4b. Stress Transformation Equations & Mohr Circle-1.pptx

A short note on Mohr circle equations in Stress and Strength of Materials.

1. Course Topics Stress-strain relationships for general dimensional cases and special cases . 2. Stress transformation equations. Graphical determination of stresses and strains using Mohr’s circle. 4. Theories of failure. 5. Stress concentration and relief of stress concentration. Moments of inertia and area. Products of inertia and area. 8 . Unsymmetrical bending and shear centers in curved beams. 9 . Torsion and stress concentration. 1

2. Content - Why transform stresses ? - Applications of 2d stress transformation analysis - Assumptions - Derivation of transformation equations (plane stress). - Examples - Practice Questions - Assignment 2

3. Calculating Stresses in a Solid Body   One of the key concepts taught in the Mechanics of Materials course concerns the nature of stress in a solid body. Internal forces in a body produce stresses that: tend to elongate or contract the material and stresses that tend to warp or skew the material. Students learn how to calculate stresses that arise in a variety of structural and mechanical components . In the typical problem-solving process, internal forces initially are considered individually.

4. For example, consider a simple component such as the elbow (pipe) shown in the figure below. Two forces, P and W, act on the end of the pipe. Force P: pulls on the component , causing elongation at points A and B. - also causes the pipe to bend about the vertical (y) axis. Force W: causes the pipe to twist about its longitudinal (x) axis. also causes the component to bend about the horizontal (z) axis. Each of these effects produce stresses in the pipe component.

5. Furthermore, the stresses produced in the pipe depend on which point we choose to examine. Stresses produced at point A by loads P and W are different from the stresses produced at point B . To make matters worse, the individual stresses produced at either point A or point B by the two loads are calculated in the x, y, and z directions. The combined effect of all stresses acting at a point will produce stresses in the pipe material that act simultaneously in all directions, and, in general, the largest stresses will not occur in the x, y, or z directions. Consequently , the engineer must be able to consider all possible combinations of stress acting in any direction.

6. To properly design the pipe component, the engineer must: - be able to compute each of the stresses acting at any point. from this set of stresses in the x, y, and z directions, compute the most critical stresses acting at any possible orientation. To assess the combined effect of stresses on a solid body, stress transformation relationships are used. Based on the necessity of satisfying equilibrium conditions, a set of equations can be derived that expresses the variation of two types of stress, normal stress and shear stress, for any orientation with respect to the original xyz coordinate system. Although abstract in nature, stress transformations are an important tool necessary to design components such as beams and pipes that are safe and reliable.

7. Applications of 2D Stress T ransformation

8. 8 Other Applications …..

9. Practical Applications of 2D stress transformation

10. ASSUMPTIONS Material is homogenous and isotropic. Material operates in the region of elastic regime. Assumption of state of plane stress exists within the material. Thus, normal and shearing stresses on the z-face of element is assumed equal to zero. i.e. σ z = τ zx = τ zy = 0 By implication no forces are exerted on the z faces of the element. 10

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12. Derivations  x’ ,  y’,  x’y ’ as a function of  x,  y,  xy and θ . Transformation equation Vs Equation of circle. Mohr Circle. Principal planes . Principal stresses. Maximum shearing stress planes. Maximum shearing stress. Important Notes. 12

13. STRAIN ROSSETE

14. The material stress may only be calculated from equation: σ = E ε , if the elongation in the force direction has been measured and the stress state is single-axis.

15. Class Practice 1 For the initial stress element shown, employing Mohr circle approach, determine the principal stresses. the maximum shearing stress and corresponding normal stress. principal planes, maximum shearing plane show these values on properly oriented stress elements. y x 300 MPa 400 MPa 200 MPa Initial stress element Y X 16

16. Class Practice 2 For the state of plane stress shown in the figure below, using Mohr’s circle determine the principal planes, maximum shearing stress plane, principal stresses and the maximum shearing stress. Show these values on properly oriented stress elements. y x 10 MPa 50 MPa Initial stress element B A 40 MPa 17

17. Example 1 For the given state of stress shown on the right, determine the normal and shearing stresses after the element shown has been rotated thru: 25 o clockwise (+ ve ) 10 o counter clockwise (- ve ). Using MS Excel determine the normal and shear stresses for 0 o to 9 o element rotation using a step of 1 O . Subsequently determine the i . maximum normal stress. ii. minimum normal stress. iii. maximum shearing stress. (e) What are the corresponding faces of the stresses determined in (d). 60 MPa 20 MPa 40 MPa

18. Assignment Read up slide 6 – Strain and Deformation 19

19. ( b) (c) ( a) ( d) Example 2 For the given state of stress, determine the normal and shearing stresses exerted on the oblique face of the shaded triangular elements shown. 70 o 4ksi 3 ksi 8ksi 75 o 10ksi 6 ksi 4 ksi 55 o 80MPa 40MPa 60 o 60MPa 90MPa 20

20. Example 3 For the plane stress shown below, determine the principal planes, the principal stresses, the maximum shearing stress and the maximum shearing stress plane. Also provide the corresponding normal stress. (d) Provide the orientation of the element at this stage. 21