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- 1. Oyelade Akintoye.O. (PhD) Civil and Environmental Engineering Department, University of Lagos. Mechanics of Materials 1 Tension and Compression in Bars
- 2. Content 01 四个 重要思想 Civil and Environmental Engineering, UNILAG Direct Stresses and Strains Hookes’s Law Stresses and Deformat ion (Tempera ture) Method of superpos ition Stress and Strain Transfor mation Stresses on inclined planes Principal Stresses (Mohr’s Circle) Stresses in Thin Cylinders and Spheres.
- 3. Textbooks 02 四个 重要思想 Civil and Environmental Engineering, UNILAG Mechanics of Materials 1 E. J. Hearn Engineering Mechanics 2 Mechanics of Materials Dietmar Gross · Werner Hauger ·Jörg Schröder · Wolfgang A. Wall Javier Bonet
- 4. Direct Stresses and Strains 03 四个 重要思想 Introduction Civil and Environmental Engineering, UNILAG Static: External and internal forces acting on structures can be determined with the aid of the equilibrium conditions alone. Real physical bodies were approximated by rigid bodies: if not, deformation is an important component to determine in practical problems. The geometry of deformation is given by kinematic equations; they connect the displacements and the strains.
- 5. Direct Stresses and Strains 04 四个 重要思想 Civil and Environmental Engineering, UNILAG In addition to the deformations, the stressing of structural members is of great practical importance The stress resultants alone, allow no statement regarding the load carrying ability of a structure: a slender rod or a stocky rod: state of stress The stresses and strains are connected in the Constitutive equations. These equations describe the behaviour of the material and can be obtained only from experiments. Introduction Assumption: Deformations and strains are very small
- 6. Direct Stresses and Strains 05 四个 重要思想 Civil and Environmental Engineering, UNILAG The solution of problems is based on three different types of equations: a) equilibrium conditions, b) kinematic relations and c) constitutive equations. In statically determinate system, these equations are uncoupled. In statically indeterminate systems, the equilibrium conditions, the kinematic relations and Hooke’s law represent a system of coupled equations. Introduction
- 7. Direct Stresses and Strains 07 四个 重要思想 Civil and Environmental Engineering, UNILAG The brain is wider than the sky Emily Dickinson
- 8. Direct Stresses and Strains 08 四个 重要思想 Civil and Environmental Engineering, UNILAG Normal stress and normal force N. Stress σ
- 9. Direct Stresses and Strains 09 四个 重要思想 Civil and Environmental Engineering, UNILAG In the case of a positive normal force N (tension) the stress σ is then positive (tensile stress). Reversely, if the normal force is negative (compression) the stress is also negative (compressive stress). Stress Normal Force： N N A A F A σ σ σ = → = =
- 10. Direct Stresses and Strains 10 四个 重要思想 Civil and Environmental Engineering, UNILAG The component τ which acts in the direction of the surface is called shear stress Stress cos A A ϕ ∗ = Equilibrium of forces： : cos sin 0, : sin cos 0 tan tan 0 A A F A A F A σ ϕ τ ϕ σ ϕ τ ϕ σ τ ϕ σ ϕ τ ∗ ∗ ∗ ∗ → + − = ↑ − = + = − =
- 11. Direct Stresses and Strains 11 四个 重要思想 Civil and Environmental Engineering, UNILAG Solving these two equations for σ and τ yields Stress 2 2 1 tan , 1 tan 1 tan F F A A ϕ σ τ ϕ ϕ = + + ( ) 0 0 1 cos2 , sin 2 2 2 σ σ σ ϕ τ ϕ = + = σ0 = F/A ( normal stress in a section perpendicular to the axis) It is practical to write these equations in a different form. Using the standard trigonometric relations ( ) 2 2 2 1 1 1 cos , cos 1 cos2 , sin cos sin 2 2 2 1 tan ϕ ϕ ϕ ϕ ϕ ϕ ϕ = = + = +
- 12. Direct Stresses and Strains 12 四个 重要思想 Civil and Environmental Engineering, UNILAG Stress The concentrated force produces high stresses near its point of application . This phenomenon is known as Stress concentration. The high stresses decay rapidly towards the average value σ0 as we increase the distance from the end of the bar. This fact is referred to as Saint-Venant’s principle
- 13. Direct Stresses and Strains 13 四个 重要思想 Civil and Environmental Engineering, UNILAG A bar with a constant cross-sectional area which has the undeformed length l. Under the action of tensile forces it gets slightly longer. The elongation is denoted by Δl and is assumed to be much smaller than the original length l. The ratio between the elongation and the original (undeformed) length: Strain l l ε ∇ = 1 l l ε ∇
- 14. Direct Stresses and Strains 14 四个 重要思想 Civil and Environmental Engineering, UNILAG Find the strain ? Strain ( ) du x dx ε =
- 15. Direct Stresses and Strains 15 四个 重要思想 Civil and Environmental Engineering, UNILAG If the displacement u(x) is known, the strain ε(x) can be determined through differentiation. Reversely, if ε(x) is known, the displacement u(x) is obtained through integration. The displacement u(x) and the strain ε(x) describe the geometry of the deformation. Strain ( ) du x dx ε = Kinematic relation
- 16. Direct Stresses and Strains 16 四个 重要思想 Civil and Environmental Engineering, UNILAG Stresses are quantities derived from statics; they are a measure for the stressing in the material of a structure. On the other hand, strains are kinematic quantities; they measure the deformation of the body Constitutive Law The physical relation that connects these quantities is called Constitutive law. It describes the behaviour of the material of the body under a load.
- 17. Direct Stresses and Strains 17 四个 重要思想 Civil and Environmental Engineering, UNILAG Constitutive Law proportional limit σP yield stress σY true stress or physical stress σt E σ ε = A material is said to be elastic if it returns to its original, unloaded dimensions when load is removed.
- 18. Direct Stresses and Strains 18 四个 重要思想 Civil and Environmental Engineering, UNILAG Constitutive Law The modulus of elasticity E is a constant which depends on the material and which can be determined with the aid of a tension test E σ ε = This relation is valid for tension and for compression: the modulus of elasticity has the same value for tension and compression. The proportionality factor E is called modulus of elasticity or Young’s. The constitutive law is called Hooke’s law after Robert Hooke (1635–1703).
- 19. Direct Stresses and Strains 19 四个 重要思想 Civil and Environmental Engineering, UNILAG Constitutive Law E σ ε = Changes of the length and thus strains are not only caused by forces but also by changes of the temperature. T T T ε α = ∆ T T E σ ε α = + ∆ ( ) T T x E σ ε α = + ∆ ( ) T E T σ ε α = − ∆
- 20. Single Bar under Tension or Compression 19 四个 重要思想 Civil and Environmental Engineering, UNILAG Three different types of equations to determine the stresses and the strains in a bar: • the equilibrium condition, • the kinematic relation and • Hooke’s law. 0 N dN n dx N + + − =
- 21. Single Bar under Tension or Compression 20 四个 重要思想 Civil and Environmental Engineering, UNILAG equilibrium condition 0 dN n dx + = du dx ε = kinematic relation Hooke’s law T T E σ ε α = + ∆
- 22. Single Bar under Tension or Compression 21 四个 重要思想 Civil and Environmental Engineering, UNILAG constitutive law for the bar : T du N N T dx EA A α σ = + ∆ = 0 dN n dx + = ( ) ( ) 0 0 0 l l T du N u l u l T dx dx EA α = − =∆ = + ∆ ∫ ∫ T Fl l Tl EA α ∆= + ∆
- 23. Single Bar under Tension or Compression 22 四个 重要思想 Civil and Environmental Engineering, UNILAG In a statically determinate system we can always calculate the normal force N(x) with the aid of the equilibrium condition. Subsequently, the strain ε(x) follows from σ = N/A and Hooke’s law ε = σ/E. Finally, integration yields the displacement u(x) and the elongation Δl. A change of the temperature causes only thermal strains (no stresses!) in a statically determinate system.
- 24. Single Bar under Tension or Compression 23 四个 重要思想 Civil and Environmental Engineering, UNILAG In a statically indeterminate problem the normal force cannot be calculated from the equilibrium condition alone. In such problems the basic equations (equilibrium condition, kinematic relation and Hooke´s law) are a system of coupled equations and have to be solved simultaneously. A change of the temperature in general causes additional stresses; they are called thermal stresses.
- 25. Single Bar under Tension or Compression 24 四个 重要思想 Civil and Environmental Engineering, UNILAG T du N T dx EA α = + ∆ 0 dN n dx + = ( ) T N EA u T α ′ = − ∆ ( ) T N EA u T n α ′ ′′ ′ = − ∆ = − T EAu n EA T α ′′ ′ = − + ∆ EAu n ′′ = −
- 26. Single Bar under Tension or Compression 25 四个 重要思想 Civil and Environmental Engineering, UNILAG ( ) ( ) N x x A σ = l x W W l ∗ − = ( ) 1 W x x A l σ = − 0 1 1 2 l W x Wl l dx EA l EA ∆ = − = ∫
- 27. Single Bar under Tension or Compression 26 四个 重要思想 Civil and Environmental Engineering, UNILAG 1 1 2 cos 0 , sin 0 tan sin s s S S F F S S F S α α α α + = = − = − + = Statically Determinate Systems of Bars
- 28. Single Bar under Tension or Compression 27 四个 重要思想 Civil and Environmental Engineering, UNILAG 1 1 2 2 1 2 , tan sin cos S l S l Fl Fl l l EA EA EA EA α α α ∆ = = − ∆ = = T Fl l Tl EA α ∆= + ∆ ( ) 1 3 2 2 , tan 1 cos sin tan sin cos Fl u l EA Fl l u v EA α α α α α α =∆ = + ∆ = + =
- 29. 28 四个 重要思想 Civil and Environmental Engineering, UNILAG
- 30. Single Bar under Tension or Compression 29 四个 重要思想 Civil and Environmental Engineering, UNILAG A rigid beam (weight W) is mounted on three elastic bars. Determine the angle of slope of the beam that is caused by its weight after the structure has been assembled.
- 31. Single Bar under Tension or Compression 30 四个 重要思想 Civil and Environmental Engineering, UNILAG 1 2 3 , 4cos 2 W W S S S α = = − = − 3 3 1 1 1 2 3 2 , 2 4 cos S l S l Wl Wl l l l EA EA EA EA α ∆ = ∆ = = − ∆ = = − Equilibrium eqn 1 3 , , tan cos B A B A l v v v l v a β α ∆ − = ∆ = = ( ) 3 3 cot 2cos 1 , tan , cot 4cos W l a EA α α β β β α α − = ≈ =
- 32. Thank you Civil and Environmental Engineering, UNILAG