Assignment strength of materials

2015 Assignment Questions: Unit-wise Mechanics of Materials

Hareesha N G Assistant Professor Department of Aeronautical Engineering Dayananda Sagar College of Engineering Bengaluru-78

Assignment Questions: Unit-wise

Mechanics of Materials

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, StressStrain 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) 1 b) 1 c)

2 a) 2 b)

2 c) 3 a)

3 b) 4 a) 4 b)

Define (i) Stress (ii) Hook's law (iii) Elasticity (iv) Lateral strain. Explain stress-strain relationship showing salient points on the diagram 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.

Fig. Q. 3(b) Fig. Q2 (c) Fig. Q1 (c) Define: i) True stress ii) Factor of safety, iii) Poisson’s ratio ,iv) Principle of superposition A bar of uniform thickness *t* tapers uniformly from a width of b 1 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. 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. 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. 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. Derive an expression for the deformation of a tapered bar of circular section, subjected to tensile load. 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.

4 6 10

Jul 14 Jul 14 Jul 14

4 8

Jan 14, Jul 13 Jan 14

8

Jan14

10

Jul 13

10

Jul 13

10

Jul 13

10

Jul 13

5 8

Dec 2012 Jun 10, 11 Jan 10 Dec 12

Fig. Q 4(b) Fig. Q.7 (b)

Fig.Q.8a

5 a) 5 b) 5 c)

Fig. Q.7(a) Define: i) Ductility, ii) True stress, iii) FOS iv) Young’s modulus, v) shear strain 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 Derive an expression for the extension of uniformly tapering rectangular bar subjected to axial load P.

Compiled by: Hareesha N G, Assistant Professor, DSCE, BLore

7

Page 1

Assignment Questions: Unit-wise 6 a) 6 b)

6 c) 7 a)

7 b)


Obtain an expression for elongation due to its self-weight. 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 Obtain an expression for total elongation of stepped bar with suitable sketch. 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. 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 Working E (MPa) stress (MPa) Bronze 450 0.83x105 120 Aluminium 600 Steel

8 a)

8 b)

9 a)

9 b)

300

0.70x105 2x10

5

6 8

Jun 08 Jun 08

6 6

Jun 08

14

Jul 11

10

Jun 10

10

Jun 10

10

Jul 12

10

Dec 10

10

Jun 11

10

Jun 11

80 140

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. 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? 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? 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.

Fig.Q.9a

10 a)

10 b)

Fig. Q.9b 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 E s = 2x105 N/mm2 and Ec = 1x105 N/mm2. 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 E s = 200GPa and Ec = 120GPa.


Page 2



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) 1 b)

2 a) 2 b)

3 a) 3 b)

4 a) 4 b)

Define lateral strain, linear strain and volumetric strain. Derive an expression for volumetric strain for rectangular plate and cylindrical rod of length L. 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 105N/mm2 Derive the relationship between E, G and K. 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? Prove that volumetric strain is equal to sum of the three principal strains. 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 Establish a relationship between the modulus of elasticity and the modulus of rigidity. 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 5 2 Modulus of elasticity 2x10 N/mm 1x105N/mm2 -6 Coefficient of thermal expansion 12x10 / °C 18x10-6/ °C

10

Jul 11

10

Jul 11 Jan 10

06

Jun 12 Jun 12 Jan 10

06

Jun 10 Dec 12 Jun 12

14

06 14

Jul 14, 11 Dec 10

Explain the following. 1) Youngs modulus, 2) Regidity modulus, 3) Bulk Modulus, 4) poisons ratio with help of a neat sketch. 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 Establish a relationship between the modulus of elasticity and the bulk modulus.

06

Jul 14

14

Jul 11 Jun 10

06

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 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. 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. Obtain an expression for thermal stresses in compound bars taking a suitable sketch. 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. Explain the following; i) Volumetric strain ii) Shear strain iii) Thermal stresses iv) Poisons ratio A rectangular bar of cross-section 30 mm * 60 mm and length 200 mm is restrained from

14

Dec 11 Dec 10 Jul 14 Jul 11

Fig.Q2b

Fig.Q.1b 5 a) 5 b)

6 a) 6 b)

7 a)

8 a) 8 b)

9 a) 9 b)


Fig. Q.4b

06 14

Jul 13

06 14

Dec 11

08 12

Dec 11 Dec 12 Page 3


10 a) 10 b)


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* 105N/mm2and =0.3. What are the changes if surrounding material can restrain only 50% of expansion on 30 mm x 200 mm side? 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. 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

10

Jan 14 Dec 11

Page 4



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)

1 b) 2 a)

2 b)

3 a) 3 b)

4 a) 4 b)

5 a) 5 b)

6 a)

6 b)

7

8

9

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. 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. 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. 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. 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 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 ?

Fig. Q.5b Fig.Q.1b Fig.Q.7 For a general two dimensional stress system, show that sum of normal stress in any two mutually perpendicular directions is constant. 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. 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. 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. 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. The tensile stresses at a point across two mutually perpendicular planes are 120 N/mm 2 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. 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.

Fig.Q.8 Fig.Q.9 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. 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.


08

Jan 14 Dec 10

12

Jul 11

08

Jun 12, 11 Jul 09 Jun 12 Jul 11

12

10

Jul 14, 12 Jul 08

10

Dec 11 Jul 10

6

Jan 14, 11 Jul 13

14

Jul 13 Jan 10

8 12

Dec 11 Jan 10 Jul 13

6

Jun 12

14

Jan 14

20

Jul 14 Dec 11

20

Dec 12 Dec 11

20

Jul 12

Page 5



UNIT 4: Energy Methods: Work and strain energy, Strain energy in bar/beams, Castiglinios theorem, Energy methods

Thick and Thin Cylinder: Stresses in thin cylinders, changes in dimensions of cylinder (diameter, length and volume), Thick cylinders Lame's equation 1 a) 1 b)

2 a) 2 b)

3 a) 3 b)

4 a) 4 b)

4 c) 5 a) 5 b)

6 a) 6 b)

7 a) 7 b)

8 a)

Derive an expression for longitudinal and circumferential stress in thin cylinders. List the differences between thin and thick cylinders with sketches. A thick cylinder of 500 mm inner diameter is subjected to an internal pressure of 9 MPa. Taking the allowable stress for the material of the cylinder as 40 MPa, determine the wall thickness of the cylinder. Also plot the stress distribution across the wall thickness of the cylinder State and prove castigliano’s theorem. A pipe of 400 mm internal dia and 80 mm thick contains a fluid at pressure of 80N/mm 2. Find the maximum and minimum hoop stresses across the section. Also sketch radial and hoop stresses distribution across the section. Derive lame’s equations for thick cylinders. A thick cylindrical pipe of outside diameter 300 mm and internal diameter of 200 mm is subjected to an internal fluid pressure of 20 N/mm2 and external fluid pressure of 5 N/mm2. Determine the maximum hoop stress developed and draw the variation of hoop stress and radial stress across the thickness. Indicate values at every 25mm interval. Derive an expression for strain energy stored in a beam subjected to torsion load. Calculate the i) Change in diameter: ii) Change in length and iii) Change in volume of a thin cylindrical shell 1000mm diameter. 10mm thick and 5m long when subjected to internal pressure of 3 N/mm2. Take the value of 0 = 2 x l05N/mm2and l/m = 0.3. A cantilever of uniform section carries a point load at the free end. Find the strain energy stored by the cantilever and hence calculate the deflection at the free end. Derive the equations for changes in length, diameter and volume of a thin cylinder subjected to internal pressure P. A thick cylindrical pipe of internal radius 150 mm and external radius 200 mm is subjected to an internal fluid pressure of 17.5 N/mm2. Determine the maximum hoop stress in the cross-section. What is the percentage error if it is determined from thin cylinders theory? Derive an expression for strain energy stored in a beam subjected to a bending moment M. A thin cylindrical shell 1.2 m in diameter and 3m long has a metal wall thickness of 12mm. It is subjected to an internal fluid pressure of 3.2 MPa. Find the circumferential and longitudinal stress in the wall. Determine change in length, diameter and volume of the cylinder. Assume E= 210GPa and µ=0.3 A beam of length l is simply supported at its ends. The beam carries a uniformly distributed load of w per unit run over the whole span. Find the strain energy stored by the beam. The internal and external diameters of a thick cylinder are 300mm and 500mm respectively. It is subjected to an external pressure of 4MPa. Find the internal pressure that can be applied if the permissible stress in cylinder is limited to 13MPa. Sketch radial and hoop stresses distribution across the section. Define: Strain energy and Work. Prove that volumetric strain in thin cylinder is given by

08 12

08 12

08 12

Jul 13 Jan 09 Jul 13 Jun 12 Dec 10 Dec 10

6 08

Jun 12 Jun 12

6

Jun 12

8

Jun 08

12

Jun 08

08

8

Dec 11 Jan 09 Jul 13 Dec 11 Dec 10 Jan 14

12

Jan 08

10

Jul 14 Jun 12 Jul 11 Jan 09 Jul 14 Jan 14

Pd 5  4  with usual notations. 4tE 8 b)

9 a) 9 b)

A C.I. pipe has 200 mm internal diameter and 50 mm metal thickness and carries water under a pressure of 5 N/mm2. Calculate the maximum and minimum intensities of circumferential stress and sketch the distribution of circumferential stress and radial pressure across the section. Obtain an expression for the volumetric strain of a thin cylinder, subjected to internal fluid pressure. The bar with circular cross-section as shown in Fig.Q.9(b) is subjected to a load of l0 kN. Determine the strain energy stored in it. Take E = 210 GPa.

Jan 09 Jul 13 Jul 11

10

06 08

Dec 12

5

Dec 12

5 10

Dec 12 Dec 12 Jun 12 Jun 10

9 c) 10 a)

A simply supposed beam of span L carries a point load W al mid-span. Find the Strain energy stored by the beam. 10 b) Derive an expression for circumferential stress tor thin cylinder. 10 c) A cylindrical pressure vessel has inner and outer radii of 200 mm and 250 mm respectively. The material of the cylinder has an allowable normal stress of 75 MPa. Determine the maximum internal pressure that can be applied and draw a sketch of radial pressure and circumferential stress distribution.


Page 6



UNIT 5: Bending Moment and Shear Force in Beams: Introduction, Types of beams, loads and reactions, shear forces and bending moments, rate of loading, sign conventions, relationship between shear force and bending moments. Shear force and bending moment diagrams for different beams subjected to concentrated loads, uniformly distributed load, (UDL) uniformly varying load (UVL) and couple for different types of beams. 1 a) 1 b)

Derive the relationship between load, shear force and bending moment also Briefly explain the different types of loads Draw SFD and BV1D for the loading pattern on the beam in Fig.Q1(b). Indicate the point of contra-flexure. Also locate the maximum BM with its magnitude.

8 12

Fig. Q.1b 2 a) 2 b)

3 a) 3 b) 3 c)

4 5 a) 5 b)

6 a) 6 b)

7 a) 7 b) 8

9

Fig.

Q.2b Define a beam. Explain with simple sketches, different types of beams. Draw the shear force and bending moment diagrams for the overhanging beam carrying uniformly distributed load of 2 kN/m over the entire length and a point load of 2 kN as shown in Fig.Q2(b). Locate the point of contra flexure. Explain the terms: i) Sagging bending moment; ii) Hogging bending moment. ;iii) Point of contra-flexure. What are the different types of loads acting on a beam? Explain with sketches. A simply supported beam of span 6m is subjected to a concentrated load of 25kN acting at a distance of 2m from the left end. Also subjected to an uniformly distributed load of l0 kN/m over the entire span. Draw the bending moment and shear force diagrams indicating the maximum and minimum values.

Fig.Q.4

06 14

Jan 14 Jan 14

6

Jul 13

6 8

Jul 13 Jul 13

Fig. Q.5 a

Fig. Q.5b Draw the shear force and bending moment diagram for the overhanging beam shown in Figure Q.4 and locate points of contra-flexure. Draw shear force and bending moment diagrams for a simply supported beam subjected to couple at midspan. as shown in Fig.Q.5(a) A beam ABC is 9 m long and supported at B and C, 6 m apart as shown in Fig.Q5b. The beam carries a triangular distribution of load over the portion BC together with an applied counterclockwise couple of moment 80 kN m at B and an u.d.1. of 10 kN/m over AB, as shown. Draw the S.F. and B.M. diagrams for the beam. Discuss the reaction loads at fixed, roller and hinge supports. A beam 25m long is supported at A and B and is loaded as shown in Fig.Q6(b). Draw the shear force and bending moment diagrams for the beam computing shear force and bending moments at A, E, D, B and C. Find the position and magnitude of the maximum bending moment. Also, determine the point of contraflexure.

Fig. Q.7b Fig. Q.6b Define the following; i) SF; ii) BM ; iii) SFD; iv) BMD Draw the shear force and bending moment diagrams for the beam shown in Fig.Q7(b). A horizontal beam 10 m long is carrying a uniformly distributed load of 1 kN/m. The beam is supported on two supports 6 m apart. Find the position of the supports, so that B.M. on the beam is as small as possible. Also draw the S.F. and B.M. diagrams. A beam 10 m long and simply supported at each end, has a uniformly distributed load of 1000 N/m extending from the left end upto the centre of the beam. There is also an anti-clockwise couple of 15 kNm at a distance of'2.5 m from the right end. Draw the S.F. and B.M. diagrams.


Jul 14 Dec 11 Jul 14

20

Jan 13

06

Jun 12

14

Jun 12

6 14

Dec 11 Dec 11

4 16 20

Dec 11 Dec 11 Jan 13

20

Jul 09

Page 7



UNIT 6: Bending and Shear Stresses in Beams: Introduction, Theory of simple bending, assumptions in simple bending, Bending stress equation, relationship between bending stress, radius of curvature, relationship between bending moment and radius of curvature, Moment carrying capacity of a section. Shearing stresses in beams: shear stress across rectangular, circular, symmetrical I and T sections. 1 a) 1 b)

2 a) 2 b)

3 a) 3 b)

4 a) 4 b)

5 a) 5 b)

6 a)

6 b)

7 a)

7 b)

8 a) 8 b)

9 a) 9 b)

10

Explain the concept of simple bending theory. A rectangular beam 200 mm deep and 300 mm wide is simply supported over a span of 8 m. What uniformly distributed load per metre the beam may carry, if the bending stress is not to exceed 120 N/mm2. State the assumptions made in the theory of bending. A rolled steel joist of I section has the dimensions: as shown in Fig.Q2b. This beam of I section carries a u.d.l. of 40 kN/m run on a span of 10 m, calculate the maximum tress produced due to bending.

Fig.Q.2b Derive bending equation with usual notations. A T-section is formed by cutting the bottom flange of an I-section. The flange is 100 mm x 20 mm and the web is 150 mm x 20 mm. Draw the bending stress distribution diagrams if bending moment at a section of the beam is 10 kN-m (hogging). Obtain the section modulus of the following sections. i) Square section with a square hole ii) Hollow circular section iii) Triangular section A symmetric I-section has flanges of size 180 mm x 10 mm and its overall depth is 500 mm. Thickness of web is 8 mm. It is strengthened with a plate of size 240 mm x 12 mm on compression side. Find the moment of resistance of the section, if permissible stress is 150 N/mm 2. How much uniformly distributed load it can carry if it is used as a cantilever of span 3 m? Obtain an expression for shear stresses in beam subjected to shear force F. Also draw the distribution of shear stress in rectangular cross sectional beam. Three beams have the same length, same allowable bending stress and the same bending moment. The cross-section of the beams are a square, rectangle with depth twice the width and a circle. Find the ratios of weights of the circular and the rectangular beams with respect to square beams. Show that the shear stress across the rectangular section varies parabolicaily. Also show that the maximum shear stress is 1.5 times the average shear stress. Sketch the shear stress variation across the section. A timber beam of rectangular section is simply supported at the ends and carries a point load at the centre of the beam. The maximum bending stress is 12 N/mm 2 and maximum shearing stress is 1 N/mm2, find the ratio of the span to the depth. Two circular beams where one is solid of diameter D and other is a hollow of outer dia. D o and inner dia. Di are of the same length, same material and of same weight. Find the ratio of section modulus of these circular beams. An I-section beam 350 mm x 150 mm has a web thickness of 10 mm and a flange thickness of 20 mm. If the shear force acting on the section is 40 kN. Find the maximum shear stress developed in the I-section. sketch the shear stress distribution across the section, also calculate the total shear force carried by the web. Prove that the maximum shear stress in a circular section of a beam is 4/3 times the average shear stress. The shear force acting on a section of a beam is 50 kN. The section of the beam is of T-shaped of dimensions 100 mm x 100 mm x 20 mm as shown in Fig.Q8b. Calculate the shear stress at the neutral axis and at the junction of the web and the flange. Explain: i) Section modulus ii) Flexural rigidity. A T section of flange 120 mm x 12 mm and overall depth 200 mm, with 12 mm web thickness is loaded, such that, at a section it has a moment of 20 kNM and shear force of 120 kN. Sketch the bending and shear stress distribution diagram, marking the salient values. A beam of I section consists of 180 mm x 15 mm flanges and a web of 280 mm depth x 15 mm thickness. It is subjected to a bending moment of 120 kN-m and shear force of 60 kN. Sketch the bending and shear stress distributions along the depth of the section.


5

Dec 08 Jan 14

6

Jul 14, 13

Jul 13

06

Dec 11 Dec 12

6

Jun 09

14

10

Jul 14 Jul 13 Dec 11 Jul 11 Dec 10

10

Jun 12

10

Jul 14 Jan 14, 12

Jul 13 10

Dec 09

Jun 10

Jun 12 Dec 11 Jul 11 10

Jun 12 Jul 11

4 16

Jun 12 Dec 11

20

Jul 07

Page 8



UNIT 7: Deflection of Beams: Introduction, Differential equation for deflection, Equations for deflection, slope and bending moment, Double integration method for cantilever and simply supported beams for point load, UDL, UVL and Couple, Macaulay's method 1 a)

Derive the deflection equation for the beam in the standard form:

1 b)

For the beam loaded as shown in Fig.Q1(b), find the position and magnitude of maximum deflection. Take I = 4.3xl08 and E = 200 kN/mm2.

EI

d2y M dx 2

6 14

Jul 14 Jun 12 Jul 14

4

Jan 14

16

Jan 14

10

Jul 13 Jul 11 Jul 13 Jun 12

Fig. Q.1b

2 a) 2 b)

3 a) 3 b)

4 a)

4 b)

5 a)

5 b)

6 a) 6 b) 7 a) 7 b)

Fig.Q.6b Fig.Q.2b A cantilever 120 mm wide and 200 mm deep is 2.5 m long. What is the uniformly distribution load which the beam can carry in order to a deflection of 5 mm at the free end? Take E = 200 GN/m2 A horizontal beam AB is simply supported at A and B, 6 m apart. The beam is subjected to a clockwise couple of 300 kN-m at a distance of 4 m from the left end as shown in Fig.Q2(b). If E = 2 x 10 N/mm2 and I = 2 x 108 mm4, determine: i) The deflection at the point where the couple is acting; ii) The maximum deflection. Show that for a simply supported beam of length T carrying a concentrated load W at its mid span, the maximum deflection in Wl3/48EI. A simply supported steel beam having uniform cross-section is 14m span and is simply supported at its ends. It carries a concentrated load of 120kN and 80kN at two points 3m and 4.5m from the left and right end respectively. If the moment of inertia of the section is 160 x I07 mm4 and E = 210 GPa, calculate the deflection of the beam at load points. Derive an expression for max deflection of a simply supported beam by double integration method. The beam is subjected to uniformly distributed load of W/ unit length, L being the length of the beam, E the Young's modulus and T and moment of inertia of the cross section. A simply supported beam of 6m span is subjected to a concentrated load of I8kN at 4m from left support. Calculate the position and value of maximum deflection using Mecaulay's method. Take E = 200 GPa and I = 15 x 106 mm4. A cantilever of length 2.5 m carries a uniformly distributed load of 16.4 kN/m over the entire length. If the moment of inertia of the beam is 7.95 x 10 7 mm4 and the value of E = 2GPa, determine the deflection at the free end. Derive the equation used. A beam of length 6 m is simply supported at its ends and carries two point loads of 48 kN and 40 kN at a distance of 1 m and 3 m respectively from the left support. Find: i) Deflection under each load, ii) Maximum deflection and iii) The point at which maximum deflection occurs. Take E =200GPa and I = 85 x 106 mm4 Find the expressions for the slope and deflection of a cantilever of length L carrying uniformly distributed load over the whole length. Determine the deflection under the loads in the beam as shown in Fig.Q6(b). Take flexural rigidity as EI throughout . Using the standard notations, derive an expression for deflection, slope and maximum deflection of a simply supported beam of span ‘L' subjected to a concentrated load W at its mid span. Derive an expression for the maximum deflection of a cantilever beam carrying a point load at its free end.

10

10

Jul 13

10

Jul 13

10

Dec 12

10

Dec 12 Dec 11

8

Jun 12

12

Jun 12

10

Dec 11

10

Dec 11

10

Jul 11

10

Dec 10

Fig.Q.8(a)

8 a) 8 b)

Fig.Q.8a Find the maximum deflection and the maximum slope for the beam loaded as shown in Fig.Q8(a). Take flexural rigidity EI = 15xl09 kN.mm2. A simply supported beam of length L, carries a UDL of w kN/m throughout the length. Obtain the slope at the supports and the maximum deflection.


Page 9



UNIT 8: Torsion of Circular Shafts and Elastic Stability of Columns: Introduction, Pure torsion, assumptions, derivation of torsional equations, polar modulus, torsional rigidity / stiffness of shafts, Power transmitted by solid and hollow circular shafts Columns: Euler's theory for axially loaded elastic long columns, Derivation of Euler's load for various end conditions, limitations of Euler's theory, Rankine's formula. 1 a) 1 b)

2 a) 2 b)

3 a) 3 b) 3 c)

4 a)

4 b)

5 a)

5 b)

6 a) 6 b) 6 c)

7 a) 7 b) 8 a) 8 b)

9 a) 9 b)

10 a)

10 b)

Derive the torsion formula, in the standard form and list all the assumptions made while deriving the same. Find the diameter of the shaft required to transmit 60KW at 150rpm, if the maximum torque is 25% of the mean torque for a maximum permissible shear stress of 60 MN/m 2. Find also angle of twist for length of 4m. Take G = 80 GPa.

10

Prove that a hollow shaft is stronger and stiffer than the solid shaft of the same material, length and weight.

10 10

Dec 09 Jul 14 Dec 11

4

Jul 14

4 12

Dec 09 Jul 13

10

Jun 12

10

Jul 14

10

Dec 12

10

Jan 15 Jul 13

6 6 8

Jan 14 Jul 12 Jan 15 Dec 12

8

Dec 11

12

Dec 11

10

Jul 14 Jun 12 Jul 13

A hollow shaft of diameter ratio 3/5 is required to transmit 700 kW at 110 rpm, the maximum torque being 12% greater than the mean. The shear stress is not to exceed 60 MPa and the twist in a length of 3 meters is not to exceed one degree. Calculate the c/s dimensions. G = 0.8 x 105 MPa. Find the maximum shear stress induced in a solid circular shaft of diameter 15 cm when the shaft transmits 150 kW power at 180 r.p.m. Obtain an expression for power transmitted by a solid shaft. A solid shaft transmits 250KW at 100rpm. If the shear stress is not to exceed 75MPa, what should be diameter of the shaft? If this shaft is to be replaced by a hollow shaft, whose diameter ratio 0.6, determine size and percentage having in weight, the maximum shear stress being the same. Two shafts of the same material and of same lengths are subjected to the same torque, if the first shaft is of a solid circular section and the second shaft is of hollow circular section, whose internal diameter is 2/3 of the outside diameter and the maximum shear stress developed in each shaft is the same, compare the weights of the shafts. Determine the size of the shaft which will transmit 70KW at 180 rpm. The shear stress is limited to 40MPa. The twist of shaft is not to exceed 1o in 2.5m length of shaft. Take G = 0.8 x 10 5 N/mm2. Assume torque to be uniform. Determine the ratio of power transmitted by a hallow shaft and solid shaft when both have same weight, length, material and speed. The diameter of solid shaft is 150mm and external of hollow shaft is 250mm. A hollow shaft, having an inside diameter 60% of its outer diameter, is to replace a solid shaft transmitting the same power at the same speed. Calculate the percentage saving in material, if the material to be used is also the same. Derive an exprn.for the critical load in a column subjected to comp. load, when both ends are fixed. Derive an expression for Euler’ s buckling load for a long column having one end fixed and other end hinged. A simply supported beam of length 4 metre is subjected to a uniformly distributed load of 30 kN/m over the whole span and deflects 15 mm at the centre. Determine the crippling loads when this beam is used as a column with the following conditions : (i) one end fixed and other end hinged (ii) both the ends pin jointed. Show the variation of Euler's critical load with slenderness ratio. Using the same, explain the limitations of Euler's theory. How the Rankine's formula overcomes these limitations? Determine Euler's crippling load for an I-section joist 40 cm x 20 cm x 1 cm and 5 in long which is used as a strut with both ends fixed. Take Young's modulus for the joist as 2.1 x 10 5 N/mm2. Derive an expression for the critical load in a column subjected to compressive load, when one end is fixed and the other is free. Find the Euler’s crippling load for a hollow cylindrical steel column of 38mm external diameter and 2.5mm wall thickness. Length of the column is 2.3 m and is hinged at both ends. Also estimate Rankine’s load for this column, Rankine’s parameters are 335 N/mm2 and 1/7500. Derive an expression for the critical load in a column subjected to compressive load, when both ends are hinged. A 1.5m long column has a circular cross section of 50mm diameter. One of the ends of the column is fixed in direction and position and other end is free. Take factor of safety as 3, calculate the safe load using i) Rankine's formula, take yield stress = 560N/mm2 and a = 1/600 for pinned ends. ii) Euler's formula, Young's modulus for C.I. = 1.2 x 105 N/mm2. Find the Euler crushing load for a hollow cylindrical cast iron column 20 cm external diameter and 25 mm thick if it is 6 m long and is hinged at both ends. Take E = 1.2x 106N/mm2. Compare the load with the crushing load as given by the Rankine's formula, taking c = 550 N/mm2 and =1/1600 for what length of the column would these two formulae give same same crushing load ? Determine the crippling load for a T section of dimensions 10cm x l0 cm x2cm and of length 5 m when it is used as strut with both of its ends hinged. Take Young's modulus, E=200GPa.


10

10

8

Jan 14 Jun 12 Jan 14 Jun 12

12

Jun 12 Jan 11 Jan 12

12

Jan 15

08

Jul 13

Page 10

Assignment strength of materials

Recommend Documents