ElasticityChapter2The Basic Theoryof the Plane Problem
1 Elasticity
The Basic Theory of the Plane ProblemChapter 2 TheBasic Theory of the Plane Problem$ 2-1 Plane StressProblemand Plane StrainProblemS2-2EquilibriumDifferentialEquationS 2-3 The Stress State at a Pointin a PlaneProblemS 2-4 GeometricalEquation.The Displacement of the Rigid Bodys 2-5 PhysicalEquationS2-6 BoundaryConditionsS 2-7 Saint-Venant'sPrinciple and Its Application$2-8 SolvingthePlaneProblemAccordingtotheDisplacementS 2-9 Solving the Plane ProblemAccording to the Stress.CompatibilityEquationsS 2-10The Simplification under the ConstantBodyForce.StressFunctionVExerciseLesson2
2 Chapter 2 The Basic Theory of the Plane Problem §2-1 Plane Stress Problem and Plane Strain Problem §2-2 Equilibrium Differential Equation §2-3 The Stress State at a Point in a Plane Problem §2-4 Geometrical Equation. The Displacement of the Rigid Body §2-5 Physical Equation §2-6 Boundary Conditions §2-7 Saint-Venant’s Principle and Its Application §2-8 Solving the Plane Problem According to the Displacement §2-9 Solving the Plane Problem According to the Stress. Compatibility Equations §2-10 The Simplification under the Constant Body Force. Stress Function Exercise Lesson
The Basic Theory of the Plane Problem金S2-1PlaneStress Problem andPlaneStrain problemIn actual problem, it is strictly Paying that any elastic body whose externalforce is a space force system is generally the space object. However, whenboth the shape and force circumstance of the elastic body have their owncertain characteristics, it can be concluded as the elasticity plane problem withthemechanical abstraction and appropriate simplification.The plane problem is divided into the plane stress problem and plane strainproblem.1.Plane stress problem串丰串Equal thickness lamella bears thesurface force that parallels with plateface and does not change along thethickness. At the same time, so does thevolumetric force.1T,= 00,=0 Tx=0Fig.2-13
3 1.Plane stress problem §2-1 Plane Stress Problem and Plane Strain problem In actual problem, it is strictly Paying that any elastic body whose external force is a space force system is generally the space object. However, when both the shape and force circumstance of the elastic body have their own certain characteristics, it can be concluded as the elasticity plane problem with the mechanical abstraction and appropriate simplification. The plane problem is divided into the plane stress problem and plane strain problem. Equal thickness lamella bears the surface force that parallels with plate face and does not change along the thickness. At the same time, so does the volumetric force. σz = 0 τzx = 0 τzy = 0 Fig.2－1
The Basic Theory of the Plane ProblemCharacteristics:1)The dimension of length and width is more larger than that ofthickness.2) The force along the plate edge is the surface force in parallel withplate face, and it is uniform distribution along the thickness. The bodyforce is in parallel with plate face and doesn't change along the thicknessof theplate,and thereare not external force onthefront and back oftheplane.VxT3Attention: Plane stress problem o,=O,but 8, + 0 ,this is contraryto that of the plane strain problem
4 x y Characteristics： 1) The dimension of length and width is more larger than that of thickness. 2) The force along the plate edge is the surface force in parallel with plate face, and it is uniform distribution along the thickness. The body force is in parallel with plate face and doesn’t change along the thickness of the plate, and there are not external force on the front and back of the plane. Attention: Plane stress problem z=0,but ,this is contrary to that of the plane strain problem. z 0
TheBasicTheoryofthePlaneProblem2.PlanestrainproblemVery long column bears the surface force in parallel with plate face anddoesn't change along the length of the column, at the same time, so does thebody force.8,=0 tzx=0 ty=0For example: dam, circular cylinder piping suffering from the internalpressure and the long horizontal laneway etc.Fig. 2-2Attention: Plane strain problem . = O, but , +0 , this is contrarytoplanestressproblem5
5 2.Plane strain problem Very long column bears the surface force in parallel with plate face and doesn’t change along the length of the column, at the same time, so does the body force. εz = 0 τzx = 0 τzy = 0 x Fig. 2－2 For example: dam, circular cylinder piping suffering from the internal pressure and the long horizontallaneway etc. Attention: Plane strain problem z = 0，but , this is contrary to plane stress problem. z 0 x y P