Derive the expressions for stresses around a hole in a plate (in polar coordinates) that is under biaxial loading, as shown the figure below. Assume plane stress condition. Hint: Apply the superposition principle (stress-based solutions can be additively decomposed!) 02 1
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- 2. At a point in a brocket, the stresses on two mutually perpendicular planes are 180 MN/m? (tensile) and 70 MN/m2 (Compressive). The shear stress across the planes is 65 MN/m2. Find the following by Mohr's Circle method and compare with Analytical solutions: (i)The Normal stress on a plane making an angle of 50° with the plane of first stress. (ii)Shear stress on the plane (iii) Maximum shear stress (iv) Resultant stress and it's direction. (v) Major and minor principal stresses.(a) Construct Mohr's circle for an element in a uniaxial state of stress (b) Use this Mohr's circle to derive the equation for the normal stress On and shear stress En on the n-face. The equations should depend on Ox and θ (c) Use the Mohr's circle to determine the planes on which the maximum shear stress acts. Sketch a properly oriented maximum-shear-stress element and indicate the normal and shear stresses acting on its faces. Ơn nt o, Note: for (b), the angle θ is between 0 and 90° and is positiveQ2 The state of stress at a point of an elastic solid is given in the x-yz coordinates by: (a) Using the matrix transformation law, determine the state of stress at the same point for an element rotated about the x-axis 60° clockwise from its original position;(b) Calculate the stress invariants and write the characteristic equation for the originalstate of stress;(c) Calculate the deviatoric invariants for the original state of stress;(d) Calculate the principal stresses and the absolute maximum shear stress at the point;(e) What are the stress invariants and the characteristic equation for the transformed state of stress.
- 3.For a body under three dimensional stress state, describe the procedure for obtaining the absolute maximum shearing stress at a given point. What is meant by strain transformation? Describe two procedures for obtaining stress form Mohr’s circle for strain.1. For the two cases i.e. (a) 2D and (b) 3D shown below, find the principal stresses, maximum shear stresses and directions by using Mohr’s circle. Support your answers with diagrams. 41 GOMPa 60MPR Txy=40MPA 40MPA (a) (b)At point O of a part subjected to plane stress made of steel material (E = 210 GPa and %3D ww v= 0.3) The tensile components are given below. 240 %3D - 30 MPa *yz yz a) The principal stress components (01, 02 and o3) and their angles with respect to the x-y-z axes of the planes where they are formed calculate b) Calculate the maximum shear stress (tmax) and the angles of the plane in which it is formed with respect to the x-y-z axes. c) Drawthe 2-D and 3-D Mohr circles for the stress components. d) At point O in a section plane passing through point O and having an angle of 30° counterclockwise with the +x-axis Calculate the resulting stress components. e) Calculate the strain components (Exx, EW, Ezz, YXV, YXz and Yyz) and draw the 2-D and 3-D Mohr circles for the strain components.
- For the following data, using Mohr's circle of stress and trigonometry, (a) Find the principle stresses and show their sense in properly oriented element and (b) find the maximum (principle) shear stresses with the associated normal stresses and show the results on a properly oriented element.1. Given a combination of stresses on an element shown in the figure, (a) draw a Mohr's circle representing the complete state of stress of the element, i.e. show prinicpal stresses (01,02), maximum shear stresses (Tmax), principal direction (0p), original orientation, and original stresses (Ox, Oy, Txy). (b) draw three-dimensional Mohr's circles showing the three principal stresses and absolute maximum shear stress. Oy = -50 MPa Txy 20 MPa Ox = -150 MPaRod AB is fixed at side A with a length of 500m and a diameter of 20m. On side B, the rod experiences a torque of 2kN-m a compressive force of 10KN and a transverse force of 12kN. a) Find the stress state at center point C. b) Draw mohrs circle for the stress state. (Be sure to include principal values, A, B, sigma1 and sigma2) T= d%=20m 250m ZKN-m IOKN 7. A, 500m I2KN
- A 30 ?? square plate is subjected to uniform pressure as shown in Figure.1. The 2D loading stressesare described. If the modulus of elasticity is 200 ??? and the lateral constant (Poisson's ratio ? = 0.3). Estimatethe change in length in the vertical, horizontal and diagonal sides as well as determine the volumetric strain.Illustrate the effect of two-dimensionalstrain by constructing appropriate sketches (before and after deformation).Repeat the analysis for lower value of pressure and higher value in the range of [20 -100 MPa] and comment onthe overall results.Consider the following plane stress state: Ox=30 MPa, y= -60 MPa, Txy= 30 MPa cw Calculate the following: 1. The coordinates of the center of the Mohr's circle C The location of the center of the Mohr's circle Cis ( 2. Principal normal stresses (01, 02) The principal normal stresses are σ₁ = 39.08 3. Maximum shear stress (7) The maximum shear stress is 54.08 MPa. 4. The angle from the x axis to 0₁ (p) The angle from the x axis to 0₁ (p) is -16.85 5. The angle from the x axis to 7 (s) The angle from the x axis to T (s) is 28.15 6. The radius of the Mohr's circle The radius of the Mohr's circle is 54.08. ✰ MPa. MPa and 02 = -69.08 MPa. O -15 MPa, CW CCW O MPa).What are the equation used for part a)? for the axial stress, why do we not use Axial Stress = Pr/(2t)? Also, the math for the pricipal stresses is incorrect.