This document contains sample problems and information about mechanics of materials and mechanical properties of materials from a course. It includes 3 sample problems involving calculating stresses and strains in loaded structures. It also provides information about stress-strain diagrams and how they characterize different material types as ductile, brittle, elastic or viscoelastic. Key mechanical properties like modulus of elasticity, yield point, proportional limit and ductility are defined. Stress-strain behavior under tension and compression and for different materials like steel, aluminum, rubber, plastics and composites are illustrated with diagrams.

1. Mechanics of Materials – MAE 243 (Section 002)
Spring 2008

2. Problem 1.2-4
A circular aluminum tube of length L = 400 mm is loaded in compression by forces P
(see figure). The outside and inside diameters are 60 mm and 50 mm, respectively. A
strain gage is placed on the outside of the bar to measure normal strains in the
longitudinal direction.
(a) If the measured strain is 550 x 10-6
, what is the shortening of the bar?
(b) If the compressive stress in the bar is intended to be 40 MPa, what should be the
load P?

3. Problem 1.2-7
Two steel wires, AB and BC, support a lamp weighing 18 lb (see figure). Wire AB is at
an angle α = 34° to the horizontal and wire BC is at an angle β = 48°. Both wires have
diameter 30 mils. (Wire diameters are often expressed in mils; one mil equals 0.001 in.)
Determine the tensile stresses AB and BC in the two wires.

4. Problem 1.2-11
A reinforced concrete slab 8.0 ft square and 9.0 in. thick is lifted by four cables
attached to the corners, as shown in the figure. The cables are attached to a hook at a
point 5.0 ft above the top of the slab. Each cable has an effective cross-sectional area A
= 0.12 in2
.
Determine the tensile stress σt in the cables due to the weight of the concrete slab. (See
Table H-1, Appendix H, for the weight density of reinforced concrete.)

5. 1.3 Mechanical properties of materials
• In order to understand the mechanical
behaviour of materials we need to perform
experimental testing in the lab
• A tensile test machine is a typical equipment
of a mechanical testing lab
• ASTM (American Society for Testing and
Materials)

6. FIG. 1-10 Stress-strain diagram for a
typical structural steel in tension (not to
scale)
Stress (σ) – strain (ε) diagrams
• Nominal stress and strain (in the
calculations we use the initial
cross-sectional area A)
• True stress (in the calculations
we use the cross-sectional area A
when failure occurs)
• True strain if we use a strain
gauge
• Stress-strain diagrams contain
important information about
mechanical properties and
behaviour

7. Stress (σ) – strain (ε) diagrams
OA: Initial region which is linear and proportional
Slope of OA is called modulus of elasticity
BC: Considerable elongation occurs with no noticeable increase in stress (yielding)
CD: Strain hardening – changes in crystalline structure (increased resistance to
further deformation)
DE: Further stretching leads to reduction in the applied load and fracture
OABCE’
: True stress-strain curve
FIG. 1-10 Stress-strain
diagram for a typical
structural steel in
tension (not to scale)

8. FIG. 1-12 Stress-strain diagram for
a typical structural steel in tension
(drawn to scale)
Stress (σ) – strain (ε) diagrams
• The strains from zero to point A
are so small as compared to the
strains from point A to E and can
not be seen (it is a vertical line…)
• Metals, such as structural steel,
that undergo permanent large
strains before failure are ductile
• Ductile materials absorb large
amounts of strain energy
• Ductile materials: aluminium,
copper, magnesium, lead,
molybdenum, nickel, brass, nylon,
teflon

9. FIG. 1-13 Typical stress-strain
diagram for an aluminum alloy.
Aluminium alloys
•Although ductile…aluminium alloys
typically do not have a clearly definable
yield point…
•However, they have an initial linear region
with a recognizable proportional limit
• Structural alloys have proportional limits
in the range of 70-410 MPa and ultimate
stresses in the range of 140-550 MPa

10. Copyright 2005 by Nelson, a division of Thomson Canada Limited
FIG 1-14 Arbitrary yield stress determined by
the offset method
Offset method
• When the yield point is not
obvious, like in the previous
case, and undergoes large
strains, an arbitrary yield stress
can be determined by the offset
method
• The intersection of the offset
line and the stress-strain curve
(point A) defines the yield
stress

11. FIG. 1-15 Stress-strain curves
for two kinds of rubber in
tension
Rubber (elastomers)
• Rubber maintains a linear relationship
between stress and strain up to relatively,
as compared to metals, large strains (up to
20%)
• Beyond the proportional limit, the
behaviour depends on the type of rubber
(soft rubber stretches enormously without
failure!!!)
• Rubber is not ductile but elastic material
• Percent elongation = (L1-Lo)/ Lo %
• Percent reduction in area = (Ao-A1)/ Ao %
Parameters that characterize ductility
Measure of the amount
of necking

12. FIG. 1-16 Typical stress-strain
diagram for a brittle material
showing the proportional limit
(point A) and fracture stress
(point B)
Brittle materials
• Brittle materials fail at relatively
low strains and little elongation
after the proportional limit
• Brittle materials: concrete,
marble, glass, ceramics and
metallic alloys
• The reduction in the cross-
sectional area until fracture (point
B) is insignificant and the fracture
stress (point B) is the same as the
ultimate stress

13. Plastics
• Viscoelasticity
• Time and temperature dependence
• Some plastics are brittle and some are
ductile
• COMPOSITES (glass fiber reinforced
plastics) combine high strength with light
weight
Polymer
matrix
Glass
fiber

14. FIG. 1-17 Stress-strain
diagram for copper in
compression
Compression
• Stress-strain curves in compression are
different from those in tension
• Linear regime and proportional limit are
the same for tension and compression for
materials such as steel, aluminium and
copper (ductile materials)
• However, after yielding begins the
behaviour is different. The material bulges
outward and eventually flattens out (curve
becomes really steep)
• Brittle materials have higher ultimate
compressive stresses than when they are
under tension. They do not flatten out but
break at maximum load.

15. Tables of mechanical properties
Appendix H contains tables that list materials properties.
Please make sure that you use these tables when solving problems
that require input of material properties data.

16. Wednesday (23 January 2008): Quiz on Statics, I will send you
e-mail with further details…
Have a good weekend…