Strength of Material Slides PDF

THE WAY TO GET STARTED IS TO QUIT TALKING AND BEGIN DOING. Walt Disney 1 STRENGTH OF MATERIALS Abdulai Ayirebi Ankrah, MSc Dept. of Mechanical Engineering University of Mines and Technology Tarkwa-Ghana.  Assignment & Presentation (10%) Final Exams (60%)...

THE WAY TO GET STARTED IS TO QUIT TALKING AND BEGIN DOING. Walt Disney 1 STRENGTH OF MATERIALS Abdulai Ayirebi Ankrah, MSc Dept. of Mechanical Engineering University of Mines and Technology Tarkwa-Ghana.  Assignment & Presentation (10%) Final Exams (60%) Attendance (10%) Quizzes (20%) Grading Policy Contents Introduction and Simple Stress and Strain Axially Loaded Spring/Bars Concept of Stress Introductio n I n t ro d uction t o M e c h an ics Introductio n Tension and Compression R e v i ew o f St a t ics Displacement of Axially St ru c t ure F re e - B o dy D i a gram Strain Loaded Members C o n c e pt o f St re s s Stress in Composite Bars Statically indeterminate St re s s A n a l y sis Poisson Ratio &Volume Change Structures /Bars Strain Energy Strain Energy STRENGTH OF MATERIALS Contents Temperature Stress and Centroids and Moment Torsion Shear Stress of Inertia Temperature Stress Introductio n Fundamental Concepts Shearing Stresses Centroids of Shapes Non- Uniform Torsion Shear Strain Moment of Inertia Transmissio n of Power by Relation Shear stress &Strain Polar Moment of Inertia Circular Shaft /Solid Shear Strain Energy Circular Bars Bearing Stresses Radius of Gyratio n Strain Energy STRENGTH OF MATERIALS Contents Beams ,Shear Force and Bending Bending Moments Introductio n Introductio n Beams Bending Equation Shear Force and Bending Position of the Neutral Axis Moment General Bending Formula Shear Force and Bending Strain Energy of Bending Moment Diagrams STRENGTH OF MATERIALS Contents Introduction and Concept of Stress Introductio n to Mechanics Review of Statics Structure Free -Bo dy Diagram Concept of Stress Stress Analysis STRENGTH OF MATERIALS Introduction and Concept of Stress ❖ Mechanics is a branch of the physical sciences that is concerned with the state of rest or motion of bodies that are subjected to the action of forces. ❖ Mechanics of rigid-body, ❖ Mechanics of Solids /Mechanics of Deformable Bodies ❖ Mechanics of Fluids. 1 Introduction and Concept of Stress 2  Introduction and Concept of Stress ❖ In mechanics of rigid body, it is assumed that there is no internal deformation in the body and we study about the static and dynamic position of the body subjected to external forces. ❖ In mechanics of solid, it is assumed that the solid deforms subjected to external forces and we study about the nature of deformation in the body. 3  Introduction to Mechanics II ❖ In fluid mechanics, shape and size of the body is not constant and we study about statics and dynamics of flow of fluid subjected to various types of forces on it ❖ The strength of a material is defined largely by the internal stress or intensities or forces in the material. ❖ Thus the concern of this subject will be internal effect of forces acting on a body 4 Introduction and Concept of Stress Forces acting on the body results in four basic deformations or displacements of the structure or solid bodies and these are ❑ Tension ❑ Compression ❑ Bending ❑ Twisting 5 Introduction and Concept of Stress ❖ Review of Statics The structure is designed to support a 30 kN load The structure consists of a boom and rod joined by pins (zero moment connections) at the junctions and supports Perform a static analysis to determine the internal force in each structural member and the reaction forces at the supports 6 Introduction and Concept of Stress ❖ Structure Free-Body Diagram ❖ Structure is detached from supports and the loads and reaction forces are indicated. 7  Introduction and Concept of Stress ❖ Concept of Stress ❑ The main objective of the study of mechanics of materials is to provide the future engineer with the means of analyzing and designing various machines and load bearing structures. ❑ Knowledge of these stresses is essential to the safe design of a machine ,an aircraft or any type of structure. ❑ Both the analysis and design of a given structure involve the determination of stresses and deformations. 8 Introduction and Concept of Stress ❖ Stress Analysis Can the structure safely support the 30 kN load? At any section through member BC, the internal force is 50 kN with a force intensity or stress of P 50 103 N  BC = = - 6 2 = 159 MPa A 314 10 m 9 Introduction and Concept of Stress From the material properties for steel, the allowable stress is  all = 165 MPa Conclusion: the strength of member BC is adequate ❑ Stress is a quantity that describes the magnitude of forces that causes deformation and generally defined mathematically as force per unit area. P ( Load / force) = = stress A(Cross − sec tional. Area) 10 Contents Simple Stress and Strain Introductio n Tension and Compression (Direct Stress ) Strain Stress in Composite Bars Poisson Ratio &Volume Change Strain Energy STRENGTH OF MATERIALS Simple Stress and Strain ❖ Stress ❑ The fundamental concepts of stress and strain can be illustrated by considering a straight metal bar loaded at it ends by a collinear or axial force P coinciding and acting through the centroid of each cross-section as shown 11 Simple Stress and Strain ❑ When the bar is stretched by a fo rc e P, the resulting stresses are t ensile. If the fo rc es are reversed in direction at each end of the bar i.e. direc ted to wards the bar, the bar is said to be in a state o f co mpressio n and the resulting stresses are compressive stresses ❑ Tensile and c o mpressive stresses are together referred to as Direct o r No r mal (Perpendic ular) stresses. They are referred to as no rmal stresses bec ause the stresses act in a direction perpendicular to the cut surface 12 Simple Stress and Strain ❑ Tensile stresses are defined to be positive and the compressive stresses to be negative. It is important to realize that the stress equation gives the average normal stress acting on the cross-sectional area. The maximum stress will depend on the bar’s geometry and the type of discontinuity. Tables and graphs are available which enable stress concentration factor K to be determined. The maximum stress is then being K times the average stress. 13 Simple Stress and Strain  max K=  ave 14 Simple Stress and Strain  max K=  ave 15 Simple Stress and Strain Question 1 A steel bar of rectangular cross -section 25 mm x 20 mm carries an axial tensile load of 3 kN. Estimate the average tensile stress in the cross -sectio n. Question 2 A steel bolt 25 mm in diameter carries a load of 4 kN in tension. Estimate the tensile stress of the section A and at the screw section B when the diameter of the thread is 20 mm. 16 Simple Stress and Strain ❖ Strain Strain is a measure of the deformatio n produced in the member by the load. strain is denoted by ε (Epsilon) and is given by the equation: ε = δ/L where δ = elongatio n (change in length) L = original/ initial length ε = strain. 17 Simple Stress and Strain ❑ If the bar is in tensio n the strain is called a tensile strain representing an elo ngation o r stretc hing o f the material. If the bar is in compression the strain and the bar shortens. Tensile strain is taken as posit ive and the compressive, negative. Question 1 A cy l i n d r i c al b lo ck o f co n cr e te i s 3 0 0 m m l o n g an d h as a ci r cu l ar cr o s s s e cti o n o f 1 0 0 m m i n d i am et er. I t c a r r i es a to t al co m p r es s iv e l o ad o f 6 7 k N an d u n d er th i s l o ad co n t r a c ts 0. 2 m m. Es ti m at e th e c o m p r es s i v e s t r e s s a n d t h e c o m p r e s si v e s t r a i n. Question 2 A b ar w i th a r e c t an g u l ar cr o s s s e cti o n 2 0 m m x 4 0 m m an d l e n g th L = 2. 8 m i s s u b j ec t ed t o a n a xi a l t en s i l e fo r ce o f 7 0 k N. Th e m e as u r e d e l o n g ati o n o f th e b ar δ i s 1. 2 m m. C al cu l ate th e t en s i l e s tr es s an d th e strain in the bar. 18 Simple Stress and Strain ❖ Tensile-Test 19 Simple Stress and Strain ❖ Tensile-Test Mild-Steel or Low -carbon Steel 20 Simple Stress and Strain 21 Simple Stress and Strain PERCENTAG E ELONG ATION is defined as: where L f – Distance between the gauge marks at fracture. L o – Original gauge length. PERCENTAGE REDUCTION is defined as: where A f – final area of the fracture section. A o – Original cross-sectional area. 22 Simple Stress and Strain A material is generally c lassified as b rit t le if the percentage elongatio n is less than 5 in a gauge length of 50 mm. ❖ Factor of Safety ❑ When designing a structure a safety facto r has to be taken into account in order to ensure the working stresses keep within safe limits. ❑ Fac to r o f safety is no rmally defined as the ratio between the failure stress and the working stress. Factor of Safety = ❑ To avoid failure, the following conditio ns must be satisfied 23 Simple Stress and Strain ❑ Fo r duc tile materials subjec ted to static lo ading, the allowable stress is often defined as: Allowable Stress = ❑ Fo r those ductile materials with no well -defined yield stress, the allowable stress is defined as: Allowable Stress = ❑ For brittle materials it is often defined as: Allowable Stress = 24 Simple Stress and Strain ❖ Hooke's Law Within the elastic region, there is a linear relationship between stress and strain. The linear relationship between stress and strain for a bar in simple tension or compression can be expressed as: σ = Eε where σ – stress ε – strain E – constant of proportionality known as Modulus of Elasticity or Young’s Modulus for the material. 25 Simple Stress and Strain Typical Values of E 200 GPa (steel) 70 GPa (aluminum) 11 GPa (wood) ❑ From the relationship between stress and Strain: δ = PL/AE 26 Simple Stress and Strain Question 1 A concrete cube 150 mm x 150 mm x 150 mm is loaded in a co mpressio n -testing mac hine. If the c o mpressive fo rc e acting no rmal to o ne fac e o f the cube is 250 kN, c alculate the compressive stress in the concrete. 27 Simple Stress and Strain Question 2 A tensile test is carried out on a bar of mild steel of diameter 20mm. The bar yields under a load of 8 kN. It attains a maximum load of 15 kN and breaks finally at a load of 7 kN. Estimate: the tensile stress at the yield point. the ultimate stress. the average stress at the breaking point if the diameter of the neck is10 mm. 27a Simple Stress and Strain ❖ Stresses in Composit e Bars ❑ A composite bar may be defined as a bar made up of two or more different materials, joined together in such a manner that the system extends or contracts as one unit, equally when subjected to tension or compression. ❑ The extension or contraction of the bar being equal, the strain is also equal. The total external load on the bar is equal to the sum of the loads carried by the different material. 28 Simple Stress and Strain Question 1 Three equally spaced rod in the same vertical plane support a rigid bar AB. Two outer rod of brass each 600mm long and 25mm in diameter. The central rod is of steel that is 800mm long and 30mm diameter. Determine the forces in the rods due to the applied load of 120kN through the midpoint of the bar. The bar remains horizontal after the application of the load. Take E s /E b =2. 29 Simple Stress and Strain Question 2 At sho rt, reinfo rced c ement c onc rete co lumn 600 mm x 600 mm has eight steel ro ds of 25mm diameter as reinfo rcement. Find the stresses in steel and c onc rete, and the elastic sho rtening of the co lumn if E = 200,000 N/mm 2 fo r steel and 10,000 N/mm 2 for concrete. Load on column = 3000 kN and length = 3 m. 30 Simple Stress and Strain ❖ Poisson’s Ratio (ν) ❑ When a bar is lo aded in tensio n, the axial elo ngatio n is acco mpanied by a lateral c ontractio n (no rmal to the directio n of the applied lo ad),as sho wn in whic h the dashed lines represent the shape befo r e loading and so lid lines give the shape after loading. ❑ The lateral strain is pro po rtional to the axial strain in the linear elastic range. 31 Simple Stress and Strain ❑ The ratio of the strain in the lateral directio n to the strain in the axial direction is known as Poisson’s ratio, and is denoted by ν (xu). V = - ❑ The lateral strain ε y is also given as ε y = ∆d/d or ∆t/t ❑ Poisso n’s ratio ran ges f rom 0.25 t o 0.35 fo r many metals. 0.3 is commo nly used f o r met als. C o nc re te has v alu es f rom 0.1 t o 0.2 , a nd c o rk app ro ximat ely 0.0. Th e th eo retical ma ximum v alue is 0.5. 32 Simple Stress and Strain ❖ Volume Change The dimensions of the bar in tension o r c o mpr essio n are changed when the load is applied, the vo lume of the bar also changes. The c hange in vo lume c an be calculated fro m the axial and lateral strains. The unit volume change ℮ is defined as e = ∆v = V f – V o = V o (1 – 2ν) ℮ = (1– 2ν) e = ε (1– 2ν) 33 Simple Stress and Strain ❑ As the bar is lo aded alo ng eac h of the three co o rdinate axes, the to tal strains in the bar in the respective direc tio ns will be given as; 34 Simple Stress and Strain ❑ The ratio of the direct stress (σ) to the c o rrespo nding vo lumetric strain (e) is known as Bulk Modulus. It is usually denoted by K. K = = ❑ The Bulk Modulus is related to Modulus of Elasticity (E) as; = 35 Simple Stress and Strain Question 1 A bar of circular cross -sectio n is loaded by tensile forces P = 85 kN. The bar has length L = 3.0 m and diameter, D = 30 mm. It is made of aluminum with Modulus of Elasticity E = 70GPa and Poisson’s ratio, ν = 1/3. Calculate the elongation δ, the decrease in diameter ∆d, and the increase in volume ∆V of the bar. Assume the propo rtio nal limit of the material is not exceeded. 36 Simple Stress and Strain Question 2 A steel pipe of length L = 1.2 m, o utside diameter d 2 = 150 mm, and inside diameter d 1 = 110 mm is c o mpressed by and axial fo rce P = 620 kN. The m aterial has modulus o f elastic ity E = 200 GPa and Poisson’s ratio ν = 0.3. Determine the following quantities for the pipe: the shortening δ the lateral strain the increase ∆d 2 in the outer diameter and the increase ∆d 1 in the inner diameter the increase ∆t in the wall thickness the increase ∆V in the volume of the material and, the dilatio n ℮. 37 Simple Stress and Strain ❖ Strain Energy As a tensile specimen extends under load, the forces applied to the ends of the test specimen move through small distances. These forces perform work in stretching the bar. To evaluate the work done by the load we use the load deflection diagram. W = ∫ Pdδ 38 Simple Stress and Strain U= U= The strain energy per unit vo lume of a material is also called the strain energy density o r Resilience. The resilience o f a material is its capac ity to abso rb potential energy within the elastic range. 39 Simple Stress and Strain Question 1 Calculate the strain energy of the bolt shown in the figure below under a tensile load of 10 kN. Show that the strain energy is increased, for the same maximum stress by turning down the shank of the bolt to the root diameter. Take E = 205000 N/mm 2 40 Unannounced Test (UT) Q1.A bar 3 m long is made of two bars, one of copper having E = 105 GN/m 2 and the other of steel having E = 210 GN/m 2. Each bar is 25 mm broad and 12.5 mm thick. This compound bar is stretched by a load of 50 kN. Find the increase in length of the compound bar and the stress produced in the steel and copper. The length of copper as well as steel is 3 m each. Unannounced Test (UT) Q1.A compression member 0.3 m long has a rectangular cross section 150 mm by 100 mm. It passes through a slot in a rigid block as shown in Figure such that there is complete restraint in the x direction. Therefore no change of dimension can take place in the x direction. There is however no restriction of movement in the y direction. If a load of 3800 kN is applied to the member as shown. Calculate: (i) the stress in the x direction (ii) the strain in the z direction (iii) the strain in the y direction Assume that E = 200 kN/mm 2 and Poisson’s ratio ν = 0.3. Unannounced Test (UT) Q1 A c onc rete co lumn, 50 c m square, is reinfo rc ed with fou r steel ro ds, each 2.5c m in diameter, embedded in the co nc rete near the co mers o f the square. If Young's mo dulus fo r steel is 200 GN/m 2 and that for concrete is 14 GN/m 2 Estimate the c o mpressive stresses in the steel and co nc rete when the total thrust on the column is 1 MN. Unannounced Test (UT) Q1 A high -strength steel wire, 3.2 mm diameter, stretches 35 mm when a 15 m length of it is stretched by a force of 3800 N. (a) What is the modulus of elasticity E of the steel? (b) If the diameter of the wire decreases by 0.0022 mm, what is Poisson’s ratio? (c) What is the unit volume change for steel? Contents Axially Loaded Spring/Bars Introductio n Displacement of Axially Loaded Members Statically indeterminate Structures /Bars Strain Energy STRENGTH OF MATERIALS  Axially Loaded Spring/Bars ❖ Introduction Axially lo aded members, whic h are structural elements having straight lo ngitudinal axes and carrying only axial fo rces (tensile or compressive) is our concern in this section. Their cross - sections may be so lid, hollow o r tubular o r thin -walled and open. 41  Axially Loaded Spring/Bars ❖ Displacement of Axially Loaded Members ❑ An axially lo aded spring is analogo us to a bar in tensio n. Consider a spring axially loaded by a force P. ❑ Under the ac tion o f the fo rce, P, the spring elo ngates by an amount δ so that its total length becomes (L+ δ), where L is the original length. The spring const ant , K is given as; K = P/ δ 42  Axially Loaded Spring/Bars ❑ The terms stiffness and flexibilit y are commonly used rather than spring constant and compliance in structural analysis. The stiffness, K of an axially loaded bar below is defined as force required to produce a unit deflection, hence stiffness of the bar, K is; K = P/ δ ❑ The flexibility, f is defined as the deflection given to a unit load. Thus the flexibility of an axially loaded bar is L f= EA 43 Axially Loaded Spring/Bars PL δ= EA 44  Axially Loaded Spring/Bars ❑ Suppose, that a prismatic bar is loaded by o ne o r m o re axial loads acting at intermediate points along the axis of the bar. n Pi L i δ=  i =1 E i A i 45 Axially Loaded Spring/Bars ❑ Consider a tapered bar sho wn and subjected to a continuo usly distributed lo ad, as sho wn by the arro ws in the Figure below; P( x )dx dδ = EA ( x ) 46  Axially Loaded Spring/Bars ❑ This same procedure is applied to a prismatic bar hanging vertically under its own weight as shown P( x)dx δ=  EA 47  Axially Loaded Spring/Bars Question 1 A steel bar 2.5 m lo ng has circular c ross -section of diameter d 1 = 20 mm o ver o ne-half of its length and diameter d 2 = 13 mm o ver the other half. Ho w much will the bar elongate under a tensile lo ad P = 22 kN? If the same v olume of the material is made into a bar of constant diameter d and length 2.5 m, what will be the elongatio n under the same load P? (Assume E = 210 GPa). 48  Axially Loaded Spring/Bars Question 2 A steel bar AD (see figur e) has a c ro ss-sec tio nal area o f 260 mm 2 and is lo aded by fo rc es P 1 = 12 kN , P 2 = 8 k N, and P 3 = 6 kN. The lengths o f the segments o f the bar are a = 1.5 m, b = 0.6 m, and c = 0.9 m. Assuming that the mo dulus of elasticity E = 210 G Pa, calculate the change in length δ of the bar. Does the bar elongate or shorten? By what amo unt P should the lo ad P 3 be inc reased s o that end D of the bar does not move when the loads are applied? 49  Axially Loaded Spring/Bars ❖ Statica lly Indeterminate Structures/ Bars ❑ Consider a prismatic bar AB of cross -sectio nal area A attached to rigid supports at both ends and axially loaded by a force P at an intermediate point C. ❑ From the free -bo dy diagram the reactions R A and R B cannot be found by statics alone, because only one equation of equilibrium is available. 50  Axially Loaded Spring/Bars Question1 The axially loaded bar ABCD shown in the figure is held between rigid supports. The bar has cross -sectio nal area A o from A to C and 2A o from C to D. I. Obtain formula's for the reactions R A and R D at the ends of the bar. II. Determine the displacement δ B and δ C at points B and C respectively. 51  Axially Loaded Spring/Bars Question 2 A steel ro d, 10 mm diameter, and an aluminium r od, 20 mm diameter, are jo ined together fixed between suppo rts as shown below.E for steel = 200 GPa and E for aluminium = 70 GPa. Find the reactio ns at the suppo rts and the stresses in the metals. 52  Axially Loaded Spring/Bars ❖ Strain Energy ❑ The strain energy of an axially lo aded bar of a linearl y elastic material is given as P2L U= ---------------------- (1) 2 EA ❑ Fo r linearly elastic spring the strain energy is obtained by replac ing the stiffness EA/L o f the prismatic bar by the stiffness k of the spring, and is given as; P2 U= 2K 53  Axially Loaded Spring/Bars ❑ The to tal strain energy U o f a bar consisting of several segments is equal to the sum of strain energies of the individual segments. Co nsidering the figure below the to tal strain energy is the strain energy o f segment AB plus the strain energy of segment BC. U= 54  Axially Loaded Spring/Bars ❑ For a non-prismatic bar with continuously varying axial force, equation (1) is applied to a differential element shown in the figure and integrating along the length of the bar gives the strain energy of the bar as; U= 55  Axially Loaded Spring/Bars ❑ The same pr ocedure is applied to a bar hanging vertically under its own weight. For such a bar shown below the strain energy is also given as; U= 56  Axially Loaded Spring/Bars Question 1 The truss ABC shown in the figure supports a horizontal load P 1 = 9 kN and a vertical load P 2 = 18 kN. Both bars have cross - sectional area A = 1,290 mm 2 and are made of steel with E = 210 GPa. Determine the strain energy U 1 of the truss when the load P 1 acts alone (P 2 = 0). Determine the strain energy U 2 when the load P 2 acts alone (P 1 = 0). Determine the strain energy U when both loads act simultaneously. 57 Contents Temperature Stress and Shear Stress Temperature Stress Shearing Stresses Shear Strain Relation Shear stress &Strain Shear Strain Energy Bearing Stresses STRENGTH OF MATERIALS  Temperature Stress and Shear Stress ❖ Temperat ure Stresses ❑ External loads are not the o nly sourc es o f stresses and strains in a structure. When the temperature o f a body is raised o r lowered the material expands o r c ontracts. If the expansion and c ontraction is wholly o r partially restric ted stresses and strains are set up in the structure. ❑ Consider a lo ng bar AB of a material at a temperature θ℃. The bar is now subjected to an inc rease ∆θ℃ in temperature. ε t = α (∆θ) 58  Temperature Stress and Shear Stress α is a pro perty o f the material kno wn as the c oeffic ient of thermal expansion or coefficient of linear expansivity. α has unit of 1/K or 1/ o C. Conventio nally the thermal strain resulting fro m expansio n is taken to be positive and that resulting fro m co ntrac tio n is taken to be negative. If the expansio n of the bar is prevented, it is as if the bar is co mpressed to its o riginal shape and dimensio ns abo ve. The co mpressive strain is given by equatio n 1 abo ve. The correspo nding stress is; σ = Eε = α E (∆θ) 59  Temperature Stress and Shear Stress Consider a prismatic bar of length L subjected to a temperature difference ∆θ℃. δ t is the elongation of the bar due to the temperature change ∆θ. δ t = ε t L = α (∆θ) L 60  Temperature Stress and Shear Stress Question 1 A steel bar o f length 200 mm is at a temperature o f 10 o C. If the material properties are; E =210 GPa and α=12 x 10 -6 /K. Find: The thermal strain induced in the bar The new length of the bar when it is heated to 23 o C. 61  Temperature Stress and Shear Stress Question 1 A 150 mm diameter steam pipe is laid in a trench at a temperature of 13 o C. When steam passes through the pipe, its temperature rises to 120 o C; (a) What is the increase ∆d in the diameter of the pipe if the pipe is free to expand in all directions? (b) What is the axial stress σ in the pipe if the trench restrains the pipe so that it lengthens only one -third as much as it would if it could expand freely? ( Note: The pipe is made of steel with modulus of elasticity E = 200 GPa and coefficient of thermal expansion α = 12 x 10 -6 / o C.) 62  Temperature Stress and Shear Stress ❖ Statica lly Indeterminate Bars Consider a prismatic bar AB of linear elastic material, rigidly constrained at the two ends and subjected to a temperature differenc e of ∆θ. The free -body diagrams are shown the increase and the decrease in the length of the bar due to temperature change ∆θ and external load R A respectively. 63  Temperature Stress and Shear Stress Question 1 In the assembly sho wn belo w the co pper bar on the left has a diameter o f 30mm and the brass bar o n the right o ne of 60 mm. The bars are stress -free at 25°C. Find the stresses and the change in length of the bars, when the temperature rises to 100°C if (i) the suppo rts are unyielding and (ii) the right support yields by 0.2 mm. For copper, E=120 GPa and a=18 x 10-6/℃, and for brass E=90GPa and a=20x 10-6/℃. 64  Temperature Stress and Shear Stress Question 1 SIMULATION  Temperature Stress and Shear Stress Question 1 A composite bar made up of aluminum bar and steel bar, is firmly held between two unyielding supports as shown in figure below. An axial load of 200 kN is applied at B at 47 0 C. Find the stress in each material, when the temperature is 97 0 C. Take E a = 70 GPa; E s = 210 GPa; α a = 24 x 10 -6 / 0 C and α s =12 x 10 -6 / 0 C. 65  Temperature Stress and Shear Stress ❑ C r eep o f M a ter i a l s u n d er Su s ta i n ed Str es s es A t o rd i n ary t e m pe rat ure s, m o s t m e t als w i l l s u s ta in s t re s ses b e l o w t h e l i mit o f p ro p o rt iona lit y f o r l o n g p e ri o ds w i t h ou t s h o w ing a d d i tion al m e a sura ble s t ra i n s. A t t h e s e t e m p era ture s m e t als d e f o rm c o n tin uo usly w h e n s t re s sed a b o v e t h e e l a st ic ra n ge. T h i s p ro c e ss o f c o n t in uou s i n e lastic s t ra i n i s c a l led c r eep. A t h i gh t e m pe rat ure s m e t als l o s e s o m e o f t h e i r e l a s tic p ro p e rt ies a n d c re e p u n d e r c o n s tan t s t re s s t a k e s p l a c e m o re ra p i d ly. 66 Temperature Stress and Shear Stress 67  Temperature Stress and Shear Stress ❑ Fatigue under Repeated Stresses When a material is subjected to repeated tensile stresses within the elastic range, it is found that the material tires and fractures rather suddenly after a large but finite number of repetitio ns of stress; the material is said to fatigue. 68  Temperature Stress and Shear Stress ❑ Shearing Stresses A part from the direct stresses (tensile and compressive stresses), and thermal stresses, there is another type of stress which plays a vital role in the behaviour of the materials and structures, especially metals. This stress acts parallel or tangential to a surface. 69  Temperature Stress and Shear Stress Shear stress is denoted by the symbo l τ (thaw), and shearing stress o n any surfac e is defined as the intensity of shearing fo rce tangential to the surfac e. The average shear stress is obtained by dividing the total shear fo r ce V by the area A o ver which it acts. τ = 70 Temperature Stress and Shear Stress 71  Temperature Stress and Shear Stress Question 1 A punch of diameter 19 mm is used to punch a hole in a 6 mm steel plate. A force P= 116 kN is required. What are the average shear stress in the plate and the average compressive stress in the punch? 72  Temperature Stress and Shear Stress ❖ Shear Strain Consider a rectangular block of material, subjected to shear stresses τ in one plane. The shearing stresses distort the rectangular face of the block into a parallelogram as shown above. If the right angles at the corners of the faces change by amounts ɣ, then ɣ is the shear strain. The shear strain ɣ can be defined as the change in the right angle. 73  Temperature Stress and Shear Stress A face is said to be po sitive if it has its o utward no rmal directed in the positive directio n o f a c oo rdinate axis. The opposite faces are negative faces. 74  Temperature Stress and Shear Stress ❑ Sign Convect ion The shear stress acting on a po sitive face of an element is positive if it acts in the direc tio n of o ne of the c oo r dinate axes and negative if it acts in the negative direction of the axis. The shear stress acting on a negative face of an element is positive if it acts in the negative direction of an axis, and negative if it acts in the positive directio n of an axis. Positive shear strain Negative shear strain 75  Temperature Stress and Shear Stress The sign conventio n for shear strain is related to that of shear stresses. Shear strain in an element is positive when the angle between two positive or (two negative) faces is reduced. The strain is negative when the angle between two positive or (two negative) faces is increased. 76  Temperature Stress and Shear Stress ❖ Relat ionship between Shear Stress and Shear Strain The τ versus γ diagram is similar in shape to the σ versus ε diagram. Fo r many materials the initial part o f the shear stress–strain diagram is a straight line. Within the linear elastic region, the shear stress and shear strain are directly proportional. That is; τ=Gγ where G is the shearing mo dulus of elastic ity (Modulus of Rigidity). G i s the ratio o f the shear stress to the shear strain, and it is also the slo pe of the linear po rtio n of the shear stress - strain diagram. E G is related to E as: G = 2 (1 + ) 77  THANK YOU Abdulai Ayirebi Ankrah [email protected] /[email protected]