32 2 General Theory of Influence Lines
Influence Line Forms a Polygon
A dangerous position of the set of moving concentrated loads occurs when one
or more loads stand over vertex of the influence line. Both, the load and the apex
of the influence line over which this load must stand to induce a maximum (or
minimum) of the function under consideration, are called critical. The critical apex
of the influence line must be convex.
In case of Uniformly distributed moving load, the maximum value of the function
Zcorresponds to the location of a distributed loadq, which covers maximum one-
sign area of influence line. The negative and positive portions of influence line must
be considered in order to obtain minimum and maximum of functionZ.
The special case of uniformly distributed moving load happens, if load is dis-
tributed within thefixed length l. In case of triangular influence line, the most
unfavorable location of such load occurs when the portionabDland baseAB
will be parallel (Fig.2.13b).
Example 2.2.Simply supported beam with two overhangs is presented in Fig.2.14.
Determine the most unfavorable position of load, which leads to maximum (posi-
tive and negative) values of the bending moment and shear at sectionk. Calculate
corresponding values of these functions. Consider the following loads: uniformly
distributed loadqand two connected loadsP 1 andP 2 (a twin-axle cart with differ-
ent wheel loads).
Solution.Influence lines for required functions Z are presented in Fig.2.14.
Action of a uniformly distributed loadqD1 : 6 k N=m. Distributed load leads to
maximum value of the function if the area of influence lines within the distributed
load is maximum. For example, the positive shear at the sectionkis peaked if
loadqcovers all portions of influence line with positive ordinates; for minimum
shear in the same section the loadqmust be applied within portions with negative
ordinates.
Qk.maxC/D1:61
2.0:3 3 C0:44/D 2 kNIQk.max/D1:61
2.0:6 6 C0:33/D3:6kN;Mk.maxC/D1:61
2 10 2:4D19:2kN mIMk.max/D1:61
2.1:2 3 C1:83/D7:2kN m:Positive value ofMkmaxmeans that if load is located betweenAB, the tensile
fibers of the beam at sectionkare located below longitudinal axis of the beam.
If load is located within the overhangs, then bending moment at sectionkis nega-
tive, i.e., the tensile fibers at sectionkare located above the longitudinal axis of the
beam.