INST.
C,'
LIB~A Ri
INVESTIGATION OF THE STRESSES IN A
CONTINUOUS TWO-SPAN HIGHWAY BRIDGE.
BY
CLARENCE C.T. LGO
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 27, 1929.
V
ACKNOWLEDGMENT
167433
May 27,
1929.
Professor A. L. Merrill,
Secretary of the Faculty,
Massachusetts Institute of Technology,
Cambridge, Mass.
Dear Sir:
In accordance with the rules of
the Faculty, I hereby submit a thesis entitled "Investigation of the Stresses in
a Continuous Two-Span Highway Bridge" in
partial fulfillment of the conditions required for the degree of Bachelor of
Science.
Respectfully submitted,
TABLE
OF
CONTENTS
Introductory - Object
Pages 1-2
Description of Bridge
3-6
Method used - Deviation of formulae
7-15
Results
15-18
Discussion
19-2/
Appendix - Calculations
2/ -53
-/1-
Up to about a quarter of a century ago,
structural
engineers have looked upon continuous bridges with disfavor.
tice.
Continuous structures were considered bad prac-.
But since that time, the prevailing attitude has
changed, and continuous bridges are now being quite
generally accepted as the full equivalent of other types
where field requirements in erection,
in material,
or where a saving
justifies their use.
Several factors are responsible for this changed
attitude.
In the first place, the uncertainties of
analysis and the apprehension concerning the initial
adjustment, have been both removed; the former by a more
thorough grasp of structural relations, and the latter
by increased constructive skill.
Then too,
practice
has served to emphasize the proper economies of con-.
strained types of structures; and this, coupled with
the increased recognition given to rigidity in service,
has helped greatly to remove the predjudice against
continuous bridges.
Further,
the popularizing of mechanical stress
analysis for use where direct calculation has proven
too complicated,
has not only removed an obstacle to
proper proportioning and practical design, but has also
supplied engineers with an eye-picture of deformations,
and thus bettered his conception of how movements at
the supports or abutments may influence the stresses.
Then again, the change from the use of pin-onnections
to that of riveted connections for bridge trusses has had
In the old days when pin-connections were
its influence.
the rule, members subject to a reversal of stress had to
be made so as to be adjustable,
up to initial
then in erection drawn
tension of an unknown amount.
The un-
certainties attendant upon such an arrangement opposed
its use for main members and so the continuous bridge,
with its reversals of stress, was looked upon with
disfavor.
However, when riveted connections replaced
pin connections, this objection was eliminated and so
the prejudice could no longer exist.
Accordingly, although steel design still
shows a
leaning toward the simpler statically determined types
of structures, the continuous structure is gradually
coming into its own,
and with the progressive weakening
of what remains of the old influence,
we may look forward
to a more rapid extension of continuous structures.
It
is then with this object in mind that the
following thesis is presented:
To illustrate the elastic
load-deflection method of stress analysis for a continuous bridge.
-3-
The bridge selected for investigation is a con.
tinuous two-span riveted Warren truss highway bridge
now under construction.
It
is
being built over the
Missouri River at St. Joseph, Mo., about 1/4 mile south
of the combined railway and highway bridge operated by
the St. Joseph and Grand Island Railway.
Among the salient features in
the design of this
structure are the application of continuous trusses to
relatively moderate span lengths, and the liberal use
of silicon steel.
A comparison of the continuous truss
with two simple end-supported spans showed a saving of
approximately $25,000 or about 7% of the cost of the
main bridge.
The use of silicon steel wherever the
size of the member justified its
of $37,000.
use resulted in a saving
A study of the relative advantages of
simple paneling as compared with subdivided paneling was
also made.
This showed some distinct advantages in favor
of the simple paneling, among which were:
(1)
(2)
(3)
(4)
Slight saving in cost.
Great simplicity in erection.
Lower secondary stresses.
Better appearance.
The bridge was designed according to the spec.
ifications of the American Association of State Nghway
Officials,
1925,
with the modifications
noted below.
Live loads:Floor - 1 15-ton truck per panel on each of three
traffic lanes spaced 9 ft.,
o. to A.
-4-
Trusses - Uniform load of 562# per ft. per truss.
Concentrated load of 26,300# per truss.
Impact:Floor - 30% of live loa d for all floor members.
Trusses - I
=
(L 250)0.8 in which I a loaded
10%+-500length, and I should never exceed 30%
Wind Pressure :Transverse wind pressure - 30# per sq. ft. on the
area of one floor, two trusses, and two
handrails.
Longitudinal wind pressure - 50J of transverse wind
pressure per lineal foot of bridge.
Temperature :Normal temp.
=
600 F.
Maximum temp.
=
1200 P.
Minimum temp.
=
-200
F
(200 below 0)
Unit Stresses:Carbon steel - Tension - 16,000 #/in. 2
Compression - 15,000
50 )
r
maximum
13,500
Silicon steel - Tension - 24,000 #/in. 2
Compression - 22,500
75 1
r
maximum : 18,750
Combinations of loading:(1)
Dead -+ live +
(2)
Dead -+ 30-lb.
(3) (1) + 15-lb.
(4) (1) + temp.
(5) (4) t 15-lb.
- temp.
(6) (2)
impact
wind)
wind)
)
wind)
Allowable unit stresses
increased 25%.
Allowable uit
t
esses
inaed
increased 40 '
The final design adopted shows an arrangement as
follows:
The floor system consists of a 6j" slab of reinforced concrete carried on transverse members consisting of 8-inch I-beams curved to the desired crown.
These beams,
spaced three feet c. to c. were carried on
two 24-inch and two 27-inch I beam stringers spaced
eight feet between centers.
The floor beams consisted
of a 50" x 3/8" web with four - 6" x 3"
angles.
x 5/8" flange
A clear roadway of 27 feet with a crown of 2"
was provided.
6" x 10" curbs were used,
and 2" pipe
handrails were installed, but no sidqwalks were provided.
The design of the trusses is shown in the acoom
panying diagram:
Jee ?/afe I (n1exI
page)
LIlve DIAGRAM
OF
TRUSS5
Plain Alympe5 .5Aowv rea S- rnmbcp%5-~
Ffure.5 In paren?'hcse-s s,ovv /cnqfh of ,nember.-5
44
U/z
jez
A/ole
Top
c-J
bof,-t,0-1 /aYe,-a/W
45l7em"s
con-,,s-# ofo 7 yo
3 o - r o - /al-e,-- a/
5 ym57 p,,er77bee5
/0,p
/A '- ,?/-5C . 91fet.;,
~// C 0,-?
-1-e e/ ,e
.5ee
jEr~j 9 .
ar-e
e/7e/,
Ndws5-iReco-
all
s///c',-
#4e126,Qt,
dao
a
57eel.
.9- 3a.,7 a /
0 aWel
a
f 4
Vol/ /02, Alo. 3, ka-,ya5 100-102,
P0L AT-E No.1I.
I
all
I
-7-
The method used in this investigation is
the elastic load method.
elastic curve)
is
known as
The deflection curve (or the
determined by the use of "elastic"
loads, and from this curve the influence lines for the
reactions may be obtained.
Once the reactions are
determined the stress analysis becomes a simple matter.
The method of computing the elastic loa do used
in the following computations is
Muller Breslau.
that proposed by H.
The theory and the derivations of
the following quotation
the formulae used are shown in
from Professor W. M. Fife's (M.I.T.) private translation
of Mr. Breslau's work:
THE DEFLECTIONS OF THE JOINTS OF A TRUSS
BY THE METHOD OF ELASTIC
WEIGHTS.
BAR CHAIN METHOD
1.
If a series of bars are connected by frictionless
joints so as to form a chain, it
is
possible to find the
deflections of the joints due to stresses in the bars by
drawing the funicular polygon for a series of imaginary
loads placed at the joints whose deflection is desired.
In determining the magnitude and direction of these
imaginary loads, hereafter called "elastic loads", it
is
necessary to consider first
certain relations between
-8-
a series of loads and the corresponding funicular polygon
and then demonstrate that the funicular polygon drawn
for a particular series of loads will have the san
ordinates from some base line as the elastic curve has.
2.
To find the loads corresponding to a particular
funicular polygon.
Fig. 1.
P
SPm
PrnPm
PPm
Fig.
1
By similar triangles:
~Sm - M-1
aC
2Xr+i
also
mxi
P
H
t1
therefore
m-bm-
cbm+i-6m+
aH
and
H
'.9m
i
m
YM)I
Irn
I
Fig. 2.
Consider any bar chain m-1,
m, m
1,..... referred to
a pair of co-ordinate axes so that the ordinate for
any joint m is
ym.
to the left of
joint
the horizontal be
Let the length of the bar immediately
f,
m be a
and let its
inclination to
the angle being positive when
measured contra-clockwise from the horizontal.
Let
be the angle between bare (m-l)-(m) and (m)-(m
1)
Om
measured on the lower side of the bar chain.
Let the chain undergo stress so that the lengths
of the barschange; their inclinations will change also
and, consequently the ordinates y will increase or
decrease.
Let the change in the ordinate ym be
JfI,
-/0-
which, consequently,
is
the vertical deflection of joint m.
9m-i~ 9m =
'm Sinf
differentiat ing,
4 y,,_
dividing by
- J yin
9
= 5,,,
m, +
m
Cos5,,,
.,,cos ,,, =/L,,
47-i A -
ynw
_m
similarly
S3qm ~ 59_rn+
5!&A'nt
$5m.
rr
4+
Subtracting
n
-
b
-/
0
a
m
4ian
-
-
r
m
or
-
-
-r
,,
Y
'
a
> fJOaml
r+/
where fm is the stress intensity in bar (m-l).-(m) and
is positive or negative according as the bar is
in
tension or compression.
Also 1800 consequently,
(0
)=
-
r,
by differentiating,
-S pmn +
=7'
01>3-7
therefore,
4"T?-/z
?22a1 9 L2
6?
+
i'wir,
The left hand side of this equation is the same as
the right hand side of the last equation in paragraph
1, consequently, if the elastic loads used are computed
by the right hand side of the equation above,
and the
funicular polygon is drawn, using a unit pole distance,
the ordinates of the joints of the funicular polygon from
-//
some base line will be the deflections of the joints.
This base line is
found from the consideration that
at the points of' support of the structure the deflections
are zero.
If
it
we examine the expression
may be seen that if
either
elastic load becomes infinite,
the above expression is
#eor
Aenis 900 the
consequently,
the use of
limited to examples where none
of the bars of the chain are vertical.
The funicular polygon may be drawn by graphic methods
or its
shape may be determined analytically.
In the
analytical method use is made of the following property
of the funicular polygon:
If
a number of vertical loads
are applied to a simple end-supported beam and the
funicular polygon is drawn, the ordinates to the funicular
polygon from the line joining the points where the out.side strings of the funicular polygon cut the lines of
action of the reactions for the simple beam, are the
bending moments for the simple beam providing that the
funicular polygon was drawn with a unit pole distance.
Consequently,
it
is
possible to find the shape of the
funicular polygon which will be the deflection curve
by imagining that the elastic loads are applied to a
simple beam whose span is
the length of the structure
and drawing the bending moment curve.
It
is
immaterial
-/Z-
whether the structure whose deflections are being in-.
vestigated is
simply end-supported or not; the deflection
curve may alvays be obtained in this way.
emphasized,
however,
It
is
to be
that this procedure leads to the shape
of the deflection curve only, and that the base line from
which the deflections are to be measured is
not necessarw
ily the base line from which the bending moment ordinates
are laid off; the base line from which the deflections
are to be measured must be such as to show zero ordinates
to the deflection curve at the points of support.
case of a structure which is
the funicular polygon is
In the
supported at one end only
obtained by considering the
elastic loads to be applied to a cantilever beam of the
same length as the structure and supported in the same
way,
but the deflections are measured from a base line
which is
tangent to the elastic curve at the end opposite
the support.
In
first
the above expression for the elastic loads the
term is
the change in the angle between adjacent
bars of the chain.
If
it is desired to find the deflect-
ions of the joints of one chord of a truss, the bars
of the chord may be considered as forming the bar chain.
In such a case the changes of angle
may be found from
the changes of the angles of the triangles of the truss
which meet at the joint in question.
This involves the
problem of finding the changes in the angles of a
triangle due to changes in
the lengths of its
sides.
-/3-
4.
To find the changes in the angles of a
to changes in the lengths of its
sides.
triangle due
Consider the
triangle in Fig. 3.
a
IC
a
hh
.
=b/roma =
h
--
Fig.
/sin
/
3.
Differentiating the expression for o.
(C = 5b-co5oc--+
a-cosp - b-sma:6oc -
: 6bcosc + &a-cop - h c&
+6)
but
+ Y = 1800
oC
and
CoX-e
+C+6Y= 0
therefore
6C =
b.coc. +
a-cop +
6by
Transposing and dividing by h
Le
Sb-cosO<.
If
the changes in
= a.sine
da(-cose
a.5Inp
, tr
h-
. b.sinoC
the lengths of the sides of a triangle
formed by the bars of a truss are the strains due to
stresses fa'
fb,
and f. (these being stress intensities)
jdb=f b
daJa a
and
c -.
Eh
bco
E b5arno
. _
_
E
acosp
amp
de = -1e-C
-/4-
[6f-f'p
~ ~EhJ
a-c
E -5
_
- fbb.coE b-5inor
.co
b-s5inv.
c fca~cop
E
cos
a
2-.b.cosac
E b.5in<~ Ea-i
-bCios
#Ec
f_.O
E
a-pi
-
namely that of a
'~fcoL
fb
similarly,
oc fb coy
fc
E
~acoil
dp
In
y
clo
the case under consideration
continuous two-span highway bridge - there is
redundant reaction.
one
The specific procedure in the
investigation will be as follows:(1)
Compute areas, weight per foot, and length of
all bars of the truss.
(2)
Replace the redundant reaction by a unit force
acting downward and calculate the stress intensity in each bar caused by the unit load.
(3) Using these stress intensities in the formula
above, compute the elastic loads.
(4)
Compute the bending moments due to the elastic
loads. The curve for the bending moments will
be the same as the deflection curve, the latter
being referred to a different base line.
(5)
In the case of a two-span structure, the
application of Maxwell's Theorem will now give
the influence line ordinates for the redundant
reaction.
(Maxwell's Theorem:If a force P at point A
produces a deflection x at point B, then the
same force P at point N will produce the same
deflection x at point A).
(6)
Having the influence table for the redundant
reaction, influence table for all
the members
of the truss may now be prepared.
(7)
Compute panel concentrations for dead load.
(8)
Compute dead stresses.
(9) Compute live stresses ard impact.
(10)
Compute reversals of stress.
-/6-
The results of the investigation show a close
agreement with the results of the design9,g as given
in the Engineering News-Record.
Influence line for end reactions
Pt.
Values found
by author
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
1.000
0.9D9
0,819
0.730
0.642
0.556
0.472
0.390
0.315
0.244
0.181
0.124
0.078
0.038
0.
.0 .033
-0.064
-0.090
-0.104
-0.113
-0.113
-0.109
-0.099
-0.086
-0.072
-0.056
-0.038
-0.020
0.
Values given in
Eng. News-Record
1.000
0.910
0,819
0.731
0.644
0.559
0.475
0.394
0.318
0.245
0.183
0.125
0.079
0.038
0.
-0.033
-0.064
-0.089
-0.103
-0.112
-0.111
-0.106
-0.096
-0.084
-0.070
-0.055
-0.038
-0.019
0.
- 17 -
DEAD STRESS
Bar
Values found
by author
LoUl
L2UI
L2U3
L4U 3
-450
211
-100
87
L5-l
L6 5
L6 U5
L8 7
LBU7
L
L
L12 U1 1
L12T 1 3
LigU13
U3 4
U5 U 6
U6U 8
U U
12
U
Ur1 1 44
L
L2 3
L4L5
L6L7
LaL
L
L1 13
Ul
-42
127
-219
-302
-393
479
-468
534
738
-466
-559
-543
-358
-369
-~11
-14
4 90
1040
309
510
556
470
204
-242
-715
467
1014
314
514
562
478
217
-222
-694
56
56
-10-10
U 6 L6
5
U7
U8L8
56
~9
U9 L
54
-t9
U - 12
U 1L
U1 L
4
-457
212
-100
87
2
-38
129
-212
298
-384
473
-463
533
-733
-472
-564
-535
U2 L 2
U3 L3
U4 L
U
Values given in
Eng. News-Record.
55
-10
58
-23
-/8-
Live Stress
Values found
by author
Bar
Values given in
Eng. News-Record
-190
-W189
103
75
LQU 1
L2 U1
L 2 U3
L4 U3
104
-75
78
78
L4 U5
63
- 69
84
-101
119
-139
160
-160
178
-218
-198
-249
-265
-235
-151
174
293
130
219
258
257
200
-153
-215
L 6 U5
L6U7
L8 U7
L8 U9
L1 0 U9
Ll2Ull
244113
11111
31
11U13
UiU2
U3U4
U5U6
UU8
UgU118
Ull 12
Ui U14
Lori
L2 L3
L6 L7
L8 L
1 1 L111
12 13
63
70
-
84
-101
119
-139
159
-154
172
-218
-199
-250
-269
-238
-153
170
293
131
221
259
560
201
-151
-212
U1 2
No stress in bars U2 L 2 ,
1 2 , and U 1 4 L1 4 .
U1 1
11
41
0 10
4L4 , U 6 1 6 , U8 L8 ' U LO'
Stress in bars UiLi, U3L3* U515. U717. U 919.
44.
, and U1313
/e ever5a/5
l
U5a
0
*/
2
-
6/4
-/9-
As will be noted, the values of the ordinates for
the influence line for the end reaction agree closely.
The
slight differencesnoted are probably due to a difference in
the precision used as the writer made no attempt to carry
any of his values beyond the fourth figure, (wherever a
fifth figure is given in the computation which follows, it
was done to keep the same number of decimal figures).
The dead stresses show the greatest discrepancy,
but even here, the maximum error did not exceed 16000#,
an
The stresses found by
error of slightly more than 10%.
the writer tended to be smaller than the designer's values
near the ends of the truss, (Lo-Lg) and greater at the
center of the truss (L9 -Ll 4 ),
The difference in the
values of the influence line ordinates probably is partly
responsible, but the chief source of error here very likely
lies in the assumptions made in computing the dead weight.
The total dead weight as found by the writer was 30000#
greater than that given in the Engineering News Record.
There was not sufficient data given to make possible a
really accurate calculation of the weight of the top lateral system and the floor system.
In the former case,
the cross struts had to be calculated approximately whereas
in the latter case, the weight of the I beams had to be
guessed at since only the nominal size of beams were given.
-20-
The writer used the figure 22% for connections and details;
the designer's figure was not known.
However, on the whole,
the differences are not serious and are about as small as
can reasonably be expected under the circumstances.
The live stresses agree almost exactly in practically every instance.
The maximum difference is 4000#
a variation of less 2-1/2%.
After the computations had
been made, the writer discovered an error in his results
due to the fact that a mistake was made in applying the
impact formula.
The writer used one panel more than he
should have (Loaded L1 4 as well as the other panel points)
in computing the loaded length for use in the impact formula.
However, as the results were entirely satisfactory,
(the difference being practically negligible), no corrections were attempted.
A check showed that in the majority
of cases, if the correction were applied, the entire difference would be eliminated.
This almost exact agreement
between the live stresses is possible because no assumptions were necessary in the computations.
The reversals of stress also agree satisfactory.
As is to be expected, the variations are not as great as
in the case of dead stresses nor as small as in that of
-2/-
the live stresses.
The reversals were relatively unim-
portant in this particular case, there being only three
bars subject to a reversal of stress.
As a result of the investigation, the writer
comesto the conclusion that the method of elastic loads
is as good as any of the other methods of analyzing a
statically indeterminate structure of the type considered.
Personally, the writer would prefer this method (elastic
loads) to any other since it eliminates the necessity for
expressions for the stresses due to applied loads and
does not involve the solution of simultaneous equations.
The one drawback to its use is that greater precision is
required in the calculations for the elastic loads.
The computations follow:
5 -re
Ler07g 4/7
LoUl
4z U
.1-ul
* 41:5L41
L6
TABLE I
wdh unri load ai
3.
Area
/50.3
67/
U
V/4 uY/J
us
L-,443
42.564
(#2.16
7342
33.03
3Z27
S4Lu
321/7
92. 7
-0.(o5
-0. 527
'74.7
7447
I'161
62.164
42.6
+o14tpJ
-0. 9II/44
74.7
216
62. 9to
LIUO
reoc-hon,
Wezhif/f?
46.31
46.81
58.62
53.62
62.86
/ef-
67/
14.4
2475
2250
3.24%
3.36
41.73
+~0.410
-. 516
-+0.596
-0596
74 0.5f96
76.7
/24k
a0 5fa
-0.596
/ /3.9
+/000
3 1377
*0.43
f0.71
71403
-0o. 06
+0.10(
-/.000
-
(./4
32./7
U7 UIL
UUs
32./7
4
32,/7
L
U+ 1-+
- 4L
L7
/- 7
* (4
32,/ 7
92.17
32./7
34,00
+0.05736
-0.047o3
+,/.000
+/.//. 4
-+0.05173
--1.000
-/./-4
-6.03/9/
/.00
144
+0.200
4-0.233
124.44
-0.097
0.200
-0.223
225.2
- .417
-[000
-/. / /Z
-t.o /6o7
38.36
44.42
44.42
/30.8
+1.5443
-0.36/
+/5170
-0.0/45
+2.496
+'./14
36./
/ 23.2
36.50
66.24
*2.504
#3.576
+4.767
3.576
/50.3
6767
/
-/.77 /
-2.110
-0.05/o2
/ 4 /.4
-4.17
/ 72o
U12 LIZ
54.00
/.4
31-2
L4z L
OL
,L
1441
1,4L,3
54,00
/ 1.10
67.1
31.2
47.8
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O
10,
INFLUENCE LINES
F
TRus, MEMBERS
PLATE -W
2. 00
INFLUENCE
TRUSS
+1.50
L /NES
FOR
MEMBER3
+1. 00
+'0.150
4
0
6
8
10
Z.z
/Z
24
26
-0.
50
-.
30
-2. 00
PLATE
Z
I.3
Dead Load Concentrations at Panel Points:
Top Chord Panel Points
Panel Point
Chord member
Chord member
Vertical
Diagonal
Diagonal
Lateral System
Total for Truss
22% details
Total
U1
U2
U3
U4
US
5
U6
U7
2.16
2.16
2.16
1.04
2.16
2.42
0.96
2.42
2.42
1.74
2.42
2.42
1.06
2.42
2.42
1.84
2.42
1.98
1.06
-
0.67
3.52
1.57
2.40
2.09
4.32
2.09
8.28
2.09
8.28
10.32
2.27
12.59
7.45
1.52
8.97
14.04
3.09
17.13
8.67
1.72
10.39
18.87
4.15
23.02
8.77
1.73
10.50
18.50
4,07
22.57
U8
U9
1.98
1.98
1.84
1.98
1.98
1.06
Panel Point
Chord member
Chord member
Vertical
Diagonal
Diagonal
Lateral System
2.19
1.96
-
U 10
1.98
1.98
1.84
2.35
2.34
-
U12
U1 1
2.11
2.65
U1 3
U1 4
2.09
8.28
2.09
1.98
2.00
1.06
3.58
4.46
8.28
Total for Truss
22% details
7.89
1.53
19.61
4.31
7.89
1.53
21.36
4.70
7.97
1.75
29.28
6.44
19.08
4.20
Total
9.42
23.92
9.42
26.06
9.72
35.72
23.28
-
2.41
3.90
-
2.00
2.00
1.88
2.09
9.45
3.71
3.71
2.21
9.45
-
2.00
3.71
1.29
4.57
8.26
Note: Dead Loads at Lo and L29 neglected since they do not affect the
stresses in the truss members.
ii
II
Dead Load Concentrations at Panel Points:
Bottom Chord Panel Points
Panel Point
Chord member
Chord member
Vertical
Diagonal
Diagonal
Lateral System
Total for Truss
22% details
Flooring, etc.
Total
Panel Point
Chord member
Chord member
Vertical
Diagonal
Diagonal
Lateral System
Total for Truss
22% details
Flooring, etc.
Total
Note:
L
2.06
2.05
0.67
L2
L3
L4
L5
L6
2.06
2.09
1.04
2.09
2.09
0.96
..
2.09
2.27
1.74
2.27
2.28
1.06
2.27
2.09
1.84
1.57
2.19
-
-
1.96
2.35
2.34
2.11
-
2.09
2.09
1.06
-
0.77
0.70
0.64
0.64
0.64
0.64
0.64
5.55
1.22
48.96
9.65
2.01
48.96
5.78
1.27
48.96
12.05
2.24
48.96
6.25
1.38
48.96
11.39
2.28
48.96
5.88
1.29
48.96
55.73
60.62
56.01
63.25
56.59
62.53
56.13
L
L
89
2.09
1.32
1.84
2.65
2.41
0.64
1.32
1.31
1.06
1
Ll
L10
1.32
1.45
1.84
1.45
1.45
1.06
1
10
1.45
2.77
1.89
4.46
4.57
2.77
2.77
1.29
2.77
2.77
2.21
8.26
8.26
-
3.90
-
0.70
3.58
0.77
0.77
0.85
0.92
0.92
10.95
2.21
48.96
4.39
0.97
48.96
12.86
2.63
48.96
4.73
1.04
48.96
15.99
3.52
48.96
7.75
1.71
48.96
25.19
5.54
48.96
62.12
54.32
64.45
54.73
68.47
58.42
79.69
-
-
Dead Loads at Lo and L 2 9 neglected since they do not affect the
stremes in the truss members.
U
Dead Load Concentrations at Panel Points:
Total Concentration at each Panel Point
Panel Point
1
2
Top load
Bottom load
12.59
55.73
8.97
60 *62
68.32
total
Panel Point
Top load
Bottom load
Total
note:
4
5
6
7
17.13
56.01
10.39
62.25
23.02
56.59
10. 50
62.53
22.57
56.13
69.59
73.14
72*64
79.61
73.03
78.70
a
9
10
12
13
14
9.42
62.12
23.92
54.32
9.42
64.45
26.06
54.73
9.72
68.47
35.72
58.42
23.2B
79.69
71.54
78.24
73.87
80 * 79
78.19
94.14
102.97
11
Dead Loads at Lo and L29 neglected since they do not affect
the stresses in the truss members.
(0
CO MPWJrAr o"S
Panel
Por+
/ & 27
In f luence Ltne
1._
Ordiiates
3 & 25 '73.1
5 &-23
6 &Z2
7&Z/
8&20
(-o.Shl)#(0.03)
Lz U,
Coie
-54-3 (+a. 5 3)f(--a024) +t34, s
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ComrOrd Inates
+o
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In~Huence Lbne
Ordna-e5
(-o.12')
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(-0.0/6) -.
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(-.z44)#(o.! ) -/0.3
(0.156) -(-0.072-)
73.1
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--
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/o & /8
To i
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A
(0.W 7)+(-o.0S) f33.3
796 (-o.556)+(o.0 86)-37.4 (0.355)+&-055) +23.1 (-a355) (o. a5) -2 35, (o.144) + (-o.t(-1) +/30.z
73.0 (-0.472)+(0.019q)
(0.390) +-.1)2.
7S.7 ( -.39)+(o./ oq)
(-o.,49)+(0-070) -14. (0. 314)
1 (-O.l)#/7.
7/.5 (-0.315)+ (o- /13) -/44 (.901) f (-o.07 2) +1.-2 (-0.Z0) +(o.o7z) -72
(0. 2 54-) +(-0.o91) -/ I.4
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7.2 9)
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AR.
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huerce Ljie Ver.
Ordinates
CoMrn.
Ver.
65.3 (-0.109) t (Q .o 2o) -60.7
2 5 26 69.6
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Lme
4,S 3 -0.073
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49-4
3 &- .25 73,/
-os-4t
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40.107 -+? 3.3
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Computations for Live Stresses:
Conc.Ld. - 26.3
Un.Ld. :.562x32.2 = 18.08 per panel
Bar
Lo U,
Uniform Load over
Concentrated load at
Un. Ld.
Conc.Ld.
Vertical Comp.or 8'
Impact
Total Live Stress
Bar
Uniform Load over
L2
-. 807 x 18.08
-0.296 x 26.3
-198.09
U5 U6 &
6 U7
Lo - L 1 4 inclusive
L6
-14.3
-
3.29
-17.988 X -18.0b
-325.26
2.832 x 26.3
- 74.48
-
17.32
Vertical Comp.or S'
2
0.56
-223.63 x---17.32
7.78
-19.95
Ll
-0.125 x 26.3
-
- -178.14
=
U3 U4 &4
US
Lo - L 1 4 inclusive
Conc.Ld.
-32.67
a
-40.45
189.28
-0.776 x 18.08
Total Live Stress
Lo - L 1 4 inclusive
S-170.22 -40.45x
190-78
0.8(456'250)
-170.22xTT0x450)
Un.Ld.
Impact
0e8(L 25)where L -loaded
M
500length
U1 U2 & U2 U3
Lo - L 1 4 inclusive
L1
-5.498 x 18.08 : -99.44
-0.909 x 26.3 a -23.91
-123.35
Concentrated load at
Stress
I
a
-399.74
-223.63 -397.74 x 3225-238.13 x 0.56
-25.06
-238.13
-248.69
-264.80
--
26.67
Computations for Live Stresses:
Un.Ld. : .562 x 32.2 . 18.08 per panel
Bar
Un.Load over
Conc.Ld. at
Un. Ld.
Conc.Ld.
Conc.Ld. - 26.3
I:
.97 U8- & US U9
Lo -L 1 4
i
L8
-15.984 x 18.08
2.52 x 26.3
-
Vertical Comp. org
-355.27 x
Impact
-211.74 x~~~
54
Ix0.56
L 10
-
288.99
66.28
-
355.27
-
211.74
-228.04
x
23.71
-135.91
x
Bar
Un.Load over
Conc.Ld. at
Un.Ld.
Conc.Ld.
--
U1
Impact
Stress
Total Live Stress
=
3
0.56
5
157.87 x 0 , 8 ( 7 4 0 250)
Lo - L 2 8
LP
28.06 x 18.08
1.59 x 3.57
173.66
173.66
135.91
151.13
-
506.53
41.82
548.35
548.35
15.79
-
14
28.95
28.95 x 3157.87
228.04
-
U1 3
Lo - L9 & L1 4 - L 2 8
L2 0
1.404 x 18.08
25.38
0.1356x 26.3
3.57
-
15.22
*
235.45
U1 2 & U1 2 U1 3
Vertical Comp.orst
Stress
-
-9.980 x 18.08
-1.810 x 26.3
32.2
Total Live Stress
loaded
length
U9 U10 & U10 U11
L
32.2
Stress
0.8(L 250)
10L 500 where L
267.05
32 * 2
0.8(900 250)
9000 500
267.05
"-
25.86
292*91
Computations for Live Stresses:
Un.Ld. -. 562 x 32.2 : 18.08 per panel
Bar
Uniform Ld.over
Conc.Ld. at
Un. Ld.
Conc.
Ld.
Conc.Ld. : 26.3
L2 U
L 2 - L14
L2
2.93 x 18.08
0.523 x26.3
Vertical CompStress
Impact
Total Live Stress
Bar
Un. Ld. over
Conc.Ld. at
Un. Ld.
Conc. Ld.
Impact
Total Live Stress
L 2 U3
L3 - L14
13
52.97
7
.*
.
.(
66.72
46.81
34
0.8(386 250)
:
500
92.07 x 3860
66.72 x
92.07
10.77
-55.78
58.62
49-55.78 x
0.8(354 250)
66.94 x 3540 500
-66.94
8.01
102.84
-74.95
L 4U3
L4 U.5
L4 - L1 4
L4
2.450 x 18.08
0.517 x 26.3
L 6 -- q
44.30
13.60
&
Lq-L
2.016 x 18.08
0.483 x 2.63
52.*62
-26
0.7 322 250)
69.48x
3220 500 :
57.9 x
69.48
8.28
77.76
28
36.45
12.70
49.15
57.90
Vertical Comp.
Stress
0.8(1 250)
I. 10 500
where L : loaded
length
62.86
49.15 x 6
O. 580 250)
87.21 x 5800
500
57.21
6.00
63.21
Computations for Live Stresses:
Bar
Un. Ld. over
Conc.Ld. at
Un.Ld.
Conc.Ld.
L6 U 5
L6 L7
Lo-L5 & L 1 4 -L2 8
L5
Lo - L6 & L14
-2.341 x 18.08
-0.444 x 26.3
42.33
11.68
Vert. Comp*
.. -
-54.01 x0.6610 250) 500
-62.86 x 6100
Un.Ld.over
Conc.Ld. at
=
_
62.86
62.86765
6
-
6.55
76.58 x
0(640
6400
250)
500
:
-
-69.41
Lo-L 7
51.87
13.92
76.58
7.90
84.48
L8 U9
L8U7
& L1 4
-
Lo -L8 & L1 4- L
28
L8
L2 8
L7
Un. Ld.
Conc.Ld.
Vert. Comp.
-3.479 x 18.08
-0.610 x 26.3
Stress
-78.94 x54':
Impact
-91.89 x
- 62.90
- 16.04
4.164 x 18.08
0.685 x 26.3
:
78.94
(675
Total
-
65.79
x
Total
Bar
2.869 x 18.08
0.528 x 26.3
L2 8
54.01
62.86
Stress
Impact
-
-
91.89
93.31
0.8(710
9.37
-101.*16
:
93.31 x354
250)0.8
6750 ~500
75.29
18.02
108.51 J
7100
108.51
250)
500
10.95
119.46
Computations for Live Stresses:
BAR
Un. Ld. over
Conc.Ld. at
Un. Ld.
Cone. Ld.
L
10
9
-I10 ul11
LO-L9 & L14- L28
Lg
-4.920 x 18.08
-0.756 x 26.3
Lo - L10 & L14
L10
: -88.85
: -19.88
62.86
-108.83 x 54
0.8(740 250)
-126.68 x 7400 500 -
Impact
Total
.12
BAR
Un. *Ld over
Conc.Ld. at
Un. Ld.
Cone. Ld.
Vert. Comp.
Stress
Impact
Total
103.76
21,54
5.739 x 18.08
0.819 x 26.3
125.30
-108.83
Vert. Comp.
Stress
L 28
L
62.86
-126.68
- 12.67
-139.35
125.3 x -54
0.8(770 250)
500
145.85x 7700
U
ll
L 12
Lo - L14
Li1
-5.50 x 18.08
-0.925x 26.3
n
115.85
a
14.59
160.44
-
117.52
26.72
13
Lo - L14
L12
-
99.44
24.33
6.50 x 18.08
1.016x 2.63
..123.77
62.86
-144.07
-123.77 x*54
0.8(450 250)
-144.07 x 4500 50 : - 16.14
-160.21
144.24_
144.24 x
73.42
66-
160.39 x .112
160.39
17.96
178.35
f
~fl~i[ii~i
Computations for Live Stresses:
L. U
14 13
Bar
Un. Ld. over
Cone. Ld. at
Un. Ld.
Conc.Ld.
w 1
Lo - L2 8
L14
-8.508 x 18.08
-0.962 x 26.3
ST
L,~
Vert. Comp
X73.42
.0890
-199.18 x 9000
Impact
& T
l
Lo-L 1 4
L
=-153.82
=- 25.30
5.498 x 18.08
0.909 x 26.3
=
99A4
2391
123.35
123.35 x
.
116.69
z-179.12
-199.18
-179.12
Stress
L
250) -500
Total
19.30
116.7
x
0.8(450 x 250)
4500 500
-218.48
Bar
Un. Ld. over
Cone. Ld. at
Un. Ld.
Cone. Ld.
St
at
Vert. Comp
L4 LS5
& L5 L6
Lo - L1 4
- 243.79
57.60
:
17.490 x 18.08
2.78 x 26.3
g 300.57
Stress
32.2
300.6 x ~9
197.34
Impact
197.34x .112
22.10
Total
129.76
L2 L3 &L 3 L 4
Lo - L1 4
L,3
13.494 x 19.08
2.19 x 26.3
13.07
219.44
316.22
73.11
389.33
:52.2220
389.33 x ~5232.04
232.04
x
.112
25.99
258.03
pu1rrr~
Computations for Live Stresses
Bar
L L7 & L
Un.Ld.over
Conc.Ld.eat
Un. Ld.
Gone. Ld.
RL9 & LgL 1
Lo-L1g
L
Lo-L 14
L9
+17.486x18.08 a + 316.15
+ 2.73 x26.3 a + 71.80
+13.48 x 18.08
+ 2.196 X 26.3
-
+ 387.95
SI
+301.50
m + 231.25
Stress
+3887
Impact
+231.25x0.112 :
+
+179.69
+3015x
25.90
+179.7
=
x.J2
+ 257.15
Total
Bar
L 10L
Un.Ld over
Cone.Ld. at
In. Ld.
Conc.Ld.
L
2
L
L20
-10.967 x 18.08 = -198.28
- 1.243 x 26.3
: - 32.69
S'
L2 0
L
& LZL14
Lo-L 1
& L
14
L2
-19.-995 x 18.08
- 1.469 x 26.3
-230.97
-230.97
X
S2.2
54
= -137.68
-400.2
x
Impact
-137.7
x .112
: - 15.42
-195.3
x
-153.10
-
L 28
-361.60
38.63
-400.23
Stress
Total
+ 20.13
+199.82
L12
L 1 4 -L2 8
+243.75
57.75
32 2
-e',
-195.26
0.8(805+250) - -19.25
80 50 +500
-214.51
-j
Computations for Live Stresses
Reversals:
U 0 U10
Un.Ld.Over
L6 U 5
L4-L28
Conc.Ld.at
L6
-
L.*
LS
L 14
U.5
- L4
L g
Un.Ld.
Conc.Ld.
+9.97x18.08
+1.13x26.3
S'
0 180.26
29.72
+1.842±18.08
0.472x26.3
+33.30 -1.933x18.08
+12.41 -0.448x26.3
-4-11.78
+209.98
+45.?
Stress
32.2
+209.98x-54
.+125.10
+54
Impact
0056
+125.10x S-
+ 14.01
+53.21x(260+250)0.S E
2600+500
-46.7
62.86
+53.21 -46.73x
7.00
3
-34.95
62.86
54
54.39
-54.39x0.8(290+500)2900+500
6. 96
'
Total
+139.01
+60.21
Dead Stress
-61.35
-
11.00
-42.00
1.00
+18.21
-62.35
Reversal
+128.00