Theory of Failure Question 1: The uniaxial yield stress of a material is 866 MPa. According to von Mises criterion, the shear yield stress (in MPa) of the material is -
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- Q.5 A component of machine made of steel which resist a biaxial state of stress o,= -120 MPa and t = 100 MPa. Check whether this part is safe or fail using 1. Max. principal stress theory 2. Max principal strain theory 3. Max. distortion energy theory.The stresses at a point in a bar are o1 = 28N/mm2 (tensile) and o2=12N/mm2 (compressive) as shown in Figure (a) Determine the normal, tangential and resultant stress in magnitude and direction on a plane inclined at 55° to the axis of the major stress. (b) Determine the maximum intensity of shear stress in the material at the point. (c) Also, solve the same problem using Mohr's circle method and compare the results. O2 Axis of Major Stress 55°Q. No .3 On the free surface of an aluminum [E = 10,000 ksi; v = 0.33] component, the strain rosette shown in the figure was used to obtain the following normal strain data: = -500 µɛ, E = -220 µe, and &e = +600 µɛ. Determine the normal stress that acts along an axis that is rotated at an angle of 0= 45° coun- terclockwise from the positive x axis. 60 60 60
- 2. A copper rod of diameter 12 mm is subjected to a compressive force of 8 kN. The change in diameter happens to be 4.5 x 103 mm. Determine the strain along the length of the material and the Poisson's ratio when E = 110 GPa.A certain material has a yield stress limit in simple tension of 387 N/mm². The yield limit in compression can be taken to be equal to that in tension. The material is subjected to three stresses in mutually perpendicular directions, the stresses being in the ratio 3 : 2: -1.8. Determine the stresses that will cause failure according to the von Mises and Tresca theories of elastic failure. Ans. Tresca: o1 = 241.8 N/mm², ơ1=161.2 N/mm², o=-145.1 N/mm?. von Mises: o1= 264.0 N/mm², ơn=176.0 N/mm², Om=-158.4 N/mm?.A homogenous isotropic material of size 20 cm x cm x 20 cm is placed inside a rigid box of internal dimension 20 x 20 x 20 cm. The top of the box is open and the material inside is subjected to a uniform compressive stress of 10 N/mm² from the top. Assuming there is no friction between the surface of contact, find the stresses on the side faces of the material. Also find maximum shear stress, the normal stresses on the plane of maximum shear stress and the change of volume of the material. Young's modulus E is 1 x 105 N/mm² and Poisson's ratio, m = 0.3 for the material. S 10 N/mm² 20 cm 20 cm-
- PROBLEM 1 A circular aluminum tube of L=600mm is loaded in by compressive forces at its ends. The outside and inside diameters are 75mm and 63mm respectively. A strain gage is placed on the outside of the tube to measure nomal strains in the longitudinal direction. E=73 GPa and Poisson's is 0.33. a. If the normal stress in the tube is 57 MPa, what is the load P? b. if the strain is 781 X 10-6, what is the shortening of the tube? What is the percent change in its cross-sectional area.3 - A strain gauge is attached to the side of a beam, as shown by the cross-section view in the figure. When the beam is subjected to a specified uniform moment, the longitudinal strain measured by the gauge is -300μ. Plot the axial stress profile over the height of the cross- section. What is the magnitude of the moment that produced this strain gauge reading? Assume E = 75 GPa. 10↑ Strain gauge dimensions in mm 30 30 beamcross-sectionFigure Q4 shows a piece of triangular thin plate ABC with thickness 5 mm. A linear pressure as shown in the figure is being applied to edge AB with maximum intensity of 100 kN/m² at the middle of edge AB. The plate is made from stainless steel, with a Young's modulus of 200 GPa and Poisson's ratio of 0.3. The deformations in the plate are to be obtained. For such purposes, the finite element method is being employed. (a) (b) (c) (d) (e) State the type of problem involved in this case (to determine the stiffness matrix to be used). Model this problem optimally using the least number of orthogonal triangular elements only. Label the nodes and elements and indicate clearly the status of all the relevant degrees-of- freedom in this model. Set up the appropriate system of simultaneous equations and solve for the unknown displacements. Displacements at point T. Without any calculation, explain on the value of strain at points S and T.
- A square plate of side 300 mm is subjected to pure shear of intensity 100 MPa as shown in figure. Young's modulus of the material is 2 x 105 MPa and Poisson's ratio is 0.2. Find the algebraic sum of principal strains. 100 MPa B 100 MPa A DQ.70 ox= X The state of stress at a point is given by : = 80 MPa, oy=100 MPa = and txy = 60 MPa. If the yield strength for the material is 150 MPa, as determined in a uniaxial test, then the maximum shear stress is, nearly A 75.5 MPa B с D 127.4 MPa 119.3 MPa 104.0 MPa14. For a linear elastic material with Young's modulus E=20 MPa: a) Calculate the amount of longitudinal strain ewhen a specimen made of this material is subjected to uniaxial tension at the stress level o =6 КРа. b) Draw the stress-strain diagrams corresponding to the respective loading protocols e1(t) and ɛ 2(t) given in the figure below Loading e (1) Loading e:(t) Ei (t) (1) 0.04 10 20 t(s)