2Components of StressElementary Cube: A infinitesimally small cube (dimensions: dxxdyxdz) is considered to get an insider’s view on the stresses acting inside the material of the beam.Nine components of stress are acting. (A stress tensor can be defined.)For homogenous, isotropic material,Hence, three normal stress components and three shear stress components can be considered.
4Typical cases of stress STRESS IS DIRECTIONAL Uniaxial Stress: An cube element of the body when subjected only to normal stresses along one direction only, the one-dimensional state of stress is referred to as uniaxial tension or compression. (Everything that has been done so far) (or in other words only is acting on the elementary cube. Biaxial Stress: An cube element of the body when subjected only to normal stresses along two perpendicular directions. In this case,
5Typical Cases of Stresses (that are modelled) Tri-axial Stress: An cube element of the body when subjected only to normal stresses, acting in mutually perpendicular directions it is said to be in a state of triaxial stress. The absence of shearing stresses indicates that the preceding stresses are the principal stresses for the element.
6Typical cases of stress Pure Shear: In this case, the element is subjected to plane shearing stresses only. The planes ab and cd are in pure shear. Typical pure shear occurs over the cross sections and on longitudinal planes of a circular shaft subjected to torsion.
7Typical Cases of Stresses (that are modelled) Plane Stress: An elementary cube of a body under stress when subjected only to biaxial stresses only on the x and y faces of the element, then the body is in plane stress. The stresses can be both normal stresses and/or shear stresses. This set of equation represent a state called plane stress and also Hooke’s law for 2D stress. (Extend it to 3D!!!)
8Stress ElementStress element can be defined as stress gauge. Similar to strain gauge it helps to measure what is happening inside the body under stress.We will be concentrating only on 2D stress element for simplicity (and sanity!)2D stress elements are actually an infinitesimally small layer cut out of the elementary cube under plane stress conditions.This will help us to know what is going on with stresses inside!!
9Stresses on Inclined Planes Consider an inclined section of a prismatic bar. For equilibrium, Assume no shearing stress working on the inclined plane. Let’s name the inclined plane pq. The force can be resolved to a component perpendicular to the plane pq, called And, parallel to the plane pq, The area of the inclined plane pq,
10Stresses on Inclined Planes Normal stress on pq, Shear Stress on pq,
11Stresses on Inclined Planes So, what is the maximum normal stress?And maximum shear stress,The formulae are valid in tension (positive) and compression (negative).
13Shear vs. Normal Stress For , 𝜎 𝑥 = 1MPa theta (degree) normal stress shear stress1.000.00150.930.25300.750.43450.5060750.0790105-0.25120-0.43135-0.50150165180
14Shear vs. Normal Stresstheta (degree)normal stressshear stress1.000.00150.930.25300.750.43450.5060750.0790105-0.25120-0.43135-0.50150165180On the circle, the highlighted point is the topmost point on the circle. From the right most point (1,0) if we move anticlockwise, then the points are located at 2 angle.
15This circle is called Circle of Stress X-axis normal stress, Y-axis shear stressPt. A Surface pq normal to axis of bar,Pt. O Surface parallel to the axis of bar,To obtain a point in this circle corresponding to the stresses on a plane at an angle of inclination it is only necessary to measure in counter-clockwise direction from point A an arc subtending an angle
16Circle of Stress Take a plane normal to pq, called mm The sum of normal stresses acting on two perpendicular planes remain constant and shearing stresses acting on two perpendicular planes are equal but opposite in direction.
26Plane Stress Hence, on the inclined plane now the stresses are, There is an longitudinal stress , a transverse stress , and a shear stress.We have worked so far with axial stresses in two directions, now add a shear stress component to it.Hence, on the inclined plane now the stresses are,
27Stress Transformation So now add the part for shearing force, Mohr’s Circle for Planes Stress
28Principal StressThe maximum and minimum normal stresses are called the principal stresses. On the x-axis of Mohr’s circle, the maximum and minimum x-ordinate values are the principal stress values. (the red dots on the circle) In case of plane stress situation, the principal stresses are,
29Maximum Shear StressFrom Mohr’s circle of plane stress, we can say, the maximum shear stress is at the top of the Mohr’s circle. Which is equal to the radius of the circle,
30Principal planesThe angle of the shaded triangle is double the angle of the principal plane (with maximum normal stress).The principal planes are perpendicular to each other.The most important stress element is the one that is along the principal plane showing principal stresses.