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

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THE INTEgRATION METHOd
A
B
C
A
C
A
B
C
A y B y C y
A y
C y
A y
B y
FIGURE 11.5a Continuous
beam on three supports.
FIGURE 11.5b Released
beam created by removing
redundant B y .
FIGURE 11.5c Released
beam created by removing
redundant C y .
In the sections that follow, three methods for analyzing statically indeterminate beams
will be discussed. In each case, the initial objective will be to determine the magnitude of
the redundant reactions. After the redundants have been determined, the remaining reactions
can be determined from equilibrium equations. After all of the reactions are known,
the beam shear forces, bending moments, bending and shear stresses, and transverse
deflections can be determined by the methods presented in Chapters 7 through 10.
11.3 The Integration method
For statically determinate beams, known slopes and deflections were used to obtain boundary
and continuity conditions, from which the constants of integration in the elastic curve
equation could be evaluated. For statically indeterminate beams, the procedure is identical.
However, the bending-moment equations derived at the outset of the procedure will contain
reactions (or loads) that cannot be evaluated with the available equations of equilibrium.
One additional boundary condition will be needed for the evaluation of each such unknown.
For example, consider a transversely loaded beam with four unknown reactions that are to
be solved simultaneously by the double-integration method. Two constants of integration
will appear as the bending-moment equation is integrated twice; consequently, this statically
indeterminate beam has six unknowns. Since a transversely loaded beam has only two
nontrivial equilibrium equations, four additional equations must be derived from boundary
(or continuity) conditions. Two boundary (or continuity) conditions will be required in order
to solve for the constants of integration, and two extra boundary (or continuity) conditions
will be needed in order to solve for two of the unknown reactions. Examples 11.1 and 11.2
illustrate the method.
ExAmpLE 11.1
A propped cantilever beam is loaded and supported as shown.
Assume that EI is constant for the beam. Determine the reactions
at supports A and B.
v
w 0
Plan the Solution
First, draw a free-body diagram (FBD) of the entire beam and A
write three equilibrium equations in terms of the four unknown
reactions A x , A y , B y , and M A . Next, consider an FBD that cuts
through the beam at a distance x from the origin. Write an equilibrium equation for the sum
of moments, and from this equation, determine the equation for the bending moment M as
it varies with x. Substitute M into Equation (10.1) and integrate twice, producing two
constants of integration. At this point in the solution, there will be six unknowns, which
L
B
x