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VARDHAMAN COLLEGE OF ENGINEERING, HYDERABAD (AUTONOMOUS) COURSE PACK FOR MECHANICS OF SOLIDS Course Title MECHANICS OF SOLIDS (MOS) Course Type Integrated Course Code A4305 Credit s 5 Class II Year I Semester Course Structure TLP Credits Contact Hours Work Load Total Number of Classes Per Semester Assessment in Weightage Theory 3 3 3 Practi ce 1 2 1 Theor y Prac tica l CIE SEE Tutori al 1 1 1 Total 5 6 5 42 28 30% 70% Course Instructors Course Lead: Mr. D.V. Ramana Reddy Theory Practice A. Mr. D. V. Ramana Reddy B. Dr. B. Subbaratnam Dr. B. Subbaratnam Mr. K. Sairam Dr. Shakti Jana Prasanna Mr. D. V. Ramana Reddy COURSE OVERVIEW Mechanics of solids is one of the important courses in Mechanical Engineering. This course will provide the fundamental background needed to understand the material behavior under various types of loadings. This deals stress and strain concepts, axially loaded members, thermal, shear and bending stresses, deflections in beams, stresses in thin and thick cylinders under internal and external pressure. The course is an integrated course having theory and practical components that integrates hands on experience to observe materials behavior under different types of loading by various mechanical testing machines. This course forms the basis for the study of advanced courses like Design of Machine Members, Finite Element Methods and Advanced Mechanics of Solids. It is imperative that these concepts are well understood. Dept. of ME Page 1 of 29
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VARDHAMAN COLLEGE OF ENGINEERING, HYDERABAD(AUTONOMOUS)

COURSE PACK FOR MECHANICS OF SOLIDS

Course Title MECHANICS OF SOLIDS (MOS) Course Type Integrated

Course Code A4305 Credits 5 Class II Year I Semester

Course Structure

TLP CreditsContact Hours

Work Load

Total Number of ClassesPer Semester

Assessment in Weightage

Theory 3 3 3

Practice 1 2 1Theory

Practical

CIE SEETutorial 1 1 1

Total 5 6 5 42 28 30% 70%

Course Instructors

Course Lead: Mr. D.V. Ramana Reddy

Theory Practice

A. Mr. D. V. Ramana ReddyB. Dr. B. Subbaratnam

Dr. B. Subbaratnam Mr. K. Sairam Dr. Shakti Jana Prasanna Mr. D. V. Ramana Reddy

COURSE OVERVIEW

Mechanics of solids is one of the important courses in Mechanical Engineering. This course will provide the fundamental background needed to understand the material behavior under various types of loadings. This deals stress and strain concepts, axially loaded members, thermal, shear and bending stresses, deflections in beams, stresses in thin and thick cylinders under internal and external pressure.

The course is an integrated course having theory and practical components that integrates hands on experience to observe materials behavior under different types of loading by various mechanical testing machines. This course forms the basis for the study of advanced courses like Design of Machine Members, Finite Element Methods and Advanced Mechanics of Solids. It is imperative that these concepts are well understood.

COURSE OBJECTIVE

The objective of this course is to make the students understand the concept of stresses and strains in various types of structures/machines under different loading conditions.

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COURSE OUTCOMES (COs)

After the completion of the course, the student will be able to:

CO# Course Outcomes POs PSOs

A43051 Understand the concepts of stress and strain in structural members - -

A43052 Construct SF & BM diagrams for beams 2 -

A43053 Solve numerical problems on structural members to find deformations and deflections.

1,2 -

A43054 Analyze stresses in bars, beams and cylindrical shells 2 1

A43055 Test for mechanical properties of the material and its behavioral analysis.

5,9,12 1

BLOOM’S LEVEL OF THE COURSE OUTCOMES

CO#Bloom’s Level

Remember(L1)

Understand(L2)

Apply(L3)

Analyze(L4)

Evaluate(L5)

Create(L6)

A43051✔

A43052✔

A43053✔

A43054✔

A43055✔

COURSE ARTICULATION MATRIX

CO#/POs P

O1

PO

2

PO

3

PO

4

PO

5

PO

6

PO

7

PO

8

PO

9

PO

10

PO

11

PO

12

PSO

1

PSO

2

A43052 2 3

A43053 2 3

A43054 2 3 1

A43055 2 3 2 3 3 1

Note: 1-Low, 2-Medium, 3-High

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VARDHAMAN COLLEGE OF ENGINEERING, HYDERABAD(AUTONOMOUS)

COURSE ASSESSMENT

S No

ComponentDuration in Hours

Component Wise

Marks

Total Marks

Weightage Marks

1Continuous Internal Evaluation (CIE)

Theory: Test-1 1.5 20

100 0.3 302 Theory: Test-2 1.5 20

3Alternate Assessment* - 20

4 Practical Exam 2 40

5 Semester End Exam (SEE) 3 100 100 0.7 70

Total Marks 100

* Assignment, Quiz, Class test, SWAYAM/NPTEL/MOOCs and etc.

COURSE CONTENT

THEORY

Contents

SIMPLE STRESSES AND STRAINS: Mechanical properties of materials, Types of stresses and

strains, Hooke’s law, stress strain diagram for mild steel, Working stress, Factor of safety,

Lateral strain, Poisson’s ratio and volumetric strain, Elastic module and the relationship

between them, Bars of varying section, composite bars, Temperature stresses.

SHEAR FORCE AND BENDING MOMENT: Definition of beam, Types of beams, Concept of

shear force and bending moment, Relation between Shear Force and Bending Moment, and

rate of loading at a section of a beam. Shear Force and Bending Moment diagrams for

cantilever simply supported and overhanging beams subjected to point loads, U.D.L.,

uniformly varying loads and combination of these loads.

FLEXURAL STRESSES: Theory of simple bending, Assumptions, Derivation of bending

equation, Neutral axis, Determination bending stresses, section modulus of rectangular and

circular sections (Solid and Hollow), I, T, Angle and Channel sections, Design of simple beam

sections. SHEAR STRESSES: Derivation of formula, Shear stress distribution across various

beams sections like rectangular, circular, I, T, angle and channel sections.

DEFLECTION OF BEAMS: Bending into a circular arc slope, deflection and radius of curvature,

Differential equation for the elastic line of a beam, Double integration and Macaulay’s

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methods, Determination of slope and deflection for cantilever and simply supported beams

subjected to various loads.

THIN AND THICK CYLINDERS: Thin seamless cylindrical shells, Derivation of formula for

longitudinal and circumferential stresses hoop, longitudinal and volumetric strains, changes in

diameter, and volume of thin cylinders, Thin spherical shells. A thick cylinder lame’s equation,

cylinders subjected to inside and outside pressures, compound cylinders.

PRACTICE:

No Title of the ExperimentTools and

TechniquesExpected Skill

/Ability

1Determine tensile strength of mild steel specimen using Universal Testing Machine.

Vernier calipers, Steel rule and Universal Testing Machine

Determine the Tensile strength of material subjected to tensile load

2Determine modulus of rigidity given specimen using Torsion Testing Machine.

Vernier calipers, Steel rule and Torsion Testing Machine

Determine the modulus of rigidity of material subjected to Torsion

3Determine Young’s modulus and stiffness of Simple supported beam.

Measuring tape, Vernier calipers, Cantilever and simply supported Beam setups

Verify the modulus of elasticity and stiffness of given beams 4

Determine Young’s modulus and stiffness of Cantilever beam.

5Determine Hardness of given specimen using Brinell Hardness Testing Machine.

Vernier calipers and Hardness testing machine setups

Measuring hardness values of different materials

6Determine Hardness of given specimen using Rockwell Hardness Testing Machine.

7Determine Hardness of given specimen using Vickers’s Hardness Testing Machine.

8Determine impact strength of given specimen by Charpy Testing Machine.

Vernier calipers, Charpy and Izod testing machine setup

Find out the impact strength of different materials

9Determine impact strength of given specimen by Izod Testing Machine.

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VARDHAMAN COLLEGE OF ENGINEERING, HYDERABAD(AUTONOMOUS)

No Title of the ExperimentTools and

TechniquesExpected Skill

/Ability

10Determine stiffness and modulus of rigidity of given spring using Spring Testing Machine.

Vernier calipers and Spring testing machine setup

Determine the stiffness and modulus of rigidity of a given test springs

11Determine compressive strength of given specimen using Compression Testing Machine.

Steel rule and Compression testing machine

Find out the compressive strength of given test sample block

Text Books:

1. Ramamrutham. S (2012), Strength of Materials, 17th edition, Dhanpat Rai Publications, New Delhi, India.

Reference Books:

1. S. S Rattan, (2017), Strength of Material, 3rd Edition, Tata McGraw-Hill.2. Bhavikatti S. S (2008), Strength of materials, 3rd edition, Vikas Publishing House, New

Delhi, India.3. Bansal R. K (2007), Strength of materials, 10th edition, Laxmi Publications,

Hyderabad, India.

Journals/Magazines

1. Mechanics and Physics of Solids2. Mechanics of Solids – Springer

SWAYAM/NPTEL/MOOCs:

1. http://www.nptelvideos.in/2012/11/mechanics-of-solids.html

2. https://nptel.ac.in/courses/105106116/

3. http://www.nptelvideos.in/2012/11/strength-of-materials-prof.html

Self-Learning Exercises:

a) Simple stresses and strains

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b) Shear force and bending moments diagramsc) Bending and shear stress distribution across various beams sections d) Slope and deflection for beams subjected to various loadse) Stresses in thin and thick cylindrical shell for industrial applications

LESSON PLAN

Lecture # Topics to be Covered1. Introduction to Mechanics of solids and mechanical properties of materials 2. Types of stresses and strains, Elastic limit and Hooke’s law3. Stress strain diagram for mild steel showing salient points

4.Ultimate stress, Working stress, Factor of safety, Lateral strain, Longitudinal strain Poisson’s ratio and volumetric strain

5.Volumetric strain of a bar of rectangular section and cylindrical rod, Numerical problems

6. Elastic constants, the relation between them7. Bars of varying section and numerical problems8. Stresses in composite bars and Numerical problems on composite bars9. Thermal (or) temperature stresses

10. Numerical problems on temperature stresses

11.Introduction to beams, Types of support and beams, Concept of shear force and bending moment, Relation between Shear Force and Bending Moment and Rate of loading at a section of a beam

12.Shear Force and Bending Moment diagrams for cantilever beam and simply supported beam

13.Shear Force and Bending Moment diagrams for overhanging beams and draw Shear Force and Bending Moment diagrams for given cantilever beams subjected to point loads

14.Shear Force and Bending Moment diagrams for given cantilever beams subjected to uniformly distributed loads and uniformly varying loads

15.Shear Force and Bending Moment diagrams for given cantilever beam subjected to combination of loads

16.Shear Force and Bending Moment diagrams for given simply Supported beams subjected to point loads and uniformly distributed loads

17.Shear Force and Bending Moment diagrams for given simply Supported beams subjected to uniformly varying loads and combination of loads

18.Shear Force and Bending Moment diagrams for given over hanging beams subjected to point loads and uniformly distributed loads

19.Shear Force and Bending Moment diagrams for given over hanging beams subjected to uniformly varying loads and combination of loads

20.Introduction to flexural stresses and theory of simple bending, Assumptions, Derivation of bending equation, Neutral axis, and determination bending stresses

21. Section modulus of rectangular and circular sections (Solid and Hollow)22. Section modulus of I, T angle and channel sections 23. Problems on design of simple beam sections.

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Lecture # Topics to be Covered

24.Introduction to shear stresses, derivation of shear stress formula and Shear stress distribution across various beams sections like rectangular and circular

25. Shear stress distribution across various beams I and T sections 26. Shear stress distribution across various beams angle and channel sections27. Numerical problems on rectangular, circular and I sections28. Numerical problems on T, angle and channel sections

29.Introduction to deflection of beams, Elastic curve (Bending into a circular arc slope)

30.Relationship of slope and deflection with (i) radius of curvature(ii) bending moments (iii)shear force and (iv) Load intensity , and sign convention for slope and deflection

31.Deflection and radius of curvature, Differential equation for the elastic line of a beam

32.Methods of finding slope and deflection at any section of beam and double integration method and Macaulay’s method

33. Numerical problems on double integration method and Macaulay’s method

34.Determination of slope and deflection for cantilever and simply supported beams subjected to various loads

35. Numerical problems on deflection of beams

36.Introduction to thin and thick cylinders and its importance, Thin seamless cylindrical shells

37. Derivation of formula for longitudinal and circumferential stresses hoop

38.Design of thin cylindrical shell Changes in diameter, and volume of thin cylinders and Numerical problems

39. Introduction to thick cylindrical shell and lame’s equation40. Thick cylinders subjected to inside and outside pressures

41.Introduction to compound cylinders, Stresses in compound cylinders and numerical problems

42. Revision

PROBLEM BASED LEARNING

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No. A] Problems on Simple stresses and strains (1-20)

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1.

Compute the stresses at various sections and the total elongation of the bar shown in Figure. L1 = 4.0 m, L2 = 3.0 m, L3 = 2.0 m, A1 = 100 mm2, A2 = 400 mm2 and A3 = 200 mm2.

Es = 2.05 X 105 N/mm2.

2.

Three bars made of copper, zinc and aluminium are of equal length and have cross-section 500, 750 and 1000 square mm respectively. They are rigidly connected at their ends. If this compound member is subjected to a longitudinal pull of 250kN, estimate the proportional of the load carried on each rod and the induced stresses. Take the value of E for copper= 1.3 x 105 N/mm2, for zinc = 1.0 x 105 N/mm2 and for aluminium = 0.8 x 105 N/mm2.

3.

A bar as shown in fig. is subjected to a tensile load of 150kN. If the stress in the middle portion is limited to 160 N/mm2. Determine the diameter of middle portion. Find also the length of the middle portion if the total elongation of the bar is to be 0.25cm. Take young modulus is equal to 2x105N/mm2.

4.

Calculate the modulus of rigidity and bulk modulus of a cylindrical bar of diameter of 25mm and of length 1.6m, if the longitudinal strain in a bar during a tensile test is four times the lateral strain. Find the change in volume, when the bar is subjected to a hydrostatic pressure of 100N/mm2. Take young modulus is equal to 1x105N/mm2.

5.

A steel rod of 20mm diameter passes centrally through a copper tube 40mm external diameter and 30mm internal diameter. The tube is enclosed at each end by rigid plates of negligible thickness; the nuts are tightened lightly home on the projected parts of the rod. if the temperature of the assembly is raised by 60oC,calculate the stresses developed in copper and steel . Take E for steel and copper as 200GPa,and 100GPa and coefficient of thermal expansion for steel and copper as 12x 10 -6 per oC and 18x 10-6 per oC

6. A compound bar ABC 1.5m long is made up of two parts of aluminium and steel and that cross sectional area of aluminium bar is twice that of the steel bar. The rod is

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subjected to an axial tensile load of 200 kN. If the elongations of aluminium and steel parts are equal, determine the lengths of the two parts of the compound bar. Take E for

steel as 200GPa and E for aluminium as 1/3rd of E for steel.

7.

A rectangular block 250 mm × 100 mm × 80 mm is subjected to axial loads as follows:

i) 480kN tensile in the direction of its length

ii) 900kN tensile on the 250mmx80mm faces

iii) 1000 kN compressive on the 250 mm x 100 mm faces. Assuming Poisson’s ratio as 0.25, determine in terms of the modulus of Elasticity E of the material, the strains in the direction of each force If E=2.0x10 N/mm , determine the values ⁵ of the modulus of rigidity and bulk modulus for the material of the block. Also, calculate the change in the volume of the block due to the applications of the loading specified in Fig.

8.

A reinforced concrete column 500 mm x 500 mm in section is reinforced with 4 steel bars of 25 mm diameter, one in each corner. The column is carrying a load of 1000 kN. Determine the stresses in the concrete and steel bars. Take E for steel as 210 GPa and E for concrete as 14 GPa.

9.Determine the young’s modulus and Passion’s ratio of a metallic bar of length 25cm breadth 3cm depth 2cm when the beam is subjected to an axial compressive load 240kN. The decrease in length is given by 0.05cm and increase in breath 0.002.

10. Calculate the strain energy that can be stored in a steel bar 2.4m long and 1000mm2

cross sectional area, when subjected to a tensile stress of 50MPa. Take E = 200GPa.

11.The extension in a rectangular steel bar of length 400mm and thickness 10 mm, is found to be 0.21mm.The bar tapers uniformly in width from 100mm to 50mm. If E for the bar is 2x105 N/mm2, determine the axial tensile load on the bar.

12. The ultimate tensile stress for a hollow steel column which carries an axial load of

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2MN is 500 N/mm2.If the external diameter of the column is 250mm, determine the internal diameter. Take FOS as 4.0.

13.Determine the changes in length and breadth and thickness of a steel bar which is 5m long, 40mm wide and 30mm thick and is subjected to an axial pull of 35kN in the direction of the length. Take E = 2x105 N/mm2 and Poisson’s ratio 0.23

14.

A bar 30 mm in diameter and 200mm long was subjected to an axial pull of 60 kN. The extension of the bar was found to be 0.1 mm, while decrease in the diameter was found to be 0.004 mm. Find the Young’s modulus, Poisson’s ratio, rigidity modulus and bulk modulus of the material of the bar.

15.

A reinforced concrete column 500x500 mm in section is reinforced with a steel bar of 25mm diameter, one in each corner, the column is carrying the load of 1000 kN Find the stresses induced in the concrete and steel bar. Take E for steel = 2.1x105 N/mm2

and E for concrete = 1.4x103 N/mm2

16.

A steel rod of 3 cm diameter is enclosed centrally in a hollow copper tube of external diameter 5 cm and internal diameter of 4 cm. the composite bar is then subjected to an axial pull of 45000N. If the length of each bar is equal to 15cm,determine i) The stress in the rod and tube ii) load carried by each bar Take E for steel = 2.1x105 N/mm2 and for copper =1.1x105 N/mm2

17.

A copper bar is 900mm long and circular in section. It consists of 200mm long of 40mm diameter,500mm long bar of 15mm diameter and 200mm long bar of 30 mm diameter. If the bar is subjected to a tensile load of 60 kN. Find the total extension of the bar. Take E for the bar material as 100GPa

18.A concrete column of 350mm diameter is reinforced with four bars of 25 mm diameter. Find the stress in steel when the concrete is subjected to a stress of 4.5 MPa. Also find the safe load the column can carry. Take Es / Ec= 15.

19.

A steel rod of 25 mm diameter axially passes through a brass tube of 25 mm internal diameter and 35 mm external diameter when the nut on the rod is tightened, initial stress of 10 MPa is developed in the rod. The temperature of the tube is then raised by 600. Calculate the final stresses in the rod and tube. Take Es=200GPa, Eb= 80 GPa. αs=11.7x10-6/0C and αb=19x10-6/0C

20.A round bar 40 mm diameter is subjected to an axial pull of 80KN and reduction in diameter was found to be 0.00775mm.Find poison’s ratio and young’s modulus for the material of the bar. Take value of shear modulus as 40 GPa

B] Problems on Shear force and Bending moments (21-40)

21.A cantilever beam of length 2 m carries the point loads as Shown in Figure. Draw the shear force and B.M. diagrams for the cantilever beam.

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22.A cantilever of length 2.0m carries a uniformly distributed load of 1kN/m run over a length of 1.5m from free end. Draw the shear force and bending moment diagrams for the cantilever?

23.A cantilever of length 2.0m carries a uniformly distributed load of 2kN/m run over the whole length and a point load of 3KN at the free end. Draw shear force and bending moment diagrams for the cantilever?

24.A cantilever of length 2.0m carries a uniformly distributed load of 1.5kN/m run over the whole length and a point load of 2KN at a distance of 0.5m from the free end. Draw shear force and bending moment diagrams for the cantilever?

25.

A cantilever of length 5.0m is loaded as shown in fig. Draw the S.F and B.M diagrams for the cantilever beam.

26.A cantilever beam of span 4m is subjected to udl of 2 kN/m over its entire length. Sketch the bending moment diagram for the beam.

27.

Draw the shear force and bending moment diagrams for the beam as shown in fig. and clearly mark the maximum bending moment.

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28.

Construct S. F. D & B. M. D for the S. S beam shown in Figure

29.

Draw the shear force and bending moment diagram for a simply supported beam o f length 9 m and carrying a uniformly distributed load of 10 kN/m for a distance of 6 m from the left end. Also calculate the maximum B.M. on the section.

30.A beam of length 12m is simply supported and carries point loads of 6kN each a distance of 3m and 7m from the left support and also a uniform distributed load of 1kN/m between the point loads. Draw the S.F and B.M diagrams for the beam.

31.

Draw the S.F and B.M diagrams for a simply supported beam of length 9m and carrying a uniformly distributed load of 10 kN for a distance of 6m from the left end. Also calculate the maximum B.M on the sections.

32.

Draw the shear force and B.M. diagrams for a simply supported beam of length 8m and carrying a uniformly distributed load of 10 kN/m for a distance of 4 m as shown in Figure.

33. Draw the S.F and B.M diagrams for a simply supported beam of length 7m and carrying uniformly distributed loads as shown in the figure.

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34.

Draw the S.F and B.M diagrams for a simply supported beam of length 8m and carrying a uniformly distributed load of 10kN for a distance of 4m as shown in the fig.

35.

A simply supported beam of length 5m carries a uniformly increasing load of 800N/m run at one end to 1600 N/m run at the other end. Draw the S.F and B.M diagrams for the beam. Also calculate the position and magnitude of maximum bending moment.

36.A simply supported beam of span 4m is subjected to udl of 2kN/m over its entire length. Sketch the bending moment diagram for the beam.

37.A overhanging beam of supported n 4m is subjected to udl of 2kN/m over its entire length. Sketch the bending moment diagram for the beam.

38.Draw the S.F. & B.M. diagrams for simply supported beam of length L carrying a point load W at its middle point.

39.Draw the S.F. & B.M. diagrams for overhanging beam of length L carrying a point load W at its middle point.

40.

Draw the shear force and bending moment diagram for the beam which is loaded as shown in fig. determine the point of contra flexure with the span AB.

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C] Problems on Flexural and shear stresses (41-68)

41.Define Neutral axis. Sketch the bending stress distribution across the cross section of a rectangular beam section 230 × 400 m subjected to 60kN-m moment.

42.Design the cross section for a beam acted upon by a bending moment = 50 kN-m. If width of beam is 200 mm calculate, depth. Stress = 9 N/mm2.

43.

A rectangular beam 200mm deep and 300 mm wide is simply supported over a span of 8 meters. What uniformly distributed load per meter the beam may carry, if the bending stress is not exceed 120N/mm2

44.

A rectangular beam 100 mm wide and 250 mm deep is subjected to a maximum shear force of 50kN. Determine :

a) Average shear stressb) Maximum shear stressc) Shear stress at a distance of 25mm above the neutral axis

45.

A beam is simply supported and carries a uniformly distributed load of 40 kN/m run over the whole span. The section of the beam is rectangular having depth as 500 mm. If the maximum stress in the material of the beam is 120 N/mm2 and moment of inertia of the section is 7 × 108 mm4, find the span of the beam.

46.

A timber beam of rectangular section is to support a load of 20 kN/m uniformly distributed over a span of 3.6m when beam is simply supported. If the depth of section is to be twice the breadth, and the stress in the timber is not to exceed 7 N/mm2, find the dimensions of the cross-section.

47.

A timber beam of rectangular section is to support a load of 20 kN over a span of 4m. If the depth of the section is to be twice the breadth, and the stress in the timber is not to exceed 60 N/mm2, find the dimensions of the cross-section. How would you modify the cross-section of the beam if it were a concentrated load placed at the centre with the same ratio of breadth to depth?

48.A hollow circular bar used as a beam has an outside diameter twice of the inside diameter. If it is subjected to a maximum bending moment of 40 kN-m and the allowable bending stress is 100MPa, determine the inside diameter of the bar.

49.

A beam is simply supported and carries a uniformly distributed load of 40KN/m run over the whole span. The section of the beam is rectangular having depth as 500mm. if the maximum stress in the material of the beam is 120N/mm2 and moment of inertia of the section is 7 X 108 mm4 , find the span of the beam

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50.A steel plate of width 60mm and thickness 10mm is bent into a Circular arc of radius 10m. Determine the max stress induced and The bending moment which will produce the max stress. Take E = 2x105 N/mm2

51.A copper plate of width 70mm and thickness 20mm is bent into a Circular arc of radius 2m. Determine the max stress induced and The bending moment which will produce the max stress. Take E = 1.5x105 N/mm2

52.

A square beam 20 mm × 20 mm in section and 2 m long is supported at the ends. The beam fails when a point load of 400 N is applied at the centre of the beam. What uniformly distributed load per metre length will break a cantilever of the same material 40 mm wide, 60 mm deep and 3 m long?

53.

An I-section has the following dimensions:

Flanges: 150 mm × 20 mm Web: 300 mm × 10 mm

The maximum shear stress developed in the beam is 16.8 N/mm2. Find the shear force to which the beam is subjected.

54.Two wooden planks 150 mm x 50mm each are connected to form a T section of a beam. If a moment of 3.4 kN-m is applied around the horizontal neutral axis, inducing tension below the neutral axis, find the stresses at the extreme fibers of the cross section. Also calculate the total tensile force on the cross section.

55.

A cast iron beam has an I-section with top flange 100mm × 40mm, web 140mm×20mm and bottom flange 180mm × 40mm. If tensile stress is not to exceed 35MPa and compressive stress 95MPa, what is the maximum uniformly distributed load the beam can carry over a simply supported span of 6.5m.

56.

A mild steel bracket has a cross section of T-section with top flange of 200mm × 50mm, web 150mm×20mm If tensile stress is not to exceed 45MPa and compressive stress 95MPa, what is the maximum uniformly distributed load the beam can carry over a simply supported span of 6.5m.

57.Obtain the shear stress distribution for a rectangular cross section 230 × 400 mm subjected to a shear force of 40 kN. Calculate maximum and average shear stress.

58.A 300 mm × 150 mm I –girder has 12 mm thick flanges and 8 mm thick web it is subjected to a shear force of 150kN at a particular section. Find the maximum shear stress in the web and flange.

59.Show that the maximum shear stress in a rectangular beam is 1.5 times of average shear stress when it is subjected to a bending moment.

60.A wooden beam supports udl of 40 kN/m over a simple supported span of 4m. It is of rectangular cross-section of 200mm wide and 400mm deep. Calculate average and maximum shear stress.

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61.Show that max shear stress in a solid circular shaft is 1.33 times of average shear stress when it is subjected to a bending moment.

62.Show that the maximum shear stress in a beam of square section with a diagonal horizontal is 9/8 times of average shear stress.

63.A square of 20mm side is used as a beam with diagonal horizontal subjected to a vertical shear force 2KN at a section. Determine the maximum shear stress

64.A beam of I-section is having overall depth of 700mm and overall width as 230mm. The thickness of the flanges is 25mm where as the thickness of the web is 20mm. If the section carries a shear force of 64kN, Calculate the shear stress at salient points.

65.A rectangular beam 125mm wide is subjected to maximum shear force of 110kN. Find the depth of the beam if the maximum permissible shear stress is 7Mpa

66.A wooden beam 100mm wide and 150mm deep is simply supported over a span of 4m. If shear force at a section of the beam is 4500N,find the shear stress at a distance 25mm above the N.A.

67.A timber beam of rectangular section is simply supported at the ends and carries appoint load at the center of the beam. The maximum bending stress is 12N/mm2, find the ratio of span to the depth.

68.A I section beam 350mm*150mm has a web thickness of 10mm and a flange thickness of 20mm.if the shear force acting on the section is 40KN.find the maximum shear stress developed in the I section.

D] Problems on Deflection of beams (69-82)

69.Write the expressions for max. slope and deflection of a cantilever beam with a point load at free end.

70.Write the maximum value of deflection for a cantilever beam of length L, constant EI and carrying central concentrated load?

71.

Prove that the slope and deflection of a cantilever beam carrying uniformly distributed load over the whole length are given by

Slope= wL3/6EI and deflection = wL4/8EI

72.A cantilever of 4m span length carries a load 40 kN at its free end. If the deflection at the free end is not to exceed 8mm, what must be the moment of inertia of the Cantilever section?

73.A steel Cantilever of 2.5m effective length carries a load of 25 kN at its free end. If the deflection at the free end is not to exceed 0.5 cm, what must be the I value of the section of the cantilever? Take E = 210 GN/m2.

74. Determine the deflection at the free end of a cantilever beam which is 2m long and carries a point load of 9kN at the free end and also a uniformly distributed load of

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8kN/mover a length of 1m from the fixed support.

I= 2.25X107mm4 and E= 2.2X105N/mm2

75.

A beam of 6m is simply supported at its ends and carries two point loads of 48kN and 40 kN at a distance of 1m and 3m respectively from the left support find:

i) Deflection under each loadii) Maximum deflectioniii) The point at which maximum deflection occur

E= 2X 105MPa, and I = 85X 106mm4

76.

A beam of length 20m is simply supported at its ends and carries two point loads of 4kNand 10kN at a distance of 8m and 12m from the left support. Calculate :

i) Deflection under each loadii) Maximum deflection

Take E = 2X 106MPa and I = 109mm4

77.

A beam of length 5m and of uniform rectangular section is simply supported at its ends. It carries a uniformly distributed load of 9 kN/m run over the entire span. Calculate the width to depth of the beam if permissible bending stress is 7MPa and central deflection is not to exceed 1cm. Take E for beam material 104 MPa

78.

A steel girder of uniform section, 14 meters long, is simply supported at its ends. It carries concentrated loads of 120 kN and 80 kN at two points 3 meters and 4.5 meters from the two ends respectively.

a) Calculate the deflection of the girder at the two points under the two loads:

b) The maximum deflection. Use Macaulay’s Method.

Take: I = 16 × 104 m4, and E = 210 × 106 kN/m2.

79.

Simply supported beam 5 m long carries concentrated loads of 10 kN each at a distance 1m from the ends. Calculate:

(a) Maximum slope and deflection for the beam, and

(b) Slope and deflection under each load.

Take: EI = 1.2 × 104 kN.m2.

80.What are the values of slope and deflection for a cantilever beam of length ‘L’ subjected to Moment ‘M’ at the free end?

81.Write the relation between deflection and bending moment and flexural rigidity for a beam?

82.

A beam of length L carries a uniformly distributed load w/unit length and rests on three supports, two at the ends and one in the middle. Find how much the middle support be lower than the end ones in order that the pressures on the three supports shall be equal.

E] Problems on Thin and Thick Cylinders (83-102)

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VARDHAMAN COLLEGE OF ENGINEERING, HYDERABAD(AUTONOMOUS)

83.Derive the expressions for hoop stress, longitudinal stress, change in diameter, length and capacity of the thin cylindrical shell, when subjected to an internal pressure.

84.A spherical shell of internal diameter 0.9m and of thickness 10mm is subjected to an internal pressure of 1.4N/mm2. Determine the increase in diameter and increase in volume. Take E = 2X105MPa and poisson’s ratio is 1/3

85.

A cylindrical thin drum 800mm in diameter and 3m. long has a shell thickness of 10mm. If the drum is subjected to an internal pressure of 2.5 N/mm2, determine (a) Hoop stress (b) Longitudinal stress(c) Change in diameter (d) Change in length (e)change in volume. Take E = 2 X 105 N/mm2, µ = 0.25.

86.

A cylindrical shell 3m long has 1m. internal diameter and 15mm metal thickness. Calculate the circumferential and longitudinal stresses induced and find out the changes in the dimensions of the shell, if it is subjected to an internal pressure of 1.5 N/mm2. Also find out the maximum shear stress. Take E = 2 X 105 N/mm2, µ = 0.3.

87.

A cylindrical vessel, whose ends are closed by means of rigid flange plates, is made of steel plate 4 mm thick. The length and the internal diameter of the vessel are 100 cm and 30 cm respectively. Determine the longitudinal and hoop stresses in the cylindrical shell due to an internal fluid pressure of 2 N/mm2. Also calculate the increase in length, diameter and volume of the vessel. Take E = 2 × 105 N/mm2 and µ = 0.3.

88.

A vertical gas storage tank is made of 25 mm thick mild steel plate and has to withstand maximum internal pressure of 1.5 MN/m2. Determine the diameter of the tank if stress is 240 MN/m2, factor of safety is 4 and joint efficiency is 80%.

89.

A steam boiler, 3 m in diameter, is made of 25 mm thick mild steel plates. The efficiency of longitudinal riveted joint is 88% and that of circumferential riveted joint is 70%. Determine the permissible stream pressure if the maximum tensile stress in the plate section through the rivets is not to exceed 120 N/mm2. Also calculate (a) Circumferential stress in the solid plate section, and (b) Longitudinal stress in the plate section through the rivets.

90.

A thin cylindrical shell with following dimensions is filled with a liquid at atmospheric pressure: Length = 1.2 m, external diameter = 20 cm, thickness of metal = 8 mm. Find the value of the pressure exerted by the liquid on the walls of the cylinder and the hoop stress induced if an additional volume of 25 cm3 of liquid is pumped into the cylinder Take E = 2.1 × 105 N/mm2 and 1/m= 0.33.

91.

A thick cylinder having internal radius 200mm and external radius 300mm is subjected to 4N/mm2. Find the internal pressure that can be applied if the max. Permissible stress is 15N/mm2. Find also the change in thickness of the cylinder. Take E = 200GN/m2 and Poisson ratio =0.3

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92.

A steel tube 240 mm external diameter is to be shrunk on another steel tube of 80 mm internal diameter. After shrinking, the diameter at the junction is 160 mm. Before shrinking on, the difference of diameter at the junction was 0.08 mm. Calculate the radial pressure at the junction and hoop stress developed in the two tubes after shrinking on Take E = 2×105

N/mm2.

93.

A boiler shell is made of 15 mm thick plate having a limiting tensile stress of 125 N/mm2. If the longitudinal and circumferential efficiencies are 70% and 60% respectively, determine the maximum diameter of the shell. The allowable maximum pressure is 2.2 N/mm2.

94.Calculate the thickness of metal necessary for a thick cylinder of internal diameter 80mm to withstand an internal pressure of 30N/mm2. If the maximum permissible tensile stress is 125N/mm2.

95.Compare the values of maximum and minimum hoop stresses for a cast steel cylindrical shell of 600 mm external dia. and 400 mm internal dia. Subjected to a pressure of 30N/mm2 applied (a) Internally and (b) Externally.

96.

A tube whose external and internal diameters are 360 mm and 240 mm respectively has another tube 60 mm thick shrunk on to it. The bore of the outer tube is machined to 1 mm less than the external diameter of the tube on to which it is subsequently shrunk. If the tubes are made of steel for which the value of E = 200KN/mm2, Determine expressions for the radial and hoop stresses developed in the inner tube.

97.

Determine the maximum and minimum hoop stress across the section of a pipe of 400mm internal diameter and 100 mm thick, when the pipe contains a fluid at a pressure of 8N/mm2. Also sketch the radial pressure distribution and hoop stress distribution across the section.

98.

A compound steel cylinder has a bore of 80 mm and an outside; diameter of 160 mm, the diameter at the common surface being 120 mm. Find the radial pressure at the common surface which must be provided by shrinkage if the resultant maximum hoop tension in the inner cylinder under a superimposed internal pressure of 60N/mm2 is to be half the value of the maximum hoop tension which would be produced in the inner cylinder if that cylinder alone were subjected to an internal pressure of 60N/mm2. Determine the final hoop tensions at the inner and outer surfaces of both cylinder under the internal pressure of 60N/mm2 and sketch a graph to show the hoop tension varies across the cylinder wall.

99.

A compound cylinder is formed by shrinking one tube on to another, the final dimensions being, internal diameter 120 mm, external diameter 240 mm, diameter at junction 180 mm. if after shrinking on, the radial pressure at the common surface is 8N/mm2, calculate the initial hoop stresses across the sections of the inner and outer tubes. If a fluid under a pressure of 60N/mm 2, is admitted inside the compound cylinder, calculate the final stresses set up in the sections of the pipes.

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100.

A compound cylinder is formed by shrinking a tube of external diameter 300mm over another tube of internal diameter 150 mm. After shrinking, the diameter at the junction of the tubes is found to be 250 mm and radial compression as 28 N/mm2. Find the original difference in radii at the junction. Take E for the cylinder metal as 200GPa.

101.

A compound cylinder is formed by shrinking a tube of 200 mm internal diameter and 20 mm thick over another tube of 120 mm diameter and 40 mm thick. If radial pressure at the common surface, after shrinking is 12 N/mm2, then determine the final stresses across the section when a fluid under a pressure of 45 N/mm2 is admitted into the cylinder.

102.

A compound cylinder is formed by shrinking a tube of external diameter 300mm over another tube of internal diameter 150 mm. After shrinking, the diameter at the junction of the tubes is found to be 250 mm and radial compression as 28 N/mm2. Find the original difference in radii at the junction. Take E for the cylinder metal as 200GPa.

PROJECT BASED LEARNING

To enhance the skill set in the integrated course, the students are advised to execute course-based design projects. Some sample projects are given below:

No. Suggested Projects

1. Measure Engineering stress stain curve for steel and from achieved value, theoretically calculate true stress - strain curve and plot both the curves.

2. Measure the bending strength of mild steel specimen using Universal Testing Machine.

3. Determination of shear strength of mild steel specimen using Universal Testing Machine.

4. Determination of compressive strength of mild steel specimen using Universal Testing Machine.

5. Determination of modulus of rigidity given specimen using Torsion Testing Machine.

6. Estimation of Young’s modulus and stiffness of Simple supported beam.

7. Estimation of Young’s modulus and stiffness of Cantilever beam of given material

8. Measure hardness of sample of steel in Vickers and Brinell scale and find out the correlation between them (Show in graphical representation)

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No. Suggested Projects

9. Estimation of Hardness of given specimen using Rockwell Hardness Testing Machine and compare Brinell and Vickers test results.

10. Determination of Hardness of given specimen using Vickers’s Hardness Testing Machine.

11. Evaluate the impact strength of various materials by Charpy Testing Machine.

12. Evaluate the impact strength of various materials by Izod Testing Machine.

13. Compare the stiffness and modulus of rigidity of various springs

14. Compare the compressive strength of various materials

15. What is meant by Ductile to brittle transition temperature and measure DBT of Steel

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