Presentation on theme: "Analysis of Stress and Strain"— Presentation transcript:
1 Analysis of Stress and Strain Review:- Axially loaded Bar- Torsional shaftptntsnqtxyhPPqtyxQuestions:(1) Is there any general method to determine stresses on any arbitrary planeat one point if the stresses at this point along some planes are known?(2) For an arbitrary loaded member, how many planes on which stresses areknown are required to determine the stresses at any plane at one point?
2 Analysis of Stress and Strain sysyState of stress at one point:tyxtyxtyztxytxytxyStress element:ytzysxsxsxsztzxtxztyxxsyzUse a cube to represent stress element. It is infinitesimal in size.(x,y,z) axes are parallel to the edges of the elementfaces of the element are designated by the directions of theiroutward normals.Sign Convention:Normal stresses: “+” tension; “-” compression.Shear stresses: “+” the directions associated with its subscripts areplus-plus or minus-minus“-” the directions associated with its subscripts areplus-minus or minus-plus
3 Plane StressDefinition: Only x and y faces are subject to stresses, and allstresses are parallel to the x and y axes.Stresses on inclined planestxyqsxtyxsyTransformation equations forplane stress
4 Transformation Equations angle between x1 and x axes, measured counterclockwise
5 Plane Stress – Special Cases Uniaxial Stress:sxsxtyxPure Shear:txytxytyxsyBiaxial Stress:sxsxsy
6 Plane StressExample 1: A plane-stress condition exists at a point on the surface ofa loaded structure, where the stresses have the magnitudes and directionsshown on the stress element of the following figure. Determine the stressesacting on an element that is oriented at a clockwise angle of 15o withrespect to the original element.
7 Principal StressesPrincipal stresses: maximum and minimum normal stresses.Principal planes: the planes on which the principal stresses actThe angle defines the orientation of the principal planes.
14 Construction of Mohr’s Circle Approach 1: For the given state of stresses, calculate and R. The centerOf the circle is ( , 0) and the radius is R.
15 Construction of Mohr’s Circle Approach 2: Find points corresponding to q = 0 and q = 90o and then draw a line.The intersection is the origin of the circle.
16 Applications of Mohr’s Circle Example 4: An element in plane stress at the surface of a large machineis subjected to stressesUsing Mohr’s circle, determine the following quantities: (a) the stressesacting on an element inclined at an angle of 40o, (b) the principal stressesand (c) the maximum shear stress.
17 Plane StrainDefinition: Only x and y faces are subject to strains, and allstrains are parallel to the x and y axes.Note: Plane stress and plane strain do not occur simultaneously.