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Mechanics of Materials 7th edition James M. Gere, Barry J. Goodno - Solutions

A rectangular plate in biaxial stress (see figure) is subjected to normal stresses (x = 90 MPa (tension) and (y = - 20 MPa (compression). The plate has dimensions 400 ( 800 ( 20 mm and is made of steel with E = 200 GPa and v = 0.30. (a) Determine the maximum in-plane shear strain ymax in the
Solve the preceding problem for an aluminum plate with (x = 12,000 psi (tension), (y = - 3,000 psi (compression), dimensions 20 ( 30 ( 0.5 in., E = 10.5 ( 106 psi, and v = 0.33?
A brass cube 50 mm on each edge is compressed in two perpendicular directions by forces P = 175 kN (see figure).Calculate the change (V in the volume of the cube and the strain energy U stored in the cube, assuming E = 100 GPa and y = 0.34.
A 4.0-inch cube of concrete (E = 3.0 ( 106 psi, v = 0.1) is compressed in biaxial stress by means of a framework that is loaded as shown in the figure.Assuming that each load equals 20 k, determine the change (V in the volume of the cube and the strain energy U stored in the cube.
An element of aluminum in the form of a rectangularparallelepiped (see figure) of dimensions a = 6.0., b = 4.0 in, and c = 3.0 in. is subjected to triaxial stresses (x = 12,000 psi, (y = - 4,000 psi, and (z = - 1,000 psi acting on the x, y, and z faces, respectively.Determine the following
A solid steel sphere (E = 210 GPa, v = 0.3) is subjected to hydrostatic pressure such that its volume is reduced by 0.4%. (a) Calculate the pressure p. (b) Calculate the volume modulus of elasticity K for the steel. (c) Calculate the strain energy U stored in the sphere if its diameter is d = 150
A solid bronze sphere (volume modulus of elasticity K = 14.5 ( 106 psi) is suddenly heated around its outer surface. The tendency of the heated part of the sphere to expand produces uniform tension in all directions at the center of the sphere. If the stress at the center is 12,000 psi, what is the
Solve the preceding problem if the element is steel (E = 200 GPA, v = 0.30) with dimensions a = 300 mm, b = 150 mm, and c = 150 mm and the stresses are (x = - 60 MPa, (y = - 40 MPa, and (z = - 40 MPa?
A cube of cast iron with sides of length a = 4.0 in.(see figure) is tested in a laboratory under triaxial stress. Gages mounted on the testing machine show that the compressive strains in the material are (x = 1 - 225 ( 10-6 and (y = (z = - 37.5 ( 10-6.Determine the following quantities: (a) the
Solve the preceding problem if the cube is granite (E = 60 GPa, v = 0.25) with dimensions a = 75 mm and compressive strains (x = - 720 ( 10-6 and (y = (z - 270 ( 10-6?
An element of aluminum in triaxial stress (see figure) is subjected to stresses (x = 5200 psi (tension), (y = - 4750 psi (compression), and (z = - 3090 psi (compression).It is also known that the normal strains in the x and y directions are (x = 7138.8 ( 10-6 (elongation) and (y = - 502.3 ( 10-6
Solve the preceding problem if the material is nylon subjected to compressive stresses (x = - 4.5 MPa, (y = - 3.6 MPa, and (z = - 2.1 MPa, and the normal strains are (x = - 740 ( 10-6 and (y = - 320 ( 10-6 (Shortenings)?
A rubber cylinder R of length L and cross-sectional area A is compressed inside a steel cylinder by a force F that applies a uniformly distributed pressure to the rubber (see figure).(a) Derive a formula for the lateral pressure between the rubber and the steel. (Disregard friction between the
A block R of rubber is confined between plane parallel walls of a steel block (see figure). A uniformly distributed pressure P0 is applied to the top of the rubber block by a force F.(a) Derive a formula for the lateral pressure between the rubber and the steel. (Disregard friction between the
A solid spherical ball of brass (E = 15 ( 106 psi v = 0.34) is lowered into the ocean to a depth of 10,000 ft. The diameter of the ball is 11.0 in. Determine the decrease (d in diameter, the decrease (V in volume, and the strain energy U of the ball?
A thin rectangular plate in is biaxial stress subjected to stresses (x and (y, as shown in part (a) of the figure on the next page. The width and height of the plate are b = 8.0 in. and h = 4.0 in., respectively. Measurements show that the normal strains in the and directions are (x = 195 ( 10-6
Solve the preceding problem for the following data: (x = - 1120 ( 10-6, (y = - 430 ( 10-6, (xy = 780 ( 10-6, and ( = 45o?
A steel plate with modulus of elasticity E = 30 ( 106 psi and Poisson's ratio v = 0.30 is loaded in biaxial stress by normal stresses (x and (y (see figure). A strain gage is bonded to the plate at an angle ( = 30o. If the stress is (x 18,000 psi and the strain measured by the gage is ( = 407 (
Solve the preceding problem if the plate is made of aluminum with E = 72 GPa and v = 1/3, the stress (x is 86.4 MPa, the angle ( is 21o, and the strain ( is 946 ( 10-6?
An element in plane stress is subjected to stresses (x = - 8400 psi,(y = 1100 psi, and Txy= - 1700 psi (see figure). The material is aluminum with modulusof elasticity E = 10,000 ksi and Poisson's ratio v = 0.33.Determine the following quantities: (a) the strains for an element oriented at an angle
Solve the preceding problem for the following data: (x = - 150 MPa, (y = - 210 MPa, Txy = - 16 MPa, and ( = 50o. the material is brass with E = 100 GPa and v = 0.34?
During a test of an airplane wing, the strain gage readings from a 45o rosette (see figure) are as follows: gage A, 520 ( 10-6; gage B, 360 ( 10-6; and gage C,- 80 ( 10-6 .Determine the principal strains and maximum shear strains, and show them on sketches of properly oriented elements?
A solid circular bar of diameter d = 1.5 in. is subjected to an axial force P and a torque T (see figure). Strain gages A and B mounted on the surface of the bar give reading(a = 100 ( 10-6 and (b = - 55 ( 10-6. The bar is made of steel having E = 30 ( 106 psi and v = 0.29.(a) Determine the axial
A cantilever beam of rectangular cross section (width b = 25 mm, height h = 100 mm) is loaded by a force P that acts at the midheight of the beam and is inclined at an angle a to the vertical (see figure). Two strain gages are placed at point C, which also is at the midheight of the beam. Gage A
Solve the preceding problem if the cross-sectional dimensions are b = 1.0 in. and h = 3.0 in., the gage angle is ( = 75o, the measure strains are (a = 171 ( 10-6 and (b = - 266 ( 10-6, and the material is a magnesium alloy with modulus E = 6.0 ( 106 psi and Poisson's ratio v = 0.35?
A 60o strain rosette, or delta rosette, consists of three electrical-resistance strain gages arranged as shown in the figure. Gage A measures the normal strain (a in the direction of the axis. Gages B and C measure the strains and in the inclined directions shown.Obtain the equations for the
On the surface of a structural component in a space vehicle, the strainsare monitored by means of three strain gages arranged as shown in the figure.During a certain maneuver, the following strains were recorded: (a = 1100 ( 10-6, (b = 200 ( 10-6, and (c = 200 ( 10-6.Determine the principal strains
The strains on the surface of an experimental device made of pure aluminum (E = 70 Gpa, v = 0.33) and tested in a space shuttle were measured by means of strain gages. The gages were oriented as shown in the figure, and the measured strains were (a = 1100 ( 10-6, (b = 1496 ( 10-6, and (c = - 39.44
Solve Problem 7.7 - 5 by using Mohr's circle for plane stain?
Solve Problem 7.7-6 by using Mohr's circle for plane strain?
Solve Problem 7.7 - 7 by using Mohr's circle for plane stain?
Solve Problem 7.7 - 8 by using Mohr's circle for plane stain?
Solve Problem 7.7 - 9 by using Mohr's circle for plane stain?
Solve Problem 7.7-10 by using Mohr's circle for plane strain?
A thin square plate in biaxial stress is subjected to stresses (x and (y, as shown in part (a) of the figure. The width of the plate is b = 12.0 in. Measurements show that the normal strains in the and directions are (x = 427 ( 10-6 and (y = 113 ( 10-6, respectively.With reference to part (b) of
Solve the preceding problem if b = 225 mm, (x = 845 ( 10-6, and (y = 211 ( 10-6.
An element of material subjected to plane strain (see figure) has strains as follows: (x = 220 ( 10-6, (y = 480 (10-6, and (xy = 180 ( 10-6.Calculate the strains for an element oriented at an angle ( = 50o and show these strains on a sketch of a properly oriented element?
Solve the preceding problem for the following data: (x = 420 ( 10-6, (y = - 170 ( 10-6, Yxy = 310 ( 10-6, and ( = 37.5o.
The strains for an element of material in plane strain (see figure) are as follows: (x = 480 ( 10-6, and (xy = - 350 ( 10-6. Determine the principal strains and maximum shear strains, and show these strains on sketches of properly oriented elements.
Solve the preceding problem for the following strains: (x = 120 ( 10-6, (y = - 450 ( 10-6, and (xy = - 360 ( 10-6?
A element of material in plane strain (see figure) is subjected to strains, (x = 480 ( 10-6, (y = 70 ( 10-6, and (xy = 420 ( 10-6. Determine the following quantities: (a) the strains for an element oriented at an angle , (b) the principal strains, and (c) the maximum shear strains. Show the results
A large spherical tank (see figure) contains gas at a pressure of 450 psi. The tank is 42 ft in diameter and is constructed of high-strength steel having a yield stress in tension of 80 ksi. Determine the required thickness (to the nearest 1/4 inch) of the wall of the tank if a factor of safety of
Solve the preceding problem if the diameter is 500 mm, the pressure is 18 MPa, the yield stress in tension is 975 MPa, the yield stress in shear is 460 MPa, the factor of safety is 2.5, the modulus of elasticity is 200 GPa, Poisson's ratio is 0.28, and the normal strain must not exceed 1210 ( 10-6?
A hollow pressurized sphere having radius r = 4.8 in. and wall thickness t = 0.4 in. is lowered into a lake (see figure). The compressed air in the tank is at a pressure of 24 psi (gage pressure when the tank is out of the water).At what depth D0 will the wall of the tank be subjected to a
Solve the preceding problem if the internal pressure is 3.75 MPa, the diameter is 19 m, the yield stress is 570 MPa, and the factor of safety is 3.0. Determine the required thickness to the nearest millimeter?
A hemispherical window (or viewport) in a decompression chamber (see figure) is subjected to an internal air pressure of 80 psi. The port is attached to the wall of the chamber by 18 bolts.Find the tensile force F in each bolt and the tensile stress ( in the viewport if the radius of the hemisphere
A rubber ball (see figure) is inflated to a pressure of 60 kPa. At that pressure the diameter of the ball is 230 mm and the wall thickness is 1.2 mm. The rubber has modulus of elasticity E = 3.5 MPa and Poisson's ratio v = 0.45.Determine the maximum stress and strain in the ball.
Solve the preceding problem if the pressure is 9.0 psi, the diameter is 9.0 in., the wall thickness is 0.05 in., the modulus of elasticity is 500 psi, and Poisson's ratio is 0.45?
A spherical steel pressure vessel (diameter 480 mm, thickness 8.0 mm) is coated with brittle lacquer that cracks when the strain reaches 150 ( 10-6 (see figure).What internal pressure p will cause the lacquer to develop cracks? (Assume E =205 GPa and v = 0.30.)
A spherical tank of diameter 48 in. and wall thickness 1.75 in. contains compressed air at a pressure of 2200 psi. The tank is constructed of two hemispheres joined by a welded seam (see figure).(a) What is the tensile load f (lb per in. of length of weld) carried by the weld?(b) What is the
Solve the preceding problem for the following data: diameter 1.0 m, thickness 48 mm, pressure 22 MPa, modulus 210 GPa, and Poisson's ratio 0.29?
A spherical stainless-steel tank having a diameter of 22 in. is used to store propane gas at a pressure of 2450 psi. The properties of the steel are as follows: yield stress in tension, 140,000 psi; yield stress in shear, 65,000 psi; modulus of elasticity, 30 ( 106 psi; and Poisson's ratio, 0.28.
A scuba tank (see figure) is being designed for an internal pressure of 1600 psi with a factor of safety of 2.0 with respect to yielding. The yield stress of the steel is 35,000 psi in tension and 16,000 psi in shear.If the diameter of the tank is 7.0 in., what is the minimum required wall
A cylindrical tank with hemispherical heads is constructed of steel sections that are welded circumferentially (see figure). The tank diameter is1.25 m, the wall thickness is 22 mm, and the internal pressure is 1750 kPa.(a) Determine the maximum tensile stress (h in the heads of the tank.(b)
A cylindrical tank with diameter d = 18 in. is subjected to internal gas pressure p = 450 psi. The tank is constructed of steel sections that are welded circumferentially (see figure). The heads of the tank are hemispherical. The allowable tensile and shear stresses are 8200 psi and 3000 psi,
A pressurized steel tank is constructed with a helical weld that makes an angle a = 55° with the longitudinal axis (see figure). The tank has radius r = 0.6 m, wall thickness t = 18 mm, and internal pressure p = 2.8 MPa. Also, the steel has modulus of elasticity E = 200 GPa and Poisson's ratio
Solve the preceding problem for a welded tank with a = 62°, r = 19 in., t = 0.65 in., p = 240 psi, E = 30 = 106 psi, and v = 0.30?
A tall standpipe with an open top (see figure) has diameter d = 2.2 m and wall thickness t = 20 mm.(a) What height h of water will produce a circumferential stress of 12 MPa in the wall of the standpipe?(b) What is the axial stress in the wall of the tank due to the water pressure?
An inflatable structure used by a traveling circus has the shape of a half-circular cylinder with closed ends (see figure). The fabric and plastic structure is inflated by a small blower and has a radius of 40 ft when fully inflated. A longitudinal seam runs the entire length of the "ridge" of the
A thin-walled cylindrical pressure vessel of radius r is subjected simultaneously to internal gas pressure p and a compressive force F acting at the ends (see figure).What should be the magnitude of the force F in order to produce pure shear in the wall of the cylinder?
A strain gage is installed in the longitudinal direction on the surface of an aluminum beverage can (see figure). The radius-to-thickness ratio of the can is 200. When the lid of the can is popped open, the strain changes by (0 = 170 ( 10-6.What was the internal pressure p in the can? (Assume E =
A circular cylindrical steel tank (see figure) contains a volatile fuel under pressure. A strain gage at point A records the longitudinal strain in the tank and transmits this information to a control room. The ultimate shear stress in the wall of the tank is 84 MPa, and a factor of safety of 2.5
A cylinder filled with oil is under pressure from a piston, as shown in the figure. The diameter d of the piston is 1.80 in. and the compressive force F is 3500 lb. The maximum allowable shear stress Tallow in the wall of the cylinder is 5500 psi. What is the minimum permissible thickness tmin of
Solve the preceding problem if d = 90 mm, F = 42 kN, and Tallow = 40 MPa?
A standpipe in a water-supply system (see figure) is 12 ft in diameter and 6 inches thick. Two horizontal pipes carry water out of the standpipe; each is 2 ft in diameter and 1 inch thick. When the system is shut down and water fills the pipes but is not moving, the hoop stress at the bottom of the
A cantilever beam of rectangular cross section is subjected to a concentrated load P = 17 k acting at the free end (see figure). The beam has width b = 3 in. and height h = 12 in. Point A is located at distance c = 2.5 ft from the free end and distance d = 9 in. from the bottom of the
A cantilever beam of T-section is loaded by an inclined force of magnitude 6.5 kN (see figure). The line of action of the force is inclined at an angle of 60° to the horizontal and intersects the top of the beam at the end cross section. The beam is 2.5 m long and the cross section has the
A simple beam of rectangular cross section has span length L = 62 in. and supports a concentrated moment M = 560 k-in at midspan (see figure). The height of the beam is h = 6 in. and the width is b = 2.5 in.Plot graphs of the principal stresses (1 and (2 and the maximum shear stress tmax, showing
Solve the preceding problem for a cross section mn located 0.18 m from the support if L = 0.75 m, M = 65 kN ( m, h = 120 mm, and b = 20 mm?
Solve the preceding problem for the following data: P = 130 kN, b = 80 mm, h = 260 mm, c = 0.6 m, and d = 220 mm?
A simple beam of rectangular cross section (width 4 in., height 10 in.) carries a uniform load of 1200 lb/ft on a span of 12 ft (see figure).Find the principal stresses (1 and (2 and the maximum shear stress Tmax at a cross section 2 ft from the left-hand support at each of the following locations:
An overhanging beam ABC with a guided support at A is of rectangular cross section and supports concentrated loads P both at A and at the free end C (see figure). The span length from A to B is L, and the length of the overhang is L/2. The cross section has width b and height h. Point D is located
Solve the preceding problem if the stress and dimensions are as follows: (1 = 2450 psi, L = 80 in., b = 2.5 in., h = 10 in., and d = 2.5 in?
A beam of wide-flange cross section (see figure) has the following dimensions: b = 120 mm, t = 10 mm, h = 300 mm, and h1 = 260 mm. The beam is simply supported with span length L = 3.0 m. A concentrated load P = 120 kN acts at the midpoint of the span. At a cross section located 1.0 m from the
A beam of wide-flange cross section (see figure) has the following dimensions: b = 5 in., t = 0.5 in., h = 12 in., and h1 = 10.5 in. The beam is simply supported with span length L = 10 ft and supports a uniform load q = 6 k/ft. Calculate the principal stresses (1 and (2 and the maximum shear
AW 200 ( 41.7 wide-flange beam (see Table E-1(b).Appendix E) is simply supported with a span length of 2.5 m (see figure). The beam supports a concentrated load of 100 kN at 0.9 m from support B.At a cross section located 0.7 m from the left-hand support, determine the principal stresses (1 and (2
AW12 ( 14 wide-flange beam (see Table E-1(a), Appendix E) is simply supported with a span length of 120 in. (see figure). The beam supports two anti-symmetrically placed concentrated loads of 7.5 k each.At a cross section located 20 in. from the right-hand support, determine the principal stresses
A bracket ABCD having a hollow circular cross section consists of a vertical arm AB, a horizontal arm BC parallel to the x0 axis, and a horizontal arm CD parallel to the z0 axis (see figure). The arms BC and CD have lengths b1 = 3.6 ft and b2 = 2.2 ft, respectively. The outer and inner diameters of
A cylindrical pressure vessel having radius r = 300 mm and wall thickness t = 15 mm is subjected to internal pressurep = 2.5 MPa. In addition, a torque T = 120 kN m acts at each end of the cylinder (see figure).(a) Determine the maximum tensile stress (max and the maximum in plane shear stress tmax
An L-shaped bracket lying in a horizontal plane supports a load P = 150 lb (see figure). The bracket has a hollow rectangular cross section with thickness t = 0.125 in. and outer dimensions b = 2.0 in. and h = 3.5 in. The centerline lengths of the arms are b1 = 20 in. and b2 = 30 in. Considering
A semicircular bar AB lying in a horizontal plane is supported at B (see figure). The bar has centerline radius R and weight q per unit of length (total weight of the bar equals (qR). The cross section of the bar is circular with diameter d.Obtain formulas for the maximum tensile stress (t, maximum
An arm ABC lying in a horizontal plane and supported at A (see figure) is made of two identical solid steel bars AB and BC welded together at a right angle. Each bar is 20 in. long.Knowing that the maximum tensile stress (principal stress) at the top of the bar at support A due solely to the
A pressurized cylindrical tank with flat ends is loaded by torques T and tensile forces P (see figure). The tank has radius r = 50 mm and wall thickness t = 3 mm. The internal pressure p = 3.5 MPa and the torque T = 450 N m.What is the maximum permissible value of the forces P if the allowable
A post having a hollow circular cross section supports a horizontal load P = 240 lb acting at the end of an arm that is 5 ft long (see figure).The height of the post is 27 ft, and its section modulus is S = 15 in.3 Assume that outer radius of the post, r2 = 4.5 in., and inner radius r1 = 4.243
A sign is supported by a pipe (see figure) having outer diameter 110 mm and inner diameter 90 mm. The dimensions of the sign are 2.0 m ( 1.0 m, and its lower edge is 3.0 m above the base. Note that the center of gravity of the sign is 1.05 m from the axis of the pipe. The wind pressure against the
A sign is supported by a pole of hollow circular cross section, as shown in the figure. The outer and inner diameters of the pole are 10.5 in. and 8.5 in., respectively. The pole is 42 ft high and weighs 4.0 k. The sign has dimensions 8 ft ( 3 ft and weighs 500 lb. Note that its center of gravity
A horizontal bracket ABC consists of two perpendicular arms AB of length 0.5 m, and BC of length of 0.75 m. The bracket has a solid circular cross section with diameter equal to 65 mm. The bracket is inserted in a frictionless sleeve at A (which is slightly larger in diameter) so is free to rotate
A cylindrical pressure vessel with flat ends is subjected to a torque T and a bending moment M (see figure). The outer radius is 12.0 in. and the wall thickness is 1.0 in. The loads are as follows: T = 800 k-in., M = 1000 k-in., and the internal pressure p = 900 psi.Determine the maximum tensile
A gondola on a ski lift is supported by two bent arms, as shown in the figure. Each arm is offset by the distance b = 180 mm from the line of action of the weight force W. The allowable stresses in the arms are 100 MPa in tension and 50 MPa in shear.If the loaded gondola weighs 12 kN, what is the
For purposes of analysis, a segment of the crankshaft in a vehicle is represented as shown in the figure. Two loads P act as shown, one parallel to (- x0) and another parallel to z0; each load P equals 1.0 kN. The crankshaft dimensions are b1 = 80 mm, b2 = 120 mm, and b3 = 40 mm.The diameter of the
A moveable steel stand supports an automobile engine weighing W = 750 lb as shown in figure part (a).The stand is constructed of 2.5 in. ( 2.5 in. ( 1/8 in. thick steel tubing. Once in position the stand is restrained by pin supports at B and C. Of interest are stresses at point A at the base of
A mountain bike rider going uphill applies force P = 65 N to each end of the handlebars ABCD, made of aluminum alloy 7075-T6, by pulling on the handlebar extenders (DF on right handlebar segment). Consider the right half of the handlebar assembly only (assume the bars are fixed at the fork at A).
Determine the maximum tensile, compressive, and shear stresses acting on the cross section of the tube at point A of the hitch bicycle rack shown in the figure.The rack is made up of 2 in. ( 2 in. steel tubing which is 1/8 in. thick. Assume that the weight of each of four bicycles is distributed
The hollow drill pipe for an oil well (see figure) is 6.2 in. in outer diameter and 0.75 in. in thickness. Just above the bit, the compressive force in the pipe (due to the weight of the pipe) is 62 k and the torque (due to drilling) is 185 k-in. Determine the maximum tensile, compressive, and
A segment of a generator shaft is subjected to a torque T and an axial force P, as shown in the figure. The shaft is hollow (outer diameter d2 = 300 mm and inner diameter d1 = 250 mm) and delivers 1800 kW at 4.0 Hz.If the compressive force P = 540 kN, what are the maximum tensile, compressive, and
A segment of a generator shaft is subjected to a torque T and an axial force P, as shown in the figure. The shaft is hollow (outer diameter d2 = 300 mm and inner diameter d1 = 250 mm) and delivers 1800 kW at 4.0 Hz. If the compressive force P = 540 kN, what are the maximum tensile, compressive, and
A cylindrical tank subjected to internal pressure p is simultaneously compressed by an axial force F = 72 kN (see figure). The cylinder has diameter d = 100 mm and wall thickness t = 4 mm.Calculate the maximum allowable internal pressure pmax based upon an allowable shear stress in the wall of the
A cylindrical tank having diameter d = 2.5 in. is subjected to internal gas pressure p = 600 psi and an external tensile load T = 1000 lb (see figure).Determine the minimum thickness t of the wall of the tank based upon an allowable shear stress of 3000 psi.
The torsional pendulum shown in the figure consists of a horizontal circular disk of mass M = 60 kg suspended by a vertical steel wire (G = 80 GPa) of length L = 2 m and diameter d = 4 mm.Calculate the maximum permissible angle of rotation (max of the disk (that is, the maximum amplitude of
Determine the maximum tensile, compressive, and shear stresses at points A and B on the bicycle pedal crank shown in the figure. The pedal and crank are in a horizontal plane and points A and B are located on the top of the crank. The load P = 160 lb acts in the vertical direction and the distances

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