When an axial load is applied to a bar, normal stresses are produced on a cross section perpendicular to the axis of the bar.

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1 11.1 AXIAL STRAIN When an axial load is applied to a bar, normal stresses are produced on a cross section perpendicular to the axis of the bar. In addition, the bar increases in length, as shown:

2 11.1 AXIAL STRAIN Change in length, represented by the Greek lowercase letter δ(delta), is called deformation. The change in lengthδis for a bar of length L. Change in length, per unit of length, represented by Greek lowercase letterε(epsilon), is strain. Strain is defined by: Strain is usually expressed dimensionally as inches per inch or meters per meter, though it is a dimensionless quantity. This equation gives average strain over a length L. To obtain strain at a point, let length Lapproach zero.

3 11.1 AXIAL STRAIN For an axial load, stress in the direction of the load is called axial stress. Strain in the direction of the load is called axial strain. With axial strain, comes a smaller normal or lateral strain, perpendicular to the load. When the axial stress is tensile, the axial strain is associated with an increase in length. Lateral strain is associated with a decrease in width. Tensile strain is called positive strain, and compressive strain is called negative strain. Examples and problems start on textbook page 42.

4 A common material test is the tension test. If a large enough piece is available, and can be machined, a round cross section could be used. For thin plate, a rectangular or square section. The profile for a typical round test specimen: Fillets reduce stress concentration caused by the abrupt change in section. Deformation or change in length of the specimen is measured for a specified distance known as the gauge length. Strain is the deformation divided by the gauge length.

5 The tension specimen is placed in a testing machine such as the one shown. Strain can be measured by sensors built into the testing machine or by separate gauges attached directly to the specimen.

6 The electrical resistance strain gaugeconsists of a length of metallic foil or fine-diameter wire that is generally formed into a looped configuration. The gauge is epoxied to the surface of a test specimen and connected via lead wires to a sensitive electrical circuit that measures the resistance of the foil or wire used in the gauge.

7 With no load applied, the electrical resistance has a certain value. When a load is applied, the foil or wire deforms, producing a measurable change in resistance. That can be directly correlated to specimen strain.

8 Values of stress are found by dividing the load by the original cross-sectional area. And the corresponding value of strain by dividing the deformation by the gauge length. Values obtained can be plotted in a stress strain curve, the shapeof which will depend on the kind of material tested. Temperature and speed at which the test is performed also affect the results.

9 Stress strain curves for three different kinds of material. Low-carbon steel, a ductile material with a yield point. A ductile material, such as aluminum alloy, which doesn t have a yield point. A brittle material, such as cast iron or concrete, in compression.

10 Proportional limit: maximum stress for which stress is proportional to strain. Stress at point P.

11 Yield point: stress for which the strain increases without an increase in stress. Horizontal portion of the curve ab. Stress at point Y.

12 Yield strength: the stress that will cause the material to undergo a certain specified amount of permanent strain after unloading. Usual permanent strain percent. Stress at point YS.

13 Breaking strength: stress in material based on original cross-sectional area at the time it breaks. Fracture or rupture strength. Stress at point B.

14 Compression tests are made in a manner similar to tension tests. Specimen cross section is preferably of a uniform circular shape, although a rectangular or square shape is often used. Recommended ratio of length to major crosssectional specimen dimension (diameter/side length) is 2:1. This ratio allows a uniform state of stress to develop on the cross section, while reducing the tendency of the specimen to buckle sideways.

15 The right-hand sample (a nominal 4 x 4) is 8 long & exhibits a typical compressive failure.

16 The left-hand specimen, 12 in length, failed by a combination of compression and buckling.

17 For ductile materials, values of yield-point stress are commonly used as the allowable stress for inservice applications. A good example of this is metal used in structural members, most of which are made of A36 steel, an industry designation based on the material s yield point strength of 36,000 psi(~250 MPa).

18 The steel tensile specimens, below left, exhibit the typical elongation and necking that precedes an unmistakable point of fracture. The copper (bottom right) and aluminum (top right) compression specimens deformed under load, with no clear signs of failure. Due to this behavior, ductile materials are not generally tested in compression.

19 TENSION AND COMPRESSION TEST Copyright 2011 Pearson Education South Asia Pte Ltd

20 APPLICATIONS Copyright 2011 Pearson Education South Asia Pte Ltd

21 APPLICATIONS (cont) Copyright 2011 Pearson Education South Asia Pte Ltd

22 STRESS STRAIN DIAGRAM Note the critical status for strength specification proportional limit elastic limit yield stress ultimate stress fracture stress Copyright 2011 Pearson Education South Asia Pte Ltd

23 Brittle materials, such as cast iron or concrete, often have little or no strength in tension. They are used primarily for compressive loads Allowable stresses for these materials are generally set at some percentage of the material s ultimate strength.

24 11.3 HOOKE S LAW Based on tests of various materials and on the idealized behavior of those materials Hooke s law states that stress is proportional to strain. Figs. (a) & (b) and to a lesser degree, (c), show stress is directly proportional to strain (the curve is a straight line) on the lower end of the stress strain curve.

25 11.3 HOOKE S LAW Shown here is a stress strain curve for a material that follows Hooke s law. The slope of the stress strain curve is the elastic modulusor modulus of elasticity, E. The elastic modulus, E, is equal to the slope of the stress strain curve. Hooke s law only applies up to the proportional limit of the material.

26 11.3 HOOKE S LAW Because strain is dimensionless, the elastic modulus, E, has the same units as stress. The modulus is a measure of the stiffness or resistance of a material to loads. Except for brittle materials, high values of E generally correspond to stiffer materials Low values are consistent with more elastic materials.

27 11.4 AXIALLY LOADED MEMBERS From Hooke s law: When the stress and strain are caused by axial loads, we have:

28 11.5 STATICALLY INDETERMINATE AXIALLY LOADED MEMBERS If a machine or structure is made up of one or more axially loaded members, the equations of statics may not be sufficient to find the internal reactions in the members. The problem is said to be statically indeterminate, and equations for the geometric fit of the members are required. To write the equations:

When an axial load is applied to a bar, normal stresses are produced on a cross section perpendicular to the axis of the bar.

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