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

both members will be stressed to their allowable limit. It is more probable that one member
will control the design. Using the equilibrium results and the allowable member
forces, we can determine the controlling member and, in turn, compute the maximum
load P.
SOLUTION
Equilibrium
The free-body diagram (FBD) for the truss joint is shown. From the FBD, two equilibrium
equations in terms of three unknowns—F 1 , F 2 , and P—can be written. Note: We will
assume that internal member forces F 1 and F 2 are tension forces (even though we may
expect member (2) to be in compression). We have, then,
∑ F = − F cos45° + F cos60° + P = 0
(a)
x 1 2
∑ F = − F sin45° − F sin60° = 0
(b)
y 1 2
y
45° 60° P
F 1 F 2
(1) (2)
From these two equations, expressions for the unknown load P can be derived in terms of
member forces F 1 and F 2 :
⎡ sin45°
P = cos45° +
F
sin60 cos60 ⎤
⎢
° ⎥ 1
(c)
⎣
° ⎦
⎡sin60°
P =−
F
sin45 cos45 cos60 ⎤
⎢
°+ ° ⎥ 2
(d)
⎣ °
⎦
Allowable stress: The allowable normal stress in the steel members can be computed from
Equation (4.1):
σ
σ Y 36 ksi
= = = 24 ksi
FS 1.5
allow (e)
Allowable member force: The allowable stress can be used to calculate the allowable force
in each member:
2
F = σ A = (24 ksi)(7.22 in. ) = 173.28 kips
(f)
1,allow allow 1
2
F = σ A = (24 ksi)(3.88 in. ) = 93.12 kips
(g)
2,allow allow 2
Problem-Solving Tip: A common mistake at this point in the solution would be to
compute P by substituting the two allowable forces into Equation (a). This approach,
however, does not work, because equilibrium will not be satisfied in Equation (b).
Equilibrium must always be satisfied.
Compute maximum P: Next, two possibilities must be investigated: Either member (1) controls,
or member (2) controls. First, assume that the allowable force in member (1) controls
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