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Question: question 1 for each of the planestress conditions given below...

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Question 1 For each of the plane-stress conditions given below, construct a Mohrs circle of stress, find the principal stresses and the orientation of the principal axes relative to the x, y axes and determine the stresses on an element, rotated in the x-y plane 30° anticlockwise from its original position: Ox 400 MPa oy 100 MPa Ox-500 MPa Oy 50 MPa Txy 200 MPa Question 2 For each of the plane-stress conditions given in Question 1, use the matrix transformation law to determine the state of stress at the same point for an element rotated in the x-y plane 30° anticlockwise from its original position.
Question1 The state of stress at a point of an elastic solid is given in the x-y-z coordinates by: 300 0 0 [o]=10 250 4501 MPa 0 450 320 a) Using the matrix transformation law, determine the state of stress at the same point for an element rotated about the x-axis (in the y-z plane) 60° clockwise from its original position. Calculate the stress invariants and write the characteristic equation for the original state of stress. b) c) Calculate the deviatoric invariants for the original state of stres. d) Calculate the principal stresses and the absolute maximum shear stress at the point. What are the stress invariants and the characteristic equation for the transformed state of stress? e) Question 2 The 3-D state of stress is given by: 60 10 -80 oJ 10 50 20 MPa -80 20 30 Calculate: f the total stress (magnitude and direction with x, y, z axes) on a plane described by direction cosines: I =-05, m = positive, n = 0.5. g) the magnitude of normal and shear stresses on this plane. h) principal stresses and direction cosines of the principal planes. i) maximum shear stress.
Question 1 The state of strain at a point of an elastic solid is given in the x-y-z coordinates by: 003 00 0 002 .00s 0 .005 .004 j) Using the matrix transformation law, determine the state of strain at the same point for an element rotated about the x-axis (in the y-z plane) 30 clockwise from its original position. k) Calculate the strain invariants and write the characteristic equation for the original state of strain. l) Calculate the deviatoric invariants for the original state of strain. m) Calculate the principal strains and the absolute maximum shear strain at the point. n) What are the strain invariants and the characteristic equation for the transformed state of strain? Question 2 o) Write a matrix giving the components of the hydrostatic (mean) strain tensor. P) Evaluate the first, second and third invariants of the hy drostatic strairn tensor.
Question 1 Determine the engineering strain e and the true strain ε for each of the following situations: q) extension from L to 1.02 L r) compression from h to 0.98 h s) extension from L to 3 L t) compression from 3 h to h. u) compression to zero thickness. Question 2 A 40-mm diameter forging billet is decreased in height from 100 to 40. Assuming constant volume for plastic deformations: v) determine the average axial strain and the true strain in the direction of compression w) What is the final diameter of the forging? x) What are the transverse strains? Question 3 A 50-mm thick plate is decreased in thickness according to the following schedule: 25, 10, 5 mm. Calculate the total strain once on the basis of initial and final dimensions and as the summation of the incremental strains, using (1) conventional strain and (ii) true strain. Question 4 Show that the constancy of volume results in eltete,-0 and 5+5+5-o Why is the relationship for conventional strain valid only for small strains but the relationship for true strain is valid for all strains?
Question 1 The state of stress at a point of an elastic solid is given in the x-y-z coordinates by: [300 0 0 [o- 0 250 450 MPa 0 450 320 If E - 210 GPa, v 0.33, calculate the strain tensor. Question 2 It was found experimentally that a certain material does not change in volume when subjected to an elastic state of stress. What is Poissons ratio for this material? Question 3 Determine the volume of a solid steel sphere that is subject to a fluid pressure of 80 MPa, E-210 GPa, v 0.33.
Question 1 A material was tested under the proportionally increasing state of stress: Yielding was observed at o, 400 MPa. What would be the yield stress of this material under simple tensile loading? y) z) If the same material is used under the conditions such that σ-G-203 , at what value of a, will yielding occur? Solve the problem completely using: i. Von Mises yield criterion ii. Tresca yield criterion. Question 2 The Von Mises yield criterion for a 3D state of stress can be expressed as (12-. Derive an expression for the value of k in terms of material yield stress ơy from the first principles for a material which has a yield stress-ơy in simple tension. Do not use any pre-derived formula for the yield criterion. If a material with ơ,-250 MPa is subjected to the three-dimensional state of stress given by: 「600 100-2001 [o100 500 100 MPa -200 100 400 Use the result above to: i. calculate the equivalent Von Mises stress ii. determine if yielding will have occurred.
Question 3 A long, copper strip, 400 mm wide, 4 mm thick, was found to have 340 MPa yield stress. The strip is required to be rolled in order to reduce its thickness. In the rolling process, the width remains practically unchanged while the rolls apply pressure in the thickness direction. An additional tension of 160 kN is applied in the longitudinal direction to assist the forming process. Ignoring the change in the width and any friction effects, determine what roll pressure would just cause deformation: i. according to the Von Mises yield criteriorn ii. according to the Tresca yield criterion.
Question 1 A thin-walled tube, 300 mm long, has a wall thickness of 1.0 mm and an initial mean diameter of 120 mm. The plastic stress-strain curve for the material is given by ơ-500025 MPa . The tube is subject to axial load F and internal pressure p so that the stress ratio-a-3.0 at all times. The deformation continues until the effective stress equals 200 MPa. Ignoring ơr determine: aa) the effective strain at the end of the deformation process bb)the true strain components at the end of the deformation process cc) the final dimensions of the tube dd) the final axial load and internal pressure Assume that the axial stress in not affected by the internal pressure. Question2 A short cylindrical billet 16 mm diameter and 40 mm gauge length is elongated under tensile load to 50 mm and then compressed to 30 mm. The material characteristic curve under axial loading is given by σ-300. MPa Ignoring the elastic deformations and Bauschinger effect, determine: ee) the diameter of the billet after elongation the final diameter of the billet after recompression the equivalent strain and new yield stress of the elongated billet the final equivalent strain and the yield stress after recompression to 30 mm the force required to elongate the billet the force needed to recompress the billet back to 30 mm.
Question 3 A plate of mild steel, 10 mm thick, 1000 mm long and 500 mm wide is stretched plastically 25% in the longitudinal direction. The characteristic stress-strain curve of the material is given by (0-500 MPa) Assuming that the width is restrained from contracting by lateral end grips, calculate f final dimensions of the plate gg)state of stress at the end of the stretching process hh)final stretching load.
Question1 A beam shown below is made of an elastic-plastic material for which the yield stress is 360 MPa. Find the plastic moment M, and the maximum elastic moment M. Also determine the distribution of the residual stresses in the beam after the plastic moment M, is applied and then released. 40mm 40 mn 1ト225ート 00 mm 40 nm Fig O1. Question 1 (figure from Nash, Third edition, p.192) Question 2 A beam shown below is made of an elastic-plastic material for which the yield stress is 200 MPa. Find the plastic moment Mp and the maximum elastic moment Me. Also determine the distribution of the residual stresses in the beam after the plastic moment Mp is applied and then released. Also calculate the bending moment that would cause the flanges to yield completely, while the web stays elastic. 200 mm 13 mm 200mm Fig 02. Question 2 (figure from Hibbeler, Second SI edition, p.364)
Question 3 A beam of the cross section shown in Figure is subjected to bending moment about the x-x axis. The material is elastic plastic with yield stress of 250 MPa. Find the plastic moment and the maximum elastic moment. 25 mm $0 mm 50 mm 150 mm 25 mm Fig Q3. Question 3 (figure from Nash, Third edition, p.193)
Question 1 The continuous beam shown in Figure has a plastic moment Mp. Sketch the possible mode of failure and determine the magnitude of the load P for plastic collapse in terms of Mp. Use both Equilibrium and Energy methods. L 4.0 m. Fig O1. Question 1 (figure from Nash, Third edition, p.312) Question 2 The asymmetric frame shown below in figure is having both bases pinned as shown. The beam has a plastic moment Mp. Sketch the possible mode of failure and determine the magnitude of the force P for plastic collapse of the frame. Use both Equilibrium and Energy methods. Fig 92. Question 2 (figure from Nash, Third edition, p.312)
Question 3 The continuous beam shown in Figure has a plastic moment Mp. It rests on three simple supports and is subject to the two equal concentrated loads indicated. Determine the magnitudes of these loads for plastic collapse of the beam. Use both Equilibrium and Energy methods. Fig Q3. Question 3 (figure from Nash, Third edition, p.311)
Question1 A pivot-ended solid steel bar having a diameter of 30 mm and a length of 1.0 m will be used to support a compressive load. A factor of safety of 2.0 is specified and the eccentricity ratio 0.30. Determine: ii) The maximum safe compressive load ji) The implied value of eccentricity oy 300 MPa, E 200 GPa Question 2 The steel bar AB of the frame is pin-connected at its ends. Determine the factor of safety with respect to buckling about the y-y axis due to the applied loading. E 200 GPa, Yield Stress- 500 MPa A 18 kN 4 m 6 m 50 mm 50 mm Fig 92. Question 2 (figure from Hibbeler, Second SI edition, p.684)
Question 3 An aluminium hollow box column of square cross section is fixed at the base and free at the top. The width of each side is b 150 mm and the thickness t-10 mm. A compressive load midpoint of the side. What is the longest permissible length L of the column if the stress is not to exceed 100 MPa? (E 70 GPa). P-160 kN acts on the outer edge of the column at the 4 L. Section A-A Fig 03. Question 3 (figure from Gere and Timoshenko, Third edition, p.621)
Show that the ratio of the maximum tangential stress to the average tangential stress (7) for a thick-walled cylinder subjected only to internal pressure is: Question 1 1 + λ where λ = 스 . Plot in a diagram the value of η as function of λ· Question 2 Show that no matter how large the outside diameter of a cylinder, subjected only pressure, is made, the maximum tangential stress is not less than pi. Question 3 Design a thick-walled cylinder of a 80 mm internal diameter for an internal pressure of 80 MPa such as to provide a factor of safety of 3.0 against any yielding in the cylinder and a factor of safety of 4.0 against ultimate collapse. The yield stress of the material is 400 MPa. Use both linear and non-linear stress distribution methods of calculation. Question 4 A thick-walled cylinder of mild steel with internal diameter-50 mm and external diameter 200 mm is loaded by internal pressure. The yield strength of the material is 400 MPa. kk) Find the internal pressure required initiate yielding in the cylinder using Il) Find the plastic limit pressure at which the cylinder is fully plastic using mm) Compare the results obtained from parts a and b with those obtained nn)Assuming linear distribution, calculate the residual stresses when the Lames formulation. Sketch the stress distribution. non-linear formulation. Sketch the stress distribution. when a linear stress distribution throughout the wall thickness is assumed. plastic pressure is applied and then removed.
Question 5 A 500 mm OD steel cylinder with approximately a 300-mm ID is shrunk into another steel cylinder of 300 mm, O.D. with a 150-mm I.D. Initially the internal diameter of the outer cylinder was 0.2 mm smaller than the external diameter of the inner cylinder. The assembly was accomplished by initially heating the larger cylinder in oil. For both cylinders E 210 GPa and v 0.3 oo) Determine the pressure at the boundaries between the two cylinders.(Hint: the elastic increase in the diameter of the outer cylinder added to the elastic decrease in the diameter of the inner cylinder accommodate the initial interference between the two cylinders.) Determine the initial tangential and radial stress distributions caused by the interference pressure found in (a) with no additionally applied loads. Show the results on a plot. pp) qq) Determine the internal pressure to which the composite cylinder may be subjected without exceeding a maximum shear stress of 200 MPa in the inner cylinder. (Hint: after assembly, the cylinders act as one unit, the initial compressive stress in the inner cylinder is released first.) rr) Calculate the resultant stress distributions in the composite cylinder resulting from the internal pressure found in (c). Show the results on a plot.
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