Structural Mechanics II
Lecture Note 3C: Maximum influence at a
point due to series of concentrated loads
Dr. Hasitha D. Hidallana-Gamage
1
Maximum Influence at a Point due to a Series
of Concentrated Loads
• As we discussed before, once the influence line of a function
is established for a point in a structure, the maximum effect
caused by a live concentrated force is determined by
multiplying the peak ordinate of the influence line by the
magnitude of the force.
• However, when several concentrated moving forces are
placed on a structure, it is importance to determine the
maximum effect
Trail-and error approach
0.3125
6
9
12
3
-0.625
• The influence line for the force in member BG is shown in
the following figure. Trial and error approach is used to
find the greatest negative (compressive) force in BG,
• (1) 1.5kN Load at Point C
FBG  1.5(  0.625 )  4(0)  2( 0.3125 / 3)(1)
  0.729 kN
• (2) 4 kN Load at Point C. By inspection this would be a
better case than the Case 1
  0.625 
FBG  4( 0.625 )  1.5
 4  2(0.3125 )
6 

 2.50 kN
• (3) 2 kN Load at Point C. All the moving loads create a
compressive force in BC.
  0.625 
  0.625 
FBG  2( 0.625 )  4
3  1.5
1
6 
6 


 2.66 kN
• Therefore, final case results in the maximum compressive
load.
Absolute Maximum Shear
• For a cantilevered beam, the absolute maximum shear will
occur at a point located just next to the fixed support. The
maximum shear can be found by the method of sections, with
the loads positioned anywhere on the span.
• For simply supported beams, the absolute maximum shear
will occur just next to one supports. In this case, the loads are
positioned so that the first one in sequence is placed close to
the support.
Absolute Maximum Moment
• For a cantilevered beam, the absolute maximum moment will
occur at a point located just next to the fixed support.
However, in this case, the concentrated loads must be
positioned at the far end of the beam.
• For a simply supported beam, the critical position of the
loads and the associated absolute maximum moment
cannot, in general, be determined by inspection. However,
we can determine the position analytically.
• For this purpose, consider a beam subjected to the forces
F1, F2 and F3 and position the loads on the beam by the
distance x measured from F2 to the beam’s centerline.
d1
d2
x
x1
L/2
L/2
d1
d2
d1
x
F1
M2
x1
L/2
L/2
RA
RB
MB  0
RA 
1
L

FR    x1  x 
L
2

V2
L/2-x
RA
M  0
L

M 2  R A   x   F1 d 1
2

M2 
M2 
1
L
 L

FR    x1  x    x   F1 d 1
L
2
 2

FR L
4

FR x1
2

FR x 2
L

FR xx1
L
 F1 d 1
• For the maximum M2, we require
dM 2
dx
x

 2 FR x
L

FR x1
L
0
x1
2
• Hence, we may conclude that the 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 centreline.
Thank You
11