Course Manual 2202

CARLETON UNIVERSITY

Department of Mechanical & Aerospace Engineering

MAAE2202 Mechanics of Solids I

August 2013

Car leton University

Depar tment of Mechanical & Aerospace Engineer ing MAAE2202 Mechanics of Solids I Course Outline 1. Review of Statics, and Fr iction Problems

Review of statics in two-dimensions; types of common structural components; types of support and reactions in 2D equilibrium; free body diagrams. Friction problems: types of friction; equilibrium problems involving Coulomb friction -general examples, friction between coaxial discs and flexible belts.

2 Concepts of Stress and Strain

Definition of stress: normal stress; shear stress; complementary shear stress; hydrostatic stress. Definition of strain: direct strain; engineering and natural strain; volumetric strain. Influence of material properties: stress-strain diagrams; yield stress; ultimate stress; measure of ductility; elastic deformation and plastic deformation. Hooke's law in one-dimension; Poisson's ratio; thermal effects. General 3D Hooke's law for direct stresses; bulk modulus. Shear stress-shear strain relation.

3. Statically Determinate and Indeterminate Stress Systems

Distinction between statically determinate and indeterminate stress systems. Principles of solution. Examples in uniaxial and multiaxial stress systems.

4. Torsion of Circular Sections

Torsion of circular sections: relation between stress, strain, angle of twist and torque. Non-uniform circular shafts. Torque transmission by gear wheels.

5. Symmetr ic Bending of Beams (I)

Shear force and bending moment diagrams; relationship between loads, shear force and bending moment. Deformations in pure bending; direct stresses in bending; first and second moments of area; simple bending formula; applications. Combined bending and direct stress. Shear stress in bending; shear flow.

6. Symmetr ic bending of Beams (II)

Moment-curvature relationship. Slopes and delections by direct integration. Use of Macaulay's step function. Deflections by superposition. Statically indeterminate beams problems.

7. Buckling Instability

Concepts of stable, neutral and unstable equilibrium. Euler's theory of strut buckling; end conditions. Buckling of real struts. Applicationsof Euler's theory.

Required Textbook and Course Manual: (a) Mechanics of Engineering Materials Authors: P.P. Benham, R.J. Crawford and C.G. Armstrong Publisher: Addison Wesley Longman Edition: 1996 (b) Course Manual Other References: i) Mechanics of Solids ii) Mechanics of Materials P.S. Steif R.C. Hibbeler Pearson-Prentice Hall Pearson-Prentice Hall 1st Edition (2012) 8th Edition (2011) iii) Mechanics of Materials iv) Mechanics of Materials J.M. Gere R.R. Craig, Jr. Thomson Brooks/Cole John Wiley & Sons 6th Edition (2004) 3rd Edition (2011) iv) Mechanics of Materials T.A. Philpot John Wiley & Sons 2nd Edition (2011) Assessment: Laboratory Experiments 25% Mid-Term Examination 15% Final Examination 60% Total 100%

NOTE: a) Absenteeism from any laboratory without a valid reason

automatically results in a Fail grade for the course. b) In the event of a missed laboratory session because of illness, it

must be suppor ted by a medical cer tificate dated no more than one day after the missed session. The medical cer tificate must be submitted to the instructor within 5 working days.

c) Failure in either the final examination or

the laboratory component will constitute failure in this course.

d) Students with disabilities that require academic accommodation in this course are encouraged to contact the Paul Menton Centre.

e) Every student is required to have a proper hard cover

scientific/engineer ing log book for the laboratory sessions. The log book is submitted at the end of each session for grading. Submissions in any other form will not be accepted.

Prepared by: Prof. C.L. Tan

Car leton University Depar tment of Mechanical & Aerospace Engineer ing MAAE2202 Mechanics of Solids I Study Guide and Suggested Problems for PA Classes Listed below are the various chapters in the textbook (Mechanics of Engineering Materials, P.P. Benham, R.J. Crawford and C.G. Amstrong) which cover the range of topics in Solid Mechanics taught in this course. Also listed are the problems at the end of these chapters that may be attempted by students. Students are strongly encouraged to work on as many of these problems as possible to help reinforce their understanding of the lectures. Difficulties with these problems may be resolved at the scheduled PA classes which every student is expected to attend (and attendance will be taken). Some students may also find it helpful to read the relevant topics in other textbooks listed in the Course Outline. 1. Review of Statics, and Fr iction Problems

Chapter 1, Sections 1.1 - 1.8 for Review of Statics.

For Friction Problems, see also the textbook used in ECOR Mechanics I for additional reading material if required.

Suggested Problems: Problems 1.6 to 1.19, plus supplementary problem set in this manual.

2. Concepts of Stress and Strain Chapter 2, Section 2.1; and Chapter 3. Suggested Problems: Problems 3.1, 3.2, 3.5, 3.7. 3. Statically Determinate and Indeterminate Stress Systems

Chapter 2 and Chapter 4.

Suggested Problems: Problems 2.1, 2.2, 2.7, 2.8, 2.9 and 2.13 in Chapter 2; Problems 4.1 to 4.3, 4.5, 4.6 and 4.8 to 4.13 in Chapter 4.

4. Torsion Chapter 1, Section 1.18 and Section 1.20; and Chapter 5, Sections 5.1 - 5.7.

Suggested Problems: Problems 5.1, 5.3 to 5.5, and 5.7 to 5.9. 5. Symmetr ic Bending of Beams (I)

Chapter 1, Section 1.10; and Chapter 6, Sections 6.1 - 6.9, Sections 6.11, 6.12, 6.14 and 6.15.

Suggested Problems: Problems 1.28 in Chapter 1.

Problems 6.1 to 6.4, 6.7, 6.8, 6.10, 6.11, 6.22, 6.28 and 6.29 in Chapter 6.

6. Symmetr ic Bending of Beams (II) Chapter 7, all sections except Sections 7.5 and 7.6; and Chapter 8, Sections 8.1 and 8.2.

Suggested Problems: Problems 7.1 to 7.3, 7.8, 7.10, 7.12 and 7.14 in Chapter 7. Problems 8.1 to 8.7 in Chapter 8. 7. Buckling Instability

Chapter 10, Sections 10.1 and 10.2.

Suggested Problems: Problems 10.1 to 10.5, plus supplementary problem set in this manual.

Prepared by: Prof. C.L. Tan

Carleton University



Supplementary Problems: Friction

1. Figure 1 shows two wedge blocks A and B, weighing 1000 N and 50 N, respectively, in contact. The coefficient of friction between all contacting surfaces is 0.1. What would be the smallest value of the force P required to push the block A upwards? What would be its value if there were no friction? Am.: 887.3 N; 577.4 N.

Figure 1 Figure 2

2. The top block of the system shown in Figure 2 is free to move vertically. The mass of each of the three blocks and the corresponding coefficients of friction for the pairs of contacting surfaces are as indicated in the figure. What is the minimwn horizontal force P that will cause slip to occur between the blocks? Am.: J089 N.

3. The schematic layout ofa band brake applied to a brake drum of a hoist is shown in Figure 3. The braking force P is applied at one end ofa lever which is pivoted on a fixed fulcrwn at F. The diameter of the drum is 1 m; the arc ofcontact between the belt and the drum is 225; and the coefficient of friction is 0.3. What is the force P required to give a braking torque of 5.5 kNm if the drum is rotating (a) clockwise, and (b) anti-clockwise? Am.: 294 N; 4694 N.

a = 62.5em b JOem

pe = 2.5 em

Figure 3

4. In a relatively large motorized mechanical hobby set is a belt-pulley system for transmitting power. There are two pulleys, one 18 cm diameter and the other 8 cm diameter, that are on parallel shafts 78 cm apart. They are connected by a cross belt. What is the length of the belt required and the angle of lap between the belt and each pulley? The maximum tension allowed in the belt is 200 N and the coefficient of friction between the belt and the pulleys is 0.25. What is the power that can be transmitted by the belt when the smaller pulley rotates at 450 rev/min? Ans.: 199 em: 3..177 rad; 219 W.

5. A friction clutch consists of a single steel disc fixed to the driving shaft. Against this are pressed to pressed to coaxial rings, one on each face, sliding in splines attached to the driven shaf and faced with friction material of outer diameter 300 mm and inner diameter 125 mm. On each ring there is a spring load of 2 kN pressing it against the steel disc. Calculate the maximum power which can be transmitted at 1200 rev/min, assuming (a) that the pressure between the friction surfaces is uniform, and (b) that the surfaces wear uniformly. Take the coefficient of friction as 0.35. Ans.: (a) 19.7 kW; (b) 18.7 kW.

Carleton University



Supplementary Problems: Buckling Instability

1. The leg in Figure 1(a) acts as a column and can be modelled by two pin-connected members that are attached to a torsional spring having a stiffness k (torque/rad), as shown in Figure 1 (b). Assuming that the bone material is rigid, detennine the critical buckling load P. Ans.: P =4k1L

p

rr--,..--P

I

4.57 m

(a) (b)Figure 1 Figure 2

2. Figure 2 shows a vertical rod of diameter of 50 mm, Young's modulus 207 GPa, secured in position by a guy wire when it is subjected to a horizontal load P. The guy wire has an ultimate stress of 345 MPa and a diameter of 6.35 mm. What is the maximum value of P that this system can withstand before failure as a load-carrying system can be expected? Ans.: P = 7.726 kN

3. Figure 3 shows a rock crusher; the arms AB and Be are 1.5 m in length. A pressure p of 3.45 MPa is developed on the 127 mm piston D. The smallest that ex. gets is Hr. Determine the thickness h of arms AB and Be, noting that the width is 50 mm as shown in the figure. For safety, design for a 50% overload. The Young's modulus for all members is 207 GPa. Ans.: h = 36.8 mm

Figure 3

4. Figure 4 shows a steel connecting rod, 2.4 m long and a rectangular cross section of 100 x 50 mm. The piston has a diameter of 0.635 m. What is the maximum safe pressure p in the cylinder when AB is locked horizontally? AB can be considered pin-ended for bending in the plane of the figure, and fixed ended for bendi.ng in the plane normal to the paper. Young's modulus for steel is 207 OPa. Ans.: p = 4/!7 MPa gauge.

Figure 4

5. The wing strut shown in Figure 5 of an airplane has a standard 63.5 mm streamlined section with the following area properties: second moments of area 111 = 16.911 x 104

4 4 2mm , 122=4.237 x let mm ; area of cross section A = 243 mm On a hard landing, the resultant load is 4.45 kN. What factor of safety does the wing strut have? The strut is pinned about the axis 2-2 and fixed about the axis 1-1. It is made of a steel with Young's modulus of 207 OPa. Ans.: 1.17

1--- 2. 44m-----i Resultant Frictionless shear pin

Figure 5


Laboratory Experiments

CARLETON UNIVERSITYDepartment of Mechanical & Aerospace Engineering


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Experiment B: Torsion of a Hollow Circular Shaft_______________________________________________________________________________

1. OBJECTIVES

(a) To obtain the torsional stiffness of a hollow circular shaft and the shear modulus ofthe specimen material.

(b) To verify the simple torsion formula for shear stresses in circular shafts.

2. APPARATUS

(a) A rigid supporting frame with fixtures for holding a long circular bar at one end.

(b) A shackle assembly mounted on the rigid frame with a threaded tie rod to prop upthe other end of the circular bar when it is carrying a load.

(c) Switch and Balance Unit; Strain indicator

(d) A long, hollow, circular aluminium bar specimen. It has a short, horizontal,rectangular cross-bar welded on to it at the free end which serves as the torque arm.

(e) A clamp-bracket to be mounted adjacent to the torque arm near the tip end of thespecimen.

(f) Vernier caliper, weights and dial gauges.

3. PROCEDURE

(a) Record the dimensions of the specimen. Note the orientations of the three straingauges which are bonded to the top surface of the specimen. Gauge (1) is alignedin the direction of the longitudinal axis of the circular bar; gauge (3) is perpendicularto the axis; and gauge (2) is aligned at an angle 45o to gauges (1) and (3).

(b) Mount a dial gauge to measure the vertical deflection of the horizontal arm of theclamp-bracket. Record also the distance from the fixed end along the longitudinalaxis of the bar of this position where the vertical displacement is measured.

(c) Mount another dial gauge to measure the vertical deflection at the tip of the bar inthe plane of the longitudinal axis.

(d) Place the weight hanger at an appropriate position on the torque arm of the bar.

(e) Turn on the Switch and Balance unit and the strain indicator. Set the channelreadings for the three gauges to zero before any load is applied to the specimen.Make sure that the shackle assembly is not resting on or propping up the bar whencarrying out this procedure.

(f) Set the dial gauge readings to zero.

(g) Apply incremental magnitudes of torque to the circular bar by adding weights ontothe hanger. Do not exceed 40 kgf in the weight.

(h) For each load applied in step (g), adjust the nut on the tie-rod at the top of thesupporting frame so that the vertical deflection as measured by the dial gauge at thetip of the bar is zero. Verify that the strains recorded in gauges (1) and (3) are alsozero. Then record the strain as indicated by gauge (2). Also, record the verticaldeflections of the points on the arms of the clamp-brackets as noted in (b).

(f) After the maximum load has been applied and the readings taken, remove all theweights on the hanger and adjust the nut on the tie-rod again to make sure that theshackle assembly is not resting or acting on the specimen. Check the readings of thestrain gauge channels and the dial gauge again. They should remain at zero.

4. REQUIREMENTS

(a) Explain why it is necessary in step 3(h) above to make sure the vertical deflection at the tipof the bar, or the strains in the directions of strain gauges (1) and (3) are zero again beforethe other readings are taken.

(b) Plot a graph of the applied torque, T, versus the angle of twist, , for the circular barspecimen tested. From the graph, determine the torsional stiffness, k = T/, of this bar.Also determine the shear modulus (or modulus of rigidity) of the specimen material andcompare it with published values. Comment on the values obtained.

(c) Convert the shear strains obtained at the outer surface of the circular bar to thecorresponding shear stresses, . Plot a graph of versus T showing the theoretical line forthe shear stresses on the outer surface of this bar and the experimental values. Comment onany discrepancies in the results.

Note: A strain gauge can only measure the direct strain along the axis of the gauge.The reading for gauge (2) is therefore the direct strain in the direction of this gauge.However, as you will learn later in this course, in a pure shear condition as in this loadtest, the magnitude of the shear strain in the plane of the circular cross-section can beshown to be equal to TWICE the direct strain in the 45o plane of gauge (2).

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Course Manual 2202

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