Mechanics of Materials – MAE 243 (Section 002)

Spring 2008

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?

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.

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.)

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)

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

Stress (σ) – strain (ε) diagrams

OA: Initial region which is linear and proportionalSlope 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)

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

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

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

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

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

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

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.

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.

Wednesday (23 January 2008): Quiz on Statics, I will send you

e-mail with further details…

Have a good weekend…