Advanced Strength of Materials - GDLC_8851.pdf

B. TECH. SEM III (MECHANICAL), ADVANCED STRENGTH OF MATERIALS (2012-13)

Department of Mechanical Engineering, Dharmsinh Desai University, Nadiad Page 1 of 12

SUBJECT : ADVANCED STRENGTH OF MATERIALS (MH301)

CLASS : B.TECH. II SEMESTER : III (ODD/FIRST TERM OF THE YEAR) BRANCH : MECHANICAL ENGINEERING

W.E.F. : 2011-12

Teaching Scheme (Hours/Week) Examination Scheme (Marks)

Lectures Tutorial Practical Theory (3 hrs)

Sessional (1 hr)

Practical T.W. Total

3 0 2 60 40 25 25 150

1

STRESSES IN THREE DIMENSIONS: Concept of Continuum, Homogeneity and Isotropy, Types of forces on a body, State of stress at a point, Equality of cross shear, Cauchy formula, principal stresses and planes, Stress invariants, Hydrostatic and deviatoric stress tensor, Mohrs circle for general state of stress, stress transformations, Octahedral stresses, Differential equation of equilibrium

2 STRAINS IN THREE DIMENSIONS:

Types of strain, Strain displacement relationship, Shear strain, Rigid body rotation, Principle strain and axes, Strain deviator and invariants, Compatibility conditions, Concept of Plane stress and strain, Stress strain relationship

3

THEORIES OF ELASTIC FAILURE: Concept of factor of safety, Maximum principal stress theory, maximum shear stress theory, maximum principal strain theory, Maximum strain energy theory, maximum shear strain energy theory

4 BUCKLING OF COLUMN: Concept of buckling and stability, differential equations of compression member with different boundary conditions, eccentrically loaded columns,

secant formula, column with initial imperfections, Rankine formula

5

STRESSES DUE TO ROTATION: Rotating ring, rotating thin disc, rotating thin solid and hollow disc, disc of uniform strength, rotating long solid and hollow cylinders

6

BENDING OF CURVED BARS: Introduction, Stresses in curved bars (Winkler-Bach theory) (Rectangular section, Circular section, Triangular section, Trapezoidal section, T-Section), Stresses in crane hooks

7

TORSION OF NON-CIRCULAR MEMBERS: St. Venants theory, approximate solution of rectangular and elliptical sections, rigorous solution, stress function approach, membrane analogy, torsion of thin hollow sections,

Torsional of thin and open sections 8

BENDING OF THIN PLATES: Assumptions of plate theory, governing differential equations for deflection of plates,

boundary conditions, solutions for rectangular plate

Term work / practical shall be based on the above syllabus.

TEXT BOOKS: 1. Advanced Mechanics of Solids L. S. Srinath, Tata McGraw Hill

2. Strength of Materials R. K. Rajput, S. Chand & Co. Ltd. 3. Strength of Materials D. S. Bedi, Khanna book publishing co. Pvt ltd.

REFERENCE BOOKS: 1. Solid Mechanics S. M. A. Kazimi, Tata McGraw Hill 2. Theory of Plates S. Chandrashekhara, Universities Press



ASSIGNMENT NO. 1

(1) The stress at a point is given by following stress tensor. Draw traction forces on the sections cut by positive and negative x, y and z planes.

(2) The stress at a point is given by following stress tensor. Draw traction forces on the sections cut by positive and negative x, y and z planes.

(3) The stress tensor at a point in a body has the following components:

Find; (a) the resulting stress T (b) the components of T which is normal and along the plane which is equally inclined to all the three

axes (4) The components of stress tensor at a point are known as

What will be the normal and shear stress components on a plane that is equally inclined to all the

three axes? (5) The components of stress tensor at a point are known as

What will be the normal and shear stress components in the direction N with l=0.5, m=0.33 and n=0.8

(6) The components of stress tensor at a point are known as

What will be the normal and shear stress components on the plane whose normal is in the direction with l=0.866, m=0.3 and n=0.4

(7) Given the components of stress tensor at a point,

What are the normal and shear stress components in the direction with l=0.6, m=0.8 & n=0

(8) The state of stress a point for reference xyz and xyz are given as ij

and ns

respectively. Check

whether the invariants of stress tensor I1, I2 and I3 really remain unchanged.

10 4 -15

4 0 -5

-15 -5 11

-2 -5 3

-5 0 2

3 2 1

69 1.5 0

1.5 -5 -2.8 10 Pa

0 -2.8 0

5 -2 0

-2 6 8

0 8 9

64 8 -3

8 5 0 10 Pa

-3 0 6

650 20 10

20 100 60 10 Pa

10 60 50

,ij ns

200 100 0 136.6 -136.6 0

100 0 0 -136.6 63.4 0

0 0 500 0 0 500

610 Pa

100 100 100

100 -50 100

100 100 -50



(9) The state of stress at a point for a given reference xyz, ij

is given below. If a new set of axes xyz is

formed by rotating xyz through 600 about z-axis in anti-clockwise direction,

a) Find the stress tensor for a set of coordinate axes x,yz b) Find the invariants of stress tensor in both the reference axes and compare them c) Find the hydrostatic and deviatoric stress tensor in both the axes.

(10) The stress at a point with reference to X = (x,y,z) are

(a) Find the stress tensor for a set of coordinate axes X = (x,y,z) rotated 300 about the x-axis anticlockwise

(b) Find the invariants of stress tensor in both the reference axes and compare them (c) Find the hydrostatic and deviatoric stress tensor in both the axes.

(11) The stress at a point with reference to X = (x,y,z) are

(a) Find the stress tensor for a set of coordinate axes X = (x,y,z) rotated 300 about the x-axis anticlockwise

(b) Find the invariants of stress tensor in both the reference axes and compare them

(c) Find the hydrostatic and deviatoric stress tensor in both the axes.

(12) Given the following state of stress find the principle stresses and the principal axes.

(13) Given the following state of stress at a point, what are the principle stresses and the principal planes?

(14) Given the following state of stress at a point, find the principle stresses and the principal axes.

(15) Given the following state of stress at a point, find the principle stresses and their axes.

(16) Given the following state of stress at a point, find the principle stresses and their axes.

(17) If the principal stresses at a point are 100, 100 and -200 MPa, find the octahedral normal and shear stress at this point.

(18) A sample is tested under tri-axial compression and the values of principal stresses are 20, 5, 5 MPa. Fin the value of octahedral normal and shear stress.

0 20 30

20 50 0

30 0 60

100 50 150

50 0 0

150 0 0

10 0 0

0 5 5

0 5 5

5 5 5

5 5 5

5 5 5

300 0 0

0 133.3 67.67

0 67.67 133.33

200 100 0

100 0 0

0 0 500

18 0 24

0 -50 0

24 0 32

j

j

3 -10 0

-10 0 30

0 30 -27



ASSIGNMENT NO. 2

1. A wheel 800 mm in diameter has a thin rim. If density is 7700 kg/m3 and E = 200 GN/m2

Calculate :

(a) How many revolutions per minute may it make without the hoop stress exceeding 130 MN/m2? (b) Change in diameter. Neglect the effect of spokes.

2. Calculate the hoop stress in a thin rim, 0.6m mean diameter revolving about its axis at 800

r.p.m. Steel weighs 7700 kg/m3.

3. A mild steel thin ring is 1m in diameter. Neglecting the effect of spokes, find the maximum

speed in r.p.m. at which it can be rotated without the stress going beyond 155 MN/m2. The steel weighs 7700 kg/m3. Also find the increase in diameter of the ring at this speed if E = 200 GN/m2.

4. Figure 2.1 shows a built up ring. If the ring rotates at 2000 r.p.m. find the stresses set up in

steel and copper rings. Assume For steel E = 200 GN/m2; = 7800 kg/m3

For copper E = 100 GN/m2; = 8900kg/m3

Figure 2.1 5. Calculate the stress in the rim of a pulley when linear velocity of the rim is 80 m/s. Assume

density of material of the pulley as 7800 kg/m3. If the speed of the pulley is increased by 20%

what will be the stress.

6. Find the limiting peripheral speed of a cast iron wheel if the allowable stress in cast iron is 6.6

N/mm2. Take density of material as 7212 kg/m3.

7. A composite ring is made of an inner copper ring and outer steel ring. The diameter of the

surface of contact of the two rings is 600 mm. If the composite ring rotates at 2500 r.p.m.

determine the stresses set up in the steel and copper rings. Both the rings are rectangular cross section 15 mm in the radial direction and 20 mm in the direction perpendicular to the plane of the ring. Take Es = 200 GN/m

2; Ecu = 100 GN/m2; s = 7800 kg/m

3; cu= 8900 kg/m3.

8. Determine the maximum angular velocity at which the disc can be rotated if the hoop stress is limited to 20 MN/m2. The ring has a mean diameter of 260 mm. What will be the change in

diameter?

Take = 7470 kg/m3 E = 207 GN/m2.

9. Determine the intensities of principal stresses in a flat steel disc of uniform thickness having a

diameter of 1 m and rotating at 2400 r.p.m. What will be stresses if the disc has a central hole of 0.2 m diameter? Take Poissons ratio = 1/3 and = 7850 kg/m3.

10. A steel disc of uniform thickness and of diameter 400 mm is rotating about its axis at 2000

r.p.m. The density of the material is 7700 kg/m3 and Poissons ratio 0.3. Determine the variations of circumferential and radial stresses.

11. A disc of uniform thickness having inner and outer diameters 100 mm and 400 mm respectively is rotating at 5000 r.p.m. about its axis. The density of the material of the disc is 7800 kg/m3

and Poissons ratio is 0.28. Determine the stress variations along the radius of the disc.

12. Determine the greatest values of radial and hoop stresses for a rotating disc in which the outer and inner radii are 0.3m and 0.15m. The angular speed is 150 rad/sec. Take Poissons ratio as 0.304 and density 7700 kg/m3.

13. Solve problem 12 if it were a long cylinder.



14. A thin uniform disc of inner radius 25 mm and outer diameter 125 mm is rotating at 10000 rpm Calculate the maximum and minimum values of circumferential and radial stresses. Take density of material 8830 kg/m3 and Poissons ratio = 0.33.

15. A steam turbine rotor (to run at a speed of 3500 r.p.m.) is to be designed so that the radial and

circumferential stresses are to be same and constant throughout and is equal to 80 MN/m2. If the

axial thickness at the centre is 15 mm what is the thickness at a radius of 500 mm? Assume

density of material as 7800 kg/m3.

16. A disc having inner and outer radius 75 mm and 150 mm respectively is rotating at an angular

speed of 150 rad/sec. Calculate the greatest values of radial and circumferential stresses. Assume density of material 7700 kg/m3 and Poissons ratio= 0.304.

17. A steel ring of outer diameter 300 mm and internal diameter 200 mm is shrunk onto a solid steel shaft. The interference is arranged such that the radial pressure between the mating surfaces will not fall below 30 MN/m2 whilst the assembly rotates in service. If the maximum circumferential stress on the inside surface of the ring is limited to 240 MN/m2, determine the

maximum speed at which the assembly can be rotated. It may be assumed that no relative slip occurs between the shaft and the ring.

For steel, = 7470 kg/m3, Poissons ratio = 0.3, E = 208 GN/m2.

18. Determine the maximum stress and the stress at the outside of a 250mm diameter disc which rotates at 12000 r.p.m.

19. A turbine rotor, 0.4m external diameter and 0.2m internal diameter, is revolving at 1000 r.p.m.

Taking the weight of rotor as 7700 kg/m3 and Poissons ratio 0.3, find the maximum hoop and radial stresses assuming (i) rotor to be a thin disc and (ii) rotor to be a long cylinder. Calculate

the percentage error in assuming it to be a thin disc.

20. Solve example 16 if the rotor were either a solid disc or solid cylinder.

21. A grinding wheel is 300mm diameter with the bore at the centre 25mm diameters. If the

thickness of the wheel at the outer radius is 25mm, what should be the thickness at the bore

diameter for a uniform allowable stress of 10MN/m2 at 2800 r.p.m.? Take density of the wheel material as 2700 kg/m3.

22. The disc of a turbine rotor is 0.5m diameter. At the blade ring its thickness is 55mm. It is keyed

to a shaft of 50mm diameter. If the uniform stress in the rotor disc is limited to 200 MN/m2 at 900 r.p.m., find the thickness of the disc at the shaft. Take density of the rotor material at 7700 kgt/m3.

23. A long cylinder of 300 mm radius is rotating at 4500 r.p.m. The density of material is 7800

kg/m3 and Poissons ratio is 0.3. Calculate the maximum stress in the cylinder. Draw the variation of radial and circumferential stress along the radius.

24. A hollow cylinder 200 mm external radius and 100 mm internal radius is rotating at 3000 r.p.m.

The density of material is 7800 kg/m3 and Poissons ratio 0.3. Calculate the maximum stress in the cylinder. Draw the variation of radial and circumferential stress in the cylinder.

25. A long cylinder of outer radius 375 mm and inner radius 125 mm is rotating about its axis at

4000 r.p.m. What are the maximum and minimum values of circumferential stress? What is the maximum radial stress and where it occurs? Take density of material = 7800 kg/m3 and Poissons ratio is 0.3.

26. A rotor of a turbine having inner and outer radii 100 mm and 200 mm respectively is rotating at

1000 r.p.m. Find the maximum radial and circumferential stresses. Assuming: (a) Rotor to be a thin (b) Rotor to be a long cylinder.



ASSIGNMENT NO. 3 1. In a metallic body the principal stresses are +35 MPa and -95 MPa, the third principal stress being

zero. The elastic limit stress in simple tension as well in simple compression is equal and is 220 MPa. Find the factor of safety based on the elastic limit if the criterion of failure for the material is

the maximum principal stress theory. 2. In a Cast Iron body the principal stresses are +40 MPa and -100 MPa the third principal stress

being zero. The elastic limit stresses in simple tension and in simple compression are 80MPa and

400MPa respectively. Find the factor of safety based on the elastic limit if the criterion of failure is the maximum principal stress theory.

3. A mild steel shaft, 120mm diameter is subjected to a maximum torque of 20kNm and a maximum

bending moment of 12 kNm at a particular section. Find the factor of safety according to the maximum shear stress theory if the elastic limit in simple tension is 220 MPa.

4. A shaft is subjected to a maximum torque of 10kNm and a maximum bending moment of 7.5kNm at a particular section. If the allowable equivalent stress is simple tension is 160 MPa, find the

diameter of the shaft according to the maximum shear stress theory. 5. Solve example 4 using the strain energy theory. Take Poissons ratio 0.24. 6. Solve example 4 using the shear strain energy theory.

7. In a steel member, at a point the major principal stress is 180MPa and the minor principal stress is compressive. If the tensile yield point of the steel is 225MPa, find the value of the minor principal stress at which yielding will commence according to each of the following criteria of failure:

(i) Maximum shearing stress (ii) Maximum total strain energy and (iii) Maximum shear strain energy. Take Poissons ratio = 0.26

8. In a material, the principal stresses are 60MPa, 48MPa and -36MPa. Calculate:

(i) Total strain energy per unit volume (ii) Volumetric strain energy per unit volume (iii) Shear strain energy per unit volume

(iv) Factor of safety on the total strain energy criterion if the material yields at 120MPa. Take E = 200GN/m2 and 1/m=0.3

9. A bolt is under an axial thrust of 9.6kN together with a transverse force of 4.8kN. Calculate its

diameter according to: (i) Maximum principal stress theory (ii) Maximum shear stress theory. Use F.O.S. = 3, yield strength of bolt = 270 MPa and

Poissons ratio = 0.3 10. A solid shaft transmits 1000kW at 300 r.p.m. Maximum torque is 2 times the mean. The shaft is

subjected to a bending moment which is 1.5 times the mean torque. The shaft is made of a ductile material for which the permissible tensile and shear stresses are 120 MPa and 60 MPa respectively.

Determine the shaft diameter using a suitable theory of failure. 11. A hollow mild steel shaft having 100mm external diameter and 50mm internal diameter is

subjected to a twisting moment of 8 kNm and a bending moment of 2.5 kNm. Calculate the

principal stresses and find direct stress which, acting alone would produce the same (i) maximum elastic strain energy (ii) maximum elastic shear strain energy as that produced by the principal stresses acting together. Take Poissons ratio equal to 0.25

12. The direct stresses on two mutually perpendicular planes in a two-dimensional stress system are

and 144MPa. In addition these planes carry a shear stress of 48MPa. Assuming factor of safety on elastic limit as 3:

(i) find the value of at which the shear strain energy is least and

(ii) if the failure occurs at this value of the shear strain energy, estimate the elastic limit of the

material in simple tension. 13. A cylindrical shell made of mild steel plate and 1.2m in diameter is to be subjected to an external

pressure of 1.5 MPa. If the material yields at 200 MPa, calculate the thickness of the plate on the

basis of the following three theories assuming a factor of safety 3 in each case: (i) Maximum principal stress theory (ii) Maximum shear stress theory and (iii) Maximum shear strain theory

14. A solid circular shaft is 100mm in diameter and subjected to combined bending and twisting moments, the bending moment being 3 times the twisting moment. If the direct tensile yield point of the material is 350MPa and factor of safety on the yield is to be 4, calculate the allowable twisting moments by the following three theories of elastic failure:

(i) Maximum principal stress theory (ii) Maximum shear stress theory and (iii) Shear strain energy theory 15. A shaft of 100mm diameter is subjected to a bending moment of 5kNm. Find the value of the

maximum torque which can be applied to the shaft for each of the following conditions: (i) maximum direct stress not to exceed 120MPa (ii) maximum shearing stress not to exceed 60MPa (iii) maximum shear strain energy per unit volume not to exceed that induced by simple

shear stress of 80MPa. 16. The stresses induced at a critical point in a machine component made of steel (yield

strength=380MPa) are x=100MPa, y=40MPa and xy=80MPa. Calculate the factor of safety by (i)

maximum normal stress theory (ii) maximum shear stress theory and (iii) maximum distortion

energy theory.



ASSIGNMENT NO. 4 1. At a point in a body, the displacement field is linear and is given by the following expressions. Find

all the strains.

0.06 0.05 0.01u x y z , 0.01 0.03v y x , 0.02 0.01w x z

2. The displacement field for a body is given by;

2 2(3 ) ( 2 )x y z x y u i j k What is the deformed position of a point originally at (3, 1, -2)?

3. Two points P and Q in the undeformed body have coordinates (0, 0, 1) and (2, 0, -1) respectively. Assuming that the displacement field given is;

2 2(3 ) ( 2 )x y z x y u i j k What is the distance between points P and Q after deformation?

4. Consider the displacement field; 2 2 23 (4 6 ) 10y yz x u i j k

What are the rectangular strain components at the point P (1, 0, 2)? Use only linear terms.

5. The general displacement field in a body, in Cartesian coordinates, is given as, 20.015 0.03u x y , 20.005 0.03v y xz , 20.003 0.001 0.005w z yz

Find the strain and rotation tensors ij

and ij

for point (1, 0, 2)

6. The state of strain at a given point in a body is given by the strain tensor;

0.002 0 0.004

0 0.06 0.001

0.004 0.001 0

ij

Find the invariants of strain tensor and the isotropic and deviatoric components of strain tensor. 7. Following state of strain exist at a point P

0.02 0.04 0

0.04 0.06 0.02

0 0.02 0

ij

What is the cubical dilatation at point P? 8. The displacement field in micro units for a body is given by;

2 2(3 ) ( 2 )x y z x y u i j k Determine the principal strains at (3, 1, -2) and the directions of the maximum and minimum principal strain.

9. The displacement field at a point is given by;

2 2 2 2 -5[ 2 (3 ) (3 ) ] 10x y y yz x z u i j k Determine the principal strains at (1, 0, 2) and the directions of the maximum and minimum principal strain.

10. Following state of strain exist at a point P

0.02 0.04 0

0.04 0.06 0.02

0 0.02 0

ij

Determine the principal strains and the directions of the maximum and minimum principal strains. 11. Following state of strain exist at a point P

0.02 0 0

0 0.1 0.15

0 0.15 0.2

ij

Determine the principal strains and the directions of the maximum and minimum principal strains.

12. The rectangular components of a small strain at a point are given by the following matrix.

1 0 0

0 0 4

0 4 3

ijp

where p = 10-4

Determine the principal strains and the direction of the maximum unit strain. 13. For the following plane strain distribution, verify whether the compatibility condition is satisfied:

23xx

x y , 24yy

y x , 32 2xy

xy x

14. For the following plane strain distribution, verify whether the compatibility condition is satisfied: 2 5

xxy xy , 2 2

yyy x x , 22

xyxy y

15. Verify whether the following strain field satisfies the equations of compatibility. p is a constant.

xxpy ,

yypx , 2 ( )

zzp x y , ( )

xyp x y , 2

yzpz , 2

zxpz



ASSIGNMENT NO. 5

1. A straight cylindrical bar is 16mm diameter and 1.2m long. It is freely supported at its two ends in a

horizontal position and loaded at the centre with a concentrated load of 90N. The central deflection is

found to be 5mm. If placed vertical and loaded along its axis, what load would cause it to buckle? What is the ratio of the maximum stresses in the two cases? Assume ends to be hinged.

2. A cast iron column 200mm external diameter is 20mm thick and has a length of 4.5m. Assuming it can

be treated as rigidly fixed at each end, calculate the safe load by Rankine's formula, using the following empirical constants c, = 550 MN/m

2, a (for hinged ends) =1/1600 factor of safety = 4.

3. A hollow cast iron column with-fixed ends supports an axial load of 1MN. If the column is 4.5m long and has an external diameter of 250 mm, find the thickness of metal required. Use the Rankine formula, taking Rankines constant of 1/6400 for pinned ends and a working stress of 80 MN/m2.

4. In the experimental determination of the buckling loads for 12.5 mm diameter mild steel pin-ended

struts of various lengths, two of the values obtained were: (i) length 0.5 m, load 9.25 kN, (ii) length 0.2 m, load 25 kN.

(a) Make the necessary calculations and then state whether either of the values conforms to the Euler formula for the critical load.

(b) Assuming that both values are in agreement with the Rankine formula, find the two constants

for this formula. 5. Compare the crippling loads given by Euler's and Rankine's formulae for a tubular steel strut 2.25m

long, having outer and inner diameters of 38mm and 33mm respectively, loaded through pin joints at both ends. Take the yield stress as 325MN/m2, the Rankine constant (for pinned ends) = 1/7500 and E = 200 GN/m2. For what length of strut of this cross-section does the Euler formula cease to apply?

6. A strut 3m long is constructed of steel tube 75mm outside diameter and 3 mm thick. The ends are pin

jointed, but the end load of 50kN is applied eccentrically through a line parallel and 2.5mm away from the axis of the strut, which is initially straight. Find the deflection and the maximum stress at the

centre of the length. E= 200 GN/m2. 7. A steel bar 25mm diameter and 1.8m long is tested as a pin-jointed strut. Calculate the crippling load

if the bar is initially straight and is centrally loaded. Find the load which will produce the yield stress of 300 MN/m2 (3300 kg/cm2) in this bar, if it is centrally loaded, but has initial curvature with an eccentricity of 9mm at the centre of its length. E = 200 GN/m2.

8. A vertical strut of uniform section is fixed rigidly at the base and carries a vertical load W at the top acting with an eccentricity e. In addition there is a horizontal force at the top, H, acting so as to produce bending in the same plane as Wand tending to increase the deflection. Obtain a formula for

the maximum bending moment. If the column is a tube 50 mm outside diameter and 44 mm inside diameter of free length 1.5 m and if W = 9 kN acting with an eccentricity of 25mm, find H to produce a maximum stress of 275 MN/m2, E = 200 GN/m2.

9. A rolled steel joist 300mm by 125mm and 6m long is used as a strut with hinged ends. It carries an

axial load of 300kN together with a lateral load of 16 kN/m uniformly distributed along one flange over the entire length. Determine the maximum stress produced. 1= 86 x 10-6 m4, A = 5.89 x 10-3 m2; E=

200 GN/m2.

10. A straight bar of alloy 1m long and 12.5mm x 5mm in section is mounted in a strut testing machine

and loaded axially till it buckles. Assuming the Euler formula for pinned ends to apply, estimate the maximum central deflection before the material attains its yield point at 280 MN/m2, E = 75 GN/m2.



ASSIGNMENT NO. 6

1. Figure 6.1 shows a circular ring of rectangular section, with a slit and subjected to load P. Calculate the magnitude of force P if the maximum stress along the section 1-2 is not to exceed 225 MPa.

Figure 6.1 Figure 6.2 Figure 6.3 2. Figure 6.2 shows a ring carrying a load of 30kN. Calculate the stresses at 1 and 2.

3. A curved bar is formed of a tube of 12cm outside diameter and 7.5mm thickness. The centre line

of this beam is a circular arc of radius 225mm. a bending moment of 3kNm tending to increase curvature of the bar is applied. Calculate the maximum tensile and compressive stresses set up in

the bar.

4. Figure 6.3 shows a crane hook lifting a load of 150kN. Determine the maximum compressive and

tensile stresses in the critical section of the crane hook. 5. A central horizontal section of hook is a symmetrical trapezium 60mm deep, the inner width being

60mm and the outer being 30mm. Estimate the extreme intensities of stress when the hook

carries a load of 30kN, the load line passing 40mm from the inside edge of the section and the centre of curvature being in the load line. Also plot the stress distribution across the section.

6. A curved bar of rectangular section 60mm wide x 75mm deep in the plane of bending initially unstressed is subjected to bending moment of 2.25kNm which tends to straighten the bar. The mean radius of curvature is 150mm. Find (i) position of neutral axis and (ii) greatest bending

stress.

7. A bar of rectangular section 40mm x 60mm is subjected to a bending moment of 2 kNm, its centre line is curved to a radius of 200mm. If the bending moment tends to increase the curvature,

determine (i) maximum tensile and compressive stresses in beam and (ii) stress at c.g. of the section

8. A steel bar 38mm in diameter is bent into a curve of mean radius 31.7mm. If a bending moment of 4.6Nm tending to increase the curvature, acts on the bar, find the intensities of maximum tensile and compressive stresses.

9. A curved bar of rectangular section 60mm x 40mm is bent in the shape of a horse shoe having a

mean radius of 70mm. Two equal and opposite forces of 10kN each are applied at a distance of 120mm from the centre line of the middle section so that they tend to straighten the rod.

Calculate the maximum tensile and compressive stresses.

10. A bar of circular cross-section is bent in the shape of a horse shoe. The radius of the section is

40mm and the mean radius is 80mm. two equal and opposite forces of 15kN each are applied so as to straighten the bar. Find maximum tensile and compressive stresses and position of neutral axis with the stress at c.g. of the section. Take the distance between c.g. and line of application of

load as 196mm.

11. A curved bar is formed of a tube 40mm outside radius and 5mm thickness. The centre line of this beam is a circular arc of radius 150mm. a bending moment of 2kNm tending to increase curvature

of the bar is applied. Calculate the maximum tensile and compressive stresses set up in the bar.



12. At the critical section of crane hook, trapezium in section, the inner and outer sides are 4cm and

2.5cm respectively and depth is 7.5cm. The centre of curvature of the section is at a distance of 6cm from the inner fibres and the load line is 5cm from the inner fibres. If the maximum stress is not to exceed 120MPa, what maximum load the hook can carry?

FORMULAE

(1) 2

2

M R y1

AR R yh

(2) Neutral axis position from c.g. axis = 2

n 2 2

Rhy

R h

SECTION FORMULA FOR h2

RECTANGULAR SECTION

32 2

e

R 2R Dh log R

D 2R D

A=BD

CIRCULAR SECTION

2 42

2

d dh

16 128R

2dA

4

TRIANGULAR SECTION

32 2

e

2R 3R 2d 3R 2dh log 1 R

d 3d 3R d

bdA

2

TRAPEZOIDAL SECTION

3

2 22 2e 2 e

1 1

R RR B bh blog R log B b R

A R d R

B bA d

2

1

d B 2bd

3 B b



ASSIGNMENT NO. 7

1. Derive the expression of shear centre for channel section as shown in Figure 7.1.

Figure 7.1 Figure 7.2

2. Derive the expression of shear centre for unequal I-section as shown in Figure 7.2.

Figure 7.3

3. Derive the expression of shear centre for channel section as shown in Figure 7.3.

4. Classify different types of plates and write Kirchhoffs hypothesis (assumptions) which are made in

classical small deflection theory of thin homogeneous elastic plates.

5. Derive the relationship between slopes and curvatures of a bent plate.

6. Derive the relationship between strain and curvature of a bent plate.

7. Derive the relationship between moment and curvature of a bent plate.

8. Derive the differential equilibrium equation in terms of bending and twisting moments for a thin

rectangular plate subjected to bending.

9. Explain different boundary conditions for a rectangular plate with neat free hand sketches.



ASSIGNMENT NO. 8 1. Evaluate the maximum value of the shearing stress by using a bar of rectangular section under

torsion T in approximate solution of rectangular section.

2. Evaluate the maximum value of the shearing stress by using a bar of elliptical cross section under torsion T in approximate solution of elliptical section.

3. Explain membrane analogy.

4. Which are the point in the process of finding the torsion stress distribution and the warping

displacement for solid sections of singly connected section?

5. Applying the stress function approach for the given bar of triangular section shown in Figure 8.1

Find the value of the angle () at the base and thus obtain the value of C.

Figure 8.1

6. Find the strength of an elliptical section in torsion as given by the torsional section modulus of the

section by using equation2 2

2 2

x y+ = 1

a b.

A

h

a G

C

B

.

Documents

Advanced Strength of Materials - GDLC_8851.pdf