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

Plan the Solution
Consider a free-body diagram (FBD) of flange B, and write the equilibrium equation
for the sum of forces in the x direction. This equation will have three unknowns: F 1 ,
F 2 , and P.
Determine the geometry-of-deformation equation and write the force–deformation relationships
for members (1) and (2). Substitute the force–deformation relationships into the
geometry-of-deformation equation to obtain the compatibility equation. Then, use the
allowable stress and area of member (1) to compute a value for P. Repeat this procedure,
using the allowable stress and area of member (2), to compute a second value for P. Choose
the smaller of these two values as the maximum
force P that can be applied to flange B.
SOLUTION
Step 1 — Equilibrium Equations: The
F 1 (1) (2)
F 2
free-body diagram for joint B is shown.
C
A
B
Notice that internal tensile forces are
— P
2
assumed in both member (1) and member
(2) [even though one would expect
to find that member (1) would actually be in compression].
The equilibrium equation for joint B is simply
∑ Fx = F2 − F1
− P = 0
(a)
Step 2 — Geometry of Deformation: Since the compound axial member is attached
to rigid supports at A and C, the overall deformation of the structure must be zero. In
other words,
δ1 + δ2
= 0
(b)
Step 3 — Force–Deformation Relationships: Write generic force–deformation
relationships for the members:
FL 1 1
FL 2 2
δ1
= and δ2
=
(c)
AE 1 1
AE 2 2
Step 4 — compatibility Equation: Substitute Equations (c) into Equation (b) to obtain
the compatibility equation:
FL 1 1 FL 2 2
+ = 0
(d)
AE 1 1 AE 2 2
Step 5 — Solve the Equations: First, we will substitute for F 2 in Equation (a). To
accomplish this, solve Equation (d) for F 2 :
F F L 1 A2
E2
2 =− 1
(e)
L2
A1
E1
Now substitute Equation (e) into Equation (a) to obtain
F L 1 A2
E2
⎡
F F L 1 A2
E2
⎤
− 1 − 1 =− 1 + 1 P
L2
A1
E
⎢
1
⎣ L2
A1
E
⎥
1 ⎦
=
— P
2
109