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Theory of Elasticity

Theory of Elasticity. Chapter 11 Bending of Thin Plates 薄板弯曲. Theory of Elasticity. Page. Chapter. Content. Introduction Mathematical Preliminaries Stress and Equilibrium Displacements and Strains Material Behavior- Linear Elastic Solids Formulation and Solution Strategies

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Theory of Elasticity

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  1. Theory of Elasticity Chapter 11 Bending of Thin Plates 薄板弯曲

  2. Theory of Elasticity Page Chapter Content • Introduction • Mathematical Preliminaries • Stress and Equilibrium • Displacements and Strains • Material Behavior- Linear Elastic Solids • Formulation and Solution Strategies • Two-Dimensional Problems • Three-Dimensional Problems • Bending of Thin Plates（薄板弯曲） • Plastic deformation - Introduction • Introduction to Finite Element Method 1 11

  3. Theory of Elasticity Page Chapter Bending of Thin Plates • 11.1 Some Concepts and Assumptions （有关概念及假定） • 11.2 Differential Equation of Deflection （弹性曲面的微分方程） • 11.3 Internal Forces of Thin Plate （薄板截面上的内力） • 11.4 Boundary Conditions（边界条件） • 11.5 Examples（例题） 2 11

  4. Theory of Elasticity Page Chapter 11.1 Some Concepts and Assumptions Thin plate（薄板） One dimension of which (the thickness)is small in comparison with the other two.（1/8－1/5）>/b≥（1/80-1/100） Middle surface(中面) The plane of Z=0 Bending of thin plate（薄板弯曲） Only transverse loads act on the plate. （垂直于板面的载荷，横向） Longitudinal loads: Plane stress State Similar with Bending of elastic beams 3 11

  5. Theory of Elasticity Page Chapter 11.1 Some Concepts and Assumptions Review: bending of beams 4 11

  6. Theory of Elasticity Page Chapter 11.1 Some Concepts and Assumptions Assumptions(beam): 1, The plane sections normal to the longitudinal axis of the beam remained plane (平面假设) 2, In the course “elementary strength of materials”: simple stress state :only normal stress exists, no shearing stress. Pure bending （单向受力假设） 5 11

  7. Theory of Elasticity Page Chapter 11.1 Some Concepts and Assumptions Assumptions for bending of thin plate ( Kirchhoff) Besides of the basic assumptions of “Theory of elasticity” 1,Straight lines normal to the middle surface will remain straight and the same length.变形前垂直于中面的直线变形后仍然保持直线，而且长度不变。 2,Normal stresses transverse to the middle surface of the plate are small and the corresponding strain can be neglected.垂直于中面方向的应力分量z, τzx， τzy远小于其他应力分量，其引起的变形可以不计. 3,The middle surface of the plate is initially plane and is not strained in bending.中面各点只有垂直中面的位移w，没有平行中面的位移 6 11

  8. or Theory of Elasticity or Page Chapter 11.1 Some Concepts and Assumptions 1,Straight lines normal to the middle surface will remain straight and the same length.变形前垂直于中面的直线变形后仍然保持直线，而且长度不变。 7 11 Physical Equation Reduced to 3

  9. Theory of Elasticity Page Chapter 11.1 Some Concepts and Assumptions 2,Normal stresses transverse to the middle surface of the plate are small and the corresponding strain can be neglected.垂直于中面方向的应力分量z, τzx， τzy远小于其他应力分量，其引起的变形可以不计. 8 11

  10. Theory of Elasticity Page Chapter 11.1 Some Concepts and Assumptions 3,The middle surface of the plate is initially plane and is not strained in bending.中面各点只有垂直中面的位移w，没有平行中面的位移 uz=0=0， vz=0=0， w=w(x, y) 9 11

  11. Theory of Elasticity Page Chapter 11.2 Differential Equation of Deflection弹性曲面的微分方程 Displacement Formulation The equilibrium equation is expressed in terms of displacement. w Besides w, the unknowns include Displacement: u, v Primary strain Components: Primary stess Components: Secondary stess Components: 10 11

  12. Theory of Elasticity Page Chapter 11.2 Differential Equation of Deflection u, v in terms of w uz=0=0， vz=0=0 u-ε Relations εx ,εy ,γxyin terms of w 11 11

  13. Physical Equations Theory of Elasticity Page Chapter 11.2 Differential Equation of Deflection x ,y ,τxyin terms of w 12 11

  14. Theory of Elasticity Page Chapter 11.2 Differential Equation of Deflection τxz,τyzin terms of w The equilibrium equation 13 11

  15. Theory of Elasticity Page Chapter 11.2 Differential Equation of Deflection zin terms of w If body force fz≠0: 14 11

  16. Theory of Elasticity Page Chapter 11.2 Differential Equation of Deflection The governing equation of the classical theory of bending of thin elastic plates: Flexural rigidity of the plate 15 11

  17. Geometrical Equations Physical Equations Theory of Elasticity Equilibrium Equations Boundary Cond. (load:q) Page Chapter 11.2 Differential Equation of Deflection +edges B.C. 薄板的弹性曲面微分方程 16 11

  18. Theory of Elasticity Page Chapter 11.2 Differential Equation of Deflection Another method to get the equation 17 11

  19. Beam Thin plate Theory of Elasticity Page Chapter 11.2 Differential Equation of Deflection History of the Equation Bernoulli, 1798: Lagrange, 1811: 18 11

  20. Theory of Elasticity Page Chapter 11.3 Internal Forces of Thin Plate Internal Forces: Stress resultants: It is customary to integrate the stresses ovet the constant plate thickness defining stress reslultants.薄板截面的每单位宽度上，由应力向中面简化而合成的主矢量和主矩。 Design requirement(薄板是按内力来设计的；) Dealing with the Boundary Conditions(在应用圣维南原理处理边界条件，利用内力的边界代替应力边界条件。) 19 11

  21. x z y Theory of Elasticity Page Chapter 11.3 Internal Forces of Thin Plate 20 11

  22. Theory of Elasticity Page Chapter 11.3 Internal Forces of Thin Plate Stress distribution 21 11

  23. Theory of Elasticity Page Chapter 11.3 Internal Forces of Thin Plate 22 11

  24. Theory of Elasticity Page Chapter 11.3 Internal Forces of Thin Plate 23 11

  25. 应力分量 和内力、载荷关系 名称 数值 最大 最大 较小 最小 Theory of Elasticity Page Chapter 11.3 Internal Forces of Thin Plate 24 11

  26. Theory of Elasticity Page Chapter 11.4 Boundary Conditions +edges B.C. Simply Supported edge简支边界 Free edge自由边界 Built-in or clamped edge固定边界 25 11

  27. Theory of Elasticity Page Chapter 11.4 Boundary Conditions Built-in or clamped edge固定边界 At a clamped edge parallel to the y axis: 26 11

  28. Theory of Elasticity Page Chapter 11.4 Boundary Conditions Simply Supported edge简支边界 Free to rotate The bending moment and the deflection along the edge must be zero. 27 11

  29. Theory of Elasticity Page Chapter 11.4 Boundary Conditions Free edge自由边界 Only 2 are allowed for an equation of 4th order 28 11

  30. Theory of Elasticity Page Chapter 11.5 Examples: Simple supported rectangular plate An application of plate theory to a specific problem Problem: Calculating the deflection w of a simply supported rectangular plate as shown in the fig., which is loaded in the z direction by a load of q(x,y) Solution: Boundary conditions: 29 11

  31. Theory of Elasticity Page Chapter 11.5 Examples:Simple supported rectangular plate The plate deflection must satisfy the following equation and the boundary conditions. Choose to represent w by the double Fourier series: All the boundary conditions are satisfied. Substituted into we obtain: 30 11

  32. Theory of Elasticity Page Chapter 11.5 Examples: Simple supported rectangular plate If q(x,y) were represented by Fourier series, It might be possible to match coefficients. Expand q(x,y) in a Fourier series. W 31 11

  33. Theory of Elasticity Page Chapter Homework • 9-1 32 11

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