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

(b) Magnitude of the largest eccentric load P: The axial and bending stresses in the allowable-stress-method
equation can be expressed in terms of an unknown P:
P Pec ⎡ 1 ec⎤
σ x = + = P⎢
+ ⎥
A I y ⎣ A I y ⎦
(e)
The largest load magnitude can be calculated by setting Equation (e) equal to the allowable
compression stress from Equation (c) and solving for P:
σ
x
⎡ 1 ec⎤
= σ = = ⎢ +
⎣ ⎦
⎥ = ⎡ 1
⎣⎢ + (14in.)(5.0 in.) ⎤
allow ⎦⎥ =
−2
12.04 ksi P P
P[0.71303 in. ]
A I y 17.0 in.
2 107 in.
4
∴ P = 16.89 kips
Ans.
(c) Is the column safe for P = 25 kips, according to the interaction method? In the interaction
method, the axial stress is divided by the allowable compression stress, the bending
stress is divided by the allowable bending stress, and the sum of the resulting two terms
must not exceed 1:
PA /
( σ )
allow
a
McI y / y
+ = 1
( σ )
allow
b
The axial and bending stresses were computed in Equations (a) and (b). The allowable
compression stress (σ allow ) a was computed in Equation (c), and the allowable bending
stress is specified as (σ allow ) b = 24 ksi. With these values, the interaction equation for the
eccentrically loaded W12 × 58 column is
1.47 ksi 16.36 ksi
+ = 0.1221 + 0.6817 = 0.8038 < 1 O.K. Ans.
12.04 ksi 24 ksi
Since the value of the interaction equation is less than 1, the column is safe for a load of
P = 25 kips, according to the interaction method.
(d) Magnitude of the largest eccentric load P: The sum of the compressive axial and bending
stresses for the eccentrically loaded W12 × 58 column can be expressed in terms of an
unknown P:
P Pec ⎡ 1 ec
+ = P⎢
+
A( σallow) I ( σallow)
⎣ A ( σallow) I ( σallow)
a y b a y b
Equation (f) can be solved for the largest load magnitude P:
⎤
⎥ = 1
(f)
⎦
⎡ 1 ec ⎤
⎢ +
⎣ σ σ ⎦
⎥ = ⎡ 1
⎢
⎣
+ (14in.)(5.0 in.) ⎤
−1
P P
⎥ = P[0.032144 kip ]
A
2 4
( allow) a I y( allow)
b (17.0 in. )(12.04 ksi) (107 in. )(24 ksi) ⎦
∴ P = 31.1 kips
Ans.
711