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physics
mechanics
Mechanics of Materials 7th edition James M. Gere, Barry J. Goodno - Solutions
Consider the sleeve made from two copper tubes joined by tin-lead solder over distance s. The sleeve has brass caps at both ends, which are held in place by a steel bolt and washer with the nut turned just snug at the outset. Then, two "loadings" are applied: n = 1/2 turn applied to the nut; at the
A polyethylene tube (length L) has a cap which when installed compresses a spring (with undeformed length L1 > L) by amount δ = (L1 L). Ignore deformations of the cap and base. Use the force at the base of the spring as the redundant. Use numerical properties in the boxes given.(a)
Prestressed concrete beams are sometimes manufactured in the following manner. High-strength steel wires are stretched by a jacking mechanism that applies a force Q, as represented schematically in part (a) of the figure. Concrete is then poured around the wires to form a beam, as shown in part
A polyethylene tube (length L) has a cap which is held in place by a spring (with undeformed length L1 (a) What is the resulting force in the spring, Fk?(b) What is the resulting force in the tube, Ft?(c) What is the final length of the tube, Lf?(d) What temperature change ΔT inside the tube
A rigid bar of weight W = 750 lb hangs from three equally spaced wires, two of steel and one of aluminum (see figure). The diameter of the wires is 1/8 in. Before they were loaded, all three wires had the same length. What temperature increase ΔT in all three wires will result in the entire load
A steel rod of 15-mm diameter is held snugly (but without any initial stresses) between rigid walls by the arrangement shown in the figure. (For the steel rod, use α =12 × 106 / °C and E = 200 GPa.)(a) Calculate the temperature drop ΔT (degrees Celsius) at
A bar AB of length L is held between rigid supports and heated nonuniformly in such a manner that the temperature increase ΔT at distance x from end A is given by the expression ΔT = ΔTBx3/L3, where ΔTB is the increase in temperature at end B of the
A plastic bar ACB having two different solid circular cross sections is held between rigid supports as shown in the figure. The diameters in the left- and right-hand parts are 50 mm and 75 mm, respectively. The corresponding lengths are 225 mm and 300 mm. Also, the modulus of elasticity E is 6.0
A circular steel rod AB (diameter d1 = 1.0 in., length L1 = 3.0 ft) has a bronze sleeve (outer diameter d2 = 1.25 in., length L2 = 1.0 ft) shrunk onto it so that the two parts are securely bonded (see figure).Calculate the total elongation δ of the steel bar due to a temperature rise
A brass sleeve S is fitted over a steel bolt B (see figure), and the nut is tightened until it is just snug. The bolt has a diameter dB = 25 mm, and the sleeve has inside and outside diameters d1 = 26 mm and d2 = 36 mm, respectively.Calculate the temperature rise ΔT that is required to
Rectangular bars of copper and aluminum are held by pins at their ends, as shown in the figure. Thin spacers provide a separation between the bars. The copper bars have cross-sectional dimensions 0.5 in. × 2.0 in., and the aluminum bar has dimensions 1.0 in. × 2.0
A steel bar of rectangular cross section (1.5 in. × 2.0 in.) carries a tensile load P (see figure). The allowable stresses in tension and shear are 14,500 psi and 7,100 psi, respectively. Determine the maximum permissible load Pmax.
A plastic bar of diameter d = 32 mm is compressed in a testing device by a force P = 190 N applied as shown in the figure.(a) Determine the normal and shear stresses acting on all faces of stress elements oriented at (1) an angle θ = 0°, (2) an angle θ = 22.5°, and
A plastic bar of rectangular cross section (b = 1.5 in. and h = 3 in.) fits snugly between rigid supports at room temperature (68°F) but with no initial stress (see figure). When the temperature of the bar is raised to 160°F, the compressive stress on an inclined plane pq at midspan becomes
A copper bar of rectangular cross section (b = 18 mm and h = 40 mm) is held snugly (but without any initial stress) between rigid supports (see figure). The allowable stresses on the inclined plane pq at midspan, for which θ = 55°, are specified as 60 MPa in compression and 30 MPa
A circular brass bar of diameter d is member AC in truss ABC which has load P = 5000 lb applied at joint C. Bar AC is composed of two segments brazed together on a plane pq making an angle α = 36° with the axis of the bar (see figure). The allowable stresses in the brass are
Two boards are joined by gluing along a scarf joint, as shown in the figure. For purposes of cutting and gluing, the angle α between the plane of the joint and the faces of the boards must be between 10° and 40°. Under a tensile load P, the normal stress in the boards is 4.9
Acting on the sides of a stress element cut from a bar in uniaxial stress are tensile stresses of 10,000 psi and 5,000 psi, as shown in the figure.(a) Determine the angle and the shear stress τθ and show all stresses on a sketch of the element.(b) Determine the maximum
A prismatic bar is subjected to an axial force that produces a tensile stress σθ = 65 MPa and a shear stress τθ = 23 MPa on a certain inclined plane (see figure). Determine the stresses acting on all faces of a stress element oriented at
The normal stress on plane pq of a prismatic bar in tension (see figure) is found to be 7500 psi. On plane rs, which makes an angle β = 30° with plane pq, the stress is found to be 2500 psi.Determine the maximum normal stress σmax and maximum shear stress Ï„max
A tension member is to be constructed of two pieces of plastic glued along plane pq (see figure). For purposes of cutting and gluing, the angle must be between 25° and 45°.The allowable stresses on the glued joint in tension and shear are 5.0 MPa and 3.0 MPa, respectively.(a) Determine
A nonprismatic bar 1-2-3 of rectangular cross section (cross sectional area A) and two materials is held snugly (but without any initial stress) between rigid supports (see figure). The allowable stresses in compression and in shear are specified as σa and τa, respectively.
A circular steel rod of diameter d is subjected to a tensile force P = 3.5 kN (see figure). The allowable stresses in tension and shear are 118 MPa and 48 MPa, respectively. What is the minimum permissible diameter dmin of the rod?
A standard brick (dimensions 8 in. × 4 in. × 2.5 in.) is compressed lengthwise by a force P, as shown in the figure. If the ultimate shear stress for brick is 1200 psi and the ultimate compressive stress is 3600 psi, what force Pmax is required to break the brick?
A brass wire of diameter d = 2.42 mm is stretched tightly between rigid supports so that the tensile force is T = 98 N (see figure). The coefficient of thermal expansion for the wire is 19.5 × + 10-6 / °C and the modulus of elasticity is E = 110 GPa(a) What is the maximum
A brass wire of diameter d = 1/16 in. is stretched between rigid supports with an initial tension T of 37 lb (see figure). Assume that the coefficient of thermal expansion is 10.6 × 10-6 / °F and the modulus of elasticity is 15 × 106 psi.)(a) If the temperature is
A steel bar with diameter d = 12 mm is subjected to a tensile load P = 9.5 kN (see figure).(a) What is the maximum normal stress σmax in the bar?(b) What is the maximum shear stress Ï„max?(c) Draw a stress element oriented at 45° to the axis of the bar and show all stresses
During a tension test of a mild-steel specimen (see figure), the extensometer shows an elongation of 0.00120 in. with a gage length of 2 in. Assume that the steel is stressed below the proportional limit and that the modulus of elasticity E = 30 × 106 psi.(a) What is the maximum normal
A copper bar with a rectangular cross section is held without stress between rigid supports (see figure). Subsequently, the temperature of the bar is raised 50°C.Determine the stresses on all faces of the elements A and B, and show these stresses on sketches of the elements. (Assume
The bottom chord AB in a small truss ABC (see figure) is fabricated from a W8 × 28 wide-flange steel section. The cross-sectional area A = 8.25 in.2 (Appendix E, Table E-1 (a)) and each of the three applied loads P = 45 k. First, find member force NAB; then, determine the normal and
A prismatic bar AD of length L, cross-sectional area A, and modulus of elasticity E is subjected to loads 5P, 3P, and P acting at points B, C, and D, respectively (see figure). Segments AB, BC, and CD have lengths L/6, L/2, and L/3, respectively.(a) Obtain a formula for the strain energy U of the
A compressive load P is transmitted through a rigid plate to three magnesium-alloy bars that are identical except that initially the middle bar is slightly shorter than the other bars (see figure). The dimensions and properties of the assembly are as follows: length L = 1.0 m, cross-sectional area
A block B is pushed against three springs by a force P (see figure). The middle spring has stiffness k1 and the outer springs each have stiffness k2. Initially, the springs are unstressed and the middle spring is longer than the outer springs (the difference in length is denoted s).(a) Draw a
A bungee cord that behaves linearly elastically has an unstressed length L0 = 760 mm and a stiffness k = 140 N/m. The cord is attached to two pegs, distance b = 380 mm apart, and pulled at its midpoint by a force P = 80 N (see figure).(a) How much strain energy U is stored in the cord?(b) What is
A bar of circular cross section having two different diameters d and 2d is shown in the figure. The length of each segment of the bar is L/2 and the modulus of elasticity of the material is E.(a) Obtain a formula for the strain energy U of the bar due to the load P.(b) Calculate the strain energy
A three-story steel column in a building supports roof and floor loads as shown in the figure. The story height H is 10.5 ft, the cross-sectional area A of the column is 15.5 in.2, and the modulus of elasticity E of the steel is 30 × 106 psi.Calculate the strain energy U of the column
The bar ABC shown in the figure is loaded by a force P acting at end C and by a force Q acting at the midpoint B. The bar has constant axial rigidity EA.(a) Determine the strain energy U1 of the bar when the force P acts alone (Q = 0).(b) Determine the strain energy U2 when the force Q acts alone
Determine the strain energy per unit volume (units of psi) and the strain energy per unit weight (units of in.) that can be stored in each of the materials listed in the accompanying table, assuming that the material is stressed to the proportional limit.
The truss ABC shown in the figure is subjected to a horizontal load P at joint B. The two bars are identical with cross-sectional area A and modulus of elasticity E.(a) Determine the strain energy U of the truss if the angle β = 60°.(b) Determine the horizontal displacement
The truss ABC shown in the figure supports a horizontal load P1 = 300 lb and a vertical load P2 = 900 lb. Both bars have cross-sectional area A = 2.4 in.2 and are made of steel with E = 30 × 106 psi.(a) Determine the strain energy U1 of the truss when the load P1 acts alone (P2 =
The statically indeterminate structure shown in the figure consists of a horizontal rigid bar AB supported by five equally spaced springs. Springs 1, 2, and 3 have stiffnesses 3k, 1.5k, and k, respectively. When unstressed, the lower ends of all five springs lie along a horizontal line. Bar AB,
A slightly tapered bar AB of rectangular cross section and length L is acted upon by a force P (see figure). The width of the bar varies uniformly from b2 at end A to b1 at end B. The thickness t is constant.(a) Determine the strain energy U of the bar.(b) Determine the elongation δ of
A sliding collar of weight W = 150 lb falls from a height h = 2.0 in. onto a flange at the bottom of a slender vertical rod (see figure). The rod has length L = 4.0 ft, cross-sectional area A = 0.75 in.2, and modulus of elasticity E = 30 × 106 psi.Calculate the following quantities:
A bumping post at the end of a track in a railway yard has a spring constant k = 8.0 MN/m (see figure). The maximum possible displacement d of the end of the striking plate is 450 mm.What is the maximum velocity vmax that a railway car of weight W = 545 kN can have without damaging the bumping post
A bumper for a mine car is constructed with a spring of stiffness k = 1120 lb/in. (see figure). If a car weighing 3450 lb is traveling at velocity v = 7 mph when it strikes the spring, what is the maximum shortening of the spring?
A bungee jumper having a mass of 55 kg leaps from a bridge, braking her fall with a long elastic shock cord having axial rigidity EA = 2.3 kN (see figure).If the jumpoff point is 60 m above the water, and if it is desired to maintain a clearance of 10 m between the jumper and the water, what length
A weight W rests on top of a wall and is attached to one end of a very flexible cord having cross-sectional area A and modulus of elasticity E (see figure). The other end of the cord is attached securely to the wall. The weight is then pushed off the wall and falls freely the full length of the
A rigid bar AB having mass M = 1.0 kg and length L = 0.5 m is hinged at end A and supported at end B by a nylon cord BC (see figure). The cord has cross-sectional area A = 30 mm2, length b = 0.25 m, and modulus of elasticity E = 2.1 GPa.If the bar is raised to its maximum height and then released,
Solve the preceding problem if the collar has mass M = 80 kg, the height h = 0.5 m, the length L = 3.0 m, the cross-sectional area A = 350 mm2, and the modulus of elasticity E = 170 GPa.
aSolve Problem 2.8-1 if the collar has weight W = 50 lb, the height h = 2.0 in., the length L = 3.0 ft, the cross-sectional area A = 0.25 in.2, and the modulus of elasticity E = 30,000 ksi.
A block weighing W = 5.0 N drops inside a cylinder from a height h = 200 mm onto a spring having stiffness k = 90 N/m (see figure).(a) Determine the maximum shortening of the spring due to the impact, and (b) determine the impact factor.
Solve the preceding problem if the block weighs W = 1.0 lb, h = 12 in., and k = 0.5 lb/in.
A small rubber ball (weight W = 450 mN) is attached by a rubber cord to a wood paddle (see figure). The natural length of the cord is L0 = 200 mm, its crosssectional area is A = 1.6 mm2, and its modulus of elasticity is E = 2.0 MPa. After being struck by the paddle, the ball stretches the cord to a
A weight W = 4500 lb falls from a height h onto a vertical wood pole having length L = 15 ft, diameter d = 12 in., and modulus of elasticity E = 1.6 × 106 psi (see figure).If the allowable stress in the wood under an impact load is 2500 psi, what is the maximum permissible height h?
A cable with a restrainer at the bottom hangs vertically from its upper end (see figure). The cable has an effective cross-sectional area A = 40 mm2 and an effective modulus of elasticity E = 130 GPa. A slider of mass M = 35 kg drops from a height h = 1.0 m onto the restrainer.If the allowable
Solve the preceding problem if the slider has weight W = 100 lb, h = 45 in., A = 0.080 in.2, E = 21 × 106 psi, and the allowable stress is 70 ksi.
A hollow circular tube having an inside diameter of 10.0 in. and a wall thickness of 1.0 in. (see figure) is subjected to a torque T = 1200 k-in. Determine the maximum shear stress in the tube using (a) the approximate theory of thin-walled tubes, and (b) the exact torsion theory. Does the
A thin-walled rectangular tube has uniform thickness t and dimensions a ( b to the median line of the cross section (see figure).How does the shear stress in the tube vary with the ratio ( = a/b if the total length Lm of the median line of the cross section and the torque T remain constant?From
A tubular aluminum bar (G = 4 ( 106 psi) of square cross section (see figure) with outer dimensions 2 in. ( 2 in. must resist a torque T = 3000 lb-in.Calculate the minimum required wall thickness tmin if the allowable shear stress is 4500 psi and the allowable rate of twist is 0.01 rad/ft.
A thin tubular shaft of circular cross section (see figure) with inside diameter 100 mm is subjected to a torque of 5000 N ( m.If the allowable shear stress is 42 MPa, determine the required wall thickness t by using (a) the approximate theory for a thin-walled tube, and (b) the exact torsion
A long, thin-walled tapered tube AB of circular cross section (see figure) is subjected to a torque T. The tube has length L and constant wall thickness t. The diameter to the median lines of the cross sections at the ends A and B are dA and dB, respectively.Derive the following formula for the
A solid circular bar having diameter d is to be replaced by a rectangular tube having cross-sectional dimensions d ( 2d to the median line of the cross section (see figure).Determine the required thickness tmin of the tube so that the maximum shear stress in the tube will not exceed the maximum
A thin-walled aluminum tube of rectangular cross section (see figure) has a centerline dimensions b = 6.0 in. and h = 4.0 in. The wall thickness t is constant and equal to 0.25 in.(a) Determine the shear stress in the tube due to a torque T = 15 k-in.(b) Determine the angle of twist (in degrees) if
A thin-walled steel tube of rectangular cross section (see figure) has centerline dimensions b = 150 mm and h = 100 mm. The wall thickness t is constant and equal to 6.0 mm. (a) Determine the shear stress in the tube due to a torque T = 1650 N ( m. (b) Determine the angle of twist (in degrees) if
A thin-walled circular tube and a solid circular bar of the same material (see figure) are subjected to torsion. The tube and bar have the same cross-sectional area and the same length.What is the ratio of the strain energy U1 in the tube to the strain energy U2 in the solid bar if the maximum
Calculate the shear stress T and the angle of twist ( (in degrees) for a steel tube (G = 76 GPa) having the cross section shown in the figure. The tube has length L = 1.5 m and is subjected to a torque T = 10 kN ( m.
A thin-walled steel tube having an elliptical cross section with constant thickness t (see figure) is subjected to a torque T = 18 k-in.Determine the shear stress T and the rate of twist (in degrees per inch) if G = 12 ( 106 psi, t = 0.2 in., a = 3 in., and b = 2 in. (See Appendix D, Case 16, for
A torque T is applied to a thin-walled tube having a cross section in the shape of a regular hexagon with constant wall thickness t and side length b (see figure).Obtain formulas for the shear stress T and the rate of twist (.
Compare the angle of twist (1 for a thin-walled circular tube (see figure) calculated from the approximate theory for thin-walled bars with the angle of twist (2 calculated from the exact theory of torsion for circular bars.(a) Express the ratio (1/(2 in terms of the nondimensional ratio ( =
A stepped shaft consisting of solid circular segments having diameters D1 = 2.0 in. and D2 = 2.4 in. (see figure) is subjected to torques T. The radius of the fillet is R = 0.1 in.If the allowable shear stress at the stress concentration is 6000 psi, what is the maximum permissible torque T max?
A stepped shaft with diameters D1 = 40 mm and D2 = 60 mm is loaded by torques T = 1100 N ( m (see figure). If the allowable shear stress at the stress concentration is 120 MPa, what is the smallest radius Rmin that may be used for the fillet?
A full quarter-circular fillet is used at the shoulder of a stepped shaft having diameter D2 = 1.0 in. (see figure). A torque T = 500 lb-in. acts on the shaft. Determine the shear stress Tmax at the stress concentration for values as follows: D1 5 0.7, 0.8, and 0.9 in. Plot a graph showing Tmax
The stepped shaft shown in the figure is required to transmit 600 kW of power at 400 rpm. The shaft has a full quarter-circular fillet, and the smaller diameter D1 = 100 mm. If the allowable shear stress at the stress concentration is 100 MPa, at what diameter D2 will this stress be reached? Is
A stepped shaft (see figure) has diameter D2 = 1.5 in. and a full quarter-circular fillet. The allowable shear stress is 15,000 psi and the load T = 4800 lb-in. What is the smallest permissible diameter D1?
A copper rod of length L = 18.0 in. is to be twisted by torques T (see figure) until the angle of rotation between the ends of the rod is 3.0°.If the allowable shear strain in the copper is 0.0006 rad, what is the maximum permissible diameter of the rod?
A plastic bar of diameter d = 56 mm is to be twisted by torques T (see figure) until the angle of rotation between the ends of the bar is 4.0°. If the allowable shear strain in the plastic is 0.012 rad, what is the minimum permissible length of the bar?
A circular aluminum tube subjected to pure torsion by torques T (see figure) has an outer radius r2 equal to 1.5 times the inner radius r1. (a) If the maximum shear strain in the tube is measured as 400 ( 10-6 rad, what is the shear strain y1 at the inner surface? (b) If the maximum allowable rate
A circular steel tube of length L = 1.0 m is loaded in torsion by torques T (see figure). (a) If the inner radius of the tube is r1 = 45 mm and the measured angle of twist between the ends is 0.5°, what is the shear strain (1 (in radians) at the inner surface? (b) If the maximum allowable shear
Solve the preceding problem if the length L = 56 in., the inner radius r1 = 1.25 in., the angle of twist is 0.5°, and the allowable shear strain is 0.0004 rad.
A prospector uses a hand-powered winch (see figure) to raise a bucket of ore in his mine shaft. The axle of the winch is a steel rod of diameter d = 0.625 in. Also, the distance from the center of the axle to the center of the lifting rope is b = 4.0 in.If the weight of the loaded bucket is W = 100
The steel axle of a large winch on an ocean liner is subjected to a torque of 1.65 kN . m (see figure). What is the minimum required diameter dmin if the allowable shear stress is 48 MPa and the allowable rate of twist is 0.75°/m? (Assume that the shear modulus of elasticity is 80 GPa.)
A hollow steel shaft used in a construction auger has outer diameter d2 = 6.0 in. and inner diameter d1 = 4.5 in. (see figure). The steel has shear modulus of elasticity G = 11.0 ( 106 psi.For an applied torque of 150 k-in., determine the following quantities:(c) Rate of twist ( (degrees per unit
Solve the preceding problem if the shaft has outer diameter d2 = 150 mm and inner diameter d1 = 100 mm. Also, the steel has shear modulus of elasticity G = 75 GPa and the applied torque is 16 kN ( m.
A vertical pole of solid circular cross section is twisted by horizontal forces P = 1100 lb acting at the ends of a horizontal arm AB (see figure). The distance from the outside of the pole to the line of action of each force is c = 5.0 in.If the allowable shear stress in the pole is 4500 psi, what
Solve the preceding problem if the horizontal forces have magnitude P = 5.0 kN, the distance c = 125 mm, and the allowable shear stress is 30 MPa.
A solid brass bar of diameter d = 1.25 in. is subjected to torques T1, as shown in part (a) of the figure. The allowable shear stress in the brass is 12 ksi.(a) What is the maximum permissible value of the torques T1?(b) If a hole of diameter 0.625 in. is drilled longitudinally through the bar, as
A hollow aluminum tube used in a roof structure has an outside diameter d2 = 104 mm and an inside diameter d1 = 82 mm (see figure). The tube is 2.75 m long, and the aluminum has shear modulus G = 28 GPa.(a) If the tube is twisted in pure torsion by torques acting at the ends, what is the angle of
A circular tube of inner radius r1 and outer radius r2 is subjected to a torque produced by forces P = 900 lb (see figure). The forces have their lines of action at a distance b = 5.5 in. from the outside of the tube.If the allowable shear stress in the tube is 6300 psi and the inner radius r1 =
When drilling a hole in a table leg, a furniture maker uses a hand-operated drill (see figure) with a bit of diameter d = 4.0 mm.(a) If the resisting torque supplied by the table leg is equal to 0.3 N.m, what is the maximum shear stress in the drill bit?(b) If the shear modulus of elasticity of the
While removing a wheel to change a tire, a driver applies forces P = 25 lb at the ends of two of the arms of a lug wrench (see figure). The wrench is made of steel with shear modulus of elasticity G = 11.4 ( 106 psi. Each arm of the wrench is 9.0 in. long and has a solid circular cross section of
An aluminum bar of solid circular cross section is twisted by torques T acting at the ends (see figure). The dimensions and shear modulus of elasticity are as follows:L = 1.4 m, d = 32 mm, and G = 28 GPa.(a) Determine the torsional stiffness of the bar.(b) If the angle of twist of the bar is
A high-strength steel drill rod used for boring a hole in the earth has a diameter of 0.5 in. (see figure). The allowable shear stress in the steel is 40 ksi and the shear modulus of elasticity is 11,600 ksi.What is the minimum required length of the rod so that one end of the rod can be twisted
The steel shaft of a socket wrench has a diameter of 8.0 mm. and a length of 200 mm (see figure). If the allowable stress in shear is 60 MPa, what is the maximum permissible torque Tmax that may be exerted with the wrench?Through what angle ( (in degrees) will the shaft twist under the action of
A circular tube of aluminum is subjected to torsion by torques T applied at the ends (see figure). The bar is 24 in. long, and the inside and outside diameters are 1.25 in. and 1.75 in., respectively. It is determined by measurement that the angle of twist is 4° when the torque is 6200 lb-in.
A propeller shaft for a small yacht is made of a solid steel bar 104 mm in diameter. The allowable stress in shear is 48 MPa, and the allowable rate of twist is 2.0° in 3.5 meters.Assuming that the shear modulus of elasticity is G = 80 GPa, determine the maximum torque Tmax that can be applied
Three identical circular disks A, B, and C are welded to the ends of three identical solid circular bars (see figure). The bars lie in a common plane and the disks lie in planes perpendicular to the axes of the bars. The bars are welded at their intersection D to form a rigid connection. Each bar
A stepped shaft ABC consisting of two solid circular segments is subjected to torques T1 and T2 acting in opposite directions, as shown in the figure. The larger segment of the shaft has diameter d1 = 2.25 in. and length L1 = 30 in.; the smaller segment has diameter d2 = 1.75 in. and length L2 = 20
The bar shown in the figure is tapered linearly from end A to end B and has a solid circular cross section. The diameter at the smaller end of the bar is dA = 25 mm and the length is L = 300 mm. The bar is made of steel with shear modulus of elasticity G = 82 GPa. If the torque T = 180 N ( m and
The nonprismatic cantilever circular bar shown has an internal cylindrical hole from 0 to x, so the net polar moment of inertia of the cross section for segment 1 is (7/8)Ip. Torque T is applied at x and torque T/2 is applied at x = L. Assume that G is constant.(a) Find reaction moment R1.(b) Find
A uniformly tapered tube AB of hollow circular cross section is shown in the figure. The tube has constant wall thickness t and length L. The average diameters at the ends are dA and dB = 2dA. The polar moment of inertia may be represented by the approximate formula IP ( (d3 t/4 (see Eq.
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