y
T=^W (- 1 / - [-5 cos. v\
~W I cos. v
2 h hi sin.^ Vi)
EXAMPLE.
Let W= 8,000 Ibs. y == 50. h = 10 feet. I = 20 feet.
v = 2630 / . w 1 =920 / . Ai=2feet. S = 22.36 feet.
(7= 0.8325X8000 ^ ^ 0.625 X 8000 X 0.446 = 19,517 Ibs.
Compression.
= 0.8125 X 8000 ~- = 21,747 Ibs. Compression,
u.^yo
= 0.366 x 8000 %~. = 4,070 Ibs. Compression.
STRAINS IN EOOF TRUSSES.
165
2- = 0.125 X 8000 (6.5 . + 5 . 0.894) = 11,050 Ibs.
Tension.
7i= 19486 X 0.986 7421 x 0.723 4930 X 0.446 = 10,000 Iba.
Tension.
S94
Tf= 0.812 X 8000 -^-^ = 19,486 Ibs. Tension.
0.29o
Truss No. 12.
Fig. 248.
Let P7= 8000 Ibs.
Z = 20 feet.
EXAMPLE.
A = 10 feet.
/S = 22.36 feet.
20
C7 1= 0.366 X 8000 = 5,856 Ibs. Compression.
8000 22 sr
(7 2 = 0.366 X -- 5 ~ = 3,280 Ibs. Compression.
10
= 0.866 X
10
20,992 Ibs. Tension.
3;= 1.23 X 8000 = 9,840 Ibs. Tension.
166
STRAINS IN EOOF TRUSSES.
Truss No. 13.
Fig. 249.
?i=*T-
cos. v t
C,^ f| W : -.
- - 1 J 01 r> It,
e 4 = ft x
sin.(D L
A.
A.
k
T- Wk * W
2 ~T i? w
EXAMPLE.
Let TF= 20,000 Ibs. h = 20 feet, v = 21 40 X .
2 = 50 feet. 1 2 = 53.8 feet. v = 0.
CO O
C = 0.5 X 20000 - = 26,900 Ibs. Compression.
(7 1= 0.683 X 20000 ~ = 37,018 Ibs. Compression.
C 2 = 0.866 X 20000 * = 46,937 Ibs. Compression.
(7 3 =0.55 x 20000 -^-i- = 11,770 Ibs. Compression.
Q=0.55 x 20000 4^ =9,900 Ibs. Compression.
STRAINS IN EOOF TRUSSES.
167
T= 0.683 X 20000 X 2.517 = 34,382 Ibs. Tension.
T^= 0.866 X 20000 X 2.517 = 43,594 Ibs. Tension.
20000 X 20 = 14)666 lbg> Tensiont
20
T 3 = 0.183 X 20000 = 3,660 Ibs. Tension.
Truss No. 14.
Fig. 250.
h h
rp n ^2
J-b O 6 T
EXAMPLE.
Let TF= 24,000 Ibs. Span = 100 feet 1= Z 1= Z 2 = ? 3 = 1.25 feet.
A = 20 feet. JJ^O. ^^53.85 feet.
168
STRAINS IN ROOF TRUSSES.
53 85
Q= 12000 x ~ = 32,310 Ibs. Compression.
<7 2 = 49088 - 0.228 X 24000 ^L = 41,728 Ibs. Com.
C 8 = 58320 0.286 x 24000 ^L = 49^88 Ibs. Com.
^ X 20
4 = 21600 -^0-- = 58,320 Ibs. Compression.
C 5 = (5801 + 0.286 X 24000) ~^-- = 12,493 Ibs. Com.
Q= 3432 + 5484 ~~- = 9,282 Ibs. Compression.
13 47
0,= 0.286 X 24000 ~^~ = 9,245 Ibs. Compression.
T,= (24000 0.1 x 24000) -^- = 54,000 Ibs. Tension.
7!,= 5 1000 0.286 x 21000 -~- = 45,420 Ibs. Tension.
7!,= 45120 9282 --f\ 3 8> 170 Ibs. Tension.
In
7^=24000 -121000= 19,200 Ibs. Tension.
r 5 = 9282 J- = 5,801 Ibs. Tension,
lo
T G = 0.286 x 24000 ~ = 3,432 Ibs. Tension.
Truss No. 15.
Fig. 251.
y i F
STRAINS IN ROOF TRUSSES. 169
C= if W --- L^_^ -- | TFsin. v JJ TFcos. v cotg. (v vj
__
sin. (v ^) 2 (A
--.- tang. yi T 5 = \$W . ,
(h -A x ) sm.( y -yj)
EXAMPLE.
Let W= 20,000 lbs. h = 20 feet. v 1= = 0.
I = 50 feet, v = 21 40 . -v 2 = 46 30 .
tf = 0.866 X 20000 * 0.733 X 20000 X 0.369 0.183 X
0.369
20000 X 0.929 x 2.517 = 32,959 lbs. Compression.
C 1= = 0.866 X 20000 X - A ~T^ 0.366 X 20000 x 0.369
0.369
= 44,236 lbs. Compression.
C. 2 = 0.866 X 20000 X 7^7- = 46,937 lbs. Compression.
Q= 0.55 x 20000 X 0.929 = 10,219 Ibs. Compression.
C 4 = 0.366 X 20000 X 0.929 = 6,800 Ibs. Compression.
40
T= 20000 X -T^- X tang, v = Null.
= 1 0,9201bs. Ten SIO n.
Tf= 0.183 X 20000 X 2.5 = 9,150 Ibs. Tension.
TZ= 10000 X -2Q^ = 25 -00 lbs - Tension.
T,= 0.683 X 20000 X 2.5 = 34,150 lbs. Tension.
2^= 0.866 X 20000 X 2.5 = 43,300 lbs. Tension.
170
STRAINS IN ROOF TRUSSES.
Truss No. 16.
Fig. 252.
C 4 =f JFcos.v.
Q= j|> JFcos. v + f- IF cos. v = JfTFcos. v
-
sm.(2v
W_ __l_
~T h
F 9 TJ7"
^4= TO W ~Z
sin.^ Vi)
cos. v
sin. (v
EXAMPLE.
Let TT= 20,000 Ibs. A = 20 feet. AI= 0.
I = 50 feet. T; = 21 40 7 . vi.= 0.
0= 41885 0.286 X 20000 x 0.369 = 39,774 Ibs. Compression.
Cj= 43567 0.228 X 20000 X 0.369 = 41, 885 Ibs. Compression.
STEAINS IN ROOF TRUSSES. 171
<7 2 = 48780 5213 = 43 ; 567 Ibs. Compression.
(7 3 = 0.9 X 20000 = 48,780 Ibs. Compression.
<?== 0.286 X 20000 X 0.929 = 5.213 Ibs. Compression.
Q= 0.514 X 20000 X 0.929 = 9,550 Ibs. Compression.
T== ( 0.9 X 20000 -~ - 0.8 X 20000 X 369* 0.1 X
\ 0.369
20000) * = 20,000 Ibs. Tension.
/ 0.686
2i= T T 5 = 20000 7188 = 12,812 Ibs. Tension.
_
2 20
99
T 3 = T T 5 = 0.757 X 20000 = 38,1 18 Ibs. Tension.
U.oo 3
D99
T= 0.9 X 20000 - - = 45,306 Ibs. Tension.
f= T fi = T T l= 7,188 Ibs. Tension.
6 = T^= 7,188 Ibs. Tension.
Truss No. 17.
Fig. 253.
When the^ rafter is resting on joint A:
n _ W _ TFcos. v cos.^ -y)
-- - r - Go -% -- ; - - -
4 sin. v sm. v l
W
C? 1 =- r -T - r
4 sin.-v
, TT cos.
l= 0, cos.
sm. ^
Bending moment at point B C! 2 sin. v, . J.
172 STRAINS IN ROOF TRUSSES.
When rafter is fixed at joint A:
_
C -
W
- r -
4 sin. v
_ W cos. v cos. (i\ v)
Go - % - : --
sm. v 1
T= JPFcotg. VI+TI
T } = ^- cotg. v
IF
Bending moment at B = - .
Truss No. 18.
Fig. 254.
TFcos.
ri i
T=Q
Ti= C 3 cos. v 4- C 2 cos.
STRAINS IN ROOF TRUSSES.
173
Truss No. 19.
Fig. 255.
C = \W cosec. v
. v
C 2 = Jf W cosec. v
a= w cotg. v
^i= f TFcotg. v + JTF tang,
T 2 = flF cotg. v
EXAMPLE.
Let W= 20,000 Ibs. v = 21 40 . t> 1= = 56
C 27,000 Ibs.
C } = 36,900 Ibs.
<7 2 = 46,800 Ibs.
C 3 = 33,466 Ibs.
Q= 6,867 Ibs.
Compression.
Compression.
Compression.
Compressian.
Compression.
(7 5 = 3,533 Ibs.
(7 6 = 6,666 Ibs.
T= 6,666 Ibs.
7\= 37,000 Ibs.
r 2 = 41,831 Ibs.
Compression.
Compression.
Tension.
Tension.
Tension.
174
STRAINS IN ROOF TRUSSES.
M
<S 2
II 8?
>S
0>
o
5.S
bJD j:
s
w
"p
g
c
r
5
H
|
O
r I
CO O
1^
CD to
CD CM 10
OS CD Ci
Ci
TO O
Ci O O
r- ( to 1^
^
1
1 1
1
cq (M
II
O O O
II II II
OC5BH
^ CM co
ii ii r
O^fH
C
j
a
I
o
co
1>- O
rH IO
Tin CM
OO 04 10
O
T 1 Ci r 1
^ O to
iO 1C 1^
r^ CM co
^
-<
CM
|
CM CM
11
O
II II II
OC5^
CO CM CO
II II II
O<o^
a
^
D
I
-<
v
1C
O
^
1 1
1
38
T 1 O
CM CM
11
CO O Ci
CM ^( T I
T i Ci T i
odd
II II II
O<o^
iO O O
r-H O O
CO O O
CO CM CO
it H II
O^^H
^
I
1
o
Ci O
$8
CD tO r-l
co CM re
T ! C2 -H
^ o >o
t^ O r^i
Ci 1- CD
o
.<
o
1
JIJI
>s
O
II II II
OC5^i
<M T-I CM
II II II
oo^
c:
"1
K^
)
I
8
o
CO
]j
CD O
^H O
r- 10
33
Jr- CM Ci
to o ^
T 1 Ci i-H
odd
II II II
ocS 1 ^
CD
I>- O O
tQ to Csj
CM T-H CM
II II II
OCJS-i
tr
1
1
1
V
>0
^t 1
o o
CO to
tO CM
to co c>q
co co lr-
rH OO T 1
MH O to
Ci to r-
r-.(N 00
S
1
r-(
(M
H
53
O O
II II II
^>C5^
CM r-l r-l
Ii II II
OOS-H
T^
I
-
o
CO
CO O
CO O
CO
CO O O
CM O Ci
CM OO r- 1
O O O
CM O O
X) O O
^
<
CD
01
n
34
O O O
II II II
OO^i
TiTiTT
OtfS-H
o-
|
i
Tfl
VO O
1^10
T 1 J>-
r^ CM i i
i^ Ci CD
CM CD CM
tO O uO
CD tO CM
rt i"- T I
^
.<
CO
CO
1!
i
i-( O
13
000
II II II
OO^
rH O r-H
II II II
OO^
05
\
^
I
<
o
tO
Tf
II
5^>
CD O
to
11
co O O
to o to
CO tO CM
d d o
II II 1!
GCj&i
CD O O
to o to
Tti to t^
111
OC5TSs
g
i-i t-T
s?~~
O r-J"
REFERENCE
"FlfiTTRTCR
;
6 ^o
to .^-2
s^s.
S ci
|e a
il|S
i?
25 s -
d ^ci
fe^S
S^S)
Ssa
as
STRAINS IN EOOF TRUSSES.
175
O 1O OC tO
1O CM 00 CM
rH r-l rH CD
CM CM tC CM
CO r-H 01 CO
rH CO CO O
CM CO <M O rH
CO r-H CD O rH
rH CO 1C 1C rH
CO CM rH 1C CO
CM iC ^H 1^- CO
r^ oo co co oq
oo ro oo T i
II II II II
0^5^^
r}H T-H O rH
II IL" Ii
^5^^H ^
HH O rH CM rH
II II I 1 Ii II
^Sjgjiiii
CO rH O CO rH
II II II II II
o^cS^s^
rh CM O 1C
(M ^H 01 01
1C OC CO CO
TTI ^rr LO CO
1C xH Ol LQ
1.^ *5f CO CO
rf O CD O rH
1C rH O 1C CD
i CO rH CM CO
o T i m o oo
CO 1C rH 1C CO
o cc GO i^ oq
1- <M 1> rH
II 1! II II
^CJ^H^
CO CO
II II II II
^^^^r
CO O r-i (M CO
II II II I! II
OO^K ^i
CO rH O LC rH
II II II II II
OCJO^iS7
CO O O 1C
rH o co oq
LC ic rH co
CD Oq CO rH
II II II 1!
o^J 1 ^^
co UC UC
JL^ 71 IO
CO 01 CO CM
co -i O co
ii 11 il IL
. -^j^e-i^
CO CO O O !>
r-i O 5 Q 1
co co CM o oq
TO O CM CO
II II II II II
>^Ji2li
CO CO I- 1C CO
CO rH rH 04 CO
CO rH 1- rH oq
1C rH O 1C rH
II II II II II
ojg^liS
rH CO OO CM
O r-< F-
"^ ^f iC co
i- ^ CM rH
O r-l CO OC
T 1 O rH O O
1:- O CO 1C CO
O CO O 1^ OO
O O O O CO
oo co r- o co
X^ Oq CD 1C CM
1C CM l-C rH
il i! ii ii
t>cS^^
<n -- i o CM
II II II II
55 O&f6>i
CM O rH rH 01
II 1! 1 II II
OO^^^T
rH rH o rH rH
II II II II II
O C?0^ s
iC iC O
CO - i - Ol
r-i CO 1 CO
C r-H rH r- 1
II II II II
06^
ic i- uc r-
1^- CO CM CO
1C Ot> CO rJH
01 o o oi
II 11 II Ii
<oQj-i E>T
1C rH 1 O 1C
1- CO CO O rH
<C iC CO 1C rH
01 o o T-H oq
II II II II II
Q^Mj^E?
O O O iC CO
i- O C^ J>- CO
rH rH 1C X) Oq
rH rH O CO rH
II II II II II
o^cS ^^T
"-0 CO O 1C
CO CO 00 0]
CO 1C i i CO
^ -^f to r- 1
CO rH Ol CO
r- OC COO
rri rH O T i
CO CO GO 1C CO
r- LC i oq o
CO 1C rH O CO
1C rH CO 1C CO
ic o rH oq oq
rH 1 rH T 1
II II Ii II
^CJ^^H
CM O O CM
i! II II II
C C^^
CM o o r-i oq
II II II II II
ercSNesw
CO O CO rH
II II II II II
ooo^^ 4
?6 o ft cl
o 01 oq co
CO r- 1 OO T 1
il II II II
C0~^^
O O iC iC
01 o CM o~i
00 L- CO CO
T O T-H
II II II II
^^?^^
O CO iC O 1C
CM iC CM O CM
CO 1C CO O CO
r-- O O rH rH
II II II II II
c5XS^^Ts
CO 01 O O CO
CO CO rH 01 CO
CO 1>- rH CO Oq
oq o o oq T-H
II II II II II
^^cJ^^
. I CO 1C
1C CO Cl 01
0> Ci rH CO
CM O CM r-4
II II II II
oo^^r
01 o ic co
CO CO CM (
rf o CO CM
T-H O CD T-H
11 II, II il
t5>v>5-i ^
Cl O CO O CO
co oq CD ic I-H
rH 1C rH t^- CM
T O O rH
II II II II \l
CJ^NK w
co c; co o co
co rH oq o co
oo ic co o CM
01 O O CM rH
II II II II II
OOCD^^T
rt< ic ic ic
co CM oq oq
CM CO CO CO
iC C * C 0^
rr rH 1 r- 1
i. ^ CO CO
iC Oj Ol O CM
rH rH rH O rH
r rH CO iC OO
CM CO CO 1C CO
OO CO 1C i^ CO
co co oq co CM
<M O r- 1 r-i
II !LJI 11
O ^5^ E>T
1111
cc ^K"
r- O O
II II II II II
^j^j ^^r^
T-H O O rH rH
II II II II II
?iS25iS
co eT
1- ill
II $
"- 381
H&
^ 2^
5 51
II fe ^
1 S^S:
> M Oi c3
S A
Hi
C5 5
l.lt
m ^ ?n
II*
SStf
fi-*3
II s^l
4, r ^ 0,
-, V
176
STRAINS IN ROOF TRUSSES.
J _V (.1.) U.J +JJ TiJ L
H GO CO TtH CO 1^ r
O CO T+i O O
^H O
* r-H T^H O CO CO
3 OO rH O 1C Tt<
) CO CM CO CM r-H
"fOOO^OOCOO
O O O O -N~t
n O O CO CO O O
II II 11-11 ILILIL
*
Cxi O O O
I I I
CO O O CM 00
O O O 00 C.I CM O O
H, li, IL 1 1 IL i IL IL 1
O 11 r-1 TTI CO -Hi CO
1-^ OO CO O -^ 00 CO
rH 1^ CO CO O 1- r-1
co o o cq co o o
COOOOCOCMCMOOO
I iL i UUL UULJl
O O i c_; co -^r oo
^ O CO O O CO CO
r- o o c: co i^ T-H
00 CO CO rH O O CO O CO CO
CM O O r-i 01 O O
II \\\\ II
CMOOOCMCMr-HOOO
IL IL I ULJLUl UL
CO "ti O i-^ O -m CO
CO O5 Ci O CO CO -00
CO IO ^ !> i > i^ T >
N Q Q rH o4 O O
01 TO *... r
O O o tx t
O T ( CO CO O CO CO O CO CO
CO CM CO CO O CO CO O lO ^t 1
r^i jo co co CM co o co CM -H
CMOOOCMr-Hr-HOOO
CMOOOr-H -HT-HOOO
lULUL UUL LUl
* " " "
CO O O rf CO TI 00
LO CO OO CM CO CO >0
lO -^ CO O CM t
T-H O O i T-^ O O
O CO O J^- O CO i O CO CO
ci j>- o o o co CM o ic ^f
CO rt" CC CM OO T iCOCOCMr-H
11111111
CM
1
r-1 ^ r CO CO ^T CO
r-H O CO CO CC CO X)
CM -^ OJ CO CO 1- r-1
OO O O CO O 1 < i O CO CO
CO CM CO O O lO 00 O O rti
CM T*H CM 04 OO I - CO CO OJ T i
r-^oo oooo ooo
ll, IL ILJLJL IL ILJLJL
REFE
FI
iH *
Si to a; c
p
^, CO
H ^
STRAINS IN ROOF TRUSSES.
CD CM iO O <M tO iO iO 1>- 1 -
J>- O^> LO iO O O CO
CO lO CO Ol 1C OO O
LO CO CO CO CM O Ci
CO^fCDociOOiOO
CO CO iO CO CM O O CO
lit 1 till
O GO (M i i-^fO-^HCC
!> CM lO T 4 -rf lO O -rH
T IIOCOI>-CD1^^O
co o o o o T-H oi co
^H (M ^H Cvj LO O iO O
Oq O O O O rH Cq 01
(MOOOOr-l iCI
1! II II 11 II II II 1!
" "
T ( O O O O i i i (
II II II II II II II II
-r^w^
I
1
T-HCMlOOOrHCM^HOCDCC
^OO^ IT ((MCO^OO
II 1! II II II II II II H II
OOicoOOOOO^OC.
iO I>- CO O 1^ O O O CO TO
COOO^OT-ICVICMOO
II II II II II II II II II II
CO CD I>- O -^H IO >O IO IO i ."
<M o o r-J o T-^ r-i oi o o
- ILOCDOr I O O O CO CO
C^OOOOr-HT-i^OO
II II II II II II II II II II.
qOOO
I II II II
OiCMcOOIr-OOOcO
CDOCDOiOOOO^-
d C^J CO TjH Ol lO t Ot) T I T
r-iOOOOOOOOc
IIILUJMI .11,11 " -
12
177
178
PRESSURE OF WIND ON ROOFS.
EXAMPLE TO TABLE OF CONSTANTS. (Tniss No. 13.)
What is the amount of strain in the various members of a truss,
according to Fig. 249, of the following dimensions, viz: Span 60
feet, distance between trusses 10 feet, height at center 10 feet,
weight to be carried, including weight of construction, 66| Ibs.
per square foot horizontally; hence total weight on one rafter
= 30 X 10 X 66} = 20,000 Ibs.?
L = 60 feet. r 6Q v = 18 20 .
h = 10 feet. = = 6. W = 20,000 Ibs.
^
o"
Member. Constant. W Strains.
C 2 = 2.745 x 20,000 = 54,900 Ibs.
C, = 0.660 X 20,000 = 13,200 Ibs. J. Compression.
C = 0.567 X 20,000 = 11,340 Ibs. }
T = 1.956 x 20,000 = 39,120 Ibs.
T. = 2.606 x 20,000 == 52,120 1
K = 0.734 x 20,000 = 14^680 Ibs. ,
T a 0.183 X 20,000 = 3,660 Ibs. J
Tension.
[NOTE. la the foregoing table the proportion of 7i to L is approximate.
The constants are based on the angles.]
PRESSURE OF WIND ON ROOFS.
In the following table the maximum pressure of wind is taken
at 50 Ibs. per square foot:
The angle between horizontal and direction of wind is generally
10 00 r . (See diagram.)
Fig. 256.
Reference.
F = Force of wind in Ibs. = 50.
w, = Pressure at right angles to surface per square foot in Ibs.
w // = Pressure, vertical, per square foot in Ibs.
w / = F sin. 2 (v + 10)
w,
w^=
COS. V
PRESSURE OF WIND ON ROOFS.
Proportion of
height h to
span I.
Angle v.
Pressure w,
in Ibs.
Pressure w tl
in Ibs.
90 00
50.00
0.00
*4
45 00
33.53
47.40
&=4-
33 41 50"
23.80
28.60
z
26 33 50"
17.64
19.70
*=-g-
21 48
13.83
14.80
;t =4
18 26
11.23
11.80
z
"7
15 54 40"
9.46
9.80
A=-L
14 02 10"
8.56
8.80
* = -r
12 31 40"
7.29
7.40
i-i
11 18 40"
6.51
6.60
180
PEESSUEE OF SNOW ON EOOFS.
PRESSURE OF SNOW ON ROOFS.
The average pressure of snow on a level surface, in the United
States, is about 15 Ibs. per square foot.
The following table gives the pressure per square foot on
various inclined surfaces :
Reference.
P = Pressure per square foot in Ibs. = 15.
p 1 = Vertical pressure in Ibs.
p z = Pressure at right angles to surface in Ibs.
v = Angle between surface and horizontal.
p 1 = P cos. v.
Proportion of
height h to
span I.
Angle v.
Pressure PI
in Ibs.
Pressure P 2
in Ibs.
I
~2~
45 00
10.60
7.49
--r
33 41 50"
12.48
10.38
* T
26 33 50"
13.42
12.00
*=4-
21 48
13.93
12.94
*=4-
18 26
14.23
13,50
*=4
15 54 40"
14.41
13.86
*=i
14 02 10"
14.52
. 14.05
*=4-
12 31 40"
14.64
14.29
h== Jo
11 18 40"
14.71
14.43
I
00 00"
15.00
15.00
TIE RODS AND BARS.
TIE BODS AND BARS.
Capacity and Proportional Dimensions of Wrought-iron Tie Rods
Tie Bars, and Pins or Bolts.
Ultimate resistance to tearing = 60,000 Ibs. = 30 tons pei
square inch.
Ultimate resistance to shearing = 50,000 Ibs. = 25 tons pe:
square inch. (See Fig. 258.)
Capacity of tie or bar.
t
d
inches,
d.
Dimension of
flat bars in in.,
uniform thick
ness.
Diamete
D of pii
or bolt.
2 1
d C
en
4
o>
O SJO
CO b
3 times safety.
5 times safety.
"ol C
g.rH
O
** 2
Q.,-1
Jj!
rO
O p
ci
Lbs.
Tons.
Lbs.
Tons.
1
CO
1
3
I-
I" 8
OJ
a cc
PH c
C X
-^
5,000
2.50
3,000
1.50
0.25
0.56
1
0.75
0.62
0.4
6.200
3.10
3.720
1.86
0.31
0.62
\\/
0-93
0.69
0.4
7,400
3.70
4,440
2.22
0.37
0.70
<<
11 A
1.12
0.75
0.5
8,000
4.30
5,160
2.58
0.43
0.74
"
1^4
1.31
0.80
0.5
10,000
5.00
6,000
3.00
0.50
0.79
"
2
1.50
0.88
0.0
11.200
5.00
6,720
3.36
0.56
0.84
"
2i/
1.68
0.92
0.0
12,400
6.2D
7,440
3.72
0.62
0.89
2\s
1.87
0.97
o.o
13,000
6.80
8.160
3.88
0.68
0.93
"
2 ^4
2.06
1.01
0.7
15,000
7.50
9,000
4.50
0.75
0.97
"
3
2.25
1.08
0.7
7,400
3.70
4,440
2.22
0.37
0.68
%
1
0.75
0.75
0.6
9,200
4.60
5,520
2.76
0.40
0.76
\\s
0.93
0.83
O.o
11,200
5.00
6,720
3.36
0.56
0.84
"
l 1 ^
1.12
0.92
0.0
13,000
0.50
7,800
3.90
0.65
0.91
"
]^
1.31
0.99
0.7
15,000
7.50
9,000
4.50
0.75
0.97
"
2
1.50
1.08
0.7
10,800
8.40
10,080
5.04
0.84
1.04
"
2/4
1.68
1.13
0.8
18,000
9.30
11,100
5.58
0.93
1.09
"
2/^
1.87
1.19
0.8
20,000
10.30
12,300
6.18
1.03
1.15
"
2^4
2.06
1.24
0.8
22,400
11.20
13,440
6.72
1.12
1.19
"
3
2.25
1.29
0.9
10,000
5.00
6,000
3.00
0.50
0.79
1 A
1
0.75
0.88
0.0
12,400
6.20
7,440
3.72
0.02
0.88
tf
0.93
0.97
0.0
15,000
7.50
9,000
4.50
0.75
0.97
|i/
1.12
1.08
0.7
17,400
8.70
10,440
5.02
0.87
1.05
M
l^i
1.31
1.16
0.8
20,000
10.00
12,000
6.00
1.00
1.13
"
2
1.50
1.24
O.S
22,400
11.20
13,440
6.72
1.12
1.20
"
214
1.68
1.32
0.9
25,000
12.50
15,000
7.50
1.25
1.26
"
2M
1.87
1.39
0.9
27.400
13.70
16,440
8.22
1.37
1.32
"
2%
2.00
1.45
1.0,
30^000
15.00
18,000
9.00
1.50
1.39
"
3
225
1.52
1.0
12,400
6.20
7,440
3.72
0.62
0.90
%
1
0.75
0.98
0.0
15.600
7.80
9,300
4.68
0.78
1.00
"
\\s
0.93
1.09
0.7
18,600
9.30
11,160
5.58
0.93
1.09
*
\\
1.12
1.20
0.8,
21,800
10.90
13,080
6.54
1.09
1.18
M
]%/
1.31
1.29
0.91
25,000
12.50
15,000
7.50
1.25
1.26
M
2
1.50
1.39
0.9!
28,000
14.00
16,800
8.40
1.40
1.34
"
2/4
1.08
1.47
1.0
30,533
15.27
18,720
9.36
1.56
1.41
"
2V
1.87
1.54
l.OJ
TIE RODS AND BARS.
Capacity of tie or bar.
g
ft
a
c .
rt C
Dimension of
flat bars in in.,
uniform thick
ness.
Diameter
D of pin
or bolt.
7. jn
.5 o
GO
* bb
3 times safety.
5 times safety.
ol C
"
fl
V H
* .:
* c
||
If
Lbs.
Tons.
Lbs.
Tons.
I
s
li
H
t*
^ c
ol
l|
1
34,200
17.10
20,520
10.26
1.71
1.48
~%
2%
2.06
1.62
1.14
37,500
18.75
22,440
11.22
1.87
1.54
" 8
2.25
1.69
1.20
15,000
7.50
9,000
4.50
0.75
0.98
%
1
0.75
1.08
0.76
18,600
9.30
11,160
5.58
0.93
1.09
M
0.93
1.20
0.85
22,400
11.20
13,440
6.72
1.12
1.19
"
\\/
1.12
1.31
0.93
26,200
13.10
15,720
7.86
1.31
1.30
"
1&
1.31
1.41
1.00
30,000
15.00
18,000
9.00
1.50
1.39
"
2
1.50
1.52
1.08
33,600
16.80
20,160
10.08
1.68
1.46
"
1.68
1.62
1.14
37,400
18.70
22,440
11.22
1.87
1.54
"
2V"
1.87
1.69
1.20
41,200
20.60
24,720
12.36
2.06
1.62
"
2%
2.06
1.77
1.26
45,000
22.50
27,000
13.50
2.25
1.69
"
3
2.25
1.86
1.32
17,400
8.70
10,440
5.22
0.87
1.05
%
1
0.75
1.16
0.82
21,800
10.90
13,080
6.54
1.09
1.18
0.93
1.29
0.91
26,200
13.10
15,720
7.86
1.31
1.29
1||
1.12
1.41
1.00
30,600
15.30
18,360
9.18
1.53
1.40
1.31
1.53
1.08
34,800
17.40
20,880
10.44
1.74
1.49
"
2 4
1.50
1.63
1.16
39,200
19.60
23,520
11.76
1.96
1.58
"
2*4
1.68
1.73
1.23
43,600
21.80
26,160
13.08
2.18
1.66
11
2/lz
1.87
1.82
1.29
48,000
24.00
28,800
14.40
2.40
1.75
"
2%
2.06
1.89
1.34
52,400
26.20
31.440
15.72
2.62
1.83
"
3
2.25
2.00
1.42
20,000
10.00
12,000
6.00
1.00
1.13
1
1
0.75
1.39
0.80
25,000
12.50
15,000
7.50
1.25
1.26
"
i/4
0.93
1.45
0.98
30,000
15.00
18,000
9.00
1.50
1.39
"
\\z
1.12
1.52
1.08
35,000
17.50
21,000
10.50
1.75
1.49
"
\3S
1.31
1.64
1.16
40,000
20.00
24,000
12.00
2.00
1.60
"
2
1.50
1.75
1.24
45,000
22.50
27,000
13 50
2.25
1.70
"
2/4
1.68
1.86
1.32
50,000
25.00
30,000
15.00
2.50
1.79
M
2V^
1.87
1.96
1.39
55,000
27.50
33,000
16.50
2.75
1.87
"
2%
2.06
2.05
1.45
60,000
30.00
36,000
18.00
3.00
1.96
*
3
2.25
2.15
1.52
28,000
14.00
16,800
8.40
1.40
1.34
iy &
VA
0.93
1.47
1.04
33,600
16.80
20,160
10.08
1.68
1.47
"
i/^
1.12
1.60
1.13
39,600
19.80
23,520
11.76
1.98
1.58
"
i?^
1.31
1.73
1.23
45,000
22.50
27,000
13.50
2.25
1.69
"
2
1.50
1.86
1.32
50,600
25.30
30,360
15.18
2.53
1.80
"
2/4
1.68
1.97
1.39
56,200
28.10
33,720
16.86
2.81
1.89
"
2V^
1.87
2.09
1.48
61,800
30.90
37,080
18.54
3.09
1.98
"
2%
2.06
2.18
1.54
67,400
33.70
40,440
20.22
3.37
2.08
"
3
2.25
2.26
1.60
73,000
36.50
43,800
21.90
3.65
2.16
*
3/4
2.43
2.36
1.67
78,600
39.30
47,160
23.58
3.93
2.24
"
3/^j
2.62
2.45
1.74
84,200
42.10
50,520
25.26
4.21
2.32
*
3^4
281
2.53
1.80
90,000
45.00
54,000
27.00
4.50
2.40
4
3.00
2.63
1.86
31,200
15.60
18,720
9.36
1.56
1.41
li,/
|i/
0.93
1.54
1.09
37,400
18.70
22,440
11.22
1.87
1.55
w
li^
1.12
1.69
1.20
43,600
21.80
26,160
13.08
2.18
1.67
1%
1.31
1.82
1.29
50,000
25.00
30,000
15.00
2.50
1.79
2
1.50
1.96
1.39
56,200
28.10
33,720
16.86
2.81
1.89
(t
2/4
1.68
2.09
1.48
62,400
31.20 ,
37,440
18.72
3J2
1.99
"
2 /^
1.87
2.19
1.55
TIE HODS AND BARS.
Capacity of tie or bar.
cr
CO
(J
03 ^
o
a -
Dimension of
flat bars in in.,
uniform thick
ness.
Diameter
D of pin
or bolt.
3 times safety.
5 times safety.
S o
1!
00
tn .
0> ~
*>:
*!
t
if
O
<>-.
1
g
|
"2
si
iis
O J3
Lbs.
Tons.
Lbs.
Tons.
CB
Q
~
g
2
03
G co
0=3
_,*-
^
68,600
3430
41,160
20.58
3.43
2.10
i l /4
2%
2.06
2.29
1.62
75,000
37.50
45,000
22.50
3.75
2.19
"
3
2.25
2.40
1.70
81,200
40.60
48,720
24.36
4.06
2.27
3/4
2.43
2.49
1.7G
87,400
43.70
52,440
26.22
4,37
2.36
"
31^
2.62
2.60
1.84
93,600
46.80
56,160
28.08
4.68
2.44
"
3%
2.81
2.68
1.89
100,000
50.00
60,000
30.00
5.00
2.53
"
4
3.00
2.77
1.96
41,200
20.60
24,720
12.36
2.06
1.62
Jg
^A
1.12
1.77
1.26
48,000
24.00
28,800
14.40
2.40
1.75
"
1/4
1.31
1.89
1.34
55.000
27.50
33,000
16.50
2.75
1.87
"
2
1.50
2.05
1.45
61,800
30.90
37,080
18.54
3.09
1.98
2//
1.68
2.18
1.54
68,600
31.30
41,160
20.58
3.43
2.09
<
2/<2
1.87
2.29
1.62
75,600
37.80
45.360
22.68
3.78
2.19
-%
2.06
2.41
1.71
82,400
41.20
49,440
24.72
4.12
2.29
3
2.25
2.51
1.78
89,200
44.60
53,520
26.76
4.46
2.38
3/4
243
261
1.85
96,200
4810
57,720
28.86
4.81
2.47
3^2
2.62
2.71
1.92
103,000
51.50
61,800
30.90
5.15
2.56
3%
2.81
2.81
1.99
110,000
55.00
66,000
33.00
5.50
2.65
4
3.00
2.90
2.05
45,000
22.5
27,000
13.50
2.25
1.70
1 A
\y
1.12
1.86
1.32
52,400
26.20
31.440
15.72
2.62
1.83
1^4
1.31
2.00
1.42
60,000
30.00
36,000
18.00
3.00
1.96
2
1.50
2.15
1.52
67,400
33.70
40,440
20.22
3.37
2.07
2/4
1.68
2.27
1.61
75,000
37.50
45,000
22.50
3.75
2.19
2Vo
1.87
2.40
1.70
82,400
41.20
49.440
24.72
4.12
2.29
2%
2.06
2.51
1.78
90,000
45.00
54,000
27.00
4.50
2.40
3
2.25
2.63
1.86
97,400
48.70
58.440
29.22
4.87
2.49
3/4
2.43
2.73
1.93
105,000
52.50
63,000
31.50
5.25
2.59
3/^2
2.62
2.84
2.01
113,400
56.20
67,440
33.72
5.62
2.67
3%
2.81
2.93
2.08
120,000
60.00
72,000
36.00
6.00
2.77
4
3.00
3.03
2.15
127,400
63.70
76.440
38.22
6.37
2.85
4/4
3.18
3.12
2.21
135 000
67.50
81,000
40.50
6.75
2.93
41^
3.37
3.22
2.28
142,400
71.20
85,440
42.72
7.12
3.01
4/4
3.55
3.30
2.34
150,000
7500
90,000
45.00
7.50
3.10
5
3.75
3.39
2.40
181 JOINTS OR CONNECTIONS IN IRON CONSTRUCTION.
JOINTS OR CONNECTIONS IN IRON CONSTRUCTION.
PROPORTIONS OF BOLTS, NUTS, RIVETS, &c.
Reft re nee.
A = Sectional area of bolt, rivet, or pin.
AI= Sectional area of all rivets in a joint.
A% Sectional area of one plate.
D = Diameter of bolt, rivet, or pin.
S= Ultimate resistance to shearing of material.
T = Ultimate resistance to tearing of material.
TI= Tensional strain on joint, &c.
a = Number of times that a bolt, &c., will have to be sheared-
(See 2 on Fig. 258.)
d = Distance between centres of rivets.
k = Factor of safety.
I = Overlap, approximately If d to If d.
m = Number of rivets in a joint.
n = Number of lines of rivets in a joint at right angles to strain.
t = Thickness of a plate.
RIVETS.
Fig. 257.
For tension in direction of rivet:
-J-
T 0.7854
For shearing at right angles :
If at one place D= I - Tl *__
N S 0.7854
If at two places D
= I _ ?L
>J S 1.5
__
.5708
Approximately : I = 3t D = 3t
JOINTS OR CONNECTIONS IN IRON CONSTRUCTION.
PIN, &o., IN TIE BARS.
Fig. 258.
PLATE JOINTS.
No. I. Plate Joint Overlapped, four lines of Rivets.
Fig. 259.
- *. ; d =-- D + -L (0.7854 JDn)
o . d
.# i- Approximately c? = 1.5i to 2t
i^-ct- cJ--ci->i ft A A
(t)-^-
4V
^0.7854
__
2mtS
2. Ptafe Jbm^ Overlapped, single line of Rivet
Fig. 260. (Same as No. 1.)
186
JOINTS OR CONNECTIONS IN IRON CONSTRUCTION.
No. 3. Plate Joint Overlapped, two lines of Rivets.
Fig. 261. (Same as No. 1.)
o o
No. 4. Fish Joints, two lines of Rivets.
Fig. 262.
One fish plate. (Same as No. 1.)
Two fish plates.
Thickness of each fish plate = J t.
/>_-!_ /__**.
m ** 1.570.
L.5708
(Otherwise same as No. 1.)
DIMENSIONS OF BOLTS AND NUTS.
187
DIMENSIONS OF BOLTS AND NUTS.
(Whitworth s proportions.)
Figs. 263, 264, 265, 266, 267, 268, 269, 270, and 271.
Inch.
3
21
2}
21
2
If
H
i
I
A
I
A
Dimension of Nuts and Heads.
. -^
Inch.
p ?
Inch.
Inch.
Inch.
4J
5.18
5
7.07
4J
4.76
4J
6.37
3|
4.33
4J
5.83
3|
3.89
3|
5.30
3
3.46
3|
4.76
2f
3.17
3
4.24
2f
3.03
21-
3.89
^1
2.88
2|
3.71
2J
2.59
21
3.53
2
2.30
21
3.18
H
2.16
2
2.82
11
1.87
IF
2.64
1}
1.73
If
2.29
!&
1.51
H
2.12
1*
1.38
1A
1.86
1
1.15
1A
1.67
|
1.01
i
1.41
i
0.86
1
1.23
t
0.86
f
1.06
A
0.64
I
1.06
TV
0.50
A
0.79
f
0.43
A
0.79
Dia. of No. Threads
Core. per inch*
Inch. Inch,
3
2
.57
3.
5
1
.50
2f
2
.35
3
.5
1
.75
2J
2
.13
4,
,0
2
.00
21
1
.91
4,
,0
2
.12
2
1
.69
4,
5
2
.25
If
1
.58
4
.5
2
.37
If
1
.47
5,
,0
2
.50
If
1
.36
5.
;0
2
.75
If
1
.25
6
3
.00
If
1
.14
6,
,0
5
.25
It
1
.08
7,
,0
3
.50
H
.92
7,
,0
4
.00
l
.81
8.
5
.00
i
.70
9.
6
.00
i
.59
10
,0
6
.00
i
.48
11,
,0
7
.00
9
T6
.42
11.
7
.00
J
.37
12.
8
.00
TV
.31
14,
,0
8
.00
1
.26
16.
9
.00
*
.20
18
.0
9
.00
i
.15
20
.0
10
.00
188 STRAINS IN HORIZONTAL AND SLOPING BEAMS.
Fig. 272.
Approximate proportions of bolts, nuts, and
beads in incbes:
d = 1.4 D -\- 0.25 = Inscribed circle.
h = D = Height of nut.
/*!= 0.7 D = Height of head.
COMPOUND STRAINS IN HORIZONTAL AND SLOPING
BEAMS.
(Load equally distributed or between supports.)
Area of Cross-section necessary to resist a Cross-breaking and
Compressive Strain in Beams acting as a Boom in Trusses, &c,,
or Beams acting as Rafters, &c.
Reference.
rii = Bending moment (See Page 100.)
C= Compressive strain. (See Roof and Simple Trusses.)
g = A factor depending on form of cross-section.
/= Moment of inertia of cross-section.
8 = Distance from neutral axis to most compressed fibres.
A = Sectional area of beam, &c.
h = Depth of beam, &c.
p = Resistance to compression with safety per square inch of
section.
W= Total load.
I = Length of beam, &c.
_
STRAINS IN HORIZONTAL AND SLOPING BEAMS. 189
For horizontal beams, &c. :
For sloping beams, &c., v = angle between horizontal and beam:
W r 1 / 1 \\ I cos. v~\
A = -- LT (-sTnTV + sm - v ) + T2?T J
RAFTER OF A ROOF TRUSS.
Fig. 273.
EXAMPLE.
Reference.
W = 2.5 tons. = 2.8 tons. I = 10 feet. v = 26 30
p =. 5 tons per square inch.
We will assume a Phoenix Go s six-inch beam of the following
dimensions: h = 6 inches; A = 4 inches; J= 22.5
showing that the six-inch beam has a greater sectional area than
required.
If the load is concentrated at the apex of roof, the compressive
strain C= 2.8 tons, and the area necessary to resist this strain
2 8
would be (taking p at five tons per square inch) - 0.56 sq.
D
inches, provided this is able to resist buckling.
By comparing this with the above result, it will be seen how
much greater the sectional area will have to be to resist a cross -
breaking strain, caused by the load being distributed. These
remarks also apply to simple trusses.
190 STRAINS IN HORIZONTAL AND SLOPING BEAMS.
SIMPLE TRUSS, (BEAM CONTINUOUS OVER STRUT.)
Fig. 274.
EXAMPLE.
Reference.
W = 20 tons. I = 20 feet, v = 15 p = 5 tons per sq. inch.
We will assume a Phoenix Go s twelve-inch beam of the fol
lowing dimensions:
h = 12 inches. J= 275.92
A = 12.5 inches. s = 6 inches.
275.92
=
0.5 X 12 X 12.5
m=0.0703xlXl20 2 = 84.36 (See Reaction of Supports.)
C= 23.32 tons.
84.36 \ , 46.26
0.306x12
) +23.32=- =9.25 inches.
Consequently the sectional area of the twelve-inch beam is
amply sufficient.
[NOTE. The formulas for horizontal beams are also applicable to rafters
of roof trusses, m and C being given. For the bending moments (m)
the various distances are the horizontal projections of those on the rafter
from abutment to ridge.
The loregoing formulas also apply to beams under a cross-creaking and
tensional strain. If the truss (Fig. 274) is inverted, the horizontal member
will be in tension. Hence, insert the resistance of the material to tension
instead of compression, and put tensional for compressive strain; other
wise, the formulas remain the same.]
WEIGHT OF MOVING LOADS.
191
WEIGHT OF MOVING LOADS.
Variable and Accidental Loads.
(Weight of construction not included.)
Character of
structure.
How loaded.
Weight in Ibs. per square
foot of surface.
Street bridges for
horse cars and
heavy traffic.
Crow d with per
sons.
Minimum
40 Ibs.
120 "
80 "
Maximum
Average
Street bridges for
general traffic,
foot passengers,
&c.
Persons, animals,
and wagons.
Public travel....
Private travel...
Heavy business
wagons
80 Ibs.
40 "
80 "
40 "
Light business
wagons
Floors, &c
Crowded public
places.
Dwellings
Minimum
40 Ibs.
120 "
80 "
40 "
80 "
100 "
200 "
250 "
f200
\ to "
(400
80 "
Maximum
Average
Churches, court
rooms, theatres,
and ball-rooms.
Storage of grain...
General merchan
dise
Warehouses
Factories.
Hay- lofts
192
STATIC AND MOVING LOADS ON BRIDGES.
STATIC AND MOVING LOADS ON BRIDGES OF
WROUGHT IRON.
The following table gives an approximate weight per lineal
foot in pounds of the static load or weight of construction complete
for Single-Line Railway Bridges, supported at the ends, from ten
to four hundred feet span; also the weight of the moving load
per lineal foot of span, based on the assumption that the heaviest
locomotives exert a pressure of three thousand pounds per lineal
foot between their extreme bearings.
The table is applicable in computing the strains in all trusses
with parallel booms mentioned in this work.
Weight of Construction and Moving Load of Wrought- Iron Single-
Line Railway Bridges for the heaviest traffic.
(From 20 to 400 feet span.)
Weight of construction complete,
including cross-ties and rails.
Weight of moving load equal to 3,000
Ibs. per lineal foot of load.
Weight in
d
Weight in
d
Weight in
jj
Weight
in Ibs.
.2
Ibs. per
.2
Ibs. per
.2
Ibs. per
.2
a
lineal foot
a
lineal foot
a
lineal foot
23
per lin.
1
of span.
a
r Sl
of span.
c3
A
V)
of span.
a
CO
span.
10
427
210
1,891
10
6,300
210
2,535
20
500
220
1,964
20
5,370
220
2,495
30
573
230
2,037
30
4,250
230
2,455
40
646
240
2,110
40
3,780
240
2,375
50
719
250
2,183
50
3,550
250
L ,335
60
792
260
2,256
60
3,400
260
2,290
70
865
270
2,329
70
3,300
270
2,245
80
938
280
2,402
80
3,250
280
2,200
90
1,011
290
2,475
90
3,180
290
2,160
100
1,084
300
2,548
100
3,120
300
2,120
110
1,157
310
2.621
110
3,050
310
2,080
120
1,230
320
2,694
120
3,000
320
2,045
130
1,303
330
2,767
130
2,930
330
2,010
140
1,380
340
2,840
140
2,880
340
1,975
150
1,453
350
2,913
150
2,820
350
1,940
160
1,526
360
2,986
160
2,760
360
1,910
170
1,599
370
3,059
170
2,700
370
1,880
180
1,672
380
3,132
180
2,655
380
1,850
190
1,745
390
3,205
190
2,615
390
1,820
200
1,818
400
3,278
200
2,575
400
1,890
STATIC AND MOVING LOADS ON BRIDGES.
193
The following gives the actual weight of some well-known
Bridges (single line) in America, Germany, and England:
Name of Bridge.
System.
a
Weight of con
struction per
lineal foot.
6 <jj
<4_T3~0
^
.5 *"
c O ^
CO
a
CO
Lbs.
Lbs.
Lbs.
"Brenz," near
Konigsbronn...
"Colomak"
"Iser," near Mu
nich
| Open Web, ^
{ parallel booms, f
63.0
111.0
164.7
760
1,090
1,770
3,131
3,067
3656
7,530
9,516
8,532
"Donau," near
Ingolstadt
"Elb,"nearMei-
ssen
<
178.0
179
1,954
1 324
3,312
2 783
8,532
10390
"Rhine," near
Mainz
{"Pauli s," par- ")
abolic arched >
345
2 170
1 970
11 660
"Royal Albert,"
near Saltash...
"Boyne"
booms. J
Lattice..**...
455.0
264.0
4,418
3225
2,240
9,954
" Leven "
36
566
"Kent"
M
36
580
"Harper s Ferry"
Truss
124.0
770
13
MISCELLANEOUS.
(195)
QEOMETRY.
LONGIMETEY AND PLANIMETRY.
(Lines and Areas.)
Reference.
A = Area.
- = Periphery of circle = 3.14159 when diameter = 1.
r = Radius of circle.
G = Length of cord of segment.
p = Circumference of circle for given diameter.
I = Length of circle arc, &c.
h = Height of segment.
v = Angles, expressed in decimals, as 15 30 / = 15.5.
For other designations, see Figuies.
[NOTE. Always use the same unit for dimensions.]
t
Values cj TT.
- = \14159 x
2~ = 6.28319 ~ = 1.04720
= 0.31831 TT
* = 0.78540
= 0.15915 ;r
--= 0.52360
1
__ = 0.10132 - 2 = 9.86960
-3 _ 31.00628
=0.63662 ^1= L77245
&-= 1.46459
(197)
198
LONGIMETRY.
Fig. 275.
p =
Fig. 276.
360 1 " " 360
I
180
v = - 180
TTr
180 ^
7T
r. 277.
= 2(180
Fig. 278.
8h " 2h
. 279.
LONGIMETRY.
199
Fig. 280.
Fig. 281.
Ellipse.
Fig. 282.
[7)2
1+ ^
- + 1
256 n "J
When n = , ;
a -\- b
b = \/a z c*
a = \/6 2 -f c 2
Fig. 283.
. 284.
c 2 a 2 Z> 2
~26~
200
PLANIMETRY.
85. (Circle plane.)
286. (Circle ring.)
. 287. (Sector.) li
= 0.008727 w 2 .
/ 360 .
r== N/~T~~~
288. (Segment.)
A =
?; sin.v)
(0.017453 v sin. ) -
2
Fig. 28$. (Circle ring sector)
360 V1
: 0.008727 ^(ry 2 r 2 2 )
PLANIMETRY.
Fig. 290. (Ellipse.)
A == nab
Fig. 291. (Square.)
Fig. 292. (Rectangle.)
A = a 2
Fig. 293. (Parallelogram.)
= a sn. v
294. (Triangle.)
A = = be sin. v
2 2
c 2 sin. v sin. Vj
2 sin. v 2
When the three sides are given:
Let a + b + c = s
CENTER OP GRAVITY OF PLANES.
CENTER OF GRAVITY OF PLANES.
Reference.
x = Distance from a fixed base to center of gravity.
r = Radius.
c = Chord.
b,p, h = Dimensions.
A = Area.
v = Angle.
Fig. 295. (Quadrangle.)
Fig. 296. (Triangle.)
Fig. 297. (Half circle, or
elliptic plane.)
a and b parallel. "
h h ( b a \
X 2 6" Vfi+o J
- = radius = r
l
x = 0.4244r
Fig. 298. (Concentric ring.)
4 sin. Ji; r 3 r^
"" ~3 v r 2 r, 2
CENTER OF GRAVITY OF PLANES.
203
Fig. 299. (Circle, or elliptic
arc.)
re 2 sin.
V
JF%,300. (Half circumfer
ence of circle or ellipse.)
x = r = 0.6366r
7T
Fig. 301. (Circle sector.
4 sin. -Jv
*
Fig. 302. (Circle segment.)
. = Area.
Fig. 303. (Parabola.)
204
CENTER OF GRAVITY OF PLANES.
Fig. 305. (Half parabola.)
Fig. 305.
Of any section, composed of any
number of simple figures:
Additional Reference.
A, -4,, -4//= Sectional areaof simple
figures.
X = Distance from center of
gravity of whole sec
tion to axis ran.
x t x /t // = Distance from center of
gravity of a simple
figure to a fixed axis
mn.
Y ^ X ~^~ ^ /x/ ~^~ A//x/ + &c-
TRIGONOMETRICAL FORMULAS.
205
TRIGONOMETRICAL FORMULAS.
Reference.
a, 6, c = Length of sides.
A, B t C= Angles opposite to a, &, c respectively.
Right Angle Triangle.
Fig. 306.
cos. C
b = a cos. C
b = c cot. (7
b = a sin. B
b = c tang. B
c = b tang.
c = a sin.
Tang. (7=4-== n
b cos.
"cot. (7
Cotang. C=
Secant (7 =
Cosec. C =
cos. (7
sin. C
1
cos. (7
1
sin.
~ tang. C
Cos.(7= -
a
Oblique Angle Triangle.
Fig. 307.
B)
a = \/6 2 + c 2 26c cos.
c sin. 5
Sin. (7=
Sin. 4 =
c sin.
6
a sin. C
b
c sin. A
a
TRIGONOMETRICAL FUNCTIONS.
NATURAL SINE
Minutes.
Deg.
5
10
15
20
25
30
.00000
.00145
.00291
.00436
.00582
.00727
.00873
1
.01745
.01891
.02036
.02181
.02327
.02172
.02618
2
.03490
.03635
.03781
.03926
.04071
.04217
.04302
3
.05234
.05379
.05524
.05669
.05814
.05960
.06103
4
.00976
.07121
.07266
.07411
.07556
.07701
.07846
5
.08716
.08860
.09005
.09150
.09295
.09440
.09585
6
.10453
.10597
.10742
.10887
.11031
.11176
.11320
7
.12187
.12331
.12476
.12620
.12764
.12908
.13053
8
.13917
.14061
.14205
.14349
.14493
.14637
.14781
9
.15643
.15787
.15931
.16074
.16218
.16361
.16505
10
.17365
.17508
.17651
.17794
.17937
.18081
.18224
11
.19081
.19224
.19366
.19509
.19652
.19794
.19937
12
.20791
.20933
.21076
.21218
.21360
.21502
.21644
13
.22495
.22(537
.22778
.22 )20
.23062
.23203
.23345
14
.24192
.24333
.24474
.24615
.21756
.24897
.25038
15
.25882
.20022
.25163
.26303
.20443
.20584
.26724
16
.27564
.27704
.27843
.27983
.28123
.28202
.28402
17
59237
.29376
.29515
.29654
.29793
.29932
.30071
18
.30902
.31040
.31178
.31316
.31454
.31593
.31730
19
.32567
.32694
.32832
.32969
.33106
.33244
.33381
20
,342i)2
.34339
.34475
.34612
.34748
.34884
.35021
21
.35837
.35973
.36108
.36244
.36379
.36515
.36650
22
.37461
.37595
.37730
.37865
.37999
.38134
.38268
23
.39073
.39207
.39341
.39474
.39608
.39741
.39875
24
.40674
.40806
.40939
.41072
.412 4
.41337
.41469
25
.42232
.42394
.42525
.42657
.42788
.42920
.43051
26
.43837
.43968
.44098
.44229
.44359
.44494
.44620
27
.45399
.45529
.45658
.45787
.45917
.46046
.46175
28
.46947
.47076
.47204
.47332
.47460
.47588
.47716
29
.48481
.48608
.48735
.48862
.48989
.49116
.49242
30
.50000
.50126
.50252
.50377
.50503
.50628
.50754
31
.51504
.51628
.51753
.51877
.52002
.52120
.52250
32
.52992
.53115
.53238
.53361
.53484
.53607
.53730
33
.54464
.54586
.54708
.54829
.54951
.55072
.55194
34
.55919
.56040
.56160
.56280
.56401
.56521
.56641
35
.57358
.57477
.57596
.57715
.57833
.57952
.58070
36
.58779
.58869
.59014
.59131
.59248
.59365
.59482
37
.00182
.60298
.60414
.60529
.60(545
.60761
.60876
38
.61566
.61681
.61795
.61909
.62024
.62138
.62251
39
.62932
.63045
.63158
.63271
.63383
.63496
.63608
40
.64279
.64390
.64501
.64612
.64723
.64834
.64945
41
.65606
.65716
.65825
.65935
.66044
.66153
.66202
42
.66913
.67221
.67129
.67237
.67344
.67452
. 67559
43
.68200
.68306
.68412
.68518
.68624
.68730
.68835
44
.69466
.69570
.69675
.69779
.69883
.69987
.70091
Deg.
60
55
50
45
40
35
30
Minutes.
NATURAL COSINE.
TRIGONOMETRICAL FUNCTIONS.
NATURAL SINE.
Minutes.
Deg.
35
40
45
50
55
GO
.01018
.01164
.01309
.01454
.01000
.01745
89
.02703
.02908
.03054
.03199
.03345
.03490
88
.04507
.04053
.04798
.04913
.05088
.05234
87
.06250
.00395
.06540
.06685
.06831
.00970
86
.07991
.08130
.08281
.08426
.08571
.08710
85
.09729
.09874
.10019
.10104
.10308
.10453
84
.11405
.11009
.11754
.11898
.12043
.12187
83
.13197
.13341
.13485
.13029
.13802
.13917
82
.14025
.15009
.15212
.15356
.15500
.15043
81
.10048
.10792
.16935
.17078
.17222
.17305
80
.18307
.18509
.18652
.18795
.18938
.19081
79
.20079
.20222
.20364
.20507
.20649
.20791
78
.21780
.21928
.22070
.22212
.22353
.22495
77
.23480
.23627
.23769
.23910
.24051
.24192
76
.25179
.25320
.2.5460
.25601
.25741
.25882
75
.23804
.27004
.27144
.27284
.27421
.27504
74
.28541
.28080
.28820
.28959
.29098
.29237
73
.30209
.30348
.30486
.30625
.30703
.30902
72
.31808
.32006
.32144
.32282
.32419
.32057
71
.33518
.33655
.33792
.33929
.34005
.34202
70
.35157
.35293
.35429
.35565
.35701
.35837
69
.30785
.36921
.37056
.37191
.37320
.37401
68
.38403
.38537
.38671
.38805
.38939
.39073
67
.40008
.40141
.40275
.40408
.40541
.40074
66
.41002
.41734
.41866
.41998
.42130
.422i2
65
.43182
.43313
.43445
.43575
.43700
.43837
64
.44750
.44880
.45010
.45140
.45209
.45399
03
.40304
.46433
.46561
.46690
.40819
.40947
62
.47844
.47971
.48099
.48226
.48354
.48481
61
.49309
.49495
.49622
.49748
.49874
.50000
60
.50879
.51004
.51129
.51254
.51379
.51504
59
.52374
.52498
.52821
.52745
.52809
.52992
58
.53853
.53975
.54097
.54220
.54342
.54404
57
.55315
.55436
.55557
.55678
.55799
.55919
56
.50700
.56880
.57000
.57119
.572:38
.57358
55
,58189
.58307
.58425
.58543
.58001
.58779
54
.59599
.59716
.59832
.59949
.60065
.00182
53
.60991
.61107
.61222
.61337
.61451
.61560
52
.62305
.62479
.62.395
.62706
.62819
.62932
51
.63720
.63832
.63944
.64056
.64107
.64279
50
.65055
.65166
.65276
.65386
.65496
.65000
49
.66371
.66480
.66588
.66097
.66805
.00913
48
.67600
.67773
.67880
.07987
.68093
.08200
47
.68941
.69046
.69151
.09256
.69361
.69466
46
.70195
.70238
.70401
.70505
.70008
.70711
45
23
21
15
10
5
MintitPs.
NATURAL COMNE.
TRIGONOMETRICAL FUNCTIONS.
NATURAL SINE.
Minutes.
Deg.
5
10
15
20
25
30
1
45
.70711
.70813
.70916
.71019
.71121
.71223
.71325
46
.71934
.72035
.72136
.72230
.72337
.72437
.72,337
47
.73135
.73234
.73333
.73432
.73531
.73629
.73728
48
.74314
.74412
.74509
.74000
.74703
.74799
.74890
49
.75471
.75506
.75661
.75756
.75851
.75940
.70041
50
.76604
.76698
.76791
.76884
.76977
.77070
.77102
51
.77715
.77806
.77897
.77988
.78079
.78170
.78201
52
.78801
.78891
.78980
.79069
.79158
.79247
.79335
53
.79804
.79951
.80038
.80125
.80212
.80299
.80380
54
.80902
.80987
.81072
.81157
.81212
.81327
.81412
55
.81915
.81999
.82082
.82105
.822i8
.82330
.82413
56
.82904
.82985
.83006
.83147
.83228
.83308
.83389
57
.83807
.83946
.84023
.84104
.84182
.84201
.84339
58
.84805
.84882
.84959
.85035
.85112
.85188
.85204
59
.85717
.85792
.85866
.85941
.86015
.80089
.80103
60
.8(5003
.80075
.80748
.80820
.86892
.80904
.87036
61
.87402
.87532
.87003
.87073
.87743
.87812
.87882
62
.88295
.88363
.88431
.88499
.88566
.88634
.88701
63
.89101
.89167
.89232
.89238
.89303
.89428
.89493
64
.89879
.89943
.90007
.90070
.90133
.90190
.90259
65
.90631
.90692
.90753
.90814
.90875
.90936
.90996
66
.91355
.91414
.91472
.91531
.91590
.91648
.91706
67
.92.)50
.92107
.92164
.92220
.92276
.92332
.92388
68
.92718
.92773
.92827
.92381
.92935
.92088
.93042
69
.93358
.93410
.93462
.93514
.93565
.93016
.93667
70
.93969
.94019
.94068
.94118
.94167
.94215
.94264
71
.94552
.94599
.94046
.94093
.94740
.94786
.94832
72
.95106
.95150
.95191
.95240
.95284
.95328
.95372
73
.95630
.95673
.95715
.95757
.95799
.95841
.95882
74
.96196
.96166
.90206
.90246
.90235
.90324
.96363
75
.96593
.96630
.90667
.96705
.90742
.90778
.96815
76
.97030
.97065
.97100
.97134
.97109
.97203
.97237
77
.97437
.97470
.97502
.97534
.97566
.97598
.97030
78
.97815
.97845
.97875
.97905
.97934
.97963
.97992
79
.98163
.98190
.98218
.98245
.98272
.98299
.98325
80
.98481
.98506
.98531
.98506
.98580
.98004
.98629
81
.98769
.98791
.98814
.98836
.98858
.98880
.98902
82
.99027
.99047
.99067
.99087
.99106
.99125
.99144
83
.99235
.99272
.99290
.99307
.99324
.99341
.99357
84
.99452
.99467
.99482
.99497
.99511
.99526
.99540
85
.99619
.99632
.99644
.99657
.99668
.99080
.99092
86
.99756
.99766
.99776
.99786
.99795
.99804
.99813
87
.99863
.99870
.99878
.99885
.99892
.99898
.99905
88
.99939
.99944
.99949
.99953
.99958
.99962
.99906
89
.99985
.99987
.99989
.99991
.99993
.99995
.99996
60
55
50
45
40
35
30
Deg
Minutes.
NATURAL COSINE.
TRIGONOMETRICAL FUNCTIONS.
NATURAL SINE.
Minutes.
35
40
45
50
55
60
Deg.
.71427
.71529
.71630
.71732
.71833
.71934
44
.72637
.72737
.72837
.72937
.73036
.73135
43
.73826
.73924
.74022
.74123
.74217
.74314
42
.74992
.75088
.75184
.75280
.75375
.75471
41
.76135
.76229
.76323
.76417
.76511
.76004
40
.77255
.77347
.77439
.77531
.77023
.77715
39
.78351
.78442
.78532
.78622
.78711
.78801
38
.79424
.79512
.79300
.79088
.79776
.79804
37
.80472
.80558
.89644
.80730
.80816
.80902
36
.81496
.81580
.81664
.81748
.81832
.81915
35
.82495
.82577
.82659
.82741
.82822
.82904
34
.83469
.83549
.83629
.83708
.83788
.83807
33
.84417
.84495
.84573
.84050
.84728
84805
32
.85340
.85416
.85491
.85507
.85642
.85717
31
.86237
.86317
.80384
.80457
.86530
.80003
30
.87107
,87178
.87250
.87321
,87391
.87462
29
.87959
,88020
.88089
.88158
.88226
.88295
28
.88768
.88835
88902
.88968
.89035
.89101
27
.89558
.8962:3
.89687
.89752
.89816
.89879
26
.90321
.90383
.90446
.90507
.90569
.90631
25
.91056
.91116
.91176
.91236
.91295
.91355
24
.91764
.91822
.91879
.91936
.91994
.92050
23
.92444
.92499
.92554
.92609
.92064
.92718
22
.93095
.93148
.93201
.93253
.93306
.93358
21
.93718
.93709
.93819
.93809
.93919
.93969
20
.94313
.94301
.94409
.94457
.94504
.94552
19
.94878
.94924
.94970
.95015
.95001
.95106
.95415
.95459
.95502
.95545
.95588
.95030
17
.95923
.95904
.96005
.9GC46
.90086
.90120
16
.96402
.96440
.96479
.96517
.90555
.90593
15
.96851
.90887
.96923
.96959
.90994
.97030
14
.97271
.97304
.97338
.97371
.97404
.97437
13,
.97661
.97092
.97723
.97754
.97784
.97815
12
.98021
.98050
.98079
.98107
.98135
.98103
11
.98352
.98378
.98404
.98430
.98455
.98481
10
.98652
.98676
.98700
.98723
.98746
.98709
9
.98923
.98944
.98965
.98986
.99006
.99027
8
.99163
.99182
.99200
.99219
.99237
.99255
7
.99374
.90390
.99406
.99421
.99437
.99452
6
.99553
.99567
.99580
.99594
.99007
.99019
5
.99703
.99714
.99725
.99736
.99746
.99756
4
.99822
.99831
.99839
.99847
.99855
.99863
3
.99911
.99917
.99923
.99929
.99934
.99939
2
.99969
.99973
.99976
.99979
.99982
.99985
1
.99997
.99998
.99999
1.00000
1.00000
1.00000
25*
20
15
10
5
Minutes.
NATURAL COSINE.
210
TRIGONOMETRICAL FUNCTIONS.
NATURAL TANGENT.
Minutes.
Deg.
5
10
15
20
25
30
0.0000
0.0014
0.0029
0.0044
0.0058
0.0073
0.0087
1
0.0175
0.0189
0.0204
0.0218
0.0233
0.0247
0.0262
2
0.0349
0.0364
0.0378
0.0393
0.0407
0.0422
0.0437
3
0.0524
0.0539
0.0553
0.0508
0.0582
0.0597
0.0612
4
O.OG99
00714
0.0728
0.0743
0.0758
0.0772
0.0787
5
0.0875
0.0889
0.0904
0.0919
0.0933
0.0948
0.0963
6
0.1051
0.1066
0.1080
0.1095
0.1110
0.1125
0.1139
7
0.1228
0.1243
0.1257
0.1272
0.1287
0.1302
0.1316
8
0.1405
0.1420
0.1435
0.1450
0.1465
0.1480
0.1495
9
0.1584
0.1599
0.1014
0.1029
0.1644
0.1058
0.1073
10
0.1763
0.1778
0.1793
0.1808
0.1823
0.1838
0.1853
11
0.1944
0.1959
0.1974
0.1989
0.2004
0.2019
0.2034
12
0.2120
0.2141
0.2150
0.2171
0.2186
0.2202
0.2217
13
((.2309
0.2324
0.2339
0.2355
0.2370
0.2385
0.2401
14
0.2493
0.2509
0.2524
0.2540
0.2555
0.2571
0.2586
15
0.2679
0.2695
0.2711
0.2726
0.2742
0.2758
0.2773
16
0.2867
0.2883
0.2899
0.9915
0.2930
0.2946
0.2962
17
0.3057
0.3073
0.3089
0.3105
0.3121
0.3137
0.3153
18
0.3249
0.3265
0.3281
0.3297
0.3314
0.3330
0.3346
10
0.3443
0.34(50
0.3470
0.3492
0.3508
0.3525
0.3541
20
0.3640
0.3656
0.3073
0.3089
0.3706
0.3722
0.3739
21
0.3839
3855
0.3872
0.3889
0.3905
0.3922
0.3939
22
0.4040
0.4057
0.4074
0.4091
0.4108
0.4125
0.4142
23
0.4245
0.4262
0.4279
0.4296
0.4314
0.4331
0.4348
24
0.4452
0.4470
0.4487
0.4505
0.4522
0.4540
0.4557
25
0.4663
0.4681
0.4698
0.4716
0.4734
0.4752
0.4770
26
0.4877
0.4895
0.4913
0.4931
0.4950
0.4968
0.4986
27
0.5095
0.5114
0.5132
0.5150
0.5169
0.5187
0.5206
28
0.5317
0.5336
0.5354
0.5373
5392
0.5411
0.5430
29
0.5543
0.5502
0.5581
0.5600
05619
0.5638
0.5658
30
0.5774
0.5793
0.5812
0.5832
0.5851
0.5871
0.5891
31
0.6008
0.6028
0.0048
0.6068
0.6088
0.0108
0.6128
32
0.6249
0.6269
0.0289
0.6309
0.0330
0.0350
0.6371
33
0.64!)4
0.6515
0.0535
0.0550
0.6577
0.6598
0.6619
34
0.0745
O.G70G
0.6787
0.6809
0.6830
* 0.0851
0.0873
35
0.7002
0.7024
0.7045
0.7007
0.7089
0.7111
0.7133
36
0.7205
0.7288
0.7310
0.7332
0.7355
0.7377
0.7400
37
0.7530
0.7558
0.7581
0.7604
0.7027
0.7050
0.7073
38
0.7813
0.7836
0.78GO
0.7883
0.7907
0.7931
0.7954
39
0.8098
0.8122
0.8146
0.8170
0.8195
0.8219
0.8243
40
0.8391
0.8410
0.8441
0.8466
0.8491
0.8510
0.8541
41
0.8693
0.8718
0.8744
0.8770
0.8795
0.8821
0.8847
42
0.9004
0.9030
0.9057
0.9083
0.9110
0.9137
0.9103
43
0.9325
0.9352
0.9380
0.9407
0.9434
0.9402
0.9490
44
0.9057
0.9085
0.9713
0.9742
0.9770
0.9798
0.9827
60
55
50
45
40
35
30
Deg.
Minutes.
NATCBAL COTANGENT.
TRIGONOMETRICAL FUNCTIONS.
NATURAL TANGENT.
Minutes.
Deg.
35
40
45
50
55
60
0.0102
0.0116
0.0131
0.0145
0.0160
0.0175
89
0.0276
0.0291
0.0305
0.0320
0.0335
0.0349
88
0.0451
0.0466
0.0480
0.0495
0.0509
0.0524
87
0.0626 .
0.0641
0.0655
0.0670
0.0685
0.0699
86
0.0802
0.0816
0.0831
0.0846
0.0860
0-0875
85
0.0978
0.0992
0.1007
0.1022
0.1036
0.1051
84
0.1154
0.1169
0.1184
0.1198
0.1213
0.1228
83
0.1331
0.1346
0.1361 *
0.1376
0.1391
0.1405
82
0.1509
0.1524
0.1539
0.1554
0.1569
0.1584
81
0.1688
0.1703
0.1718
0.1733
0.1748
0.1763
80
0.1868
0.1883
0.1899
0.1914
0.1929
0.1944
79
0.2050
0.2065
0.2080
0.2095
0.2110
0.2126
78
0.2232
02247
0.2263
0.2278
0.2293
0.2309
77
0.2416
0.2432
0.2447
0.2462
0.2478
0.2493
76
0.2602
0.2617
0.2633
0.2648
0.2664
0.2679
75
0.2789
0.2805
0.2820
0.2836
0.2852
0.2867
74
0.2978
0.2994
0.3010
0.3026
0.3041
0.3057
73
0.3169
0.3185
0.3201
0.3217
0.3233
0.3249
72
0.3362
0.3378 "
0.3394
0.3411
0.3427
0.3443
71
0.3558
0.3574
0.3590
0.3607
0.3623
0.3640
70
0.3755
0.3772
0.3789
0.8805
0.3822
0.3839
69
0.3956
0.3973
0.3990
0.4006
0.4023
0.4040
68
0.4159
0.4176
0.4193
0.4210
0.4228
0.4245
67
0.4365
0.4383
0.4400
0.4417
0.4435
0.4452
66
0.4575
0.4592
0.4010
0.4628
0.4645
0.4(363
65
0.4788
0.4805
0.4823
0.4841
0.4859
0.4877
64
0.5004
0.5022
0.5040
0.5059
0.5077
0.5095
63
0.5224
0.5243
0.5261
0.5280
0.5298
0.5317
62
0.5448
0.5467
0.5486
0.5505
0.5524
0.5543
61
0.5677
0.5696
0.5715
0.5735
0.5754
0.5774
60
0.5910
0.5930
0.5949
0.5969
0.5989
0.6008
59
0.6148
0.6168
0.6188
0.6208
0.6228
0.6249
58
0.6391
0.6412
0.6432
0.6453
0.6473
0.6494
57
0.6640
0.6661
0.6682
0.6703
0.6724
0.6745
56
0.6894
0.6916
0.6937
0.6959
0.6980
0.7002
55
0.7155
0.7177
0.7199
0.7221
0.7243
0.7265
54
0.7422
0.7445
0.7467
0.7490
0.7513
0.7536
53
" 0.7696
0.7720
0.7743
0.7766
0.7789
0.7813
52
0.7978
0.8002
0.8026
0.8050
0.8074
0.8098
51
0.8268
0.8292
0.8317
0.8341
0.8366
0.8391
50
I* 0.8566
0.8591
0.8617
0.8642
0.8667
0.8693
49
I > 0.8873
0.8899
0.8925
0.8951
0.8978
0.9004
48
0.9190
0.9217
0.9244
0.9271
0.9298
0.9325
47
0.9517
0.9545
0.9573
0.9601
0.9629
0.9657
46
0.9856
0.9884
0.9913
0.9942
0.9971
1.0000
45
25
20
15
10
5
Deg.
Minutes.
NATURAL COTANGENT.
212
TRIGONOMETRICAL FUNCTIONS.
NATURAL TANGENT.
Minutes.
Deg.
5
10
15
20
25
30
45
1.0000
1.0029
1.0058
1.0088
1.0117
1.0146
1.0176
46
1.0355
1.0385
1.0416
1.0446
1.0477
1.0507
1.0538
47
1.0724
1.0755
1.0786
10818
1.0850
1.0881
1.0913
48
11106
1.1139
1.1171
1.1204
1.1237
1.1270
1.1303
49
1.1504
1.1537
1.1571
1.1606
1.1640
1.1674
1.1708
50
1.1917
1.1953
1.1988
1.2024
1.2059
1.2095
1.2131
51
1.2349
1.2386
1.2423
1.2460
1.2497
1.2534
1.2572
52
1.2799
1.2838
1.2876
1.2913
1.2954
1.2993
1.3032
53
1.3270
1.3311
1.3351
1.3302
1.3432
1.3472
1.3514
54
1.3764
1.3806
1.3848
1.3891
1.39^4
1.3976
1.4019
55
1.4281
1.4326
1.4370
1.4415
1.4460
1.4505
1.4550
56
1.4826
1.4872
1.4919
1.4966
1.5013
1.5061
1.5108
57
1.5399
1.5448
1.5497
1.5547
1.5597
1.5647
1.5697
58
1.6003
1.6055
1.6107
1.6160
1.6212
1.6265
1.6318
59
1.6643
1.6698
1.6753
1.6808
1.6864
1.6920
1.6976
60
1.7320
1.7379
1.7437
1.7496
1.7556
1.7615
1.7675
61
1.8040
1.8102
1.8165
1.8228
1.8291
1.8354
1.8418
62
1.8807
1.8873
1.8940
1.9007
1.9074
1.9142
1.9210
63
1.9626
1.9697
1.9768
1.9840
1.9912
1.9984
2.0057
64
2.0503
2.0579
2.0655
2.0732
2.0809
2.0887
2.0965
65
2.1445
2.1527
2.1609
2.1692
2.1775
2.1859
2.1943
66
2.2460
2.2549
2.2637
2.2727
2.2817
2.2907
2.2998
67
2.3558
2.3654
2.3750
2.3847
2.3945
2.4043
2.4142
68
2.4751
2.4855
2.4960
2.5065
2.5171
2.5279
2.5386
69
2.6051
2.6165
2.6279
2.6394
2.6511
2.6628
2.6746
70
2.7475
2.7600
2.7725
2.7852
2.7980
2.8109
2.8239
71
2.9042
2.9180
2.9319
2.9456
2.9600
2.9743
2.9886
72
3.0777
3.0930
3.1084
3.1240
3.1397
3.1556
3.1716
73
3.2708
3.2879
3.3052
3.3226
3.3402
3.3580
3.3759
74
3.4874
3.5067
3.5201
3.5457
3.5656
3.5856
3.6059
75
3.7320
3.7539
3.7760
3.7983
3.8208
3.8436
3.8667
76
4.0108
4.C358
4.0611
4.0867
4.1126
4.1388
4.1653
77
4.3315
4.3604
4.3897
4.4194
4.4494
4.4799
4.5107
78
4.7046
4.7385
4.7729
4.8077
4.8430
4.8788
4.9152
79
5.1445
5.1848
5.2257
5.2671
5.3093
5.3521
5.3955
80
5.6713
5.7199
5.7694
5.8197
5.8708
5.9228
5.9758
81
6.3137
6.3737
6.4348
6.4971
6.5605
6.6252
6.6912
82
7.1154
7.1912
7.2687
7.3479
7.4287
7.5113
7.5957
83
8.1443
8.2434
8.3450
8.4490
8.5555
8.6648
8.7769
84
9.5144
9.6493
9.7S82
9.9310
10.0780
10.2290
10.3850
85
11.4300
11.6250
11.8260
12.0350
12.2510
12.4740
12.7060
86
14.5010
14.6060
14.9240
15.2570
15.6050
15.9690
16.3500
87
19.0810
19.6270
20.2060
20.8190
21.4700
22.1640
22.9040
88
28.6360
29.8820
31.2420
32.7300
34.3680
36.1780
38.1880
89
57.2900
62.4990
68.7500
76.3900
85.9480
98.2180
114.5900
60
55
50
45
40
35
30
Deg.
Minutes.
NATURAL COTANGENT.
TBIGONOMETKICAL FUNCTIONS.
NATURAL TANGENT.
Minutes.
Deg.
35
40
45
50
55
60
1.0206
1.0235
1.0265
1.0295
1.0325
1.0355
44
1.05G8
1.0590
1.0630
1.0661
1.0692
1.0724
43
1.0945
1.0977
1.1009
1.1041
1.1074
1.1106
42
1.1336
1.1369
1.1403
1.1436
1.1470
1.1504
41
1.1743
1.1778
1.1812
1.1847
1.1882
1.1917
40
1.2167
1.2203
1.2239
1.2276
1.2312
1.2349
39
1.2609
1.2647
1.2685
1.2723
1.2761
1.2799
38
1.3071
1.3111
1.3151
1.3190
1.3230
1.3270
37
1.3555
1.3597
1.3638
1.3680
1.3722
1-3764
36
1.4063
1.4106
1.4150
1.4193
1.4237
1.4281
35
1.4595
1.4641
1.4687
1.4733
1.4779
1.4826
34
1.5156
1.5204
1.5252
1.5301
1.5350
1.5399
33
1.5747
1.5798
1.5849
1.5900
1.5952
1.6003
32
1.6372
1.6426
1.6479
1.6534
1.6588
1.6643
31
1.7033
1.7090
1.7147
1.7205
1.7263
1.7320
30
1.7735
1.7795
1.7856
1.7917
1.7979
1.8040
29
1.8482
1.8546
1.8611
1.8676
1.8741
1.8807
28
1.9278
1.9347
1.9416
1.9486
1.9556
1.9626
27
2.0130
2.0204
2.0278
2.0353
2.0428
2.0503
26
2.1044
2.1123
2.1203
2.1283
2.1364
2.1445
25
2.2028
2.2113
2.2199
2.2286
2.2373
2.2460
24
2.3090
2.3183
2.3276
2.3369
2.3464
2.3558
23
2.4242
2.4342
2.4443
2.4545
2.4648
2.4751
22
2.5495
2.5605
2.5715
2.5826
2.5938
2.6051
21
2.6865
2.6985
2.7106
2.7228
2.7351
2.7475
20
2.8370
2.8502
2.8636
2.8770
2.8905
2.9042
19
3.C032
3.0178
3.0326
3.0475
3.0625
3.0777
18
3.1877
3.2041
3.2205
3.2371
3.2539
3.2708
17
3.3941
3.4124
3.4308
3.4495
3.4684
3.4874
16
3.6264
3.6471
3.6680
3.6891
3,7105
3.7320
15
3.8900
3.9136
3.9375
3.9616
3.9861
4.0108
14
4.1921
4.2193
4.2468
4.2747
4.3029
4.3315
13
4.5420
4.5736
4.6057
4.6382
4.6712
4.7046
12
4.9520
4.9894
5.0273
5.0658
5.1049
5,1445
11
5.4397
5.4845
5.5301
5.5764
5.6234
5.6713
10
6,0296
6.0844
6.1402
6.1970
6.2549
6.3137
9
6.7584
6.8269
6.8969
6.9682
7.0410
7.1154
8
7.6821
7.7703
7.8606
7.9530
8.0476
8.1443
7
8.8918
9.0098
9.1309
9.2553
9.3831
9.5144
6
10.5460
10.7120
10.8830
11.0590
11.2420
11.4300
5
12.9470
13.1970
13.4570
13.7270
14.0080
14.3010
4
16.7500
17.1690
17.6110
18.0750
18.5640
19.0810
3
23.6940
24.5420
25.4520
26.4320
27.4900
28.6360
2
40.4360
42.9640
45.8290
49.1040
52.8820
57.2900
1
137.5100
171.8800
229.1800
343.7700
687.5500
25
20
15
10
5
Deg.
Minutes.
NATURAL COTANGENT.
214
TRIGONOMETRICAL FUNCTIONS.
NATURAL SECANT.
Minutes.
Deg.
5
10
15
20
25
30
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1
1.0001
1.0002
1.0002
1.00C2
1.0003
1.0003
1.0003
2
1.0006
1.0007
1.0007
1.0008
1.0008
1.0009
1.0009
3
1.0014
1.0014
1.0015
1.0016
1.0017
1.0018
1.0019
4
1.0021
1.0025
1.0021
1.0027
1.0023
1.0030
1.0031
5
1.0038
1.0039
1.0041
1.0042
1.0043
1.0045
1.0046
6
1.0055
1.0057
1.0058
l.OOGO
1.0061
1.0063
1.0065
7
1.0075
1.0077
1.0079
1.0080
1.0082
1.0084
1.0086
8
1.0098
1.0 LOO
1.0102
1.0104
1.0107
1.0109
1.0111
9
1.0125
1.0127
1.0129
1.0132
1.0134
1.0136
1.0139
10
1.0154
1.0157
10159
1.0162
1.0165
1.0167
1.0170
11
1.0187
1.0190
1.0193
1.0196
1.0199
1.0202
1.0205
12
1 0223
1.022ii
1.0229
1.0233
1.02:56
1.0239
1.0243
13
1.0263
1.0266
1.0270
1.0274
1.0277
1.0280
1.0284
14
1.0306
1.0310
1.0314
1.0317
1.0321
1.0325
1.0329
15
1.0353
1.0357
1.0361
1.0365
1.0369
1.0373
1.0377
16
1.0403
1.0407
1.0412
1.0416
1.0420
1.0425
1.0429
17
1.0457
1.0461
1.0466
1.0471
1.0476
1.0480
1.0485
18
1.0515
1.0520
1.0525
1.0530
1.0535
1.0540
1.0545
19
1.0577
1.0581
1.0587
1.0592
1.0598
1.0G03
1.0608
20
1.0642
1.0647
1.0653
1.0659
1.0664
1.0ti70
1.0676
21
1.0711
1.0717
1.0723
1.0729
1.0736
1.0742
1.0748
22
1.0785
1.0792
1.0798
1.0804
1.0811
1.0817
1.0824
23
1.0864
1.0870
1.0877
1.0884
1.0891
1.0897
1.0904
24
1.0946
1.0953
1.0961
1.0968
1.0975
1.09S2
1.0989
25
1.1034
1.1041
1.1049
1.1056
1.1064
1.1072
1.1079
20
1.1126
1.1134
1.1142
1.1150
1.1158
1.11 (56
1.1174
2T
1.1223
1.1231
1.1240
1.1248
1.1257
1.1265
1.1274
28
1.1326
1.1334
1.1343
1.1352
1.1361
1.1370
1.1379
29
1.1433
1.1443
1.1452
1.1461
1.1471
1,1480
1.1489
30
1.1547
1.1557
1-1566
1.1576
1.1586
1.1596
1.1606
31
1.1666
1.1676
1.1687
1.1697
1.1707
1.1718
1.1728
32
1.1792
1.1802
1.1830
1.1824
1.1835
1.1846
1.1857
33
1.1923
1.1935
1.1946
1.1958
1.1969
1.1980
1.1992
34
1.2002
1.2074
1.2068
1.2098
1.2110
1.2122
1.2134
35
1.2208
1.2220
1.2233
1.2245
1.2258
1.2270
1.2283
36
1.2361
1.2374
1.2387
1.2400
1.2413
1,2427
1.2440
37
1.2521
1.2535
1.2549
1.2563
1.2577
1.2591
1.2605
38
1.2690
1.2705
1.2719
1.2734
1.2748
1.2763
1.2778
39
1.2867
1.2883
1.2898
12913
1,2929
1.2944
1.29GO
40
1.3054
1.3070
4.3086
1.3102
1.3118
1.3134
1.3151
41
1.3250
1.3267
1.3284
1.3301
1.3318
1.3335
1.3352
42
1.3456
1.3474
1.3492
1.3509
1.3507
1.3540
1.3563
43
1.3673
1.3692
1.3710
3,3729
1,3748
1.3767
1.3786
44
1.3902
1.3921
1.3941
1.3960
1.3980
1.4000
1.4020
60
55
50
45
40
35
30
Deg.
Minutes.
NATURAL COSECANT.
TRIGONOMETRICAL FUNCTIONS.
NATURAL SECANT.
Minutes.
L>eg.
35
40
45
50
55
60
1.0000
1.0001
1.0001
1.0001
1.0001
1.0001
89
1.0004
1.0004
1.0005
1.0005
1.0005
1.0(OG
88
1.0010
1.0011
1.0011
1.0012
1.0013
1.0014
87
1.0019
1.0020
1.0021
1.0022
1.0023
1.0024
86
1.0032
1.0033
1.0034
1.0036
1.0037
1.0038
85
1.0048
1.0049
1.0050
1.0052
1.0053
1 .0055
84
1.00G6
1.0008
1.0070
1.0071
1.0073
1.0075
83
1.0088
1.0090
1.0092
1.0094
1.0096
1.0098
82
1.0113
1.0115
1.0118
1.0120
1.0122
1.0125
81
1.0141
1.0145
1.0146
1.0149
1.0152
1.0154
80
1.0173
10176
1.0179
1.0181
1.0184
1.0187
79
1.0208
1.0211
1.0214
1.0217
1.0220
1.0223
78
1.0246
1.0249
1.0253
1.0256
1.0260
1.0263
77
1.0288
1.0291
1.0295
1.0298
1.0302
1.0306
76
1.0333
1.0337
1.0341
l.< 345
1.0349
1.0353
75
1.03S2
1.0386
1.0390
1.0394
1.0399
1.0403
74
1.0434
1.0438
1.0443
1.0448
1.0452
1.0457
73
1.0490
1.0495
1.0500
1.0505
1.0510
1.0515
72
1.0550
1.0555
1.0560
1.0565
1.0571
1.0577
71
1.0644
1.0619
1.0625
1.0630
1.0636
1.0642
70
1.0082
1.0688
1.0694
1.0699
1.0705
1.0711
69
1.0754
1.0760
1.0766
1.0773
1.0779
1.0785
68
1.0830
1.0837
1.0844
1.0850
1.0857
1.0864
67
1.0911
1.0918
1.0925
1.0932
1.0939
1.0946
66
1.0997
1.1004
1.1011
1.1019
1.1026
1.1034
65
1.1087
1.1095
1.1102
1.1110
1.1118
1.1126
64
1.1182
1.1190
1.1198
1.1207
1.1215
1.1223
63
1.1282
1.1291
1.1299
1.1308
2.1317
1.1326
62
1.1388
1.1397
1.1406
1.1415
1.1424
1.1433
61
1.1499
1.1508
1.1518
1.1528
1.1537
1.1547
60
1.1616
1.1626
1.1636
1.1646
1.1656
1.1666
59
1.1739
1.1749
1.1760
1.1770
1.1781
1.1792
58
1.1808
1.1879
1.1819
1.1901
1.1912
1.1923
57
1.2004
1.2015
1.2027
1.2039
1.2050
1.2062
56
1.2146
1.2158
1.2171
1.2183
1.2195
1.2208
55
1.2296
1.2309
1.2322
1.2335
1.2348
1.2361
54
1.2453
1.2467
1.2480
1.2494
1.2508
1.2521
53
1.2619
1.2633
1.2647
1.2661
1.2676
1.2690
52
1.2793
1.2807
1.2822
1.2837
1.2852
1.2867
51
1.2975
1.2991
1.3006
1.3022
1.3038
1.3054
50
1.3167
1.3184
1.3200
1.3217
1.3233
1.3250
49
1.3369
1.3386
1.3404
1.3421
1.3439
1.3450
48
1.3581
1.3GOO
1.3618
1.3636
1.3655
1.3073
47
1.3805
1.3824
1.3843
1.3863
1.3882
1.3902
46
1.4040
1.4056
1.4081
1.4101
1.4122
1.4142
45
25
20
15
10
5
Dee
Minutes.
NATURAL COSECANT.
216
TRIGONOMETRICAL FUNCTIONS.
NATURAL SECANT.
Minutes.
Deg.
5
10
15
20
25
30
45
1.4142
1.4163
1.4183
1.4204
1.4225
1.424G
1.4267
46
1.4395
1.4417
1.4439
1.44G1
1.4483
1.4505
1.4527
47
1.46G3
1.4G86
1.4709
1.4732
1.4755
1.4778
1.4802
48
14945
1.4969
1.4993
1.5018
1.5042
1.50G7
1.5092
49
1.5242
1.5268
1.5294
1.5319
1.5345
15371
1.5398
50
1.5557
1.5584
1.5011
1.5639
1.5GGG
1.5G94
1.5721
51
1.5890
1.5919
1.5947
1.5976
1.6005
l.GO:54
1.6064
52
1.6243
1.6273
1.6303
1.6334
1.6365
1.6396
1.6427
53
1.6616
1.6648
1.GG81
1.6713
1.6746
1.6779
1.6812
54
1.7013
1.7047
1.7081
1.7116
1.7151
1.7185
1.7220
55
1.7434
1.7471
1.7507
1.7544
1.7581
1.7018
1.7655
56
1.7883
1.7921
1.7960
1.7999
1.8039
1.8078
1.8118
57
1.8361
1.8402
1.8443
1.8485
1.8527
1.8569
1.8G11
58
1.8871
1.8915
1.8959
1.9004
1.9048
1.9093
1.9139
59
1.9416
1.9463
1.9510
1.9558
1.9606
1.9C54
1.9703
60
2.0000
2.0050
2.0102
2.0152
2.0204
2.0256
2.0308
61
2.0(527
2.0681
2.0735
2.0790
2.0846
2.0901
2.0957
62
2.1300
2.1359
2.1418
2.1477
2.1536
2.1596
2.1657
63
2.2027
2.2090
2.2153
22217
2.2282
2.2346
2.2411
64
2.2812
2.2880
2.2949
2.3018
2.3087
2.3158
2.3228
65
2.3662
2.3736
2.3811
2.3886
2.3961
2.4037
2.4114
66
2.4586
2.4G6G
2.4748
2.4829
2.4912
2.4995
2.5078
67
2.5593
2.5G81
2.5770
2.5859
2.5949
2.G040
2.6131
68
2.G695
2.G791
2.6888
2.6986
2.7085
2.7185
2.7285
69
2.7904
2.8010
2.8117
2.8225
2.8334
2.8444
2.8554
70
2.9238
2.9355
2.9474
2.9593
2.9713
2.9835
2.9957
71
3.0715
3.0S4G
3.0977
3.1110
3.1244
3.1379
3.1515
72
3.23G1
3.250G
3.2G53
3.2801
3.2951
3.3102
3.3255
73
3.4203
3.4366
3.4532
3.4G97
3.4867
3.5037
3.5209
74
3.6276
3.6464
3.GG51
3.68-10
3.7031
3.7224
3.7420
* 75
3.8G37
3.8848
3.9061
3.9277
3.9495
3.9716
3.9939
76
4.1330
4.1578
4.1824
4.2072
4.2324
4,2579
4.2836
77
4.4454
4.4736
4.5021
4.5331
4.5604
4.5901
4.6202
78
4.8097
4.8429
4.8765
4.9106
4.9452
4.9802
5.0158
79
5.2408
5.2803
5.3205
5.3G12
5.4020
5.4447
5.4874
80
5.7588
5.80G7
5.8554
5.9049
5.9554
5.99G3
6.0588
81
6.3924
6.4517
6.5121
6.5736
6.6363
6.7003
6.7655
82
7.1853
7.2604
7.3372
7.4156
7.4957
7.5776
7.6613
83
8.2055
8.3C39
8.4046
8.5079
8.6138
8.7223
8.8337
84
9.5608
9.7010
9.^391
9.9812
10.1270
10.2780
10.4330
85
11.4740
11.6680
11.8680
12.07GO
12.2910
12.5140
12.7450
86
14.3350
14.6400
14.9580
15.2900
15.6370
16.0000
16.3800
87
19.1070
19.G530
20.23(10
20.8430
21.4940
22.18GO
22.9250
88
28.6540
29.8990
31.2570
32.7450
34.3820
36.1910
38.2010
89
57.2990
62.5070
68.7570
76.3960
85.9460
98.2230
114.5900
GO
55
50
45
40
35
30
Deg
Minutes.
NATURAL COSECANT.
TRIGONOMETRICAL FUNCTIONS.
NATURAL SECANT.
Minutes.
35
40
45
50
55
60
1.4288
1.4310
1.4331
1.4352
1.4374
1.4395
1.4550
1.4572
1.4595
1.4617
1.4640
1.4663
1.4825
1.4849
1.4873
1.4897
1.4921
1.4945
1.511G
1.5141
1.5166
1.5192
1.5217
1.5242
1.5424
1.5450
1.5477
1.5503
1.5530
1.5557
1.5749
1.5777
1.5805
1.5833
1.5862
1.5890
1.G093
1.6123
1.6153
1.6182
1.6212
1.6243
1.6458
1.6489
1.6521
1.6552
1.6584
1.6616
1.6845
1.6878
1.6912
1.6945
1.6979
1.7013
1.7256
1.7291
1.7327
1.7362
1.7398
1-7434
1.7693
1.7730
1.7768
1.7806
1.7844
1.7883
1.8158
1.8198
1.8238
1.8279
1.8320
1.8361
1.8654
1.8G97
1.8740
1.8783
1.8827
1.8871
1.9184
1.9230
1.9276
1.9322
1.9369
1.9416
1.9752
1.9801
1.9850
1.9900
1.9950
2.0000
2.0360
2.0413
2.0466
2.0519
2.0573
2.0627
2.1014
2.1070
2.1127
2.1185
2.1242
2.1300
2.1717
2.1778
2.1840
2.1902
2.1964
2.2027
2.2477
2.2543
2.2610
2.2676
2.2744
2.2812
2.3299
2.3371
2.3443
2.3515
2.3588
2.3662
2.4191
2.4269
2.4347
2.4426
2.4506
2.4586
2.5163
2.5247
2.5333
2.5419
2.5506
2.5593
2.G223
2.6316
2.6410
2.6504
2.6599
2.6695
2.7386
2.7488
2.7591
2.7694
2.7799
2.7904
2.8666
2.8778
2.8892
2.9006
2.9122
2.9338
3.0081
3.0206
3.0331
3.0458
3.0586
3.0715
3.1653
3.1792
3.1932
3.2074
3.2216
3.2361
3.3409
3.3565
3.3722
3.3881
3.4041
3.4203
3.53S3
3.5559
3.5736
3.5915
3.6096
3.6279
3.7617
3.7816
3.8018
3.8222
3.8428
3.8637
4.0165
4.0394
4.0625
4.0859
4.1090
4.1336
4.3098
4.3362
4.3630
4.3901
4.4176
4.4454
4.6507
4.6817
4.7130
4.7448
4.7770
4.8097
5.0520
5.0886
5.1258
5,1636
5.2019
5.2408
5.5308
5.5749
5.6197
5.6653
5.7117
5.7588
6.1120
6.166 1
6,2211
6.2772
6.3343
6.3924
6.8320
6,8998
6.9690
7,0390
7.1117
7.1853
7.7469
7.8344
7.9240
7.9971
8.1094
8.2055
8.9479
9.0651
9.1855
9.3092
9.4362
9.5668
10.5930
10.7580
10.9290
11.1040
11.2080
11,4740
12.9850
13.2350
13.4940
13.7630
14.0430
14.3350
16.7790
17.1980
17.6390
18.1030
18.5910
19.1070
23.7160
24,5620
25.4710
26.1500
27.5080
28.6540
39.9780
42,9760
45.8400
49.1140
52.8910
57.2990
137.5100
171.8900
229.1800
343.7700
687.5500
00
25
20
15
10
5
Minutes.
NATURAL COSECANT,
218 CIRCUMFERENCE, AREA, AND CUBIC CONTENTS OF CIRCLES.
l- 5 8
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CIRCUMFERENCE, AREA, AND CUBIC CONTENTS OF CIRCLES.
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220 CIRCUMFERENCE, AREA, AND CUBIC CONTENTS OF CIRCLES.
^ H co cS o u5 o o cS CM o co o CM 56 GO TH co co co o i^
COCiTf*<X>OOCOOSCMCOOCOCOCOOCO -- CSCDfMCOTlGO
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CIRCUMFERENCE, AREA, AND CUBIC CONTENTS OF CIRCLES. 221
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222 CIRCUMFERENCE, AREA, AND CUBIC CONTENTS OF CIRCLES
T. >"~ l o on
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CIRCUMFERENCE, AREA, AND CUBIC CONTENTS OF CIRCLES.
223
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221
SPECIFIC GRAVITIES OF MATERIALS.
SPECIFIC GRAVITIES OF MATERIALS.
Weight of
a cubic foot
in Ibs.
GASES at 32 Fahr., and under the pressure of on
atmosphere of 2116.4 Ibs. on the square foot:
Air
e
0.080728
Carbonic acid 0.12344
Hydrogen 0.005592
Oxygen 0.089256
Nitrogen 078596
Steam (ideal) 0.05022
^Ether vapor (ideal) . 02093
Bisulphuret-of-carbon vapour (i
ieal) 2137
079R
LIQUIDS at 32 Fahr. (except water,
which is taken at 39.4 Fahr.):
Water, pure, at 39.4
Weight of a
cubic foot in Ibs.
avoirdupois.
Specific
gravity, pure
water = I.
62.425
64.05
49.38
57.18
44.70
848.75
52.94
58.68
57.12
57.62
54.31
54.81
187.3
125 to 135
112
117 to 174
120
100
77. 4 to 89. 9
62.43 to 103. 6
162.3
164.2
1.000
1.026
0.791
0.916
0.716
13.596
0.848
0.940
0.915
0.923
0.870
0.878
3.00
2 to 2. 167
1.8
1.87 to 2. 78
1.92
1.602
1.24 to 1.44
1.00 to 1.66
2.6
2.63
" sea ordinary
Alcohol pure
proof spirit
^Ether .
Mercury
Naphtha
Oil, linseed
" olive
" whale
" of turpentine
SOLID MINERAL SUBSTANCES, non-
metallic:
Basalt
Brick
Brickwork
Chalk . ....
Clay
Coal anthracite
" bituminous .
Coke ,
Felspar
Flint..
SPECIFIC GRAVITIES OF MATERIALS.
225
SOLID MINERAL SUBSTANCES con
Weight of a
cubic foot in Ibs.
avoirdupois.
Specific
gravity, pure
water = 1.
tinued:
Glass crown average
156
2 5
11 flint
187
3.0
" green
169
2.7
" plate
169
2 7
Granite
164 to 172
2.63 to 2.76
Gypsum
143.6
2.3
Limestone, (including marble)...
magnesian
169 to 175
178
2. 7 to 2.8
2.86
Marl ... .
100 to 119
1 6 to 1 9
Masonry
116 to 144
1.85 to 2 3
Mortar
109
1.75
Mud
102
1 63
Quartz
165
2 65
Sand (damp) ....
118
1 9
" (dry). .
88.6
1 42
Sandstone average
144
2 3
" various kinds .
130 to 157
2 08 to 2 52
Shale
162
2 6
Slate
175 to 181
2 8 to 2 9
Trap...
170
2 72
METALS, solid:
Brass, cast
487 to 524.4
7 8 to 8 4
" wire.
533
8 54
Bronze
524
8 4
Copper, cast
537
8 6
" sheet
549
8 8
" hammered
556
8 9
Gold
1186 to 1224
19 to 19 6
Iron, cast various
434 to 456
6 95 to 7 3
average
444
7 11
Iron, wrought various
474 to 487
7 6 to 7 8
average
480
7 69
Lead
712
114-
Platinum
1311 to 1373
21 to 22
Silver
655
10 ^
Steel
487 to 493
7 8 to 7 9
Tin
456 to 468
7 3 to 7 5
Zinc
424 to 449
6 8 to 7 2
TIMBER:*
Ash
47
7^3
Bamboo
25
4
Beech
43
69
15
226
SPECIFIC GRAVITIES OF MATERIALS.
TIMBER :* continued.
Birch . ..
Weight of a
cubic foot in Ibs.
avoirdupois.
Specific
gravity, pure
water = 1.
44.4
52.5
60
65.3
56.2
30.4
33.4
36.2
74.5
34
30 to 44
30 to 44
29
31 to 35
62.5
57
54
47
47
57
42 to 63
41 to 83
44
35
53
49
57
43 to 62
54
36
60
37
41 to 55
61
62 to 66
62.5
25
50
0.711
0.843
0.96
1.046
0.9
0.486
0.535
0.579
1.193
0.544
0.48 to 0.7
0.48 to 0.7
0.46
0.5 to 0.56
1.001
0.91
0.86
0.76
0.76
0.92
0.675 to 1.01
0.65 to 1.33
0.71
0.56
0.85
0.79
0.92
0.69 to 0.99
0.87
0.58
0.96
0.59
0.66 to 0.88
0.98
0.99 to 1.06
1.001
0.4
0.8
Blue-gum
Box
Bullet- tree
Cabacalli
Cedar of Lebanon
Chestnut
Cowrie
Ebony, West Indian
Elm
Fir, red pine
" spruce
" American yellow pine
Greenhart
Haw thorn
Ha*zel
Holly
Hornbeam
Laburnum
Lancewood
Larch. (See "fir".)
Lignum-vitae
Locust
Mahogany, Honduras
Spanish .
Maple
Mora
Oak European
" American red
Poon
Saul
Sy cam ore
Teak Indian
11 African
Tonka
Water-gum
Willow
Yew
*The timber in every case is supposed to be dry.
WEIGHT OF A SUPERFICIAL INCH. ETC.
227
WEIGHT OF A SUPERFICIAL INCH OF WROUGHT AND
CAST IRON.
(From one-sixteenth to one-inch thickness.)
Thickness in
inches.
WROUGHT IRON.
Cubic foot = 480 Ibs.
CAST IRON.
Cubic foot = 450 Ibs.
Weight in Ibs.
Weight in Ibs.
A
0.017356
0.0163
i
0.0347
0.0326
A
0.0520
0.0489
i
0.0694
0.0652
T 5 6
0.0867
0.0815
1
0.1041
0.0978
ft
0.1214
0.1141
i
0.1388
0.1304
T 9 *
0.1562
0.1467
i
0.1735
0.1630
0.1909
0.1793
1
0.2082
0.1956
n
0.2256
0.2119
i
0.2429
0.2282
it
0.2603
0.2445
i
0.2777
0.2608
228
WEIGHT PEE SQUARE FOOT IH POUNDS AVOIRDUPOIS.
WEIGHT PER SQUARE FOOT IN POUNDS AVOIRDUPOIS.
Thickness in
inches.
Wrought
Iron.
Cast Iron.
Copper,
sheet.
Lead.
Zinc.
480 Ibs. per
cubic foot.
450 Ibs. per
cubic foot.
549 Ibs. per
cubic foot.
712 Ibs. per
cubic foot.
436 Ibs. per
cubic foot.
TV
2.50
2.34
2.86
3.71
2.27
*
5.00
4.69
5.72
7.42
4.54
T 3 6
7.50
7.03
8.58
11.12
6.81
i
10.00
9.37
11.44
14.83
9.08
ft
12.50
11.72
14.30
18.54
11.35
1
15.00
14.06
17.16
22.25
13.62
ft
17.50
16.41
20.02
25.96
15.89
20.00
18.75
22.88
29.66
18.16
ft
22.50
21.09
25.74
33.37
20.43
1
25.00
23.44
28.60
37.10
22.70
H
27.50
25.78
31.46
40.79
24.97
f
30.00
28.12
34.32
44.50
27.24
it
32.50
30.47
37.18
48.20
29.51
*
35.00
32.81
40.04
51.91
31.78
it
37.50
35.16
42.90
55.62
34 05
i
40.00
37.50
45.75
59.33
36.33
WEIGHT OF A LINEAL FOOT, ETC.
229
WEIGHT OF A LINEAL FOOT OF FLAT AND SQUAKE
BAR IRON IN POUNDS AVOIRDUPOIS.
(480 pounds per cubic foot.)
Breadth in
inches.
Thickness in
inches.
Weight in Ibs.
d
5 2
ll
Thickness in
inches.
Weight in Ibs.
Breadth in
inches.
Thickness in
inches.
Weight in Ibs.
i
i
0.104
ij
1
5.000
2J
1
1.875
0.208
H
5.625
|
2.813
1
1
0.208
"
if
6.250
"
3.750
0.416
"
if
6.874
"
4.687
"
1
0.832
"
if
7.500
M
5.624
1
|
0.312
it
4
0.739
6.562
I
0.624
i
1.459
11
1
7.500
((
I
0.937
"
I
2.187
"
li
8.437
((
1.249
*
2.916
"
H
9.374
II
.|
1.562
<(
1
3.646
rt
If
10.310
((
j
1.874
i
4.375
"
If
11.250
1
i
0.416
"
-
5.103
"
If
12.190
"
0.833
5.833
11
If
13.120
<(
-1
1.249
6.562
l|
14.060
M
j
1.667
H
u
7.291
11
o
15.000
-1
2.089
"
if
8.020
H
2J
15.940
t
2.500
"
if
8.750
11
2i
17.810
11
7
F
2.916
"
if
9.478
2}
|
1.041
(t
1
3.333
u
it
10.930
2.089
11
i
0.521
2
1
833
"
|
3.125
|
1.041
"
I
1.667
11
1
4.166
(i
f
1.562
3
8"
2.500
"
I
5.208
2.Q89
11
|
3.333
"
I
6.250
II
2.603
11
4.166
11
i
7.291
<(
3.124
11
5.000
"
8.333
3.646
c
I
5.833
"
li
9.398
<(
1
4.166
"
6.666
11
10.410
ii
1*
4.687
11
If
7.500
"
If
11.460
<(
li
5.728
"
l|
8.333
"
IJ
12.500
i|
J
0.624
11
If
9.156
(1
If
13.540
(i
\
1.250
u
if
10.000
"
If
14.580
11
i
1.875
11
If
10.830
If
15.620
ii
2.500
11
If
11.660
11
2
16.660
3.125
M
li
12.500
"
a
17.710
ii
3.750
M
2
13.330
((
2i
18.750
ii
4.375
2i
4
0.937
"
2|
20.820
230
WEIGHT OF A LINEAL FOOT, ETC.
Breadth in
inches.
Thickness in
inches.
Weight in Ibs.
Breadth in
inches.
Thickness in
inches.
Weight in Ibs.
Breadth in
inches.
Thickness in
inches.
Weight in Ibs.
2J
2f
19.800
3J
2
21.660
4
2
26.660
2J
^
1.146
i
24.370
"
21
30.000
i
2.292
"
2
27.080
"
i
33.330
1
^
3.437
11
2|
29.790
M
2f
36.660
4
|
4.583
"
3
32.500
u
3
40.000
1
-|
5.729
41
3i
24.200
M
31
43.330
1
J
6.874
3i
i
2.916
"
3*
46.660
1
I
8.020
i
5.833
11
3|
50.000
*
1
9.154
u
i
8.750
"
4
53.330
*
H
10.310
"
11.660
$
i
3.541
1
u
11.460
11
H
14.580
7.082
if
12.600
11
U
17 500
"
|
10.620
i|
13.750
M
if
20.430
M
1
14.160
1
i|
14.900
11
2
23.330
M
It
16.800
1
ij
16.030
"
2
26.250
H
21.330
1
1J
17.190
2J
29.160
If
24.780
2
18.330
M
2f
32.080
i
2
28.330
2J
19.480
"
3
35.000
i
2|
31.870
i
20.620
M
3|
37.910
1
1
35.410
c
1
21.770
3 !
40.830
1
i
38.950
1
2^
22.910
3f
3.125
3
42.500
1
2|
24.060
j
6.250
3J
46.030
1
2|
25.200
11
j
9.375
1
3
49.570
3
i
2.500
it
i
12.500
(
3|
53.120
5.000
"
14
15.620
1
4
56.660
1
i
7.500
"
1}
18.750
60.200
1
i
10.000
M
if
21.870
IF
i
3.750
1
ia
12.500
"
2
25.000
7.500
1
i?
15.000
"
2*
28.120
y
|
11.250
if
17.500
2J
31.250
M
1
15.000
".": :,
2
20.000
2f
34.370
"
it
18.750
1
2}
22.500
"
3
37.500
!
9
22.500
1
2J
25.000
M
31
40.620
1
if
26.250
2|
27.500
"
3^
43.750
2
30.000
3
30.000
"
3f
46.860
1
^
33.750
3|
|
2.708
4
i
3.330
1
^
37.500
5 416
"
i
6.660
<
2f
41.250
C "**;
|.
8.124
"
f
10.000
f
3
45.000
!t .**.]
1
10.830
*
13.330
5i
48.750
..;_ ;,.
H
13.500
"
11
16.660
1
3J
52.500
4
1|
16.250
M
1
20.000
M
si
56.250
t
If
18.950
a
If
23.330
N
4
60.000
WEIGHT OF A LINEAL FOOT, ETC.
231
5 *
1
a/ G
t, 1 - 1
PQ
Thickness in
inches.
Weight in Ibs.
a
5s
~Q-Z
06
P
PQ
Thickness in
inches.
J5
3f
Breadth in
inches.
co
CJ <P
g-g
!s-S
^
Weight in Ibs.
4J
4}
63 . 750
5i
i
8.753
5|
i
4.788
4
67.500
-1
1-3.130
i
9.587
4f
1
3.953
I
17.500
(
1
14.370
i
7.910
1
H
21.870
i
1
19.160
"
i
11.860
1
H
26.250
i
1}
23.950
15.830
1
if
30.620
i
li
28.750
ti
I*
19.760
f
2
35.000
1
I
33.540
li
23.750
IJ
m
39.370
2
38.330
<
If
27.700
u
2i
43.750
"
2t
43.120
2
31.670
1
2|
48.110
c<
2J
47.910
i
2i
35.620
3
52.500
M
2|
52.700
1
2i
39.580
1
3J
56.680
(
3
57.500
1
2|
43.540
3i
61.250
1
3i
62.300
1
3
47.500
1
3|
65.620
4
3i
67.080
1
P
51.460
4
4
70.000
(
3f
71.860
M
31
55.410
|
4}
74.370
4
4
76.650
(i
3|
59.370
4|
78.750
4
H
81.450
4
63.330
4
4f
83.110
1
4J
86.240
4}
67.290
5
87.500
4f
91.030
M
4
71.250
!
5J
91.860
4
5
95.820
4|
75.200
5J
J
4.587
4
5t
100.600
5
1
4.166
i
9.164
(
5}
105.400
M
i
8.330
44
1
13.750
4
5|
119.700
i
1
12.500
"
1
18.330
6
J
10.000
1
1
16.660
"
H
22.900
4 -
1
20.000
1
li
20.830
u
27.500
M
li
30.000
4
li
25.000
II
If
32.080
44
2
40.000
li
29.160
M
2
36.660
44
2J
50.000
4
2
33.330
11
2J
41.250
44
3
60.000
21
37.500
u
2i
45.830
3J
70.000
1
2i
41.660
"
2}
50.310
44
4
80.000
4
2|
45.830
11
3
55.000
44
4
90.000
1
3
50.000
3i
59.570
44
5
100.000
<
3}
54.160
11
3J
64.160
tc
5J
110.000
1
31
58.330
11
3|
68.740
6
120.000
i
3f
62.500
((
4
73.330
6J
i
10.830
4
66.660
11
4J
77.910
21.660
t
4J
70.830
11
4J
82.500
44
li
32.500
1
4^
75.000
"
4|
87.080
44
2
43.330
4
4
79.160
"
5
91.560
44
2i
54.160
4
5
83.330
u
5|
96.240
44
3
65.000
H
i
4.376
M
5J
100.600
ii
3i
75.830
232
WEIGHT OF A LINEAL FOOT, ETC.
Breadth in
inches.
Q
? A
C 03
C
Jifl
Weight in Ibs.
Breadth in
inches.
Thickness in
inches.
Weight in Ibs.
c
j|
c
Jl
c- .2
H
Weight in Ib.s.
6 ; >
4
86.66
8
4
106.60
9
8}
255.00
4J
97.50
"
4.1
120.00
"
9
270.00
it
5
108.30
"
5"
133.30
9}
}
15.83
M
5
119.10
"
5}
146.60
1
31.66
6
130.00
(i
6
160.00
"
1}
47.50
"
6}
140.80
"
8}
173.30
M
2
63.33
7
i
11.66
u
7
186.60
"
2}
79.16
"
1
23.33
7}
200.00
"
S
95.00
"
u
35.00
"
8
213.30
"
3^
110.80
"
2
46.66
8|
}
14.16
"
4
126.60
2J
58.33
28.33
"
4}
142.50
11
3
70.00
u
1J
42.48
M
5"
158.30
11
3J
81 66
11
2
56.66
"
5-^r
174.10
"
4
93.33
u
2J
70.83
H
6
190.00
4}
105.00
11
3
85.00
"
6J
205.80
"
5
116.60
"
3}
99.16
"
7
221.60
u
5J
128.30
"
4
113.30
"
7i
237.60
6
140.00
"
4J
127.50
"
8
253 . 30
"
6J
151.60
"
5
141.60
"
8J
269.10
"
7
163.30
u
5}
155.80
"
9
285.00
n
i
12.50
(i
6
170.00
11
9J
300.80
1
25.00
u
6|
184.10
10
1
16.66
"
U
37.50
u
7
198.30
"
1
33.33
M
2
50.00
"
7}
212.50
"
1}
50.00
ii
2
62.50
11
8
226.60
11
2
66.66
tl
3
75.00
M
8}
240.70
"
2^-
83.33
3}
87.50
9
i
15.00
"
3"
100.00
11
4
100.00
<
i
30.00
"
3^
116.60
"
4i-
112.50
11
u
45.00
11
4
133.30
"
5
125.00
11
2
60.00
11
4J
150.00
M
5}
137.50
"
2J
75.00
M
5
166.60
11
6
150.00
11
3
90.00
"
5}
183.30
ii
6J
162.50
"
3i
105.00
(f
6
200.00
"
7
175.00
u
4
120.00
6}
216.60
M
7^
187.50
11
4
135.00
1
7
233.30
8
|
13.33
11
5
150.00
7*
250.00
11
1
26.66
11
5}
165.00
8
266.60
11
H
40.00
"
6
180.00
1
8
283.30
11
2
53.33
11
61
195.00
1
9
300.00
"
2J
66.66
"
y
210.00
(
9J
316.60
M
3
80.00
"
7*
225.00
1
10
333.30
"
31
93.33
"
8
240.00
10}
i
17.50
WEIGET OF A LINEAL FOOT, ETC.
233
Breadth in
inches.
Thickness in
inches.
Weight in Ibs.
Breadth in
inches.
Thickness in
inches.
Weight in Ibs.
Breadth in
inches.
Thickness in
inches.
Weight in Ibs.
10}
1
35.00
11
n
55.00
11}
H
57.50
1
1}
52.50
"
2 2
73.33
2
76.66
4
$
70.00
"
2}
91.56
44
2J
95.83
1
2}
87.50
11
3
110.00
(<
3
115.00
!
3
105.00
11
3J
128.30
44
3}
134.10
f
3}
122.50
"
4
146.60
"
4
153.30
1
4
140.00
M
4J
165.00
"
4J
172.50
!
4J
157.50
11
5
183.30
5
191.60
(
5
175.00
M
51
201.60
"
51
210.80
1
5}
192.50
44
6
220.00
11
6
230.00
1
6
210.00
"
6J
238.30
<
^}
249.10
!
6J
227.50
"
7
256.60
"
7
268.30
1
7
245.00
M
7}
275.00
71
287.50
1
7}
262.50
M
8
293.30
cJ :;
8
306.60
i
8
280.00
it
8J
311.60
"
8i
325.80
!
8}
297.50
11
9
330.00
M
9
345.00
!
9
315.00
44
9}
348.30
M
91
364.10
1
9J
332.50
"
10
366.60
"
10"
383.30
1
10
350.00
"
10}
385.00
M
10}
402.50
10}
367.50
11
11
403.30
M
11
421.60
11
i
18.33
11*
J
19.16
"
11}
440.70
1
36.66
1
38.33
12
12
480.00
234
WEIGHT OF A LINEAL FOOT, ETC.
WEIGHT OF A LINEAL FOOT OF ROLLED ROUND
IRON IN POUNDS AVOIRDUPOIS.
(480 pounds per cubic foot.)
Diameter in
inches.
1
Weight in Ibs.
Diameter in
inches.
Weight in Ibs-
Diameter in
ineher.
Weight in Ibs.
Diameter in
inches.
Weight in Ibs.
A
0.010
03
^8
14.77
Bf
82.79
8J
206.2
4
0.041
2}
16.36
6|
86.52
9
212.2
A
0.091
2|
18.04
5{
90.34
01
y f
218.0
I
0.163
2J
19.80
6
94.26
9V
223.9
iV
0.255
91
"8
21.64
6J
98.18
^8 L
230.1
I
0.3 68
3
23.56
B|
102.20
9}
236.2
A
0.501
3J
25.56
8|
106.40
9|
242.5
i
0.655
31-
27.64
6.V
110 60
248.9
A
0.828
3|
29.82
61
114.90
^1
255.2
I
1.022
3J
32.07
81
119.30
10
261.7
1.237
3|
34.39
6}
123 . 70
10}
268.4
I
1.473
3J
36.81
7
128.30
iQi
275.0
1 . 728
3|
39.30
n
132.90
10|
281.8
i
2.004
4
41.88
7}
137.60
10.}
288.6
2.301
45-
44.57
71
142.30
10|
295.6
i
2.618
4}
47.28
7-V
147.30
10|
302.5
if
3.310
4|
50.10
71
152.20
10^-
309.5
i}
4.094
4f
53.02
7-1
157 20
11
316.8
is
4.950
4l
56.03
n
162.40
114
323.9
ij
5.885
4J
59.05
8
167.50
lit
331.3
if
6.911
H
62.17
8i
172.80
HI
338.7
ij
8.018
5
65.49
S|
178.20
11 J
346.2
i|
9.205
5J
68.71
8J
183.60
n!
353.7
2
10.470
5}
72.13
a}
189.10
iif
3G1.5
2J
11.820
5|
75.65
81
194.80
ill
369.1
2}
13.250
5*
79.17
8f
200.40
12
376.9
BOLTS, NUTS, AND HEADS.
235
BOLTS, NUTS, AND HEADS.
(Whitworth s Proportions.)
Weight in Ibs. of Heads and Nuts.
Diameter of
bolt in in.
Hexagonal.
Square.
Hexagonal.
Square.
Head.
Nut.
Head.
Nut.
Two
Heads.
Head
&Nut.
Two
Heads.
Head
& Nut.
i
0.008
0.005
0.022
0.019
0.017
0.013
0.044
0.041
A
0.014
0.007
0.027
0.021
0.029
0.022
0.055
0.048
I
0.029
0.017
0.061
0.049
0.057
0.046
0.122
0.110
A
0.059
0.040
0.069
0.050
0.119
0.101
0.138
0.119
I
0.068
0.041
0.104
0.076
0.136
0.109
0.208
0.181
0.104
0.065
0.157
0.118
0.208
0.169
0.315
0.276
0.151
0.097
0.246
0.193
0.302
0.248
0.493
0.440
0.254
0.161
0.362
0.269
0.508
0.415
0.724
0.631
0.367
0.219
0.551
0.408
0.734
0.586
1.102
0.959
i
0.546
0.326
0.683
0.463
1.092
0.872
1.366
1.146
i
0.724
411
1.109
0.797
1.448
1.135
2.217
1.906
i!
1.060
0.630
1.400
0.971
2.120
1.690
2.800
2.371
it
1.330
0.759
1.949
1.379
2.660
2.088
3.898
3.328
i
1.840
1.098
2.625
1.883
3.680
2.938
5.250
4.508
if
2.460
1.517
3.135
2.192
4.920
3.977
6.270
5.327
if
2.920
1.742
3.704
2.532
5.840
4.662
7.409
6.236
u
3.440
1.991
4.725
3.276
6.880
5.431
9.450
8.001
2
4.370
2.611
6.384
4.625
8.740
6.981
12.77
11.00
2J
6.150
3.645
8.858
6.353
12.30
9.795
17.71
15.21
2}
8.480
5.045
11.91
8.476
16.96
13.52
23.82
20.39
2J
11.32
6.747
15.59
9.019
22.64
18.06
31.18
24.61
3
14.72
8.783
21.00
15.06
29.44
23.50
42.00
36.06
1
236
WEIGHT IN POUNDS OF HOUND IRON, ETC.
WEIGHT IN POUNDS OF ROUND IRON FOR
Diameter
in inches.
Length in inches.
N
K
%
K
%
H
y*
i
2
3
i
0.002
0.003
0.005
0.007
0.008
0.010
0.012
0.014
0.027
0.041
A
0.003
0.005
0.008
0.011
0.013
0.016
0.019
0.021
0.043
0.064
1
0.004
0.007
0.011
0.015
0.019
0.023
0.027
0.031
0.062
0.093
A
0.005
0.010
0.016
0.021
0.026
0.031
0.036
0.042
0.084
0.126
i
0.007 0.014
0.021
0.027
0.034
0.041
0.048
0.055
0.110
0.166
0.009
0.017
0.026
0.035
0.043
0.052
0.061
0.069
0.139
0.208
0.011
0022
0.032
0.043
0.054
0.065
0.076
0.087
0.174
0.261
0.015
0.031
0.046
0.062
0.077
0.093
0.108
0.124
0.249
0.373
1
0.021
0.042
0.063
0.084
0.105
0.126
0.148
0.170
0.338
0.508
0.027
0.055
0083
0.110
0.138
0.165
0.193
0.221
0.442
0.663
H
0.035
0.070
0.105
0.140
0.185
0.210
0.245
0.280
0.560
0.840
4
0.043
0.087
0.131
0.173
0.217
0.262
0.304
0.347
0.695
1.043
it
0.053
0.104
0.157
0.209
0.261
0.314
0.366
0.418
0.836
1.255
if
0.062
0.124
0.186
0.249
0.311
0.373
0.435
0.497
0.995
1.493
it
0.072
0.143
0.215
0.287
0.358
0.430
0.502
0.584
1.168
1.752
if
0.084
0.168
0.253
0.337
0.421
0.506
0.590
0.677
1.354
2.032
if
0097
0.194
0.291
0.389
0.486
0.583
0.680
0.778
1.555
2.333
2
0.111
0.221
0.332
0.442
0.553
0.663
0.774
0.884
1.770
2.654
21
0.140
0.280
0.420
0.560
0.700
0.840
0.980
1.120
2.240
3.360
2}
0.174
0.347
0.521
; 695
0.869
1.042
1.216
1.390
2.781
4.172
2|
0.209
0.418
0.627
0.836
1.045
1.254
1.463
1.673
3.346
5.019
3
0.250
0.500
0.750
1.000
1.250
1.500
1.750
1.990
3.981
5.972
EXAMPLE. Required, the weight of a bolt 1J inches diameter,
4 inches between inside of head and nut.
Weight of bolt = 1.39
Weight of square head = 1.40
Weight of hexagonal nut = 1.06 taken as a hexagonal head
Ans. 3.85 Ibs.
WEIGHT IN POUNDS OF ROUND IKON, ETO.
237
BOLTS, ETC., BETWEEN HEAD AND NUT.
Diameter 1
in inches.
Length in inches.
4
5
6 -
7
8
9
10
11
12
i
0.055
0.069
0.082
0.096
0.110
0.124
0.137
0.151
0.165
A
0.086
0.107
0.128
0.150
0.171
0.192
0.214
0.235
0.257
1
0.124
0.155
0.186
0.217
0.248
0.279
0.311
0.342
0.373
A
0.167
0.209
0.251
0.293
0.335
0.377
0.419
0.461
o.5oa v
1
0.221
0.276
0.331
0.386
0.442
0.497
0.552
0.607
0.663 :
0277
0.347
0.416
0.486
0.555
0.624
0.694
0.763
0.833
0.347
0434
0.521
0.608
0.695
0.782
0.869
0.956
1.043
0.497
0.622
0.746
0.871
0.995
1.119
1.244
1.368
1.493
0.677
0.846
1.016
1.185
1.354
1.524
1.693
1.862
2.032
i
0.884
1.105
1.326
1.548
1.769
1.990
2.211
2.432
2.654
U
1.120
1.400
1.680
1.960
2.240
2.520
2.800
3.080
3.360
li
1.390
1.738
2.085
2.433
2.781
3.128
3.476
3.823
4.172
If
1.673
2.091
2.510
2.928
3.346
3.765
4.182
4.601
5.019
li
1.990
2.488
2.985
3.483
3.981
4.478
4.976
4.973
5.972
2.336
2.920
3.504
4.088
4.673
5.257
5.841
6.425
7.010
If
2.709
3.386
4.064
4.741
5.418
6.096
6.773
7.450
8.128
ll
3.111
3.888
4.666
5.334
6.221
6.999
7.777
8.547
9.333
2
3.538
4.423
5.307
6.192
7.077
7.961
8.846
9.730
10.610
2}
4.480
5.600
6.720
7.840
8.960
10.080
11.200
12.320
13.440
2J
5.562
6.953
8.343
9.734
11.120
12.510
13.910
15.290
16.690
2f
6.692
8.365
10.040
11.710
13.380
15.060
16.730
18.400
20.070
3
7.962
9.953
11.940
13.930
15.920
17.910
19.910
21.890
23.890
WEIGHT OF MATERIALS USED IN BUILDING.
WEIGHT OF MATERIALS USED IN BUILDING.
(Per square foot from one inch thickness to a cubic foot.)
Stones, Earths, &c.
9
tt.
Brick.
0)
M
pj
<o
fl
i.3
9
cS
g"
averaj
SH
fi Jj
of Parii
1
a
1
|
1
B.C
C
73
*
i
9
2
el
8s
1
o
S
o
cT
1
1
d
jj>
e"
<J
S
a
5
a
1
1
1
i
6.58
14.58
8.50
11.41
9.33
6.12
9.08
16.5
14.08
8.16
8.5
10.83
2
13.16
29.1C
17.00
22.83
18.66
12.25
18.16
33.0
28.16
16.33
17.0
21.66
3
19.74
43.74
25.50
34.24
28.00
18.36
27.24
49.5
42.25
24.50
25.5
32.49
4
26.32
58.32
34.00
45.66
37.33
24.50
36.33
66.0
56.32
32.66
34.0
43.33
6
32.90
72.90
42.50
57.08
46.66
50.61
45.41
82.5
70.40
40.83
42.5
54.16
6
39.48
87.48
51.00
68.50
56.00
J6.74
54.50
99.Q
84.48
49.00
51.0
65.00
7
46.06
102.00
59.50
80.00
65.33
42.86
63.60
115.5
98.56
57.16
59.5
75.83
8
52.64
116.64
68.00
91.32
74.66
49.00
72.66
132.0
112.64
65.32
68.0
86.66
9
59.22
131.22
76.50
102.75
84.00
55.10
81.75
148.5
126.72
72.50
76.5
97.50
10
65.80
145.80
85.00
114.16
93.33
61.23
90.83
165.0
140.80
81.66
85.0
108.33
11
72.38
160.38
93.50
125.60
102.6(5
67.35
99.13
181.5
154.90
89.82
93.5
119.16
12
79.00
175.00
102.00
137.00
112.00
73.50
109.00
198.0
169.00
98.00
102.0
130.00
Stones, Earths, &c.
-^
i
Granite.
fl
a
o
7j
S
>
a
g
S:
o
C3
<n .
11
a>
o
tJ
a
bO
5
1
I?
1
a
o
d
1
1
73 si
OS
73
&
0?
I
5*
S
3
1
s
|
H
1
53
5
O
1
1
1
s
02
3
1
6.75
11.16
10.0
12.91
10.41
11.41
13.75
8.66
12.25
13.75
14.08
5.21
2
13.50
22.33
20.0
25.82
20.83
22.83
2750
17.33
24.50
27.50
28.16
10.42
3
20.25
33.50
30.0
38.73
31.25
34.25
41.25
26.00
36.75
41.25
42.24
15.62
4
27.00
44.66
40.0
51.64
41.66
45.66
55.00
34.66
49.00
55.00
56.32
20.83
5
33.75
55.83
50.0
6455
52.08
5708
6875
4333
61.25
68.75
70.40
26.04
6
40.50
67.00
60.0
77.46
64.50
68.50
82.50
52.00
73.50
82.50
84.48
31.24
7
47.25
78.16
7Q.O
90.37
73.00
80.00
96.25
60.66
85.75
96.25
98.56
36.45
8
54.00
89.33
800
103.28
83.32
91.32
110.00
69.22
98.00
110.00
112.64
41.66
9
60.75
100.50
90.0
116.19
93.75
102.75
123.75
80.00
110.25
123.75
126.72
4687
10
67.50
111.66
100.0
129.10
104.16
114.16
137.50
86.66
12250
13750
140.80
52.08
11
74.25
122.83
110.0
142.01
114.57
125 57
150.25
95.32
134.75
150.25
154.88
57.28
12
81.00
134.00
120.0
155.00
125.00
137.00
165.00
104.00
147.00
165.00
169.00
62.50
DIVISIONS OF A FOOT, ETC.
239
w
w
Q
52;
CDl^-r^OOOOr-ifMCOCOLOiOCDt^oOO^
co co o :N 10 *- as i co LO r- o^ r-i co o i
T-H ^ rr< Jt. <N;t>; <N !> CO OO CO CO <Q OS rt* ai -^
r
OOGOOOGOCOOOGOGOGOOOOOCOOOO^aiOi
CO " CO i i C
cococDCDcocoi t
r-
GO OC Ot) Oi O O i-H T i C^l CO CO r}H TH tO IO CD
lOiOlOOCDCDCDCDCDCDCDCDCDCDCDCD
(M^CDoOOCVJ^CDoOOCSIiOt-OirH
lOOlOOCDr-- fCDr ICD(MI^CVJ1>.C\IOO
-
COOiOOr-tCvlCOTjHKDCD
CO IO 1> O5 i iCOlOtOCvjTtiCDOOOC^lTtH
COOOCOOOrt<a5TtiailOOlOOiOi iCDrH
COCO *"T^lOiOCDCDt^OOOOaiO^OOr-l
COCOCOCOCOCOCOCOCOCOCOCOCO-^TjHrtl
gOto Ocor- CDr-icDC^r^eqt-c
CDCDJ^J>-OOOOO^CiOOT (r (C
<M<N(MC^l(M(M(MCq(MCMcOCOCOCOC
CDi li^<Mr-<Mt COCOCOOOCOOi ^Ci rJH
CDi>-t OOOOOiOiOOr- IT ((MCMCOCO^i
T _, T _ HrHr _, T _( T _i^_ i<M(M(M(M(M<M<MC^c<j
r
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<
240
TABLE FOR COMPARING MEASURES AND WEIGHTS.
TABLE FOR COMPARING MEASURES AND WEIGHTS
OF DIFFERENT COUNTRIES.
Weights.
UNITED
STATES AND
ENGLAND.
PRUSSIA.
AUSTRIA.
BADEN AND
SWITZERLAND.
FRANCE.
Pound.
Pound, Z. V.
Pound.
Pound.
Kilogra e.
1
0.9072
0.8100
0.4536
1 . 1023
1
0.8928
Same as
0.5000
1.2346
1 . 1200
1
Prussia.
0.5600
1.2346
1.1200
0.9999
0.5600
2.2046
2.0000
1.7857
1
Measures of Length.
Foot.
Foot.
Foot,
Foot,
Meter.
= 12 inches.
= 12 inches.
= 12 inches.
==10 inches.
= 100 Centi.
1
0.9711
0.9642
1..0160
0.3048
1.0297
1
0.9929
1.0462
0.3138
1.0371
1.0072
1
1.0537
0.3161
0.9843
0.9559
0.9490
1
0.3000
3.2809
3.1862
3.1635
3.3333
1
Measures of Surface Square Measure.
Square foot.
Square foot.
Square foot.
Square foot.
Sq. Meter.
1
0.9431
0.9297
1.0322
0.0929
1.0603
1
0.9858
1.0945
0.0985
1.0756
1.0144
1
1.1103
0.0999
0.9688
0.9137
0.9007
1
0.0900
10.7643
10.1519
10.0074
11.1111
1
TABLE FOB COMPARING .MEASURES AND WEIGHTS.
Cubic Measure.
UNITED
STATES AND
ENGLAND.
PEUSSIA.
AUSTRIA.
BADEN AND
SWITZERLAND.
FRANCE.
Cubic foot.
Cubic foot.
Cubic foot.
Cubic foot.
Cubic meter
1
0.9159
0.8964
1.0487
0.0283
1.0918
1
0.9787
1 . 1450
0.0309
1.1156
1.0217
1
1.1699
0.0316
0.9535
0.8733
0.8548
1
0.0270
35.3166
32.3459
31.6578
37.0370
1
Weight per Unit of Length.
Lbs. per
lineal foot.
Lbs. per
lineal foot.
Lbs. per
lineal foot.
Lbs. per
lineal foot.
Kil. per
lineal meter
1
0.9342
0.8400
0.8929
1.4882
1.0705
1
0.8993
0.9559
1.5931
1.1904
1.1120
1
1.0629
1.7716
1.1199
1.0462
1.9408
1
1.6667
0.6720
0.6277
0.5645
0.6000
1
Weight per Unit of Surface.
Lbs. per
square inch.
Lba. per
square inch.
Lbs. per
square inch.
Lbs. per
square inch.
Kil. per
square cent.
1
0.9619
0.8712
1 . 2656
0.0703
1.0396
1
0.9057
1.3157
0.0731
1.1478
1.1041
1
1.4526
0.0807
0.7902
0.7601
0.6884
1
0.0556
14.2223
13.6811
12.3910
18.0000
1
16
242
RESISTANCE TO CROSS-BKEAKING.
RESISTANCE TO CROSS-BREAKING.
To Cut the Strongest and Stiffest Rectangular Beam from a Log,
Fig. 308. (Strongest.)
The diameter aa = d, divided into three equal parts, with per*
pendiculars J d from a erected thereon, intersecting the circle at
b, will give section for greatest capacity.
Fig. 309. (Stiffeet.)
The diameter aa = d, divided into four equal parts, with per-
rndiculars J d from a erected thereon, intersecting the circle at
, will give section with least deflection, but less capacity than
Fig. 308.
INDEX.
Area, circumference, and cubic contents of circles 218
Axis, neutral 4
Bars, tie rods, &c 181
resistance of, to tearing 2
Beams, capacity and strength of 29
of rolled 39
of cast-iron 57
TFof rolled l-shaped 39
and strength of parabolic arched 153
cast-iron 53
iron ties, struts, and 3
sloping rafters and 102
strains in trussed 122
horizontal andsloping 188
strength of wooden 88
Bolts and nuts, dimensions of. 187
nuts, and heads 235
Boom derricks, strains in 114
Booms, strains in trusses with parallel 126
Bow-string girders 147
Bridges, static and moving loads, of wrought iron 192
Camber 2
Capacity 2
and strength of beams 29
W of rolled l-shaped beams 39
of rolled beams 41
of cast-iron beams 57
and strength of parabolic arched beams..... 153
Cast-iron beams 3, 53
Center of gravity of planes 202
Circumference, area, and cubic contents of circles 218
Columns, pillars, and struts, strength of 110
Composition and resolution of forces Ill
Compound strains in horizontal and sloping beams 188
Compression 1
Compressive strain and pressure on supports 102
Contraction and expansion 4
(243)
244 INDEX.
MOB,
Constants for strain in trusses 117
roof trusses 174
Connections in iron construction, joints or 184
Cross-breaking 2
and shearing, resistance to 29
Crushing, resistance to 103
direct 1
Deflection 2
Derricks, strains in boom 114
Dimensions of bolts 187
Divisions of a foot, expressed in equivalent decimals 239
Expansion and contraction 4
External forces .T^J/i
Factors of safety 29
Forces external ., ..r 1 %
internal 1
composition and resolution of. Ill
parallelogram of Ill
Frame, strains in polygonal 154
Functions, trigonometrical 207
Geometry 197
Girders, strains in parabolic and bow-string 147
Gravities of materials, specific 224
Heads, nuts, and bolts 235
Horizontal and sloping beams, compound strains in 188
Howe truss 129
Inertia and resistance o various sections, moments of 5
Internal forces 1
Iron beams, capacity of cast 57
cast 53
bridges, static and moving loads, of wrought 192
construction, joints or connections in .-. 184
ties, struts, or beams 3
Joints or connections in iron construction 184
Lattice truss 139
with vertical members 131
Longimetry and planimetry 197
Materials, &c., strength of 26
Miscellaneous , 195
IffDEX. 245
Modulus of rupture 4
Moment of inertia and resistance of various sections 5
Moving loads, weight of. 191
Natural sine, cosine, &c 306
Neutral axis 4
Nuts, heads, and bolts 235
dimensions of... 187
Parallelogram of forces..... Ill
Parallel booms, strains in trusses with 126
Parabolic arched beams, capacity and strength of. 153
curved trusses, strains in 147
Planimetry, longimetry, &c 197
Pillars, columns, and struts, strength of 110
Pins, &c., in tie bars 185
Polygonal frame, strains in ,. 154
Pressure on supports 100
compressive strain and 102
of snow on roofs 178
of wind on roofs 180
Rafters, &c., sloping beams _. 102
Reactions of supports 100
Resistance to direct crushing 1
of bars, &c., to tearing 2
to cross-breaking and shearing 29
crushing 103
Resolution of forces, composition, &c Ill
Rolled beams, capacity of. 41
l-shaped beams, capacity of. 39
Rods and bars, tie 181
Roof trusses..., 3
strains in 156
constants for strains in 174
Roofs, pressure of wind on 178
of snow on 180
Rupture, modulus of. 4
Shearing 2
and cross-breaking, resistance to 29
Sloping beams, rafters, &c 102
and horizontal beams, compound strains in 188
Specific gravities of materials 224
Static and moving loads of wrought-iron bridges 192
Strength of materials 26
wooden beams 98
columns, pillars, and struts 110
246 INDEX.
PAGE.
Strength of beams, capacity, &c 29
Strains in frames 112
boom derricks 114
trusses 115
trussed beams 122
trusses with parallel booms 126
parabolic curved trusses, or bow-string girders.... 147
polygonal frame 154
roof trusses 156
constants for 174
trusses, constants for 117
Strongest and stiffest rectangular beam from a log, to cut the.. 242
Struts and beams, iron ties 3
Supports, reaction of. ... 100
compressive strain and pressure on 1 02
Table for comparing measures and weights 240
Tearing, resistance of bars, &c., to 2
Tension 1
Tie rods and bars 181
Trigonometrical functions 207
formulas 205
Truss, Howe 129
Warren 132
Whipple 144
lattice 139
with vertical members 131
Trusses parallel booms, strains in 126
parabolic curved, or bow-string 147
constants for strains in roof 1 74
constants for strains in 117
strains in 115
roof 156
Trussed beams, strains in 122
Warren truss , 132
Weight of moving loads 191
static and moving loads of wrought-iron bridges... 192
a lineal foot of flat or square bar iron 229
rolled round iron 234
materials used in building 238
superficial inch of wrought and cast iron 227
rolled round iron for bolts 236
heads and nuts 235
per square foot of metals 228
Whipple truss 144
Wooden beams, strength of. . 98
1 388
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