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

ExAmpLE 16.8
The W12 × 58 structural steel column shown (see Appendix B for its cross-sectional
properties) is fixed at its base and free at its upper end. At the top of the column, a load
P is applied to a bracket at an eccentricity e = 14 in. from the centroidal axis of the
wide-flange shape. Use the AISC ASD formulas given in Section 16.5, and assume that
E = 29,000 ksi and σ Y = 36 ksi.
P
14 in.
x
(a) Using the allowable-stress method, determine whether the column is safe for a
load P = 25 kips. Report the results in the form of the stress ratio σ x /σ allow .
(b) Determine the magnitude of the largest eccentric load P that may be applied to the
column according to the allowable stress method.
(c) Repeat the analysis, using the interaction method, and determine whether the
column is safe for a load P = 25 kips. Assume that the allowable bending stress is
(σ allow ) b = 24 ksi. Report the value of the interaction equation.
(d) Determine the magnitude of the largest eccentric load P that may be applied to the
column according to the interaction method.
y
11 ft
Plan the Solution
The section properties can be obtained from Appendix B for the W12 × 58 structural z
steel shape. From these properties, the compressive stresses due to the axial force and
the bending moment can be determined for the specified 25 kip load, and the allowable
compression stress can be determined from the AISC ASD formulas. These values, along
with the specified allowable bending stress, can then be used in Equation (16.30) for the
allowable-stress method, and in Equation (16.31) for the interaction method, to determine
whether the column can safely carry P = 25 kips at the specified 14 in. eccentricity. To
determine the largest acceptable eccentric load, the axial and bending stresses are specified
in terms of P and the resulting equations are then solved for the maximum load magnitude.
SolutioN
Section Properties
The following section properties can be obtained from Appendix B for the W12 × 58
structural steel shape:
2 4 4
A = 17.0 in. I = 475 in. r = 5.28 in. I = 107 in. r = 2.51 in.
z z y y
The subscripts for these properties have been revised to correspond to the axes shown. In
addition to the preceding values, the flange width of the W12 × 58 shape is b f = 10.0 in.
Axial Stress Calculation
The 25 kip load will produce compressive normal stress in the column:
P 25 kips
σ axial = = = 1.47 ksi
(a)
2
A 17.0 in.
Bending Stress Calculation
The eccentric axial load P applied at an eccentricity e = 14 in. will produce a bending
moment M y = Pe about the y axis (i.e., the weak axis) of the wide-flange shape. The bending
stress can be calculated from the flexure formula σ bend = M y c/I y , where c is equal to
half the flange width: c = b f /2 = 10.0 in./2 = 5.0 in. For the specified axial load P = 25 kips,
the maximum bending stress magnitude is
Mc y Pec (25 kips)(14in.)(5.0 in.)
σ bend = = = = 16.36 ksi (b)
4
I I
107 in.
y
y
709