Mechanics of Materials – MAE 243 (Section 002)
Spring 2008
Dr. Konstantinos A. Sierros
Problem 2.4-14
A rigid bar ABCD is pinned at point B and supported by springs at A and D (see
figure). The springs at A and D have stiffnesses k1 = 10 kN/m and k2 = 25
kN/m, respectively, and the dimensions a, b, and c are 250 mm, 500 mm, and
200 mm, respectively.
A load P acts at point C. If the angle of rotation of the bar due to the action of
the load P
is limited to 3°, what is the maximum permissible load Pmax?
Problem 2.5-5
A bar AB of length L is held between rigid supports and heated nonuniformly in
such a manner that the temperature increase ΔT at distance x from end A is
given by the expression ΔT = TB (x^3/L^3), where ΔTB is the increase in
temperature at end B of the bar (see figure).
Derive a formula for the compressive stress c in the bar. (Assume that the
material has modulus of elasticity E and coefficient of thermal expansion .)
2.6: Stresses on inclined sections
• Up to now we have considered normal stresses acting on cross-sections
• Provided that the bar is prismatic, the material is homogeneous, the axial force
P acts at the centroid of the cross-sectional area and the cross-section is away
from any localized stress concentrations
(a) Bar with axial forces P
(b) Three-dimensional
view of the cut bar
showing the normal
stresses
(c) Two-dimensional view
FIG. 2-30
Prismatic bar in tension showing the
stresses acting on cross section mn
Copyright 2005 by Nelson, a division of Thomson Canada Limited
2.6: Stress elements
• Isolating a small element C (fig. 2-30 c)
• We have a stress element (fog 2-31 a)
• The only stresses acting are the normal
stresses σx
• Because it is more convenient, we
usually draw a 2-D view of the stress
element (fig. 2-31 b)
Stress element at point C of the
axially loaded bar shown in Fig. 2-30c:
(a) three-dimensional view of the element,
and (b) two-dimensional view of the element
FIG. 2-31
Copyright 2005 by Nelson, a division of Thomson Canada Limited
2.6: Stresses on inclined sections
• In order to obtain a more complete picture, we need to investigate the stresses
acting on inclined sections (plane pq fig. 2-32 a)
• Uniform distribution of stresses as shown in figs 2-32 b and c
(a) Bar with axial forces P
(b) Three-dimensional
view of the cut bar
showing the stresses
(c) Two-dimensional view
FIG. 2-32
Prismatic bar in tension showing the stresses
acting on an inclined section pq
Copyright 2005 by Nelson, a division of Thomson Canada Limited
2.6: Stresses on inclined sections
•We need to specify the orientation of the section pq. Define an angle θ
between the x-axis and the normal n to the section
• We need to find the stresses acting on the section pq
• Load P, which is the stress resultant, can be resolved with respect to N and V
• N is associated with normal stresses and V is associated with shear stresses
Prismatic bar
in tension showing
the stresses acting on
an inclined section pq
FIG. 2-33
Copyright 2005 by Nelson, a division of Thomson Canada Limited
2.6: Stresses on inclined sections
Establish standard notation and sign convention
• Normal stresses σθ are positive in tension and shear stresses τθ when they tend
to produce counterclockwise rotation of the material
where σx = P/A, in which σx is the normal stress on a cross-section
Sign convention for stresses acting
on an inclined section (Normal stresses are
positive when in tension and shear stresses
are positive when they tend to produce
counterclockwise rotation)
FIG. 2-34
Copyright 2005 by Nelson, a division of Thomson Canada Limited
2.6: Stresses on inclined sections
Graph of normal stress  and shear
stress  verses angle  of the inclined section
(see Fig. 2-34 and Eqs. 2-29a and b)
FIG. 2-35
Copyright 2005 by Nelson, a division of Thomson Canada Limited
2.6: Stresses on inclined sections
• Element A: The only stresses are the maximum normal stresses (θ = 0)
• Element B: This is a special case where all four faces have the same magnitude
normal and shear stress (σx/2)
FIG. 2-36 Normal and shear
stresses acting on stress
elements oriented at  = 0
and  = 45 for a bar in
tension
Copyright 2005 by Nelson, a division of Thomson Canada Limited
2.6: Stresses on inclined sections
The shear stress may cause failure if the material is much weaker in shear than in tension
Shear failure
along a 45 plane of a
wood block loaded in
compression
FIG. 2-37
Copyright 2005 by Nelson, a division of Thomson Canada Limited
Slip bands
(or Lüders’ bands)
in a polished steel
specimen loaded in
tension
FIG. 2-38
Copyright 2005 by Nelson, a division of Thomson Canada Limited