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

11.5 The Superposition method
461
THE SuPERPOSITION METHOd
The concepts of redundant reactions and a released beam were introduced in Section 11.2.
These notions can be combined with the principle of superposition to create a very powerful
method for determining the support reactions of statically indeterminate beams. The general
approach can be outlined as follows:
1. Redundant support reactions acting on the statically indeterminate beam are identified.
2. The selected redundant is removed from the structure, leaving a released beam that is
stable and statically determinate.
3. The released beam subjected to the applied load is considered. The deflection or
rotation (depending on the nature of the redundant) of the beam at the location of the
redundant is determined.
4. Next, the released beam (without the applied load) is subjected to one of the redundant
reactions and the deflection or the rotation of this beam-and-loading combination is
determined at the location of the redundant. If more than one redundant exists, this
step is repeated for each redundant.
5. By the principle of superposition, the actual loaded beam is equivalent to the sum of
these individual cases.
6. To solve for the redundants, geometry-of-deformation equations are written for each
of the locations where redundants act. The magnitude of the redundant can be obtained
from the particular deformation equation.
7. Once the redundants are known, the other beam reactions can be determined from the
equilibrium equations.
To clarify this approach, consider the propped cantilever beam shown in Figure 11.6a. The
free-body diagram for this beam (Figure 11.6b) shows four unknown reactions. Three
equilibrium equations can be written for the beam (ΣF x = 0, ΣF y = 0, and ΣM = 0);
therefore, the beam is statically indeterminate to the first degree. Consequently, one
additional equation must be developed in order to compute the reactions for the
propped cantilever.
v
P
v
P
A
x
B
M A
A x A
— L 2
x
B
L—
2
L—
2
A y
L—
2
B y
(a) actual beam
(b) Free-body diagram
v
P
v
A
L—
2
L—
2
(c) Released beam subjected to actual loads
FIGURE 11.6 Superposition method applied to a propped cantilever beam.
B
v B
x
A
L
(d) Released beam subjected to redundant B y
B
B y
v B ′
x