Theory of Elasticity Chapter 10 Three-Dimensional Problems

Chapter Page 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 Introduction to Finite Element Method Bending of Thin Plates 10 1

Three-Dimensional Problems 10.1 Review: stress Formulation ( 按应力求解空间问题 ) 10.2 Torsion of straight bars (Prismatical Bars) （等截面直杆的扭转） 10.3 Elliptic Bars in torsion （椭圆面直杆的扭转） 10.4 Rectangular bars in torsion （矩形截面杆的扭转） 10.5 Membrane analogy of torsion （扭转的薄膜比拟） Chapter Page 10 2

10.1 Review ： Stress Formulation （按应力求解） Chapter Page Review: Stress Formulation 10 3 Equilibrium Equations 平衡方程 Geometrical Equations 几何方程 Physical Equations 物理方程 Eliminating the displacements and strains σ ε-u σ-εσ-ε 1 2 3

10.1 Review ： Stress Formulation （按应力求解） Chapter Page Eliminating the displacements 10 4

10.1 Review ： Stress Formulation （按应力求解） Chapter Page 6 compatibility equations may also be represented by the 3 independent fourth-order equations 10 5

10.1 Review ： Stress Formulation （按应力求解） Chapter Page Eliminating the strains using Hooke’s law and eliminate the strains in the compatibility relations incorporating the equilibrium equations into the system For the case with no body forces + the necessary six relations to solve for the six unknown stress components for the general three-dimensional case. B.D Simple Connected ( 单连通域 ) 10 6

10.2 Torsion of straight bars （等截面直杆的扭转） Chapter Page Examples: 10 7

10.2 Torsion of straight bars （等截面直杆的扭转） Chapter Page Torsion of circular shaft The corss section of the bar remain plane and rotate without and distortion Coulomb Assumptions( on the torsional deformation of cylinders of circular cross-section)  Each section rotates as a rigid body about the center axis.  For small deformation theory, the amount of rotation is a linear function of the axial coordinate.  Because of symmetry, circular cross-sections remain plane after deformation. 10 8

10.2 Torsion of straight bars （等截面直杆的扭转） Chapter Page For noncircular cross sections Naivier, also applied above assumptionsArrived at the erroneous conclusion. x y τ yz τ zx The lateral surface of the bar is free form external forces Naivier’s assumption in contradiction with above 10 9

10.2 Torsion of straight bars （等截面直杆的扭转） Chapter Page assumptions the following for general cross-sections  The projection of each section on the x,y-plane rotates as a rigid body about the central axis.  The amount of projected section rotation is a linear function of the axial coordinate.  Plane cross-sections do not remain plane after deformation, thus leading to a warping ( 翘曲 ) displacement 10 The Correct solution was ginven by Saint-Venant,1855 Saint-Venant’s Principle was proposed and applied

Chapter Page Torsion of straight bars （等截面直杆的扭转） O ， center of twist, where u ＝ 0 ， v ＝ 0.  The projection of each section on the x,y-plane rotates as a rigid body about the central axis. Deformation field bases on above assumptions  The amount of projected section rotation is a linear function of the axial coordinate.  Plane cross-sections do not remain plane after deformation, thus leading to a warping ( 翘曲 ) displacement assume that the cylinder is fixed at z =0 and  is the angle of twist per unit length.

10.2 Torsion of straight bars （等截面直杆的扭转） Chapter Page Stress Formulation the strain-displacement relationsHooke’s law 10 12

10.2 Torsion of straight bars （等截面直杆的扭转） Chapter Page Stress Formulation (semi-inverse) the equilibrium equations （ with zero body forces ） compatibility equations Hook’s law governing equations for the stress formulation. (1) (2) introducing a stress function Prandtl stress function Poisson equation 10 13

10.2 Torsion of straight bars （等截面直杆的扭转） Chapter Page Stress Formulation Boundary Conditions (1) the lateral surface is free of tractions

10.2 Torsion of straight bars （等截面直杆的扭转） Chapter Page Stress Formulation Boundary Conditions (2) the ends of the cylinder 10 15

10.2 Torsion of straight bars （等截面直杆的扭转） Chapter Page the equilibrium equations （ with zero body forces ） compatibility equations Boundary Conditions Summary Simple Connected 10 16

10.3 Elliptic Bars in torsion （椭圆面直杆的扭转） Chapter Page The boundary equation a stress function

10.3 Elliptic Bars in torsion （椭圆面直杆的扭转） Chapter Page The boundary equation a stress function

10.3 Elliptic Bars in torsion （椭圆面直杆的扭转） Chapter Page 10 18

10.3 Elliptic Bars in torsion （椭圆面直杆的扭转） Chapter Page Contour lines of the stress function Contour lines of the dipsplacement a positive counterclockwise torque Solid lines correspond to positive values of w dotted lines indicate negative values of displacement Along each of the coordinate axes the displacement is zero, With a =b(circular section), the warping displacement vanishes everywhere. If the ends restrained, normal stresses  z are generated as a result of the torsion

10.4 Rectangular bars in torsion （矩形截面杆的扭转） Chapter Page 10 20

10.4 Rectangular bars in torsion （矩形截面杆的扭转） Chapter Page 10 21

10.5 Membrane analogy of torsion （扭转的薄膜比拟） Chapter Page Membrane analogy Introduced by Prandtl,1903A.A.Griffith and G.I.Taylor, Further development

10.5 Membrane analogy of torsion （扭转的薄膜比拟） Chapter Page z  10 23

Homework Chapter Page 10 24