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# Mechanics of Materials: Question Bank from old VTU Question papers

This document contains: Mechanics of Materials: Question bank from old VTU Question papers ; Pprepared by Hareesha N G, DSCE, Bengaluru. These questions are picked from last 06 years of old VTU question papers.

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### Mechanics of Materials: Question Bank from old VTU Question papers

1. 1. 2015 Hareesha N G Assistant Professor Department of Aeronautical Engineering Dayananda Sagar College of Engineering Bengaluru-78 Assignment Questions: Unit-wise Mechanics of Materials
2. 2. Assignment Questions: Unit-wise Mechanics of Materials Compiled by: Hareesha N G, Assistant Professor, DSCE, BLore Page 1 DAYANANDA SAGAR COLLEGE OF ENGINEERING DEPARTMENT OF AERONAUTICAL ENGINEERING Assignment Questions Unit wise Subject: Mechanics of Materials Sub Code: 10ME34 Faculty In charge: Hareesha N G UNIT1: Simple Stress and Strain: Introduction, Stress, strain, mechanical properties of materials, Linear elasticity, Hooke's Law and Poisson's ratio, Stress- Strain relation - behaviour in tension for Mild steel, cast iron and nonferrous metals, Extension / Shortening of a bar, bars with cross sections varying in steps, bars with continuously varying cross sections (circular and rectangular), Elongation due to self weight, Principle of super position. 1 a) Define (i) Stress (ii) Hook's law (iii) Elasticity (iv) Lateral strain. 4 Jul 14 1 b) Explain stress-strain relationship showing salient points on the diagram 6 Jul 14 1 c) A stepped bar is subjected to an external loading as shown in Fig. Ql (c). Calculate the change in the length of bar. Take E = 200 GPa for steel. E = 70 GPa for aluminium and E = 100 GPa for copper. 10 Jul 14 Fig. Q1 (c) Fig. Q2 (c) Fig. Q. 3(b) 2 a) Define: i) True stress ii) Factor of safety, iii) Poisson’s ratio ,iv) Principle of superposition 4 Jan 14, Jul 13 2 b) A bar of uniform thickness *t* tapers uniformly from a width of b1 at one end to b2 at other end, in a length of l. Find the expression for the change in length of the bar when subjected to an axial force P. 8 Jan 14 2 c) A vertical circular steel bar of length 3l fixed at both of its ends is loaded at intermediate sections by forces W and 2W as shown in Fig.Q2(c). Determine the end reactions if W= 1.5 kN. 8 Jan14 3 a) The tensile lest was conducted on a mild steel bar. The following data was obtained from the test: Diameter of steel bar= 16mm; Gauge length of the bar = 80mm; Load at proportionality limit = 72 kN; Extension at a load of 60 kN = 0.115mm; Load at failure = 80 kN; Final gauge length of bar = 104mm; Diameter of the rod at failure = 12mm Determine: i) Young's modulus; ii) Proportionality limit, iii) True breaking stress and iv) Percentage elongation. 10 Jul 13 3 b) A brass bar having cross-sectional area 300mm' is subjected to axial forces as .shown in Fig. Q. 3(b). Find the total elongation of the bar. E = 84 GPa. 10 Jul 13 4 a) Derive an expression for the deformation of a tapered bar of circular section, subjected to tensile load. 10 Jul 13 4 b) A steel bar of cross section 500mm2 is acted upon by forces shown in Fig.Q.4(b). Determine the total elongation of the bar. Take E = 200GPa. 10 Jul 13 Fig. Q 4(b) Fig.Q.8a Fig. Q.7(a) Fig. Q.7 (b) 5 a) Define: i) Ductility, ii) True stress, iii) FOS iv) Young’s modulus, v) shear strain 5 Dec 2012 5 b) Sketch the typical stress-strain curve for the following cases and explain briefly. i) Mild steel under compression ii) Brittle materials iii) Aluminium iv) Hard and soft rubber 8 Jun 10, 11 Jan 10 5 c) Derive an expression for the extension of uniformly tapering rectangular bar subjected to axial load P. 7 Dec 12
3. 3. Assignment Questions: Unit-wise Mechanics of Materials Compiled by: Hareesha N G, Assistant Professor, DSCE, BLore Page 2 6 a) Obtain an expression for elongation due to its self-weight. 6 Jun 08 6 b) A steel wire of S mm diameter is used for lifting a load of 1.5 kN at its lower end. the length of the wire being 160 m. Calculate the total elongation of the wire taking E = 2 x 105 N/mm2 and unit weight of steel = 78 kN/m3 8 Jun 08 6 c) Obtain an expression for total elongation of stepped bar with suitable sketch. 6 Jun 08 7 a) A vertical prismatic bar is fastened at its upper end and supported at the lower end by an unyielding floor as shown in Fig Q7(a). Determine the reaction R exerted by the floor of the bar if external loads P1 = 1500 N and P2 = 3000 N are applied at the intermediate points shown. 6 7 b) A compound bar consisting of Bronze, Aluminium and Steel segments is loaded axially as shown in Fig.Q7(b). Determine the maximum allowable value of P if the change in length of the bar is not to exceed 2mm and the working stresses in each material of the bar, indicated in table below is not to be exceeded. Material Area (mm2 ) Elastic modulus E (MPa) Working stress (MPa) Bronze 450 0.83x105 120 Aluminium 600 0.70x105 80 Steel 300 2x105 140 14 Jul 11 8 a) A bar of 800mm length is attached rigidly at A and B as shown in Fig. Q.8a. Forces of 30 kN and 60 kN act as shown on the bar. If E=200 MPa, determine the reactions at the two ends. If the bar diameter is 25 mm, find the stresses and change in length of each portion. 10 Jun 10 8 b) The extension of a bar uniformly tapering from a diameter of d + a to d- a in a length L is calculated by treating it as a bar of uniform cross-section of average diameter d. What is the percentage error? 10 Jun 10 9 a) A bar of length 1000 mm and diameter 30 mm is centrally bored for 400 mm, the bore diameter being 10 mm as shown in Fig. Q.9a. Under a load of 25 kN, if the extension of the bar is 0.185 mm, what is the modulus of elasticity of the bar? 10 Jul 12 9 b) A stepped bar having circular sections of diameter 1.5D and D is shown in Fig. Q.9b. If ρ and E are the density and Young's modulus of elasticity respectively, find the extension of the bar due to its own weight. 10 Dec 10 Fig.Q.9a Fig. Q.9b 10 a) Two vertical rods one of steel and the other of copper are each rigidly fixed at the top and 500mm apart. Diameters and lengths of each rod are 20mm and 4m respectively. A cross bar fixed to the rods at the lower ends carries a load of 5kN, such that the cross bar remains horizontal even after loading. Find the stress in each rod and the position of the load on the bar. Take Es = 2x105 N/mm2 and Ec = 1x105 N/mm2 . 10 Jun 11 10 b) A compound bar consists of a circular rod of steel of diameter 20 mm rigidly fitted into a copper tube of internal diameter 20 mm and thickness 5 mm. If the bar is subjected to a load of 100 KN, find the stresses developed in the two materials. Take Es = 200GPa and Ec = 120GPa. 10 Jun 11
4. 4. Assignment Questions: Unit-wise Mechanics of Materials Compiled by: Hareesha N G, Assistant Professor, DSCE, BLore Page 3 UNIT 2: Stress in Composite Section: Volumetric strain, expression for volumetric strain, elastic constants, simple shear stress, shear strain, temperature stresses (including compound bars). 1 a) Define lateral strain, linear strain and volumetric strain. Derive an expression for volumetric strain for rectangular plate and cylindrical rod of length L. 10 Jul 11 1 b) The composite bar shown in Fig. Q.1b. is 0.2 mm short of distance between the rigid supports at room temperature. What is the maximum temperature rise which will not produce stresses in the bar? Find the stresses induced when temperature rise is 40°C. Given: As : Ac = 4:3, αs = 12x10-6 /°C, αc = 17.5 x 10-6 /°C; Es= 2x105 N/mm2 , Ec= 1.2 x 105 N/mm2 10 Jul 11 Jan 10 2 a) Derive the relationship between E, G and K. 06 Jun 12 2 b) AB is a rigid bar and has an hinged support at C as shown if fig. Q.2b. A steel and an aluminium bar support it at ends A and B respectively. The bars were stress free at room temperature. What are the stresses induced, when the temperature rises by 40°C? Jun 12 Jan 10 3 a) Prove that volumetric strain is equal to sum of the three principal strains. 06 Jun 10 3 b) A compound bar is made of a central steel plate 60mm wide and 10mm thick to which copper plates 40mm wide and 5mm thick are connected rigidly on each side. The length of the bar at normal temperature is 1 meter. If the temperature is raised by 80o C, determine the stresses in each metal and change in length. Take Es = 200 GPa ; Ec = 100 GPa; αs= 12x10-6 /°C ; αc= 17x10-6 /°C 14 Dec 12 Jun 12 4 a) Establish a relationship between the modulus of elasticity and the modulus of rigidity. 06 Jul 14, 11 4 b) A rigid bar ABC is pinned at A and is connected by a steel bar CE and a copper bar BD as shown in Fig. Q4.b. If the temperature of the whole assembly is raised by 40°C, find the stresses induced in steel and copper rods. Given: For steel bar For copper bar Area 400mm2 600 mm2 Modulus of elasticity 2x105 N/mm2 1x105 N/mm2 Coefficient of thermal expansion 12x10-6 / °C 18x10-6 / °C 14 Dec 10 Fig.Q.1b Fig. Q.4b 5 a) Explain the following. 1) Youngs modulus, 2) Regidity modulus, 3) Bulk Modulus, 4) poisons ratio with help of a neat sketch. 06 Jul 14 5 b) A steel tube of 25mm external diameter and 18mm internal diameter encloses a copper rod of 15mm diameter. The ends are rigidly fastened to each other. Calculate the stress in the rod and the tube when the temperature is raised from 15° to 200°C. Take Es = 200 GPa ; Ec = 100 GPa; αs= 11x10-6 /°C ; αc= 18x10-6 /°C 14 Jul 11 Jun 10 6 a) Establish a relationship between the modulus of elasticity and the bulk modulus. 06 Dec 11 Dec 10 6 b) A 12 mm diameter steel rod passes centrally through a copper tube 48 mm external diameter and 36 mm internal diameter and 2.50 m long. The tube is closed at each end by 24 mm thick steel plates which are secured by nuts. The nuts are tightened until the copper tube is reduced in length by 0.508 mm. The whole assembly is then raised in temperature by 60°C. Calculate the stresses in copper and steel before and after raising the temperature, assuming the thickness of the plates remain to be unchanged. Take Es = 210 GPa ; Ec = 105 GPa; αs= 1.2x10-5 /°C ; αc= 1.75x10-5 /°C 14 Jul 14 Jul 11 7 a) Obtain the expressions for stresses in the bars when i) Both ends of the bar are rigidly fixed, ii) some gap is left in one support and other is fixed. 06 The modulus of rigidity of a material is 51 GPa. A 10 mm diameter rod of this material was subjected to an axial pull of 10 kN and the change in diameter was 0.003 mm. Calculate Poisson's ratio and Young's modulus. Given an elongation of 0.1 mm on a gauge length of 100 mm. 14 Jul 13 8 a) Obtain an expression for thermal stresses in compound bars taking a suitable sketch. 06 8 b) A 500 mm long bar has rectangular cross-section 20 mm * 40 mm. This bar is subjected to i) 40 kN tensile force on 20mm * 40 mm faces, ii) 200 kN compressive force on 20 mm * 500 mm faces, and iii) 300 kN tensile force on 40 mm * 500 mm faces. Find the change in volume if E= 2 * 105 N/mm2 and  = 0.3. 14 Dec 11 9 a) Explain the following; i) Volumetric strain ii) Shear strain iii) Thermal stresses iv) Poisons ratio 08 Dec 11 9 b) A rectangular bar of cross-section 30 mm * 60 mm and length 200 mm is restrained from 12 Dec 12 Fig.Q2b
5. 5. Assignment Questions: Unit-wise Mechanics of Materials Compiled by: Hareesha N G, Assistant Professor, DSCE, BLore Page 4 expansion along its 30 mm * 200 mm sides by surrounding material. Find the change in dimension and volume when a compressive force of 180 kN acts in axial direction. Take E=2* 105 N/mm2 and =0.3. What are the changes if surrounding material can restrain only 50% of expansion on 30 mm x 200 mm side? 10 a) A bar of rectangular cross section is subjected to stresses x, y and z, in x, y and z directions respectively. Show that if sum of these stresses is zero, there is no change in volume of the bar. 10 Jan 14 10 b) Rails are laid such that there is no stress in them at 24°C. If the rails are 32 m long. determine: i) The stress in the rails at 80°C, when there is no allowance for expansion ii) The stress in the rails at 80°C. when there is an expansion allowance of 8 mm per rail. iii) The expansion allowance for no stress in the rails at 80°C. Coefficient of linear expansion  = 11 x 10-6 /°C and Young's modulus E = 205 GPa. 10 Jan 14 Dec 11
6. 6. Assignment Questions: Unit-wise Mechanics of Materials Compiled by: Hareesha N G, Assistant Professor, DSCE, BLore Page 5 UNIT 3: Compound Stresses: Introduction, Plane stress, stresses on inclined sections, principal stresses and maximum shear stresses, Mohr’s circle for plane stress. 1 a) Obtain an expression for normal and tangential stresses on an inclined plane when an element subjected to bi-axial direct stresses. Also obtain the expressions for resultant stress and their direction. 08 Jan 14 Dec 10 1 b) The direct stresses at a point in a strained material are 100 N/mm2 compressive and 60 N/mm2 tensile as shown in Fig. Q.1b. Find the stresses on the plane AC. 12 Jul 11 2 a) Obtain an expression for normal and tangential stresses on an inclined plane when an element subjected to bi-axial direct stress along with shear stress. Also obtain the expressions for maximum and minimum principal stresses and their direction and maximum shear stress and their direction. 08 Jun 12, 11 Jul 09 2 b) The stresses at a point in a bar are 200 N/mm2 (tensile) and 100 N/mm2 (compressive). Determine the resultant stress in magnitude and direction on a plane inclined at 60° to the axis of the major stress. Also determine the maximum intensity of shear stress in the material at the point. 12 Jun 12 Jul 11 3 a) Explain the construction of Mohr’s circle diagram with an example. And hence derive the expressions for principal stresses and their directions and shear stresses and directions 10 Jul 14, 12 Jul 08 3 b) At a point in a strained material, the principal stresses are 100 N/mm2 tensile and 40 N/mm2 compr. Detn. the resultant stress in magnitude and direction on a plane inclined at 60° to the axis of the major principal stress. What is the max intensity of shear stress in the material at the point ? 10 Dec 11 Jul 10 Fig.Q.1b Fig. Q.5b Fig.Q.7 4 a) For a general two dimensional stress system, show that sum of normal stress in any two mutually perpendicular directions is constant. 6 Jan 14, 11 Jul 13 4 b) A rectangular block of material is subjected to a tensile stress of 110 N/mm2 on one plane and a tensile stress of 47 N/mm2 on the plane at right angles to the former. Each of the above stresses is accompanied by a shear stress of 63 N/mm2 and that associated with the former tensile stress tends to rotate the block anticlockwise. Find : i) the direction and magnitude of each of the principal stress and (ii) magnitude of the greatest shear stress. 14 Jul 13 Jan 10 5 a) What are the principal stresses and principal planes? Obtain an expression for normal and tangential stresses in an inclined plane when a body subjected to uni-axial stress. 8 Dec 11 Jan 10 5 b) At a certain point in a material under stress the intensity of the resultant stress on a vertical plane is 1000 N/cm2 inclined at 30° to the normal to that plane and the stress on a horizontal plane has a normal tensile component of intensity 600 N/cm2 as shown in Fig. Q.5b. Find the magnitude and direction of the resultant stress on the horizontal plane and the principal stresses. 12 Jul 13 6 a) Determine the expressions for normal and tangential stresses on a plane at  to the plane of stress in x-direction in a general two dimensional stress system and show that Principal planes are planes of maximum normal stresses also. 6 Jun 12 6 b) The tensile stresses at a point across two mutually perpendicular planes are 120 N/mm2 and 60 N/mm2 . Determine the normal, tangential and resultant stresses on a plane inclined at 30° to the axis of the minor stress by graphical method. 14 Jan 14 7 A point in a strained material is subjected to stresses shown in Fig. Q.7. Using Mohr's circle method, determine the normal and tangential stresses across the oblique plane. Check the answer analytically. 20 Jul 14 Dec 11 Fig.Q.8 Fig.Q.9 8 For the state of plane stress already considered in Fig. Q.8, construct Mohrs circle, (b) determine the principal stresses, (c) determine the maximum shearing stress and the corresponding normal stress. Verify your answers analytically. 20 Dec 12 Dec 11 9 For the state of plane stress shown, determine (a) the principal planes and the principal stresses, (b) the stress components exerted on the element obtained by rotating the given element counterclockwise through 30°. Verify your answers analytically. 20 Jul 12