Mechanics of Materials – MAE 243 (Section 002)
Spring 2008
Dr. Konstantinos A. Sierros
Problem 4.3-14
The simply-supported beam ABCD is loaded by a weight W = 27
kN through the arrangement shown in the figure. The cable
passes over a small frictionless pulley at B and is attached at E to
the end of the vertical arm. Calculate the axial force N, shear force
V, and bending moment M at section C, which is just to the left of
the vertical arm.
(Note: Disregard the widths of the beam and vertical arm and use
Center line dimensions when making calculations.)
Problem 4.5-14
The cantilever beam AB shown in the figure is subjected to a
uniform load acting throughout one-half of its length and a
concentrated load acting at the free end.
Draw the shear-force and bending-moment diagrams for this
beam.
5.5: Normal stresses in beams (Linearly elastic materials)
• Since longitudinal elements of a beam are subjected only to
tension/compression, we can use the stress-strain curve of the material to
determine the stresses from the strains
• The most common stress-strain relationship encountered in engineering
is the equation for a linearly elastic material
Resultant of the normal stresses
Normal stresses in a beam of linearly
elastic material: (a) side view of beam
showing distribution of normal stresses, and
(b) cross section of beam showing the z axis
as the neutral axis of the cross section
FIG. 5-9
1) A force acting in the x-direction
2) A bending moment acting about
the z-axis
Copyright 2005 by Nelson, a division of Thomson Canada Limited
5.5: Location of neutral axis
• Consider an element of area dA in the cross-section. The element is located at
distance y from the neutral axis
• This equation states that the z-axis must pass through the centroid of the
cross-section
The neutral axis (z-axis) passes through the
centroid of the cross-sectional area when the
material follows Hooke’s law and there is
no axial force acting on the cross-section
Normal stresses in a beam of linearly
elastic material: (a) side view of beam
showing distribution of normal stresses, and
(b) cross section of beam showing the z axis
as the neutral axis of the cross section
FIG. 5-9
Copyright 2005 by Nelson, a division of Thomson Canada Limited
5.5: Moment curvature relationship
• Moment of inertia of the cross-sectional area with respect to the z-axis
•The moment-curvature equation shows that the curvature is directly
proportional to the bending moment M and inversely proportional to the
quantity EI, which is called the flexural rigidity
FIG. 5-10
Positive bending moment produces positive
curvature and a negative bending moment
produces negative curvature
Relationships
between signs of
bending moments
and signs of
curvatures
Copyright 2005 by Nelson, a division of Thomson Canada Limited
5.5: Flexure formula
• We can determine the stresses in terms of the bending moment
• This equation is called the flexure formula and shows that the stresses are
directly proportional to the bending moment M and inversely proportional to
the moment of inertia I of the cross-section
• Stresses that are calculated from the flexure formula are called bending
stresses or flexural stresses
5.5: Maximum stresses at a cross section
• The maximum tensile and compressive bending stresses acting at any given
cross-section occur at points located furthest for the neutral axis
• The corresponding maximum normal stresses σ1 and σ2 (from the flexure
formula)
Relationships between signs of
bending moments and directions of normal
stresses: (a) positive bending moment, and
(b) negative bending moment
FIG. 5-11
section moduli
Copyright 2005 by Nelson, a division of Thomson Canada Limited
5.5:Doubly symmetric shapes
• If the cross-section of a beam is symmetric with respect to the z-axis as well
as the y-axis then we have
or
For a beam of rectangular cross-section with
width b and height h
FIG. 5-12
Doubly symmetric
cross-sectional
shapes
For a circular cross-section
Copyright 2005 by Nelson, a division of Thomson Canada Limited
Wednesday: QUIZ on Chapter 4
Duration 20 minutes
1 question to answer