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 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM 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 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM 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 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM 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 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM 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 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM 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 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM 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 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM Proportional limit: maximum stress for which stress is proportional to strain. Stress at point P.

11 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM Yield point: stress for which the strain increases without an increase in stress. Horizontal portion of the curve ab. Stress at point Y.

12 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM 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 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM 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 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM 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 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM The right-hand sample (a nominal 4 x 4) is 8 long & exhibits a typical compressive failure.

16 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM The left-hand specimen, 12 in length, failed by a combination of compression and buckling.

17 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM 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 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM 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 11.2 TENSION TEST AND STRESS STRAIN DIAGRAM 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:

29 11.6 POISSON S RATIO When a load is applied along the axis of a bar, axial strain is produced. At the same time, a lateral strain, perpendicular to the axis, is also produced. If the axial force is in tension, the length of the bar increases. The cross section contracts or decreases. A positive axial stressproduces a positive axial strain and a negative lateral strain. For negative axial stress, axial strain is negative, and the lateral strain is positive.

30 11.6 POISSON S RATIO The ratio of lateral strain to axial strain is called Poisson s ratio. It is constant for a given material provided: The material is not stressed above the proportional limit. The material is homogeneous. The material has the same physical properties in all directions.

31 11.6 POISSON S RATIO Poisson s ratio, represented by Greek lowercase letter ν(nu), is defined by the equation: The negative sign ensures that Poisson s ratio is a positive number. The value of Poisson s ratio,ν, varies from 0.25 to 0.35 for different metals. For concrete, it may be as low as ν= 0.1, and for rubber, as high asν= 0.5.

32 11.8 ADDITIONAL MECHANICAL PROPERTIES OF MATERIALS Elastic Limit: The highest stress that can be applied without permanent strain when the stress is removed. To determine elastic limit would require application of larger & larger loadings and unloadings of the material until permanent strain is detected.

33 11.8 ADDITIONAL MECHANICAL PROPERTIES OF MATERIALS Elastic Range: Response of the material as shown on the stress strain curve from the origin up to the proportional limit P.

34 11.8 ADDITIONAL MECHANICAL PROPERTIES OF MATERIALS Plastic Range: Response of the material as shown on the stress strain curve from the proportional limit P to the breaking strength B.

35 11.8 ADDITIONAL MECHANICAL PROPERTIES OF MATERIALS Necking Range: Response of the material as shown on the stress strain curve from the ultimate strength U to the breaking strength B. Beyond the ultimate strength, the cross-sectional area of a localized part of the specimen decreases rapidly until rupture occurs. Referred to as necking, it is a characteristic of low-carbon steel brittle materials do not exhibit it at usual temperatures. It is part of the plastic range.

36 11.8 ADDITIONAL MECHANICAL PROPERTIES OF MATERIALS Percentage Reduction in Area: When a ductile material is stretched beyond its ultimate strength, the cross section necks down, and the area reduces appreciably. Defined by: where A o is the original, and A f the final minimum cross-sectional area. It is a measure of ductility.

37 11.8 ADDITIONAL MECHANICAL PROPERTIES OF MATERIALS Percentage Elongation: a comparison of the increase in the length of the gauge length to the original gauge length. Defined by: where L o is the original, and L f the final gauge length.. It is also a measure of ductility.

38 11.8 ADDITIONAL MECHANICAL PROPERTIES OF MATERIALS Modulus of Resilience: The work done on a unit volume of material from a zero force up to the force at the proportional limit. This is equal to the area under the stress strain curve from zero to the proportional limit. At right, Area A1. Units of in.-lb/in. 3 or N ( m/m 3.)

39 11.8 ADDITIONAL MECHANICAL PROPERTIES OF MATERIALS Modulus of Toughness: The work done on a unit volume of material from a zero force up to the force at the breaking point. This is equal to the area under the stress strain curve from zero to the breaking strength. At right, Areas A1 and A2. Units of in.-lb/in. 3 or N ( m/m 3.)

40 11.8 ADDITIONAL MECHANICAL PROPERTIES Recent Developments in Materials Technology Research in development &applications of new materials is a continual process. Today, much work takes place at the microscopic level and involves nanotechnology. The strongest material ever tested is a carbon material called graphene. Said to be 200 times stronger than structural steel, it consists of a single layer of graphite atoms that may be rolled into tiny tubes (called nanotubes). The tubes can be used as the basis of graphite fibers found in products requiring high strength & light weight.

41 11.8 ADDITIONAL MECHANICAL PROPERTIES Recent Developments in Materials Technology A process has been developed in which ceramic particles are added to molten aluminum, and a gas is blown into the mixture. When solidified, a honeycombed cellular aluminum material is formed that is lightweight yet strong. The use of nano-sized additives has been found to double the effective life of concrete. By preventing penetration of chloride and sulfate ions from sources such as road salt, seawater, and soils.

42 11.8 ADDITIONAL MECHANICAL PROPERTIES Recent Developments in Materials Technology A new structural material similar to concrete has been developed that uses a mixture of fly ash and organic materials. This new material has good insulating properties & fire resistance, as well as high & light weight. Several forms of enhanced steels have recently been developed. Sometimes referred to as super steels, they are stronger than traditional counterparts, can withstand extreme levels of heat and radiation, and generally have much higher resistance to corrosion.

43 11.8 ADDITIONAL MECHANICAL PROPERTIES Recent Developments in Materials Technology A structural coating has been developed using a combination of carbon nanotubes & polymers. When applied to a bridge, the coating can detect internal cracks long before they become visible. The U.S. Environmental Protection Agency recently approved public health claims that copper, brass, and bronze are capable of killing potentially deadly bacteria, including methicillinresistant Staphylococcus aureus (MRSA).

44 11.9 STRAIN AND STRESS DISTRIBUTIONS: SAINT-VENANT S PRINCIPLE In our discussion of the uniform axially loaded bar, we have assumed a uniform distribution of normal stress on any plane section near the middle of the bar away from the load. We ask here what effect a concentrated compressive load has on the stresses and strains near the load.

45 11.9 STRAIN AND STRESS DISTRIBUTIONS: SAINT-VENANT S PRINCIPLE Consider a short rubber bar of rectangular cross section. A grid or network of uniformly spaced horizontal and perpendicular lines is drawn on the side of the bar, as shown, to form square elements. (Half of the bar is shown in the figure.)

46 11.9 STRAIN AND STRESS DISTRIBUTIONS: SAINT-VENANT S PRINCIPLE Compressive loads are applied to the bar. Close to the load, the elements are subjected to large deformations or strain, while other elements on the end of the bar, away from the load, remain virtually free of deformations. Moving axially away from the load, we see a gradual smoothing of the deformations of the elements, and thus a uniform distribution of strain and the resulting stress along a cross section of the bar.

47 11.9 STRAIN AND STRESS DISTRIBUTIONS: SAINT-VENANT S PRINCIPLE Observations are verified by results from the theory of elasticity. Showing distribution of stress for various cross sections of an axially loaded short compression member.

48 11.9 STRAIN AND STRESS DISTRIBUTIONS: SAINT-VENANT S PRINCIPLE Sections are taken at distances of b/4, b/2, and b from the load where bis the width of the member. It appears the concentrated load produces a highly nonuniform stress distribution & large local stresses near the load.

49 11.9 STRAIN AND STRESS DISTRIBUTIONS: SAINT-VENANT S PRINCIPLE Sections are taken at distances of b/4, b/2, and b from the load where bis the width of the member. Notice how quickly the stress smoothes out to a nearly uniform distribution. Away from the load at a distance equal to the width of the bar, the maximum stress differs from the average stress by only 2.7 percent.

50 11.9 STRAIN AND STRESS DISTRIBUTIONS: SAINT-VENANT S PRINCIPLE The smoothing out of the stress distribution is an illustration of Saint-Venant s principle. Barre de Saint-Venant, a French engineer and mathematician, observed that near loads, high localized stresses may occur, but away from the load at a distance equal to the width or depth of the member, the localized effect disappears and the value of the stress can be determined from an elementary formula. Such as:

51 11.9 STRAIN AND STRESS DISTRIBUTIONS: SAINT-VENANT S PRINCIPLE This principle applies to almost every other type of member and load as well. It permits us to develop simple relationships between loads and stresses and loads and deformations.

52 11.10 STRESS CONCENTRATIONS Stress near a concentrated load is several times larger than the average stress in the member. Similar conditions exist at discontinuities in a member. Shown here is stress distribution in flat bars under an axial load, with a circular hole, semicircular notches, and quartercircular fillets.

53 11.10 STRESS CONCENTRATIONS Whether stress concentrations are important in design depends on the nature of the loads and the material used for the member. If the loads are applied statically on a ductile material, stress concentrations are usually not significant. For impact or repeated loads on ductile material, or static loading on brittle material, it cannot be ignored. Stress concentrations are usually not important in conventional building design. They may be important in the design of supports for machinery and equipment and for crane runways.

54 11.10 STRESS CONCENTRATIONS Stress concentrations should always be considered in the design of machines. In most machine part failures, cracks form at points of high stress. The cracks continue to grow under repeated loading until the section can no longer support the loads. The failure is usually sudden and dangerous.

55 11.11 REPEATED LOADING, FATIGUE Many structural and most machine members are subjected to repeated loading and the resulting variations of the stresses in the members. These stresses may be significantly less than the static breaking strength of the member, but if repeated enough times, failure due to fatigue can occur. The mechanism of a fatigue failure is progressive cracking that leads to fracture. If a crack forms from repeated loads, the crack usually forms at a point of maximum stress.

56 11.11 REPEATED LOADING, FATIGUE In a fatigue test, a specimen of the material is loaded and unloaded until failure occurs. The repeated loading produces stress reversals or large stress changes in either tension or compression. The lower the stress level the greater the number of cycles before failure. In the test, the stress level is lowered in steps until a level is reached where failure does not occur. That stress level is the fatigue strength/endurance limit.

57 11.11 REPEATED LOADING, FATIGUE Nonferrous metals such as aluminum and magnesium do not exhibit a fatigue strength. They must be tested until the number of cycles (service life) the metal will be subjected to is reached. For example, if an aluminum member has a service life of 10^7 cycles, the maximum stress would be 17.5 ksi (121 MPa) without stress concentrations.

58 11.11 REPEATED LOADING, FATIGUE If handbook data are not available and stress concentrations are not present, the fatigue strength for ferrous materials may be taken at approximately 50% of the tensile strength. For nonferrous metals, it must be lowered to approximately 35%

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