10.4 Kinematical Method for Construction of Influence Lines 363
is located within portions 2-3 and 3-4, then ordinates of influence line are negative,
i.e., the extended fibers at the support 3 is located above the longitudinal axis of
the beam.
Influence line for bending momentMkin the span. In this case we need to include
the hinge at the sectionk, to apply two positive couplesMkand show the elastic
curve; for this curve a mutual angle of rotation at sectionkis equal to unity (see
Sect. 10.1.1). The positive ordinates of influence line show that extended fibers at
the sectionkare located under of neutral axis of the beam.
For each span we can determine two specific points is called the left and right
foci points. They are labeled asFleftandFright. Figure10.19presents these points
for span 3-4 only. Formulas for computation of locationFLandFRare presented
in Appendix, Section A. Foci Points.
Influence lines for bending moment at the span may be of three different shapes
depending on where section is located, i.e., between two foci points, between sup-
port and focus, and for section which coincides with focus.
If the sectionkis located between two fociFLandFR, then ordinates of influ-
ence line within corresponding span are positive. If a sectionnis located between
left support and focusFL, then ordinates of influence line within the corresponding
span are positive and negative (influence line forMn). The same conclusion will
be done if the section is located betweenFRand right support. If the section under
consideration coincides withFRthen bending momentdoes not arisein the section
FR, when loadPis located in the first and second spans (influence line forMFR).
Therefore, for construction of the model of influence line for bending moment at the
any section within the span it is necessary first of all, to find location of foci points
for given span and then to define which case takes place.
Influence line for shear forceQk. In this case we need to eliminate the constraint,
which corresponds to the shear force at the sectionkand apply two positive shears
Qk. Elastic curve in the new system due to forcesQkD 1 is a model of influence
line forQk. Shape of influence line for shear forces at the sections, which are in-
finitely closed to the support, may be obtained as limiting cases, when the section is
located within the span.
It is obvious that the construction of influence lines for statically determi-
nate multispan beams (Ch. 3) using interaction scheme reflects the M ̈uller–Breslau
principle.
Summary
The purpose of influence lines and their application for statically determinate struc-
tures in case of the fixed and moving load have been discussed in Chaps. 2–4. Of
course, this remains also for statically indeterminate structures. However, in case of
the statically indeterminate structures, the importance of influence lines and their
convenience are sharply increased. Although the construction of influence lines for
statically indeterminate structures is not as simple as for statically determinate ones,
the consumption of a time is paid off by their advantages. Additional and very