东南大学：《建筑力学 Architectural Mechanics》课程教学课件（英文讲稿）A16 Stress State

Stress Statemi@see.cn

Stress State mi@seu.edu.cn

ContentsTheStress State ofa Point（点的应力状态）·Simple,General&Principal StressState（简单、一般和主应力状态·OrderingofPrincipal Stresses（主应力排序）·On the DamageMechanisms of Materials（材料失效机制）·Plane Stress States（平面应力）·2-DCauchy'sRelation（平面柯西关系）·PlaneStressTransformation（平面应力转换）·Mohr's Circle（莫尔圆）·MaximumShearingStress（最大切应力）·Constructionof Mohr'sCircle（莫尔圆的构建）2

• The Stress State of a Point（点的应力状态） • Simple, General & Principal Stress State（简单、一般和主应力状 态） • Ordering of Principal Stresses（主应力排序） • On the Damage Mechanisms of Materials（材料失效机制） • Plane Stress States（平面应力） • 2-D Cauchy’s Relation（平面柯西关系） • Plane Stress Transformation（平面应力转换） • Mohr’s Circle（莫尔圆） • Maximum Shearing Stress（最大切应力） • Construction of Mohr’s Circle （莫尔圆的构建） Contents 2

Contents·3-DCauchy'sRelation（三维柯西关系）·StressTransformationofGeneral3-DStresses（空间应力转换）·ApplicationofMohr'sCirclein3-D（莫尔圆在三维应力状态中的应用）·GeneralizedHookesLaw（广义胡克定律）·TheRelationamongE,vandG（杨氏模量、泊松系数及剪切模量间的关系）·Fiber-reinforcedComposites（纤维增强复合材料）·Strain Energy Density under Generalized Stress States（一般应力状态下的应变能）·VolumeStrain,Mean Stress and Bulk Modulus（体应变、平均应力和体积模量）·Volumetric&DistortionEnergyDensity（体积应变能密度和畸变能密度）3

3 • 3-D Cauchy’s Relation（三维柯西关系） • Stress Transformation of General 3-D Stresses（空间应力转换） • Application of Mohr’s Circle in 3-D（莫尔圆在三维应力状态中的 应用） • Generalized Hooke’s Law（广义胡克定律） • The Relation among E, ν and G（杨氏模量、泊松系数及剪切模量 间的关系） • Fiber-reinforced Composites（纤维增强复合材料） • Strain Energy Density under Generalized Stress States（一般应力状 态下的应变能） • Volume Strain, Mean Stress and Bulk Modulus（体应变、平均应 力和体积模量） • Volumetric & Distortion Energy Density（体积应变能密度和畸变 能密度） Contents

The Stress State of a Point. The stress state at a point- is defined as the collection of the stress distributionson all planes passing through the point- can be analyzed via the stress distributions acting on adifferential cube, arbitrarily oriented with referencecoordinates. Method of differential cubes with:- infinitesimal side length- uniformly distributed stress on each of the six surfaces- equal and opposite stress on parallel sides4

• The stress state at a point - uniformly distributed stress on each of the six surfaces The Stress State of a Point - is defined as the collection of the stress distributions on all planes passing through the point. - can be analyzed via the stress distributions acting on a differential cube, arbitrarily oriented with reference coordinates • Method of differential cubes with: - infinitesimal side length - equal and opposite stress on parallel sides 4

Stresses on Obligue Planes. The stress distribution atLa point depends on theplane orientation alongwhich the point isexamined.=OCOS0Q0sin 2α25

F F A F F    • The stress distribution at a point depends on the plane orientation along which the point is examined. 2 cos sin 2 2           Stresses on Oblique Planes 5

Examples of the Stress State of a Point下H十6

F A F Me Me A F A B C A  A  A  B  C   Examples of the Stress State of a Point 6

Stress Under General Loadings: A member subjected to a generalcombination of loads is cut into4Atwo segments by a plane passingAFxthrough a point of interest QThedistributionofinternal stresscomponents may be defined as.AFxox= limAA4A->0PAVAVAAVlimlim=Txz=TxyAAAA->0△A->0AVAF.For equilibrium, an equal andoppositeinternalforceand stressdistributionmust be exerted ontheother segmentofthemember7

Stress Under General Loadings • A member subjected to a general combination of loads is cut into two segments by a plane passing through a point of interest Q • For equilibrium, an equal and opposite internal force and stress distribution must be exerted on the other segment of the member. A V A V A F x z A xz x y A xy x A x                lim lim lim 0 0 0    • The distribution of internal stress components may be defined as, 7

GeneralStress State of a Pointy·StresscomponentsaredefinedfortheplanesO,AAAAcut parallel to the x, y and z axes. ForFAATAAequilibrium, equal and opposite stresses areTuAAexerted on the hidden planes.to.OAA0.AA.The combination of forces generated by thestresses must satisfy the conditions forT-AATAAequilibrium:ZFx=ZF,=ZF_=0ZMx=ZM,=ZM,=0y: Consider the moments about the z axis:O.AAZM, = 0 = (txyA)a-(t xA)aWAATxy = TyxxyAAsimilarly, Ty- =Tay and Ty =tayG.AA:Itfollowsthatonly6components of stress areTxAArequiredtodefinethecomplete stateof stressAA楼0.8

• Stress components are defined for the planes cut parallel to the x, y and z axes. For equilibrium, equal and opposite stresses are exerted on the hidden planes. • It follows that only 6 components of stress are required to define the complete state of stress • The combination of forces generated by the stresses must satisfy the conditions for equilibrium: 0 0             x y z x y z M M M F F F     xy yx Mz xy A a yx A a        0     yz zy yz zy similarly,    and    • Consider the moments about the z axis: General Stress State of a Point 8

Principal Stress State of a Point. The most general state of stress at apoint may be represented by 6components,normal stressesOx,Oy,OzTxy,Tyz,Tzxshearingstresses(Note: Txy = Tyx, Ty- = Tay, Tzx = Tx). Same state of stress is represented by adifferent set of components if axes arerotated.: There must exist a unique orientation(Principal Directions) of thedifferential cube, having only normalstresses (Principal Stresses) acting oneach of its six faces9

• The most general state of stress at a point may be represented by 6 components, (Note : , , ) , , shearing stresses , , normalstresses xy yx yz zy zx xz xy yz zx x y z                • Same state of stress is represented by a different set of components if axes are rotated. • There must exist a unique orientation (Principal Directions) of the differential cube, having only normal stresses (Principal Stresses) acting on each of its six faces. Principal Stress State of a Point 9

Ordering of Principal Stress State03≥o:>00minintermax-9i ≥ 2 ≥Q3aPrincipalaxes & stresses. Stress states classified based on the number of zero principalstresses:- Tri-axial stress state: none zero principal stresses:- Biaxial stress state: one zero principal stresses;- Uniaxial stress state: two zero principal stresses;10

• Stress states classified based on the number of zero principal stresses: A 2 1 3 Principal axes & stresses    max inter min      1 2 3   Ordering of Principal Stress State - Tri-axial stress state: none zero principal stresses; - Biaxial stress state: one zero principal stresses; - Uniaxial stress state: two zero principal stresses; 10