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

H
y
F = 30 kips
e = 8 in. 30 kips
M = 240 kip·in.
K
x
the structural member. Consequently, the concentrated load is located 8 in. to the right
of the centroidal axis of the structural member. The distance between the line of action
of the load and the centroidal axis of the member is commonly termed the eccentricity e.
In this instance, the load is said to be located at an eccentricity e = 8 in.
Since its line of action does not coincide with the centroidal axis of the
structural member, the 30 kip load produces bending in addition to axial compression.
The equivalent force at section a–a is simply equal to the 30 kip load.
The moment at section a–a that is required for equivalence is equal to the product
of the load and the eccentricity e. Therefore, an internal force F = 30 kips
and an internal bending moment M = F × e = (30 kips)(8 in.) = 240 kip · in. acting
at the centroid of section a–a are together equivalent to the 30 kip load applied
to the top of the structural member.
z
Section Properties
The centroid location is known from symmetry. The area of the cross section is
A = (10 in.)(6 in.) = 60 in. 2 . The bending moment M = 240 kip · in. acts about the
z axis; therefore, the moment of inertia about the z axis must be determined in order to
calculate the bending stresses:
I
z
3
(6 in.)(10in.)
= = 500 in.
12
4
Axial Stress
On section a–a, the internal force F = 30 kips (which acts along the y centroidal axis)
produces a normal stress of
F 30 kips
σ axial = = = 0.5 ksi(C)
2
A 60 in.
which acts vertically (i.e., in the y direction). The axial stress is a compressive normal
stress that is uniformly distributed over the entire section.
1.9 ksi (T)
Combined
stress
2.4 ksi (T)
284
Bending
stress
Axial
stress
F = 30 kips
H
2.9 ksi (C)
2.4 ksi (C)
0.5 ksi (C)
M = 240 kip·in.
K
Bending Stress
The magnitude of the maximum bending stress on section a–a can be determined
from the flexure formula:
Mc (240 kip⋅in.)(5in.)
σ bend = =
= 2.4 ksi
I 500 in.
4
z
The bending stress acts in the vertical direction (i.e., in the y direction) and increases
linearly with distance from the axis of bending. In the coordinate system defined for this
problem, distance from the axis of bending is measured in the x direction from the z axis.
The sense of the bending stress (either tension or compression) can be readily
determined by inspection, based on the direction of the internal bending moment M.
In this instance, M causes compressive bending stresses on the K side of the structural
member and tensile bending stresses on the H side.
Combined Normal Stresses
Since the axial and bending stresses are normal stresses that act in the same direction
(i.e., the y direction), they can be directly added to give the combined stresses acting on
section a–a. The combined normal stress on side H of the structural member is
σH = σaxial + σ bend =− 0.5 ksi + 2.4 ksi = + 1.9 ksi = 1.9 ksi(T) Ans.