Transcript Document

Mechanics of Materials – MAE 243 (Section 002)
Spring 2008
Dr. Konstantinos A. Sierros
Problem 2.6-7
During a tension test of a mild-steel specimen (see figure), the extensometer
shows an elongation of 0.00120 in. with a gage length of 2 in. Assume that the
steel is stressed below the proportional limit and that the modulus of elasticity E
= 30 (10^6) psi.
(a) What is the maximum normal stress σmax in the specimen?
(b) What is the maximum shear stress τmax?
(c) Draw a stress element oriented at an angle of 45° to the axis of the bar and
show all stresses acting on the faces of this element.
Problem 2.6-15
Acting on the sides of a stress element cut from a bar in uniaxial
stress are tensile stresses of 10,000 psi and 5,000 psi, as shown
in the figure.
(a) Determine the angle θ and the shear stress τ θ and show all
stresses
on a sketch of the element.
(b) Determine the maximum normal stress σmax and the maximum
shear stress τmax in the material.
2.6: Strain energy
• Strain energy is a fundamental concept in applied mechanics
• Consider axially loaded structural members subjected to static loads
• Consider a prismatic bar subjected to a static load P
• During the loading process, the load P moves slowly though the distance δ
and does a certain amount of work
• To find the work done by load P, we need to know the manner in which the
force varies. Therefore we need to use a load-displacement diagram
FIG. 2-41
Prismatic bar
subjected to a
statically applied
load
Copyright 2005 by Nelson, a division of Thomson Canada Limited
2.6: Strain energy: load – displacement diagram
‘ The work done by the load is equal to the area below the load – displacement curve’
FIG. 2-42
Load-displacement
diagram
Copyright 2005 by Nelson, a division of Thomson Canada Limited
Strain energy: The energy absorbed by the bar during
the loading process
2.6: Elastic and inelastic strain energy
• Recall loading – unloading of a prismatic bar
• The strain energy that recovers during unloading is called the elastic strain
energy (triangle BCD)
• Area OABDO represents energy that is lost in the process of permanently
deforming the bar. This energy is called inelastic strain energy
FIG. 2-43
Elastic and inelastic strain energy
Copyright 2005 by Nelson, a division of Thomson Canada Limited
2.6: Linearly elastic behaviour
• If the material is linearly elastic (i.e. follows Hooke’s law)
• Load – displacement curve is a straight line and the strain energy stored in the
bar is:
we also
know…
• Therefore we can express the strain energy of
a linearly elastic bar in either of the following
forms:
Load-displacement diagram
for a bar of linearly elastic material
FIG. 2-44
Copyright 2005 by Nelson, a division of Thomson Canada Limited
…and for linearly elastic springs (replacing EA/L by k)…
2.6: Nonuniform bars
• The total strain energy U of a bar consisting of several segments is equal to
the sum of the strain energies of the individual segments
• The strain energy of a nonprismatic bar with continuously varying axial force
can be calculated by using equation 1 for the differential element dx and then
integrating for the whole length of the bar.
(1)
FIG. 2-45
Bar consisting of
prismatic segments
having different crosssectional areas and
different axial forces
Copyright 2005 by Nelson, a division of Thomson Canada Limited
FIG. 2-46
Nonprismatic bar
with varying axial
force
Copyright 2005 by Nelson, a division of Thomson Canada Limited
2.6: Comments
• Strain energy is not a linear function of the loads applied
• Therefore, we cannot obtain the strain energy of a structure supporting
more than one load by combining the strain energies obtained from the
individual loads acting separately
• Instead we must evaluate the strain energy with all the loads acting
simultaneously (see example 2-13 page 124)
2.6: Displacements caused by a single load
• The displacement of a linearly elastic structure supporting only one
load can be determined from its strain energy.
Condition 1: Structure must behave in a linearly elastic manner
Condition 2: Only one load may act on the structure
FIG. 2-47
Structure
supporting a
single load P
Copyright 2005 by Nelson, a division of Thomson Canada Limited
2.6: Strain – energy density
divide by
volume of bar
(V = AL)
Replace P/A
by σ
Replace δ/L
by ε
In many situations it is convenient to use a quantity called strain
energy density, defined as the strain energy per unit volume of
material
Quiz 2 this Friday 15 February 2008
(1 question – 20 minutes) during class
Statically indeterminate structures
Homework 2 due next Wednesday 20
February 2008
It is already posted on the website
…plus 1st
midterm Friday
22nd February…