STRESS AND STRAIN AXIAL LOADING

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1 Chapter 1 STRESS ND STRIN XIL LDING 1.1 INTRDUCTIN The purpose to study the mechanics of materials is to acquire analytical tools necessary for analysis and design of load taking members of practical utility structures. These members could be parts of some machine or members of a building. The role of a design engineer is to ensure strength and rigidity of a member considering its physical dimensions, mechanical properties of the used material and the economy so that the member and the structure as a whole should neither break nor deform excessively in the life span of the structure while in service. The analytical tools are developed based on the laws of physics and application of mathematical methods. The properties of materials are known from systematically carried out experiments on the materials in the laboratories. In the study of mechanics of materials, we determine the stresses, strains and displacements in structures and their components caused by different loads acting on them. urther, we may compute these quantities for different combination and magnitude of loads until the failure is caused. Such a study gives a complete picture of behaviour of these structures. 1.2 STRESS The first fundamental concept in mechanics of materials is stress. The strength against an axially loaded member of a structure is assessed on the magnitude of stress, the intensity of the axial force, i.e., the force per unit area and is denoted by the Greek letter σ (sigma). σ =...(1.1) where, = xial load (a load that acts along the axis of the member) and = rea of cross-section measured perpendicular to the load direction. The stress may be assumed to be uniform over the area when the cross-sectional area is small and load acts on the centroid of the geometry of the cross-section. The stress that acts in the direction perpendicular to the area, is called normal stress. Sign convention: positive sign is assigned to indicate a tensile stress (the member is in tension) and a negative sign to indicate a compressive stress (the member is in compression). If a bar is pulled outward by the force, the stress is tensile stress, if the force is inward (into the area) causing the bar to be compressed, the stress is called compressive 1

2 2 MECHNICS MTERILS stress (Refer ig. T-1.1). Elementary form of a member is a prismatic bar, a structural member having same cross-section throughout its length. In S.I. units, the force () is expressed in newton (N) and area () in square meter (m 2 ), the stress in (N/m 2 ), that is, pascals (a). large value of stress is sometimes indicated as: 1 ka (kilopascal) = N/m 2 1 Ma (megapascal) = N/m 2 1 Ga (gigapascal) = N/m 2 (+) (+) (+) ( ) (+) ( ) (Bar under tension) (Bar under compression) ig. T SHERING STRESS There could be a situation where a body is subjected to a transverse force in the direction normal to its axis. If the transverse force is quite large, the body may split into two parts. This phenomenon may be well understood with the making of slices out of a bread using a knife. The force on the knife makes it to pass through the cross-section of the bread. The transverse force is along/tangential to the plane of area of the body. While split takes place one part slides over its other part. The transverse or tangential force divided by the associated area of cross-section, we get average shear stress. Denoting shearing stress by the Greek letter τ (tau), we write τ =...(1.2) This is known as direct shear or simple shear as well. In a situation when a member is split at one plane, the member is in single shear. There could be another situation where split occurs at two planes, the member is said to be in double shear, (determine shear stress considering the area twice of the area in single shear), (Refer ig. T-1.2). Direct shear arises in the design of bolts, pins, welds and glued joints. τ = 2 τ = (Single shear) ig. T-1.2 (Double shear)

3 STRESS ND STRIN XIL LDING 3 Shear stresses also arise in an indirect manner when members are subjected to tension, torsion, and bending. 1.4 BERING STRESS The bearing stress occurs at the surface of contact of two members experiencing external load. Consider two plates of thickness t each is joined together with a pin of diameter d. The two plates, when are pulled by force, the outer surface of the pin comes in contact with the plate surface. The pin surface will tend to crush the plate, while in turn, the plate through its surface in contact with the pin would crush the pin. The intensity of stress thus produced would depend on the contact area which is rectangular and its value is t d. Thus, the stress called bearing stress is given by: t σ b = td in...(1.3) CHTER 1 late t d Bearing area (Rectangle) ig. T UNCHING STRESS We know about a paper punching machine. When pressed, a solid pin pierces through the paper in making a hole in the paper and a piece of paper circular in shape drops down. Such a process is adopted to mint coins from a metallic plate. unch d d late t t τ Metal piece punched out ig. T-1.4

4 4 MECHNICS MTERILS Here the cylindrical surface of the metal piece bears the stress, the corresponding punching stress is given by: τ =...(1.4) π dt where π dt is the area of the curved surface of the metal piece that offers resistance to the punching load. 1.6 SHER STRESS This occurs when each of two equal forces is applied in opposite direction along two parallel planes of a body (here we consider a rectangular parallelopiped). The shear stress: x τ =...(1.5) a c 1.7 LLWBLE LD ND LLWBLE STRESS; z CTR SETY ig. T-1.5 n element of rom safety consideration, that is avoiding material subjected to shear stress structural failure, a structural member or a machine part is loaded less than its strength. Here, the strength is meant as the maximum load that can be sustained by a member. The maximum load is termed as ultimate load as well. While a smaller load than this is termed as allowable load (also called the permissible load or the safe load). The stress corresponding to allowable load is called as working stress or allowable stress. Then we have, Ultimate load actor of safety (.S.) =...(1.6) llowable load n alternative definition based on stress could be Ultimate stress or Yield stress actor of safety (.S.) =...(1.7) llowable stress Selection of magnitude of factor of safety depends on many factors, such as properties of a material, loading conditions, uncertainity factors related to manufacturing process, construction techniques, the perfectness in the analysis and computational techniques and importance of the structure under consideration. There are design codes (In India these are known as IS codes) available for use, these are prepared by experienced engineers. They recommend appropriate values of factor of safety which could be any thing more than STRESSES N INCLINED SECTINS n an axially loaded prismatic bar, we easily determine the normal stress, dividing the axial load by the area of bar normal to the load. or complete analysis, we need to determine the stresses that act on inclined sections of the bar. Consider an axial load acting along x-axis, passing through the centroid of the bar. The direct-normal stress on x-plane of the bar is: c y a b σ x =...(1.8)

5 τ θ STRESS ND STRIN XIL LDING 5 n inclined plane mn passes through the bar ig. T-1.6, its inclination is angle θ, measured in counter-clockwise direction with respect to y-axis. The interest, here, is to investigate stress values on this plane. We consider the free body diagram and condition of static equilibrium for the cut-out part containing the plane mn of the bar (say the left part). n the inclined plane, a force has to act in opposite direction to that it acts on the negative x-plane. Resolving this force, into two components, a normal force N that is perpendicular to the inclined plane mn and a shear force V that is tangential to it. These are: N = cos θ, V = sin θ The corresponding normal and shear stresses are obtained if dividing them by area θ, of the inclined section. Thus, N σ θ =, τ = V θ θ where θ = θ cosθ or, σ θ =. Ê cos2 θ, τ θ = ˆ Á Ë sin θ cos θ Writing, cos 2 θ = 1 + cos 2 θ, sin 2θ = 2 sin θ cos θ, 2 CHTER 1 we get σ θ = σ x 2 (1 + cos 2θ), τ θ = σ x 2 (sin 2θ)...(1.9) Sign convention: The subscript θ is used to indicate that the stresses act on a plane (section) inclined at an angle θ. Normal stresses σ θ are positive in tension and shear stresses τ θ are positive when they tend to rotate the part in counter-clockwise direction. y + θ m ositive x-plane x Centroid of area Negative x-plane n rea = Clockwise rotation σ θ ( ) V N θ τ θ rea = cos θ ig. T-1.6 Stresses acting on an inclined plane mn (prismatic bar in tension) rom Equation (1.9), we may conclude: the maximum normal stress occurs at θ = 0 and is σ max = σ x...(1.10)

6 6 MECHNICS MTERILS and maximum shear stress by magnitude τ max = σ x...(1.11) 2 The largest positive shear stress is at θ = 45 and the largest negative value of shear stress is at θ = STRIN ND STRESS-STRIN DIGRM The other important concept in mechanics of materials is strain. body subjected to a change in temperature or to an external load undergoes some change in shape, termed as deformation. specific relationship between stress and deformation is characteristic of the material and helps a design engineer in selecting a proper material to meet the requirement of the structural member. Consider a prismatic bar of a homogeneous material is subjected to an axial force, the bar of length L shows a change in length by δ (deformation). The change in length per unit length is defined a strain, and is denoted by the Greek letter ε (epsilon). ε = δ...(1.12) L If the bar is in tension, the strain is called a tensile strain representing an elongation of the material. If the bar is in compression, the strain is a compressive strain, the bar shortens. Sign convention: Tensile strain is taken as positive and compressive strain as negative. The strain ε is called a normal strain as it is associated with normal stresses. It is a dimensionless quantity. In the design and construction of structures, the knowledge of mechanical behaviour of materials in use is important. ne such behaviour of the material is stress-strain relationship. The best way to obtain this is to conduct experiments on the material in the laboratory and to derive its stress-strain diagram. specimen of the material under study is taken and put on the special machine. By applying loads in a specified manner, parameters like change in length, diameters, and nature of breaking are recorded and the results are interpreted. 12 mm Gauge length 50 mm L n extensometer or a strain gauge is attached to measure elongation ig. T-1.7 Test specimen with tensile load The specimen is cylindrical in shape of diameter 12 mm and a gauge length of 50 mm between gauge marks, which are the points where the extensometer arms are attached to the specimen ig. T-1.7. The test specimen is placed in a testing machine (tensile-test machine or universal-test machine) which is used to apply an axial load. s the load is increased the gauge length L is measured, and the elongation δ is determined for each δ

7 STRESS ND STRIN XIL LDING 7 value of. or each pair of readings and δ, the stress σ is computed by dividing by the original cross-sectional area of the specimen, and the strain ε by dividing the elongation δ by the original distance between the two gauge marks. The stress-strain diagram may then be obtained by plotting ε as an abscissa and σ as an ordinate. stress-strain diagram is a characteristic of the particular material being tested and provides important informations about the mechanical properties and type of behaviour. Stress-strain diagram (ig. T-1.8): We shall discuss results of the material known as structural steel (also known as mild steel or low carbon steel). Structural steel is one of the most commonly used materials in the construction of buildings, bridges, cranes, ships automobiles, locomotives and in many other constructions roportional Limit The diagram begins from the origin, is a straight line. This shows a linear relationship between stress and strain and the two have proportionality relationship. Beyond point, the proportionality no longer exists, hence the stress at, is called the proportional limit. or structural steels, this limit is in the range of Ma and for high tensile steels the value is around 550 Ma. In many of designs, the stress corresponding to proportional limit is the maximum stress that is allowed in the structural component Elastic Limit This is a stress value at beyond which if the load is removed the material will not return to its original shape. t this state of material a permanent deformation in the structural member sets in. This state of material is known as permanent set Yield oint This is a point B on the graph from where excessive elongation begins in the material. The elongation of the mild-steel in the region B to C is about times elongation that takes place in the linear portion. rom B to C, the behaviour of the material is that of a plastic, the slope of the curve is nearly zero, which means that it deforms without an increase in the applied load (in some case, the load may decrease while yielding occurs) Yield Stress This is the stress at yield point. The stress at yield point is also known yield strength of the material denoted by σ Y. This is a very important value of the material, on many instances it helps in deciding the working stress in the designs Strain Hardening When the loading continues to increase beyond yield, the restructuring of crystalline arrangement begins within the material and it starts resisting the impressed load and deformation rate is slowed and load carrying capacity increases. This process is known as strain-hardening. In the graph the line from C to D has positive slope and is about 1/50th of the slope of line between the origin, and Ultimate Stress In the process of increasing load on the specimen, the load bearing capacity of the material eventually reaches its maximum value at point D. It is highest ordinate of the curve. The corresponding stress is ultimate stress, σ U or ultimate strength. CHTER 1

8 8 MECHNICS MTERILS The Rupture Strength fter having reached the point D, the load starts falling (this will be observed as the back motion of the pointer on the dial of the machine) while elongation continues to increase. Eventually the specimen breaks at point E on the graph. The stress σ B corresponding to rupture is called the breaking strength. This value is smaller than the ultimate strength. Stress σ = Ultimate stress Yield stress roportional limit Elastic behaviour Yielding B Elastic limit σ Y erfect plasticity C Strain hardening D σ U Necking (a) Stress-strain diagram for a typical structural steel in tension (not to scale) σ E σ B racture point racture strength Strain ε = δ L σ U = σ B racture Cup Cone roportional limit ε (b) Necking of steel-specimen at failure (c) Stress-strain diagram for a typical brittle material ig. T Necking t loading condition corresponding to point D on the graph, the diameter of the test specimen gets progressively reduced (thinning or narrowing). The phenomenon is called

9 STRESS ND STRIN XIL LDING 9 necking. Thereafter the material breaks into two parts. The shape of two parts have special appearance, one is of cup shape and the other is of a cone. The cup-shaped surface forms an angle of approximately 45 with the original surface of the specimen Ductility This is a special property of the material. The ductility is a measure to the extent a material undergoes plastic deformation (a permanent strain) during increasing load. The material that undergoes greater deformation more ductile it is. Ductility is measured either by %age reduction in area or by the %age elongation of a known measured length of specimen. ercent reduction in area = or, ercent elongation = i i L L i f f (1.13)...(1.14) Li subscript i is for initial value and f for the final value at the point of fracture. Some of the known ductile materials are: structural steel, aluminium, copper, magnesium, lead, molybdenum, nickel, brass, bronze, monel metal, nylon. or structural steel the per cent reduction in area is of the order 60 to 70%. The property of ductility of the material enables to give some arbitrary shape to suit the construction requirement Brittle roperty There are materials which fail at low strain when subjected to a load to cause tension, at elongation little beyond its proportional limit,. The reduction in area at point of failure is insignificant absence of any necking. Such materials are known as brittle. typical stress-strain curve for a brittle material has been shown in the ig. T-1.8. Highcarbon steels (high-strength steels) have high value of proportional limit about 550 Ma very high yield stresses about 700 Ma but they behave in a brittle manner. rdinary glass is an ideal brittle material, because it exhibits almost no ductility. Its ultimate stress is about 70 Ma. Glass fibres have ultimate stress upto about 7 Ga. The prominent brittle materials are: concrete, stone, cast iron, glass, ceramics Comparative Stress-Strain Curves for Different Materials The relative strength and behaviour of different materials under axial loading (tension) can best be understood by plotting, stress-strain values on the same σ-ε graph as shown in the ig. T-1.9. s High carbon steel s Tempered alloy steel CHTER 1 Cast iron luminium Concrete High-strength, low-alloy steel Carbon steel ure iron e (a) (b) (a) Comparative stress-strain diagrams for different materials (b) Stress-strain diagrams for iron and different grades of steel ig. T-1.9 e

10 10 MECHNICS MTERILS The structural steel has about 0.2% carbon. With increasing carbon content, steel becomes less ductile but possesses higher yield stress and higher ultimate stress. That is, some of the physical properties of structural metals, such as strength, ductility, and corrosion resistance, can be greatly affected by alloying (adding other metals), heat treatment (tempering), and the manufacturing process (rolling, forging) used. We may find from the ig. T-1.9 (b) that the pure iron and three different grades of steel have large variations in the yield strength, ultimate strength and final strain (ductility) roof Stress luminium and many other ductile materials do not exhibit a distinct yield point on stress-strain diagram, instead the stress keep increasing-not linearly-until the ultimate strength is reached and eventually rupture takes place. The yield strength is determined by the offset method. straight line is drawn on the stress-strain diagram parallel to the initial part of the curve but offset by some standard strain, such as ε = on ε-axis. The intersection of the offset line and the stress-strain curve, the point defines the yield stress. This value is distinguished from a true yield stress by referring to it as the proof stress Compression Test or compression tests, the specimens of the metals ig. T-1.10 rbitrary yield stress determined by the offset method are in the shape of cubes 50 mm a side or circular cylinders-diameter 25 mm and 25 mm to 300 mm. Both the load applied by the machine and the shortening of the specimen are measured. In order to eliminate end effects, the gauge length is selected to be smaller than the total length of the specimen. Concrete is tested in compression on every important construction project to ensure that required strength has been achieved. The test specimen could be a cube of 150 mm a side or a cylinder 150 mm dia. and 30 cm long and is subjected to curing for 28 days. Smaller specimens are used when performing compression test on stones. concrete of M20 grade has compressive strength of 20 N/mm 2 and that of sand stone is σ 65 N/mm 2. Stress-strain curves for materials, in general, in compression differ from those in tension. Ductile metals-steel, aluminum, and copper have proportional limits nearly same to those in tension and the stress-strain diagrams are also same. Beyond yield point, the behaviour is quite different when in compression. With compressive loading, the specimen bulges outward (ig. T-1.11). ε Brittle materials loaded in compression, initially have σ-ε relationship a straight line, further the shortening increases at a higher rate. ig. T-1.11 Stress-strain diagram for a ductile material in compression The stress-strain curves for compression and tension often have similar shapes. The ultimate stresses are higher than those in tension. σ offset ε

11 STRESS ND STRIN XIL LDING ELSTICITY ND LSTICITY RERTIES MTERIL The elasticity and plasticity are the properties exhibited by a material during loading, unloading and then loading in a sequence and in an extended manner. The entire behaviour of the material is best understood on the σ-ε diagram developed through experimental results of a specimen. σ E σ E B CHTER 1 ε C D ε Elastic lastic Residual strain Elastic recovery (a) (b) ig. T-1.12 Consider a specimen subjected to tensile load (increasing). The stress and strain go from the origin to point on the stress-strain curve of ig. T-1.12(a). n removal of load at this point, the material traces back to the origin. This property of a material, by which it returns to its original dimensions/shape during unloading, is called elasticity. The material is known as elastic. The largest value of the stress at point E for which the material behaves elastically is called the elastic limit of the material. Suppose the load is increased from point E and taken to point B on the curve and the unloading takes place, the stress and strain decrease along a straight line BC parallel to the initial portion of the loading process. The point C represents on the curve where the load is no more on the specimen and ε is not zero-ig. T-1.12(b). The magnitude of C indicates a residual strain, or permanent strain or plastic deformation of the material. When the material is subjected to loads beyond elastic limit the strain values are inelastic the strain is known as plasticity. Suppose the specimen is loaded again with tensile load from the point C on the graph, the new curve is along the old line CB and further, it follows the stress-strain curve of the original curve toward point. or reloading, B is proportional limit, which is at a higher stress than the original elastic limit, E. The point of rupture, remains same. The ductility is reduced it is measured as B which is smaller than that of original one way loading measured as E HKE S LW, MDULUS ELSTICITY Most of engineering structures are designed for deformation which corresponds to a point on the proportionality line, the stress is normally not allowed to exceed the limit of proportionality, where the stress σ is directly proportional to the strain ε, we may write σ = E. ε

Chapter 4 MECHANICAL PROPERTIES OF MATERIAL. By: Ardiyansyah Syahrom

Chapter 4 MECHANICAL PROPERTIES OF MATERIAL By: Ardiyansyah Syahrom Chapter 2 STRAIN Department of Applied Mechanics and Design Faculty of Mechanical Engineering Universiti Teknologi Malaysia 1 Expanding