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

wooden beam. Determine an expression for the upward deflection of the simply supported
steel beam due to a concentrated load acting at midspan. Next, consider the wooden beam.
The upward reaction force exerted on the steel beam by the wooden beam causes the wooden
beam to deflect downward. Determine an expression for the downward deflection of the
wooden beam due to this unknown reaction force. Combine the three expressions for the
deflection at D in a compatibility equation, and solve for the reaction force. Once the magnitude
of the reaction force is known, the deflection at point D can be computed.
SolutioN
Case 1—Simply Supported Steel Beam with uniformly
Distributed load
Remove wooden beam (2), and consider simply supported steel
beam (1) subjected to a uniformly distributed load of 1,500 lb/ft.
The deflection of this beam at point D must be determined.
From Appendix C, the deflection of beam (1) at midspan is
given by
v
D
4
5wL1
=-
384EI
1 1
(a)
A
v
1,500 lb/ft
D
12 ft 12 ft
v D
(1)
B
x
Case 2—Simply Supported Steel Beam with
Concentrated load
Wooden beam (2) exerts an upward reaction force on the steel
beam at D. Consider steel beam (1) subjected to this upward
reaction force D y . From Appendix C, the midspan deflection of
a simply supported beam due to a concentrated load applied at
midspan is given by
v
D
3
PL
=- =- - 3
3
1
( Dy)
L1
DL y 1
=
48EI
48EI
48EI
1 1
1 1
1 1
(b)
A
v
v D (1)
D
B
12 ft 12 ft
D y
x
Case 3—Simply Supported Wooden Beam with
Concentrated load
Wooden beam (2) supplies an upward force to the steel beam at
D. In reaction, steel beam (1) exerts a force of equal magnitude
on the wooden beam, causing it to deflect downward. The
downward deflection of beam (2) that is produced by reaction
force D y is given by
C
v
D y
(2)
x
v D
E
D
5 ft 5 ft
v
D
DL y 3
2
=-
48E I
2 2
(c)
Compatibility Equation
The sum of the downward deflection of the steel beam due to the distributed load [Equation
(a)] and the upward deflection produced by the reaction force supplied by the wooden beam
[Equation (b)] must equal the downward deflection of the wooden beam [Equation (c)].
These three equations for the deflection at D are combined in a compatibility equation:
4
3
3
5wL1
DL y 1 DL y 2
- + =-
384EI
48EI
48EI
1 1
1 1
2 2
(d)
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