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

Beam Deflection and Slope at A
The deflection and slope of the beam at A are obtained by setting x = 0 in Equations (e)
and (f). The beam deflection and slope at the free end of the cantilever are
v
A
PL
=−
3EI
and
⎛ dv ⎞ PL
⎝
⎜
⎠
⎟ =
dx 2EI
3 2
A
Ans.
mecmovies
ExAmpLE
m10.2 Derive the equation for the elastic curve, and determine
expressions for the slope and deflection of the beam
at B. Use the double-integration method.
ExAmpLE 10.2
A
v
400
L—
2
B
w 0
Elastic curve
L—
2
1 2w 0 w x 2
x x 0
L
=
2
L
2w
w=
0 x
L
M
a
A
a
— x V
3
w 0 L
x
4
C
x
A simply supported beam is subjected to the linearly distributed
load shown. Determine the equation of the elastic curve.
Also, determine the deflection of the beam at midspan B and
the slope of the beam at support A. Assume that EI is constant
over the entire span of the beam.
Plan the Solution
Generally, two moment equations would be needed to define
the complete variation of M over the entire span. However, in
this case, the beam and loading are symmetrical. On the basis of symmetry, we
need only solve for the elastic curve in the interval 0 ≤ x ≤ L/2. The boundary
conditions for this interval will be found at the pin support A and at midspan B.
SolutioN
Support Reactions
Since the beam is symmetrically supported and symmetrically loaded, the
beam reactions at A and C are identical:
A
y
wL 0
= Cy
=
4
No loads act in the x direction; therefore, A x = 0.
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 direction. The equilibrium equation for the sum of moments about
section a–a is