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

414
beam deflections
integration gives EIv(x), and the resulting constant has the value C 2 = EIv(0). For some
beams, the slope or deflection or both may be known at x = 0, making it effortless to determine
either C 1 or C 2 . More typically, boundary conditions such as pin supports, roller supports,
and fixed supports occur at locations other than x = 0. For such beams, it will be necessary
to use two beam boundary conditions to develop equations containing the unknown
constants C 1 and C 2 . These equations are then solved simultaneously for C 1 and C 2 .
The application of discontinuity functions to compute beam slopes and deflections is
illustrated in Examples 10.6–10.8.
ExAmpLE 10.6
v
25 kN
A B C
D
v
25 kN
2 m 2.5 m 2.5 m
A B C
D
2 m 2.5 m 2.5 m
B y
D y
x
x
For the beam shown, use discontinuity functions to compute the
deflection of the beam
(a) at A.
(b) at C.
Assume a constant value of EI = 17 × 10 3 kN · m 2 for the beam.
Plan the Solution
Determine the reactions at the simple supports B and D. Using
Table 7.2, write w(x) expressions for the 25 kN concentrated
load as well as the two support reactions. Integrate w(x) four
times to determine equations for the beam slope and deflection.
Use the boundary conditions known at the simple supports to
evaluate the constants of integration.
SolutioN
Support Reactions
An FBD of the beam is shown. On the basis of this FBD, the
beam reaction forces can be computed as follows:
Σ M = (25 kN)(2 m) + D (5 m) = 0
B
∴ D
y
y
= −10 kN
Σ F = B + D − 25 kN = 0
y y y
∴ B = 35 kN
y
Discontinuity Expressions
25 kN concentrated load: Use case 2 of Table 7.2 to write the following expression for the
25 kN concentrated load:
−
wx ( ) =−25 kN x − 0m 1
Reaction forces B y and D y : The upward reaction forces at B and D are also expressed by
using case 2 of Table 7.2:
−1 −1
wx ( ) = 35 kN x − 2m −10 kN x − 7m
Note that the term for the reaction force D y will always have a value of zero in this example,
since the beam is only 7 m long; therefore, this term may be omitted here.
Integrate the beam load expression: Integrate the load also expression,
−1 −1
wx ( ) =−25 kN x − 0m + 35 kN x − 2m