Transcript Document

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 heigh
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
FIG. 5-26
Shear stresses
in a beam of
rectangular
cross section
Copyright 2005 by Nelson, a division of Thomson Canada Limited
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
FIG. 5-28 Shear stresses in a beam of rectangular cross section
Copyright 2005 by Nelson, a division of Thomson Canada Limited
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.
FIG. 5-30
Where A = bh is the cross-sectional area
Distribution of shear
stresses in a beam of
rectangular cross
section:
(a) cross section of
beam, and
(b) diagram showing
the parabolic
distribution of shear
stresses over the
height of the beam
Copyright 2005 by Nelson, a division of Thomson Canada Limited
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:
1. The edges of the cross section are parallel to the y-axis
2. The shear stress is uniform across the width of the cross section
3. The beam is prismatic
Second Midterm Test will take place this Friday
Can you do 5:00 – 6:30 ???