Mechanics of Materials – MAE 243 (Section 002)
Spring 2008
Dr. Konstantinos A. Sierros
3.5: Stresses on inclined planes
Graph of normal stresses  and shear
stresses  versus angle  of the inclined plane
FIG. 3-22
Copyright 2005 by Nelson, a division of Thomson Canada Limited
Stress elements oriented at  = 0
and  = 45° for pure shear
FIG. 3-23
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• The normal stresses acting on the 45o element correspond to an element
subjected to shear stresses τ acting in the directions shown in fig. 3-23a
3.5: Strains in pure shear
Torsion failure of a brittle material
by tension cracking along a 45° helical
surface
FIG. 3-24
Strains in pure shear: (a) shear
distortion of an element oriented at  = 0,
and (b) distortion of an element oriented at
 = 45°
FIG. 3-25
Copyright 2005 by Nelson, a division of Thomson Canada Limited
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• Shear distortion (fig. 3-25a) and γ = τ/G
• When θ = 45o (fig. 3-25b) and εmax = γ/2
• The existence of maximum tensile stresses on planes at 45o to the x-axis
explains why bars of brittle materials that are under torsion fail by cracking
along a 45o helical surface (fig. 3-24)
3.6: Relationship between moduli of elasticity E and G
• It can be shown that shear modulus of elasticity G and Young’s modulus E are
related by:
• We see that E, G and v are not independent properties of a linearly elastic
material. Instead if two of them are known the third can be calculated
• Typical values of E,G and v are listed in Table H-2, Appendix H
3.7: Transmission of power by circular shafts
• The work W done by a torque of constant magnitude is equal to the product of
the torque and the angle ψ through which it rotates
•Power is the rate at which work is done
Power is expressed in watts (W). 1W = Nm/s
Shaft transmitting a constant
torque T at an angular speed 
FIG. 3-29
Copyright 2005 by Nelson, a division of Thomson Canada Limited
3.9: Strain energy in torsion and pure shear
• Bar AB is in pure torsion under the action of the torque T. The free end rotates
through an angle φ. The strain energy U of the bar is
• Using the equation φ = TL/GIp we can express the strain energy as follows
FIG. 3-34
Torque-rotation diagram for a bar
in torsion (linearly elastic material)
Prismatic bar in pure torsion
FIG. 3-35
Copyright 2005 by Nelson, a division of Thomson Canada Limited
Copyright 2005 by Nelson, a division of Thomson Canada Limited
3.9: Strain energy density in pure shear
• The SI units for strain energy is joule per cubic meter and the USCS
unit is inch-pound per cubic inch
FIG. 3-36
Element in
pure shear
Copyright 2005 by Nelson, a division of Thomson Canada Limited
4.1: Shear forces and bending moments
• In this chapter we will study beams which are structural members subjected to
lateral loads.
• Lateral loads are forces or moments that act perpendicular to the axis of the
bar
• The beams shown in figure below are classified as planar structures because
they lie in a single plane. If all loads act in that same plane and deflections
occur in that plane, then we refer to that plane as the plane of bending
Examples of beams
subjected to lateral loads
FIG. 4-1
Copyright 2005 by Nelson, a division of Thomson Canada Limited
4.2: Types of beams
• Simply supported beam (fig 4-2a). A pin
support prevents translation at the end of the
beam but does not prevent rotation. The
roller support at end B cannot prevent
translation in the horizontal direction but can
prevent translation in the vertical direction
• A cantilever beam (fig 4-2b) is fixed at one
end and free at the other. At the fixed support
the beam can neither translate nor rotate,
whereas at the free end it may do both.
Therefore, force and moment reactions may
exist at the fixed support
FIG. 4-2
Types of beams:
(a) simple beam,
(b) cantilever beam,
and (c) beam with an
overhang
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• A beam with an overhang (fig 4-2c) is a
beam which is simply supported at points A
and B and projects beyond point B. The
segment BC is similar to a cantilever beam
but also the beam axis may rotate at point B
4.2: Types of loads
• Concentrated loads (eg. P1, P2, P3, P4 )
• When a load is spread along the axis of a
beam is a distributed load. Distributed loads
are measured by their intensity q (force per
unit distance)
•Uniformly distributed load has constant
intensity q (fig 4-2a)
• A varying load has an intensity q that
changes with distance along the axis. Linearly
varying load from q1 - q2 (fig 4-2b)
• Another kind of load is a couple of moment
M1 acting on the overhanging beam (fig 4-2c)
FIG. 4-2
Types of beams:
(a) simple beam,
(b) cantilever beam,
and (c) beam with an
overhang
Copyright 2005 by Nelson, a division of Thomson Canada Limited
4.2: Reactions
• Finding the reactions is the first step in the analysis of a beam.
• If the problem is statically determinate the reactions can be found from free
body diagrams and equations of equilibrium.
• For example consider a simply supported beam (figure below)
4.3: Shear forces and bending moments
• When a beam is loaded by forces or couples, stresses and strains are created
throughout the interior of the beam. To determine these stresses and strains we
must first find the internal forces and couples that act on cross-sections of
beams
• Shear force V and bending moment M. Action of the left-hand part of the
beam on the right-hand part (fig 4-4b).
FIG. 4-4
Shear force V and
bending moment M
in a beam
Copyright 2005 by Nelson, a division of Thomson Canada Limited
4.3: Sign convections
• A positive shear force acts clockwise against the material. A positive positive
bending moment compresses the upper part of the beam
• A positive shear force tends to deform the element by causing the right-hand
face to move downward. A positive moment compresses the upper part of the
beam
Sign conventions for shear
force V and bending moment M
FIG. 4-5
Deformations (highly exaggerated) of a
beam element caused by (a) shear forces, and
(b) bending moments
FIG. 4-6
Copyright 2005 by Nelson, a division of Thomson Canada Limited
Copyright 2005 by Nelson, a division of Thomson Canada Limited