1. Mechanics of Materials – MAE 243 (Section 002) Spring 2008
Dr. Konstantinos A. Sierros

2. Problem 3.3-3
While removing a wheel to change a tire, a driver applies forces P = 25 lb at the ends of two of the arms of a lug wrench (see figure). The wrench is made of steel with shear modulus of elasticity G = 11.4 x (10^6) psi. Each arm of the wrench is 9.0 in. long and has a solid circular cross section of diameter d = 0.5 in.
(a) Determine the maximum shear stress in the arm that is turning the lug nut (arm A).
(b) Determine the angle of twist (in degrees) of this same arm.

3. Problem 5.5-4
A simply supported wood beam AB with span length L = 3.5 m carries a uniform load of intensity q = 6.4 kN/m (see figure).
Calculate the maximum bending stress max due to the load q if the
beam has a rectangular cross section with width b = 140 mm and height
h = 240 mm.

4. 5.8: Shear stresses in beams of rectangular cross-section
Vertical and horizontal shear stresses.
We can isolate a small element mn of the beam. There are horizontal shear stresses acting between horizontal layers of the beam as well as vertical shear stresses acting on the cross-sections
At any point in the beam, these complementary shear stresses are equal in magnitude
τ = 0 where y = ±h/2

5. 5.8: Shear formula
A formula for the shear stress τ in a rectangular beam can be derived
Where V is the shear force, I is the moment of inertia and b is the width of the beam. Q is the first moment of the cross-sectional area above the level at which the shear stress τ is being evaluated.
The shear formula can be used to determine the shear stress τ at any point in the cross-section of a rectangular beam

6. 5.8:Distribution of shear stresses in a rectangular beam
We can determine the distribution of the shear forces in a beam of rectangular cross-section
The distribution of shear stresses over the height of the beam is parabolic. Note that τ = 0 where y = ±h/2
The maximum value of shear stress occurs at the neutral axis (y1 = 0) where the first moment Q has its maximum value.
Where A = bh is the cross-sectional area

7. 5.8: Limitations
A common error is to apply the shear formula to cross-sectional shapes for
which it is not applicable. It is not applicable to triangular or semicircular
cross-sections
The formula should be applied when:
The edges of the cross section are parallel to the y-axis
The shear stress is uniform across the width of the cross section
The beam is prismatic

8. Second Midterm Test will take place this Friday
Can you do 5:00 – 6:30 ???