Problems 195
Group 2This group presents methods, which are based on the concept of the strain
energy. The following precise analytical methods are presented in the second group:
strain energy method (Clapeyron and Castigliano theorems), Maxwell–Mohr inte-
gral (dummy load method), and Vereshchagin rule (graph multiplication method).
All methods of this group use a concept of generalized force and corresponding
generalized coordinate.
Work–strain energy method allows calculating displacement at specified points.
Even if a numerical procedure of this method is very simple, the area of applica-
tion of this method is limited.
Castigliano theorem allowscalculating any displacement at specified direction
as a partial derivative of the strain energy with respect to generalized force. This
theorem has a fundamental character.
Maxwell–Mohr integral presents the principal formula for computation of dis-
placement at any specified direction. This formula is presented in terms of
internal forces caused by a given load (orchange of temperature) and unit gen-
eralized force, which corresponds to required generalized coordinate. In general,
application of this method reduced to integration procedure.
The graph multiplication method is the modification of the Maxwell–Mohr
integral and presents extremely convenient procedure for computation of dis-
placements at specified points for bending structures. This approach allows us to
avoid the integration procedure and requires plotting only two bending moment
diagrams due to given loads and unit load. After that simple algebraic procedures
over them should be performed. In fact, this is the most effective method for cal-
culation of displacement of any nonuniformbeams and frames. For trusses, the
graph multiplication method coincides with Maxwell–Mohr integral.
The graph multiplication method is so much simple and effective that it is hard to
expect that new methods for calculation of displacements of elastic structures may
be developed.
Elastic load method presents the combination of conjugate beam method and
Maxwell–Mohr integral. This method allows calculating displacements at theset
of pointssimultaneously. The method is especially effective for computation of
displacements of the joints of a truss.
Problems
Problems 6.1 through 6.8 should be solved by initial parameter method.The flexural
rigidity,EI, is constant for each beam; the span of the beam isl.
6.1.The cantilevered beam is subjected to uniformly varying load as shown in
Fig.P6.1. The maximum ordinate of load isq. Derive an equation for the elastic
curve of the beam. Determine the vertical displacement and slope at the free end.