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CED, UET, Taxila 1
Influence Line Diagram -III
Theory of Structures-I
CED, UET, Taxila 2
Contents
1. Maximum Influence at a point due to a series of concentrated loads
2. Absolute Maximum Shear and Moment
CED, UET, Taxila 3
Maximum Influence at a Point Due to a Series of Concentrated Loads
General Case: Develop ILD for a function and then maximum effect is calculated by
Maximum Effect (Point Load)= Magnitude of force x Peak ordinate of ILD
In some cases, several concentrated forces must be placed on structure e.g. truck loading or train loading on a bridge
CED, UET, Taxila 4
Shear
10’ 30’
CA B
x
Vc
-0.25
0.75
10’ 40’
ILD for Shear at Point C
CED, UET, Taxila 5
Case 1
10’ 30’
CA B
1K 4K 4K
5’ 5’
CED, UET, Taxila 6
Case 1
10’ 30’
CA B
1K 4K 4K
x
Vc
-0.25
0.75
10’ 40’15’ 20’
0.625 0.5
(Vc)1 = 1(0.75) + 4(0.625) + 4(0.5) = 5.25 k
CED, UET, Taxila 7
Case 2
10’ 30’
CA B
1K 4K 4K
5’ 5’
CED, UET, Taxila 8
Case 2
10’ 30’
CA B
1K 4K 4K
x
Vc
-0.25
0.75
10’ 40’15’
0.625
(Vc)2 = 1(-0.125) + 4(0. 75) + 4(0.625) = 5.375 k
5’
-0.125
CED, UET, Taxila 9
Case 3
10’ 30’
CA B
1K 4K 4K
5’ 5’
CED, UET, Taxila 10
Case 3
10’ 30’
CA B
1K 4K 4K
x
Vc
-0.25
0.75
10’ 40’15’
(Vc)3 = 1(0) + 4(-0.125) + 4(0.75) = 2.5 k
5’
-0.125
CED, UET, Taxila 11
ComparisonCase 1:
(Vc)1 = 1(0.75) + 4(0.625) + 4(0.5) = 5.25 k
Case 2:
(Vc)1 = 1(-0.125) + 4(0. 75) + 4(0.625) = 5.375 k
Case 3:(Vc)1 = 1(0) + 4(-0.125) + 4(0.75) = 2.5 k
CED, UET, Taxila 12
Method Based on Change in Function
When many concentrated loads act on the span, the trial-and-error computations used above can be tedious.
Critical position of loads can be determined in a more direct manner by finding the change in shear v when loads move from case 1 to case 2 and case 3.
As long as each computed v is positive, the new position will yield a larger shear in the beam at C than the previous position.
CED, UET, Taxila 13
Method Based on Change in Function … Each movement is investigated until a
negative change in shear is computed.
When this occurs, the previous position of the loads will give the critical value.
CED, UET, Taxila 14
Method Based on Change in Function … The change in shear V for a load P that
moves from position x1 to x2 over a beam can be determined by multiplying P by the change in the ordinate of the
influence line, that is, (y2-y1). If the slope of the influence line is s, then (y2-
y1)=s(x2-x1)
V = Ps(x2-x1)
CED, UET, Taxila 15
Method Based on Change in Function … If the load moves past a point where there is
a discontinuity or jump in the influence line, as point c in previous examples, then change in shear is simply
V = P (y2-y1)
CED, UET, Taxila 16
Case 1-2
10’ 30’
CA B
1K 4K 4K
5’ 5’
CED, UET, Taxila 17
Case 1-2
10’ 30’
CA B
1K 4K 4K
x
Vc
-0.25
0.75
10’ 40’15’
0.625
V1-2 = 1(-1) + [1+4+4] (0.025)(5)= +0.125 k
5’
-0.125
S = 0.75/(40-10) = 0.25/10 = 0.025
CED, UET, Taxila 18
Case 2-3
10’ 30’
CA B
1K 4K 4K
5’ 5’
CED, UET, Taxila 19
Case 2-3
10’ 30’
CA B
1K 4K 4K
x
Vc
-0.25
0.75
10’ 40’15’
5’
-0.125
V2-3 = 4(-1) + [1+4+4] (0.025)(5)= -2.875 k
CED, UET, Taxila 20
Moment Use the same method to calculate the critical
position of series of concentrated forces so that they create largest internal moment at a specified position in the structure.
First Draw ILD of moment for the given point and then proceed with the calculations.
CED, UET, Taxila 21
Moment
10’ 30’
CA B
x
Mc
10’ 40’
ILD for Moment at Point C
7.5
CED, UET, Taxila 22
Critical Position of Loads
M = Ps(x2-x1)
M1-2 = -2(7.5/10)(4) + (4+3)(7.5/(40-10))(4) = 1.0 k. ft
M2-3 = -(2+4)(7.5/10)(6) + (3)(7.5/(40-10))(6) = -22.5 k. ft
CED, UET, Taxila 23
Change in MCA B
2K 4K 3K
4’ 6’Case 1
CA B
2K 4K 3K
4’ 6’Case 2
CA B
2K 4K 3K
4’ 6’Case 3
10’ 30’
10’ 30’
10’ 30’
CED, UET, Taxila 24
Change in M
CA B
2K 4K 3K
Case 1
CA B
2K 4K 3K
Case 2
CA B
2K 4K 3K
Case 3
10’ 30’
10’ 30’
10’ 30’
M1-2 = -2(7.5/10)(4) + (4+3)(7.5/(40-10))(4) = 1.0 k. ft
M2-3 = -(2+4)(7.5/10)(6) + (3)(7.5/(40-10))(6) = -22.5 k. ft
CED, UET, Taxila 25
Maximum Moment From the results we can conclude that case 2
will produce the maximum moment.
(Mc) max = 2(4.5) + 4(7.5) + 3(6.0) = 57 k. ft.
x
Mc
10’ 40’
7.5
4.56
6’ 16’
CED, UET, Taxila 26
Absolute Maximum Shear and Moment We developed the methods for computing
maximum shear and moment at a specified point due to series of concentrated moving loads.
Now to determine both the location of the point in the beam and the position of loading on the beam so that one can obtain the absolute maximum shear and moment caused by the loads.
CED, UET, Taxila 27
Shear in Cantilever Beam For a cantilevered beam the absolute
maximum shear will occur at a point located just next to the fixed support. Loads will be positioned closed to the support.
V abs max
CED, UET, Taxila 28
Moment in Cantilever Beam For a cantilevered beam the absolute
maximum moment will occur at a same point where absolute maximum shear occur but the loads will be located at far end of the beam.
M abs max
CED, UET, Taxila 29
Moment in Cantilever Beam
M abs max
CED, UET, Taxila 30
Shear in Simply Supported Beam For simply supported beams the absolute
maximum shear will occur just next to one of the supports. Loads are positioned such that first load is near the support.
A B
V abs max
CED, UET, Taxila 31
Moment at Simply Supported Beam For simply supported beam the critical
position of the loads and the associated absolute maximum moment cannot, in general, be determined by inspection.
We can determine the position analytically.
CED, UET, Taxila 32
Example
A B
CL FR F3F2F1
L/2 L/2
x x’-x
x’d1
d2
Ay By
xxLF
LA
M
Ry
b
'2
1
0
CED, UET, Taxila 33
Example…
A
F1
L/2 - x
d1
V2
M2
11
2
11
112
'2
'4
2'
21
2
0
dFLxxF
LxFxFLF
dFxLxxLFL
dFxLAM
M
RRRR
R
y
b
CED, UET, Taxila 34
Example…
A
F1
L/2 - x
d1
V2
M2
2'
0'22
xx
LxF
LxF
dxdM RR
For Maximum M2 we require
CED, UET, Taxila 35
Conclusion: Simply supported beams Absolute maximum moment in a simply
supported beam occurs under one of the concentrated forces, such that this force is positioned on the beam so that it and the resultant force of the system are equidistant from the beam’s centerline.
CED, UET, Taxila 36