Advanced Methods of Structural Analysis

2.2 Application of Influence Lines for Fixed and Moving Loads 31

In case of set of concentrated moving loads, we assume that some of loads may
be connected. This case may be applicablefor moving cars, bridge cranes, etc. We
will consider different forms of influence line.

Influence Line Forms a Triangle

A dangerous position occurs when one of the loads is locatedover the vertexof
an influence line; this load is called a critical load. (The term “critical load” for
problems of elastic stability, Chap. 13 , has a different meaning.) The problem is to
determinewhichload among the group of moving loads is critical. After a critical
load is known, all other loads are locatedaccording to the given distances between
them.
The critical load may be easy defined by agraphical approach (Fig. 2.13a). Let
the moving load be a model of two cars, with loadsPion the each axle. All distance
between forces are given.

Step 1.Trace the influence line for functionZ. Plot all forcesP 1 ,P 2 ,P 3 ,P 4 in
order using arbitrary scale from the left-most pointAof influence line; the
last point is denoted asC.
Step 2.Connect the right most pointBwith pointC.
Step 3.On the base line show the pointD, which corresponds to the vertex of in-
fluence line and from this point draw a line, which is parallel to the lineCB
until it intersection with the vertical lineAC.
Step 4.The intersected force (in our caseP 2 ) presents a critical load; unfavorable
location of moving cars presented in Fig.2.13a.
Step 5.Maximum (or minimum) value of relevant function isZD

P

Piyi.

C

B

P 1

P 1

P 2

P 2

P 3

P 3

P 4

P 4

A

D

B

q

A

a b

abl

y (^1) y
2 y^3
y 4
Fig. 2.13Graphical definition of the unfavorable position of load for triangular influence line.
(a) Set of concentrated load. (b) Uniformly distributed load of fixed lengthl