Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

11.4 Use of Discontinuity Functions for Statically
Indeterminate Beams
The use of discontinuity functions for statically determinate beam analysis has been
discussed in Chapters 7 and 10. In Section 7.4, discontinuity functions were used to derive
functions expressing the shear force and bending moment in beams. Beam deflections for
statically determinate beams were computed with discontinuity functions in Section 10.6.
In both sections, the reaction forces and moments were computed beforehand from equilibrium
considerations, making it possible to incorporate known values into the load function
w(x) from the outset of the calculation process. The added difficulty posed by statically
indeterminate beams is that the reactions cannot be determined from equilibrium considerations
alone and, thus, known values for the reaction forces and moments cannot be
included in w(x).
For statically indeterminate beams, reaction forces and moments are initially expressed
as unknown quantities in the load function w(x). The integration process then proceeds in
the manner described in Section 10.6, producing two constants of integration, C 1 and C 2 .
These integration constants, as well as the unknown beam reactions, must be computed in
order to complete the elastic curve equation. Equations containing the constants C 1 and C 2
can be derived from the boundary conditions and, along with the beam equilibrium equations,
are then solved simultaneously to evaluate C 1 and C 2 as well as to determine the
beam reaction forces and moments. The solution process is demonstrated in Examples 11.3
and 11.4.
ExAmpLE 11.3
v
A B C D
454
2 m
80 kN/m
5 m
3 m
x
For the statically indeterminate beam shown, use discontinuity
functions to determine
(a) the force and moment reactions at A and D.
(b) the deflection of the beam at C.
Assume a constant value of EI = 120,000 kN · m 2 for the beam.
Plan the Solution
The beam is statically indeterminate; therefore, the reaction
forces at A and D cannot be determined solely from equilibrium
considerations. From an FBD of the beam, two nontrivial
equilibrium equations can be derived. However, since the beam is statically indeterminate,
the reaction forces and moments can be stated only as unknowns. The distributed
load on the beam, as well as the unknown reactions, will be expressed by discontinuity
functions. The load function will be integrated twice to obtain the bending-moment
function for the beam. In these first two integrations, constants of integration will not be
necessary. The bending-moment function will then be integrated twice to obtain the
elastic curve equation. Constants of integration must be considered in these two integrations.
The three boundary conditions known at A and D, along with the two nontrivial
equilibrium equations, will produce five equations that can be solved simultaneously to
determine the three unknown reactions and the two constants of integration. After these
quantities are found, the beam deflection at any location can be computed from the
elastic curve equation.