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

The only unknown term in this equation is the force in the tie rod—that is, B y . Accordingly,
solve the equation for B y , to obtain
3
4
B ⎡ L L1
⎤
⎢ + ⎥
⎣ EI AE ⎦
= wL
y
3 1 1 8EI
(f)
Before beginning the calculation, pay special attention to the terms L 1 , A 1 , and E 1 . These
are properties of the tie rod—not the beam. A common mistake in this type of problem is
using the beam elastic modulus E for both the beam and the rod.
Now calculate the reaction force applied to the beam by the tie rod, using the following
values:
Beam Properties
w = 40 kN/m = 40 N/mm
L = 6 m = 6,000 mm
tie-Rod Properties
L 1 = 9 m = 9,000 mm
d 1 = 30 mm
v
B
I = 300 × 10 6 mm 4 A 1 = 706.858 mm 2
E = 200 GPa = 200,000 N/mm 2 E 1 = 70 GPa = 70,000 N/mm 2
Substitute these values into Equation (f), and compute B y = 78,153.8 N = 78.2 kN. Therefore,
the internal axial force in the tie rod is 78.2 kN (T).
Ans.
The deflection of the beam at B can be calculated from Equation (e) as
BL y 1 (78,153.8 N)(9,000 mm)
=− =− =− 14.22 mm = 14.22 mm ↓ Ans.
2 2
AE (706.858 mm )(70,000 N/mm )
1 1
ExAmpLE 11.9
A
C
(1)
1,500 lb/ft
D
(2)
5 ft
5 ft
E
B
A 24 ft long W12 × 30 steel beam is supported at its ends by simple pin
and roller supports and at midspan by a wooden beam, as shown in the
accompanying figure. Steel [E = 29 × 10 6 psi] beam (1) supports a
uniformly distributed load of 1,500 lb/ft. Wooden [E = 1.8 × 10 6 psi]
beam (2) spans 10 ft between simple supports C and E. The steel beam
rests on top of the wooden beam at the middle of the 10 ft span. The
wooden beam has a cross section that is 6 in. wide and 10 in. deep.
Determine
(a) the reaction force at point B applied by the wooden beam to the
steel beam.
(b) the deflection of point D.
Plan the Solution
The wooden beam acts as an elastic support for the steel beam. Thus,
the final deflection of the system will be determined by how much the
wooden beam deflects downward in response to the force exerted on it by the steel beam.
Begin by considering the steel beam. Remove the reaction force provided by the wooden
beam, so that the released beam is a simply supported span with a uniformly distributed
load. Determine an expression for the wooden beam’s downward deflection. Next, consider
the released beam with only the unknown upward reaction force at point D provided by the
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