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

P10.11 For the beam and loading shown in Figure P10.11, use
the double-integration method to determine
(a) the equation of the elastic curve for the beam.
(b) the deflection at B.
Assume that EI is constant for the beam.
P10.12 For the beam and loading shown in Figure P10.12, use
the double-integration method to determine
(a) the equation of the elastic curve for the beam.
(b) the location of the maximum deflection.
(c) the maximum beam deflection.
Assume that EI is constant for the beam.
v
w
v
w 0
A
B
C
x
A
B
x
FIGUrE P10.11
L—
2
L—
2
FIGUrE P10.12
L
10.5 Determining Deflections by Integration
of Shear-Force or Load Equations
In Section 10.3, the equation of the elastic curve was obtained by integrating the differential
equation
EI d 2
v = M
(10.1)
2
dx
and applying the appropriate bounding conditions to evaluate the two constants of integration.
In a similar manner, the equation of the elastic curve can be obtained from shear-force
or load equations. The differential equations that respectively relate the deflection v to the
shear force V and load w are thus
EI dv 3
3
dx
= V
(10.2)
and
EI d 4
v = w
(10.3)
4
dx
where both V and w are functions of x. When Equation (10.2) or Equation (10.3) is used to
obtain the equation of the elastic curve, either three or four integrations will be required
instead of the two integrations required with Equation (10.1). These additional integrations
will introduce additional constants of integration. The boundary conditions, however, now
include conditions on the shear forces and bending moments, in addition to the conditions
on slopes and deflections. The selection of a particular differential equation is usually
based on mathematical convenience or personal preference. In those instances when the
expression for the load is easier to write than the expression for the moment, Equation (10.3)
would be preferred over Equation (10.1). The following example illustrates the use of
Equation (10.3) for calculating beam deflections:
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