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F inite Element Method

F inite Element Method. for readers of all backgrounds. G. R. Liu and S. S. Quek. CHAPTER 2:. INTRODUCTION TO MECHANICS FOR SOLIDS AND STRUCTURES. CONTENTS. INTRODUCTION Statics and dynamics Elasticity and plasticity Isotropy and anisotropy Boundary conditions

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F inite Element Method

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  1. Finite Element Method for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 2: INTRODUCTION TO MECHANICS FOR SOLIDS AND STRUCTURES

  2. CONTENTS • INTRODUCTION • Statics and dynamics • Elasticity and plasticity • Isotropy and anisotropy • Boundary conditions • Different structural components • EQUATIONS FOR THREE-DIMENSIONAL (3D) SOLIDS • EQUATIONS FOR TWO-DIMENSIONAL (2D) SOLIDS • EQUATIONS FOR TRUSS MEMBERS • EQUATIONS FOR BEAMS • EQUATIONS FOR PLATES

  3. INTRODUCTION • Solids and structures are stressed when they are subjected to loads or forces. • The stresses are, in general, not uniform as the forces usually vary with coordinates. • The stresses lead to strains, which can be observed as a deformation or displacement. • Solid mechanics and structuralmechanics

  4. Statics and dynamics • Forces can be static and/or dynamic. • Statics deals with the mechanics of solids and structures subject to static loads. • Dynamics deals with the mechanics of solids and structures subject to dynamic loads. • As statics is a special case of dynamics, the equations for statics can be derived by simply dropping out the dynamic terms in the dynamic equations.

  5. Elasticity and plasticity • Elastic: the deformation in the solids disappears fully if it is unloaded. • Plastic: the deformation in the solids cannot be fully recovered when it is unloaded. • Elasticity deals with solids and structures of elastic materials. • Plasticity deals with solids and structures of plastic materials.

  6. Isotropy and anisotropy • Anisotropic: the material property varies with direction. • Composite materials: anisotropic, many material constants. • Isotropic material: property is not direction dependent, two independent material constants.

  7. Boundary conditions • Displacement (essential) boundary conditions • Force (natural) boundary conditions

  8. Different structural components • Truss and beam structures

  9. Different structural components • Plate and shell structures

  10. EQUATIONS FOR 3D SOLIDS • Stress and strain • Constitutive equations • Dynamic and static equilibrium equations

  11. Stress and strain • Stresses at a point in a 3D solid:

  12. Stress and strain • Strains

  13. Stress and strain • Strains in matrix form where

  14. Constitutive equations s = c e or

  15. Constitutive equations • For isotropic materials , ,

  16. Dynamic equilibrium equations • Consider stresses on an infinitely small block

  17. Dynamic equilibrium equations • Equilibrium of forces in x direction including the inertia forces Note:

  18. Dynamic equilibrium equations • Hence, equilibrium equation in x direction • Equilibrium equations in y and z directions

  19. Dynamic and static equilibrium equations • In matrix form Note: or • For static case

  20. EQUATIONS FOR 2D SOLIDS Plane stress Plane strain

  21. Stress and strain (3D)

  22. Stress and strain • Strains in matrix form where ,

  23. Constitutive equations s = c e (For plane stress) (For plane strain)

  24. Dynamic equilibrium equations (3D)

  25. Dynamic and static equilibrium equations • In matrix form Note: or • For static case

  26. EQUATIONS FOR TRUSS MEMBERS

  27. Constitutive equations • Hooke’s law in 1D s = Ee Dynamic and static equilibrium equations (Static)

  28. EQUATIONS FOR BEAMS • Stress and strain • Constitutive equations • Moments and shear forces • Dynamic and static equilibrium equations

  29. Stress and strain • Euler–Bernoulli theory

  30. Stress and strain Assumption of thin beam Sections remain normal Slope of the deflection curve where sxx= Eexx 

  31. Constitutive equations sxx = Eexx Moments and shear forces • Consider isolated beam cell of length dx

  32. Moments and shear forces • The stress and moment

  33. Moments and shear forces Since Therefore, Where (Second moment of area about z axis – dependent on shape and dimensions of cross-section)

  34. Dynamic and static equilibrium equations Forces in the x direction Moments about point A  

  35. Dynamic and static equilibrium equations Therefore,  (Static)

  36. EQUATIONS FOR PLATES • Stress and strain • Constitutive equations • Moments and shear forces • Dynamic and static equilibrium equations • Mindlin plate

  37. Stress and strain • Thin plate theory or Classical Plate Theory (CPT)

  38. Stress and strain Assumes that exz = 0, eyz = 0 , Therefore, ,

  39. Stress and strain • Strains in matrix form e = -z Lw where

  40. Constitutive equations • s = c e where c has the same form for the plane stress case of 2D solids

  41. z y h O fz yz yy yx xx xz xy x Moments and shear forces • Stresses on isolated plate cell

  42. z Qx Mx Mxy Qy y O Qy+dQy Myx My My+dMy Myx+dMyx Qx+dQx dx Mxy+dMxy Mx+dMx x dy Moments and shear forces • Moments and shear forces on a plate cell dxx dy

  43. Moments and shear forces s = c e  s= - c z Lw Like beams, Note that ,

  44. Moments and shear forces Therefore, equilibrium of forces in z direction or Moments about A-A

  45. Dynamic and static equilibrium equations

  46. Dynamic and static equilibrium equations  (Static) where

  47. Mindlin plate

  48. Mindlin plate , e = -z Lq Therefore, in-plane strains where ,

  49. Mindlin plate Transverse shear strains Transverse shear stress

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