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

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bEAM dEFLECTIONS
5. Boundary and continuity conditions: List the boundary conditions that are
applicable to the bending-moment equation. If the analysis involves two or more
beam segments, list the continuity conditions also. Remember that two conditions
are required in order to evaluate the two constants of integration produced in each
beam segment.
6. Evaluate constants: Use the boundary and continuity conditions to evaluate all
constants of integration.
7. Elastic curve and slope equations: Replace the constants of integration arrived
at in step 4 with the values obtained from the boundary and continuity conditions found
in step 6. Check the resulting equations for dimensional homogeneity.
8. Deflections and slopes at specific points: Calculate the deflection at specific
points when required.
The following examples illustrate the use of the double-integration method for calculating
beam deflections:
ExAMPLE 10.1
v
P
The cantilever beam shown is subjected to a concentrated
load P at its free end. Determine the equation of the elastic
curve, as well as the deflection and slope of the beam
at A. Assume that EI is constant for the beam.
v A
A
θ
A
Elastic curve
L
B
x
Plan the Solution
Consider a free-body diagram that cuts through the beam at
a distance x from the free end of the cantilever. 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. Use the boundary conditions known at the
fixed end of the cantilever to evaluate the constants of
integration.
v
P
SolutioN
Equilibrium
Cut through the beam at an arbitrary distance x from the origin, and draw a free-body
diagram, taking care to show the internal moment M acting in the positive sense. The
equilibrium equation for the sum of moments about section a–a is
a
M
Σ M = Px + M = 0
a−a
A
x
a
V
Therefore, the bending-moment equation for this beam is simply
M =−Px (a)
Notice that moment equation (a) is valid for all values of x for this particular beam.
In other words, Equation (a) is valid in the interval 0 ≤ x ≤ L. Substitute the expression for
M into Equation (10.1) to obtain
EI d 2
v
2
dx
=−Px
(b)