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

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.

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.

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

5.8: 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

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

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

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