**UFBA**

# Solution Manual - Mechanics of Materials 7th Edition, Gere, Goodno

(Parte **1** de 3)

An Instructor’s Solutions Manual to Accompany

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Contents

1. Tension, Compression, and Shear

Normal Stress and Strain1 Mechanical Properties of Materials15 Elasticity, Plasticity, and Creep21 Linear Elasticity, Hooke’s Law, and Poisson’s Ratio25 Shear Stress and Strain30 Allowable Stresses and Allowable Loads 51 Design for Axial Loads and Direct Shear69

2.Axially Loaded Members

Changes in Lengths of Axially Loaded Members89 Changes in Lengths under Nonuniform Conditions105 Statically Indeterminate Structures124 Thermal Effects151 Stresses on Inclined Sections178 Strain Energy198 Impact Loading212 Stress Concentrations 224 Nonlinear Behavior (Changes in Lengths of Bars)231 Elastoplastic Analysis 237

3. Torsion

Torsional Deformations 249 Circular Bars and Tubes252 Nonuniform Torsion 266 Pure Shear287 Transmission of Power294 Statically Indeterminate Torsional Members302 Strain Energy in Torsion319 Thin-Walled Tubes 328 Stress Concentrations in Torsion338

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4.Shear Forces and Bending Moments

Shear Forces and Bending Moments343 Shear-Force and Bending-Moment Diagrams355

5.Stresses in Beams (Basic Topics)

Longitudinal Strains in Beams389 Normal Stresses in Beams392 Design of Beams412 Nonprismatic Beams 431 Fully Stressed Beams440 Shear Stresses in Rectangular Beams442 Shear Stresses in Circular Beams453 Shear Stresses in Beams with Flanges457 Built-Up Beams466 Beams with Axial Loads475 Stress Concentrations 492

6.Stresses in Beams (Advanced Topics)

Composite Beams497 Transformed-Section Method 508 Beams with Inclined Loads520 Bending of Unsymmetric Beams529 Shear Stresses in Wide-Flange Beams541 Shear Centers of Thin-Walled Open Sections543 Elastoplastic Bending 558

7.Analysis of Stress and Strain

Plane Stress571 Principal Stresses and Maximum Shear Stresses582 Mohr’s Circle 595 Hooke’s Law for Plane Stress608 Triaxial Stress615 Plane Strain622 iv CONTENTS 00FM.qxd 9/29/08 8:49 PM Page iv

CONTENTS v

8.Applications of Plane Stress (Pressure Vessels,Beams,and Combined Loadings)

Spherical Pressure Vessels649 Cylindrical Pressure Vessels655 Maximum Stresses in Beams664 Combined Loadings 675

9.Deflections of Beams

Differential Equations of the Deflection Curve707 Deflection Formulas 710 Deflections by Integration of the Bending-Moment Equation714 Deflections by Integration of the Shear Force and Load Equations722 Method of Superposition730 Moment-Area Method 745 Nonprismatic Beams 754 Strain Energy770 Castigliano’s Theorem 775 Deflections Produced by Impact784 Temperature Effects 790

10.Statically Indeterminate Beams

1. Columns

Idealized Buckling Models845 Critical Loads of Columns with Pinned Supports851 Columns with Other Support Conditions863 Columns with Eccentric Axial Loads871 The Secant Formula880 Design Formulas for Columns889 Aluminum Columns903

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12.Review of Centroids and Moments of Inertia

Centroids of Plane Ares913 Centroids of Composite Areas915 Moment of Inertia of Plane Areas919 Parallel-Axis Theorem 923 Polar Moments of Inertia927 Products of Inertia929 Rotation of Axes932 Principal Axes, Principal Points, and Principal Moments of Inertia936

Answers to Problems944 vi CONTENTS 00FM.qxd 9/29/08 8:49 PM Page vi

Normal Stress and Strain

Problem 1.2-1A hollow circular post ABC(see figure) supports a load

(a)Calculate the normal stress ABin the upper part of the post. (b)If it is desired that the lower part of the post have the same compressive stress as the upper part, what should be the magnitude

(c)If P1remains at 1700lb and P2is now set at 2260lb, what new thickness of BCwill result in the same compressive stress in both parts?

Tension, Compression, and Shear

Solution 1.2-1 PART (a)

PART (b)

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2CHAPTER 1Tension,Compression,and Shear

sAB b sAB 2.744 sAB ABC

Problem 1.2-2A force Pof 70N is applied by a rider to the fronthand brake of a bicycle (Pis the resultant of an evenlydistributed pressure). As the hand brake pivots at A, a tension Tdevelops in

stress and strain in the brake cable.

Solution 1.2-2 inches; tBC 0.499 sAB b sAB b

NOTE: (E for cables is approx. 140 GPa)

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SECTION 1.2Normal Stress and Strain3

Problem 1.2-3A bicycle rider would like to compare the effectiveness of cantilever hand brakes [see figure part (a)] versus V brakes [figure part (b)].

(a)Calculate the braking force RBat the wheel rims for each of the bicycle brake systems shown. Assume that all forces act in

(b)For each braking system, what is the stress in the brake cable (assume effective cross-sectional area of 0.00167in.2)? (HINT: Because of symmetry, you only need to use the right half of each figure in your analysis.)

Solution 1.2-3

Acable 0.00167 in.2 (a) CANTILEVER BRAKES-BRAKING FORCE

Statics: sum forces at D to get TDC

so RB 2T vs 4.25T for V brakes (below)

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4CHAPTER 1Tension,Compression,and Shear

(same for V-brakes (below))

Apad

Acable

Apad

Problem 1.2-4A circular aluminum tube of length L 400mm is loaded in compression by forces P(see figure). The outside and inside diameters are 60mm and 50mm, respectively. A strain gage is placed on the outside of the bar to measure normal strains in the longitudinal direction.

(b) COMPRESSIVE LOAD P 40 MPa

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SECTION 1.2Normal Stress and Strain5

Problem 1.2-5The cross section of a concrete corner column that is loaded uniformly in compression is shown in the figure.

(a)Determine the average compressive stress cin the concrete if the load is equal to 3200k.

(b)Determine the coordinates xcand ycof the point where the resultant load must act in order to produce uniform normal stress in the column.

(b)

NOTE:xc& ycare the same as expected due to symmetry about a diagonal ; yc 19.2 inches

38bd

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6CHAPTER 1Tension,Compression,and Shear

Problem 1.2-6A car weighing 130 kN when fully loaded is pulled slowly up a steep inclined track by a steel cable (see figure). The cable has an effective cross-sectional area of 490 mm2, and the angle aof the incline is 30°.

Calculate the tensile stress stin the cable.

Solution 1.2-6Car on inclined track FREE-BODY DIAGRAM OF CAR

T Tensile force in cable

Angle of incline

A Effective area of cable

133 MPa ; st (130 kN)(sin 30°)

490 mm2

Wsin a A

Problem 1.2-7Two steel wires support a moveable overhead camera weighing W 25 lb (see figure) used for close-up viewing of field action at sporting events. At some instant, wire 1 is at on angle 20°to the horizontal and wire 2 is at an angle 48°. Both wires have a diameter of 30 mils. (Wire diameters are often expressed in mils; one mil equals 0.001 in.)

Determine the tensile stresses 1and 2in the two wires.

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SECTION 1.2Normal Stress and Strain7

Solution 1.2-7 NUMERICAL DATA

a cos(b) cos (a)

180 radiansa 20 p

A wire

A wire cos (a)

Problem 1.2-8A long retaining wall is braced by wood shores set at an angle of 30°and supported by concrete thrust blocks, as shown in the first part of the figure. The shores are evenly spaced, 3m apart.

For analysis purposes, the wall and shores are idealized as shown in the second part of the figure. Note that the base of the wall and both ends of the shores are assumed to be pinned. The pressure of the soil against the wall is assumed to be triangularly distributed, and the resultant force acting on a 3-meter length of the wall is F 190kN. If each shore has a 150mm 150mm square cross section, what is the compressive stress cin the shores?

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Problem 1.2-9A pickup truck tailgate supports a crate

WT 60lb andissupported by two cables (only one is shown in thefigure). Each cable has an effective cross-

(a)Find the tensile force Tand normal stress in each cable. (b)If each cable elongates 0.01in. due to the weight of both the crate and the tailgate, what is the average strain in the cable?

8CHAPTER 1Tension,Compression,and Shear Solution 1.2-8Retaining wall braced by wood shores

(190 kN)(1.5 m) C(sin 30°)(4.0 m) C(cos 30°)(0.5m) 0

5.21 MPa ;

A 117.14 kN

0.0225 m2

C compressive force in wood shore

CH horizontal component of C CV vertical component of C CH Ccos 30° CV Csin 30°

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SECTION 1.2Normal Stress and Strain9

HT v

Problem 1.2-10Solve the preceding problem if the

(a)Find the tensile force Tand normal stress in each cable. (b)If each cable elongates 0.25mm due to the weight of both the crate and the tailgate, what is the average strain in the cable?

L c

Ae

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10 CHAPTER 1 Tension, Compression, and Shear

H T v

L c

A e

Problem ★1.2-11An L-shaped reinforced concrete slab 12 ft 12 ft (but with a 6 ft 6 ft cutout) and thickness t 9.0 in. is lifted by three cables attached at O, Band D, as shown in the figure. The cables are combined at point Q, which is 7.0ft above the top of the slab and directly above the center of mass at C. Each cable has an effective cross-

(a)Find the tensile force Ti(i 1, 2, 3) in each cable due to the weight Wof the concrete slab (ignore weight of cables).

(b)Find the average stress iin each cable. (See TableH-1 in Appendix H for the weight density of reinforced concrete.)

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SECTION 1.2Normal Stress and Strain11

Solution 1.2-1 CABLE LENGTHS

(a) SOLUTION FOR CABLE FORCES USING STATICS (3 EQU, 3 UNKNOWNS)

For unit force in Z-direction

P 0 0 1Q

T1 T2 check:

NOTE: preferred solution uses sum of moments about a line as follows –

1.sum about x-axis to get T3v, then T3 2.sum about y-axis to get T2v, then T2 3.sum vertical forces to get T1v, then T1 OR sum forces in x-dir to get T1x in terms of T2x xcg 5same for ycgycg xcg Multiply unit forces by W

(b)

;s P49.0 ksi 39.0 ksi60.0 ksiQ psis

T 0.12

0.3850.589Q | TuP |

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12 CHAPTER 1 Tension, Compression, and Shear

Problem *1.2-12A round bar ACB of length 2L (see figure) rotates about an axis through the midpoint Cwith constant angular speed v (radians per second). The material of the bar has weight density g.

(a)Derive a formula for the tensile stress sxin the bar as a function of the distance xfrom the midpoint C.

(b)What is the maximum tensile stress smax?

Solution 1.2-12Rotating Bar

angular speed (rad/s)

A cross-sectional area weight density

mass density

We wish to find the axial force Fxin the bar at Section D, distance xfrom the midpoint C.

The force Fxequals the inertia force of the part of the rotating bar from Dto B.

g g

Consider an element of mass dMat distance from the midpoint C. The variable ranges from xto L.

dF Inertia force (centrifugal force) of element of mass dM

(a) TENSILE STRESS IN BAR AT DISTANCE x

(b) MAXIMUM TENSILE STRESS gv2

LLx g g gAv2

Problem 1.2-13Two gondolas on a ski lift are locked in the position shown in the figure while repairs are being made elsewhere. The distance between support towers is L 100ft. The length of each cable segment under gondola weights and DCD 20ft. The cable sag at Bis B 3.9ft and that at C( C) is 7.1ft. The effective cross-sectional area of the cable

(a)Find the tension force in each cable segment; neglect the mass of the cable. (b)Find the average stress () in each cable segment.s

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SECTION 1.2Normal Stress and Strain13

Solution 1.2-13

b | u30.363 |

b | u20.046 |

b | u10.331 |

(b) COMPUTE STRESSES IN CABLE SEGMENTS

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14 CHAPTER 1 Tension, Compression, and Shear

Problem 1.2-14A crane boom of mass 450kg with its center of mass at Cis stabilized by two cables AQand BQ

in the y–zplane.

(a)Find the tension forces in each cable: TAQand

TBQ(kN); neglect the mass of the cables, but include the mass of the boom in addition to load P.

(b)Find the average stress ( ) in each cable.

Solution 1.2-14 DataMboom 450 kg

TAQz

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SECTION 1.3Mechanical Properties of Materials15

Mechanical Properties of Materials

Problem 1.3-1Imagine that a long steel wire hangs vertically from a high-altitude balloon.

(a)What is the greatest length (feet) it can have without yielding if the steel yields at 40ksi? (b)If the same wire hangs from a ship at sea, what is the greatest length? (Obtain the weight densities of steel and sea water from Table H-1, Appendix H.)

Solution 1.3-1Hanging wire of length L

max 40ksi (yield strength)

40,0 psi 490 lb/ft3 (144 in.2/ft2)

A gSL

(b) WIRE HANGING IN SEA WATER F tensile force at top of wire

Problem 1.3-2Imagine that a long wire of tungsten hangs vertically from a high-altitude balloon.

(a)What is the greatest length (meters) it can have without breaking if the ultimate strength (or breaking strength) is 1500MPa? (b)If the same wire hangs from a ship at sea, what is the greatest length? (Obtain the weight densities of tungsten and sea water from Table H-1, Appendix H.)

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16 CHAPTER 1 Tension, Compression, and Shear

Solution 1.3-2Hanging wire of length L W total weight of tungsten wire

max 1500MPa (breaking strength)

1500MPa 190 kN/m3

A gTL

(b) WIRE HANGING IN SEA WATER F tensile force at top of wire

Problem 1.3-3Three different materials, designated A, B, and C, are tested in tension using test specimens having diameters of 0.505in. and gage lengths of 2.0in. (see figure). At failure, the distances between the gage marks are found to be 2.13, 2.48, and 2.78in., respectively. Also, at the failure cross sections the diameters are found to be 0.484, 0.398, and 0.253in., respectively.

Determine the percent elongation and percent reduction in area of each specimen, and then, using your own judgment, classify each material as brittle or ductile.

Solution 1.3-3Tensile tests of three materials where L1is in inches.

A0 b(100)

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SECTION 1.3Mechanical Properties of Materials17

Percent reduction in area (Eq. 2)

where d1is in inches.

Material (in.) (in.) (Eq. 1) (Eq. 2) Ductile?

Problem 1.3-4The strength-to-weight ratioof a structural material is defined as its load-carrying capacity divided by its weight.

For materials in tension, we may use a characteristic tensile stress (as obtained from a stress-strain curve) as a measure of strength. For instance, either the yield stress or the ultimate stress could be used, depending upon the particular application. Thus, the strength-to-weight ratio RS/Wfor a material in tension is defined as in which is the characteristic stress and is the weight density. Note that the ratio has units of length.

Using the ultimate stress Uas the strength parameter, calculate the strength-to-weight ratio (in units of meters) for each of the following materials: aluminum alloy 6061-T6, Douglas fir (in bending), nylon, structural steel ASTM-A572, and a titanium alloy. (Obtain the material properties from Tables H-1 and H-3 of Appendix H. When a range of values is given in a table, use the average value.)

Solution 1.3-4Strength-to-weight ratio

The ultimate stress Ufor each material is obtained from Table H-3, Appendix H, and the weight density is obtained from Table H-1.

The strength-to-weight ratio (meters) is

Values of U, , and RS/Ware listed in the table.

Titanium has a high strength-to-weight ratio, which is why it is used in space vehicles and high-performance airplanes. Aluminum is higher than steel, which makes it desirable for commercial aircraft. Some woods are also higher than steel, and nylon is about the same as steel.

(Parte **1** de 3)