Mechanics of Materials – MAE 243 (Section 002)
Spring 2008
Dr. Konstantinos A. Sierros
Problem 4.3-1
Calculate the shear force V and bending moment M at a cross
section just to the left of the 1600-lb load acting on the simple
beam AB shown in the figure.
Problem 4.3-12
A simply supported beam AB supports a trapezoidally distributed
load (see figure). The intensity of the load varies linearly from 50
kN/m at support A to 30 kN/m at support B. Calculate the shear
force V and bending moment M at the midpoint of the beam.
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
centerline dimensions when making calculations.)
5.1: Stresses in beams - Introduction
• Stresses and strains associated with shear forces and bending moments of
beams
• The loads acting on the beam cause it to bend. The initially straight axis (fig
5-1a) is bent into a curve which is called the deflection curve (fig 5-1b)
• All loads act in the xy plane, known as the plane of bending
• The deflection of the beam at any point along its axis is the displacement of
that point from its original position, measured in the y direction
Bending of a cantilever beam: (a) beam
with load, and (b) deflection curve
FIG. 5-1
Copyright 2005 by Nelson, a division of Thomson Canada Limited
5.2:Pure bending and non-uniform bending
• Pure bending refers to flexure of a beam under a constant bending moment
(figs 5-2 and 5-3)
•Non-uniform bending refers to flexure in the presence of shear forces, which
means that the bending moment changes as we move along the axis of the beam
FIG. 5-4
FIG. 5-2
Simple beam in pure bending (M = M1)
Copyright 2005 by Nelson, a division of Thomson Canada Limited
Simple beam with
central region in
pure bending and
end regions in
nonuniform
bending
Copyright 2005 by Nelson, a division of Thomson Canada Limited
FIG. 5-3
Cantilever beam in pure bending (M = M2)
Copyright 2005 by Nelson, a division of Thomson Canada Limited
5.3: Curvature of a beam
• When loads are applied to a beam, its longitudinal axis is deformed into a
curve. The resulting strains and stresses in the beam are directly related to the
curvature of the deflection curve
• Point O’ is the center of curvature and ρ is the radius of curvature κ
For small deflections
FIG. 5-5
Curvature of a bent
beam:
(a) beam with load,
and (b) deflection
curve
Copyright 2005 by Nelson, a division of Thomson Canada Limited
5.3: Curvature of a beam
• The sign convention for curvature depends upon the orientation of the
coordinate axis
• Positive when the beam is bent concave upward
• Negative when the beam is bent concave downward
FIG. 5-6
Sign convention
for curvature
Copyright 2005 by Nelson, a division of Thomson Canada Limited
5.4: Longitudinal strains in beams
• The longitudinal strains in a beam can be found by analyzing the curvature of
the beam and the associated deformations
• Cross-sections of the beam mn and pq remain plane and normal to the
longitudinal axis
• Surface ss is called the neutral surface of the beam. Its intersection with any
cross-sectional plane is called the neutral axis of the cross section
• Normal strains εx are created from planes that either lenghten or shorten
FIG. 5-7
Strain-curvature relation
Deformations of
a beam in pure
bending:
(a) side view of
beam,
(b) cross section
of beam, and
(c) deformed
beam
Copyright 2005 by Nelson, a division of Thomson Canada Limited