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

ExAmpLE 11.4
For the statically indeterminate beam shown, use discontinuity
functions to determine
(a) the force reactions at B, D, and E.
(b) the deflection of the beam at A.
(c) the deflection of the beam at C.
Assume a constant value of EI = 120,000 kN · m 2 for the beam.
v
90 kN/m
A B C
D
Plan the Solution
The beam is statically indeterminate; therefore, the reaction forces cannot be determined
solely from equilibrium considerations. From an FBD of the beam, two nontrivial
equilibrium equations can be derived. However, since the beam is statically indeterminate,
the reaction forces can be stated only as unknowns. The distributed load on the
beam, as well as the unknown reactions, will be expressed by discontinuity functions.
The load function will be integrated twice to obtain the bending-moment function for
the beam. In these first two integrations, constants of integration will not be necessary.
The bending-moment function will then be integrated twice to obtain the elastic curve
equation. Constants of integration must be considered in these two integrations. The
three boundary conditions known at B, D, and E, along with the two nontrivial equilibrium
equations, will produce five equations that can be solved simultaneously to determine
the three unknown reactions and the two constants of integration. After these quantities
are found, the beam deflection at any location can be computed from the elastic
curve equation.
2 m 5 m 5 m 5 m
E
x
SolutioN
(a) Support Reactions
An FBD of the beam is shown. Since no forces act in the x direction, the ΣF x equation will
be omitted here. From the FBD, the beam reaction forces can be expressed by the following
relationships:
Σ F = B + D + E − (90 kN/m)(12m) = 0 ∴ B + D + E = 1, 080 kN (a)
y y y y y y y
Σ M = −B (15m) − D (5m) + (90 kN/m)(12 m)(11m) = 0
E y y
∴ B (15m) + D (5m) = 11,880 kN⋅
m
y
y
(b)
Discontinuity Expressions
Distributed load between A and D: Use case 5 of Table 7.2 to
write the following expression for the distributed load:
v
0 0
wx ( ) =−90 kN/m x − 0m + 90 kN/m x − 12 m
Reaction forces B y , D y , and E y : Since the beam is statically
indeterminate, the reaction forces at B, D, and E can be expressed
only as unknown quantities at this time:
−1 −1 −1
y y y
wx ( ) = B x − 2m + D x − 12 m + E x −17m
90 kN/m
A B C
D
2 m 5 m 5 m 5 m
B y D y E y
E
x
Note that the term for E y will always have the value zero in this example, since the beam
is only 17 m long; therefore, this term may be omitted here.
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