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Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Mechanics of Materials (MindTap Course List)

2.11.3P A prismatic bar in tension has a length L = 2.0 m and cross-sectional area A =249 mn2. The material of the bar has the stress-strain curve shown in the figure. Determi ne t he elongation 5 of the bar for each of the following axial loads: P = 10 kN, 20 kN, 30 kN, 40 kN. and 45 kN. From these results, plot a diagram of load P versus elongation 5 (load-displacement diagram).An aluminum bar subjected to tensile Forces P has a length L = 150 in. and cross-sectional area A = 2.0 in2 The stress-strain behavior of the aluminum may be represented approximately by the bilinear stress-strain diagram shown in the figure. Calculate the elongation S of the bar for each of the following axial loads: p = 8 kips, 16 kips. 24 kips, 32 kips, and 40 kips. From these results, plot a diagram of load P versus elongation S (load-displacement diagram).A rigid bar AB is pinned al end A and is supported by a wine CD and loaded by a force P at end B (see figure). The wire is made of high-strength steel having a modulus of elasticity E = 210 GPa and yield stress ??Y= 820 MPa. The length of the wire is L = 1.0 m. and its diameter is d = 3 mm. The stress-strain diagram for the steel is defined by the riuniift-ed power taw. as Two identical bars AB and BC support a vertical load P (see figure). The bars are made of steel having a stress-strain curve that may be idealized as elastoplastic with yield stress aY. Each bar has cross-sectional area A. Determine the yield load Pyand the plastic load Pp.A stepped bar ACB with circular cross sections is held between rigid supports and loaded by an axial force P at midlength (see figure). The diameters for the two parts of the bar are d1= 20 ram and d2= 25 mm, and the material is elastoplastic with yield stress s = 250 MPs. Determine the plastic load Pp.A horizontal rigid bar AB supporting a load P is hung from Five symmetrically placed wires, each of cross-sectional area A (see figure). The wires are fastened to a curved surface of radius R. (a) Determine the plastic load Ppif the material of the wires is elastoplaslic with yield stress trr. (b) How is Pp changed if bar AB is flexible instead of rigid? (c) How is PPchanged if the radius R is increased?2.12.4P The symmetric truss ABCDE shown in the figure is constructed of four bars and supports a load P at joint E. Each of the two outer bars has a cross-sectional area of 0.307 in2, and each of the two inner bars has an area of 0.601 in2. The material is elastoplastic with yield stress ??Y= 36 ksi. Determine the plastic load PP.Five bars, each having a diameter of 10 mm. support a load P as shown in the figure. Determine the plastic load Ppif the material is clastoplastic with yield stress 2.12.7P A rigid bar ACB is supported on a fulcrum at C and loaded by a Force P at end B (see figure). Three identical wires made of an elasloplastic material (yield stress oYand modulus of elasticity E) resist tbe load P. Each wire has cross-sectional area A and length L. (a) Determine the yield load PYand the corresponding yield displacement Syat point B. (b) Determine the plastic load PPand the corresponding displacement The structure shown in the figure consists of a horizontal rigid bar ABCD supported by two steel wires: one of length L and the other of length 3L/4. Both wires have cross-sectional area A and are made of elastoplastic material with yield stress aYand modulus of elasticity E. A vertical load P acts at end D of the bar. (a) Determine the yield load PYand the corresponding yield displacement grat point D. (b) Determine the plastic load Ppand the corresponding displacement Two cables, each having a length i. of approximately 40 m, support a loaded -container of weight W (see figure). The cables, which have an effective cross-sectional area A = 48.0 mm2 and effective modulus of elasticity E = 160 GPa. are identical except that one cable is longer than the other when they are hanging separately and unloaded. The difference in lengths is d = 100 mm. The cables are made of steel having an elastoplastic stress-strain diagram with a r= 500 MPa. Assume that the weight ft' is initially zero and is slowly increased by the addition of material to the container. (a) Determine the weight Wythat lirsl produces yielding of the shorter cable. Also, determine the corresponding elongation 5 of the shorter cable. (b) Determine the weight Wpthat produces yielding of both cables. Also, determine the elongation Spof the shorter cable when the weight W just reaches the value Wp. (c) Construct a load-displacement diagram showing the weight W as ordinate and the elongation A hollow circular tube T of a length L = 15 in. is uniformly compressed by a force P acting through a rigid plate (see figure). The outside and inside diameters of the tube are 3.0 and 2.75 in., respectively. A concentric solid circular bar B of 1.5 in. diameter is mounted inside the lube. When no load is present, there is a clearance c = 0.0I0 in. between the bar B and the rigid plate. Both bar and tube are made of steel having an c[autoplastic stress-strain diagram with E = 29 X LO3 ksi and err= 36 ksi. (a) Determine the yield load Pt- and the corresponding shortening 3yof the lube. (b) Determine the plastic load Ppand the corresponding shortening Spof the tube. (c) Construct a load-displacement diagram showing the load Pas ordinate and the shortening 5 of the tube as abscissa. Hint: The load-displacement diagram is not a single straight line in the region 0 ^ P ^ Pr A circular tube is subjected to torque Tat its ends. The resulting maximum shear strain in the tube is 0.005. Calculate the minimum shear strain in the tube and the shear strain at the median line of the tube section.-2. 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°. (a) If the allowable shear strain in the plastic is 0.012 rad, what is the minimum permissible length of the bar? (b) If the length of the bar is 200 mm, what is the maximum permissible diameter of the bar?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°. (a) If the allowable shear strain in the copper is 0.0006 rad, what is the maximum permissible diameter of the rod? (b) If the rod diameter is 0.5 in., what is the minimum permissible length of the rod?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 y1(in radians) at the inner surface? (b) If the maximum allowable shear strain is 0.0004 rad and the angle of twist is to be kept at 0.45° by adjusting the torque T, what is the maxi mum permissible outer radius (r2)max?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 circular aluminum tube subjected to pure torsion by torques T(sec figure) has an outer radius r2equal 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 y1at the inner surface? (b) If the maximum a1lo-abk rate of twist is 0.125 °/m and the maximum shear strain is to be kept at 400 × 10-6 rad by adjusting the torque T, that is the minimum required outer radius ( r2)Min?A solid steel bar of circular cross section has diameter d = 2.5 in., L = 60 in., and shear modulus of elasticity G = 11.5 × 106 psi. The bar is subjected to torques T = 300 lb-ft at the ends. Calculate the angle of twist between the ends. What is the maximum shear stress and the shear stress at a distance rA=1.0 in. measured from the center of the bar?A solid copper bar of circular cross section has length L = 1.25 m and shear modulus of elasticity G = 45 GPa. The bar is designed to carry a 250 N · m torque acting at the ends. If the allowable shear stress is 30 M Pa and the allowable angle of twist between the ends is 2.5°, what is the minimum required diameter?Repeat Problem 3.3-1, but now use a circular tube with outer diameter d0= 2.5 in. and inner diameter di= 1.5 in.A copper tube with circular cross section has length L = 1.25 m, thickness t = 2 mm, and shear modulus of elasticity G = 45 GPa. The bar is designed to carry a 300 N·m torque acting at the ends. If the allowable shear stress is 25 MPa and the allowable angle of twist between the ends is 2.5°, what is the minimum required outer diameter d?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 lb, what is the maximum shear stress in the axle due to torsion? If the maximum bucket load is 125 lb and the allowable shear stress in the axle is 9250 psi, what is the minimum permissible axle diameter?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. 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? If the allowable shear stress in the drill bit is 32 MPa, what is the maximum resisting torque before the drill binds up? If the shear modulus of elasticity of the steel is G = 75 GPa, what is the rate of twist of the drill bit (degrees per meter)?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 x 106 psi. Each arm of the wrench is 9.0 in. long and has a solid circular cross section of diameter d = 0.5 in. Determine the maximum shear stress in the arm that is turning the lug nut (arm A). Determine the angle of twist (in degrees) of this same arm.-8 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 arc L = 1.4 m, d = 32 mm, and G = 28 GPa. Determine the torsional stiffness of the bar. If the angle of twist of the bar is 5º, what is the maximum shear stress? What is the maximum shear strain (in radians)? If a hole of diameter d/2 is drilled longitudinally through the bar, what is the ratio of the torsional stiffnesses of the hollow and solid bars? What is the ratio of their maximum shear stresses if both arc acted on by the same torque? If the hole diameter remains at d/2, what new outside diameter d2will result in equal stiffnesses of the hollow and solid bars?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 30º with respect to the other end without exceeding the allowable stress? If the shear strain in part (a) is limited to 3.2 × 10-3 , what is the minimum required length of the drill rod?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 Tmaxthat may be exerted with the wrench? Through what angle (in degrees) will the shaft twist under the action of the maximum torque? (Assume G = 78 GPa and disregard any bending of the shaft.)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. Calculate the maximum shear stress in the tube, the shear modulus of elasticity G, and the maximum shear strain (in radians). If the maximum shear strain in the tube is limited to 2.5 × 10-3 and the inside diameter is increased to 1.375 in., what is the maximum permissible torque?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. (a) Assuming that the shear modulus of elasticity is G = 80 GPa, determine the maximum torque that can be applied to the shaft. (b) Repeat part (a) if the shaft is now hollow with an inner diameter of 5d18. Compare values to corresponding values from part (a).Three identical circular disks A, B, and Care 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 arc welded at their intersection D to form a rigid connection. Each bar has diameter d1= 0.5 in. and each disk has diameter d2= 3.0 in. Forces P1, P2, and P3act on disks A, B, and C, respectively, thus subjecting the bars to torsion. If P1= 28 lb, what is the maximum shear stress in any of the three bars?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 dminif the allowable shear stress is 48 M Pa and the allowable rate of twist is 0.75º/m? (Assume that the shear modulus of elasticity is 80 G Pa.) Repeat part (a) if the shaft is now hollow with an inner diameter of 5d/8, Compare dminvalues to corresponding values from part (a)A hollow steel shaft used in a construction auger has an outer diameter d2= 6.0 in. and inner diameter d1= 4.5 in. (see figure). The steel has a shear modulus of elasticity G = 11.0 × 106 psi. For an applied torque of 150 kip-in., determine the following quantities: shear stress at the outer surface of the shaft, shear stress at the inner surface, and rate of twist (degrees per unit of length). Also, draw a diagram showing how the shear stresses vary in magnitude along a radial line in the cross section.Solve the preceding problem if the shaft has an outer diameter d2=150 mm and inner diameter d1= 100 mm. Also, the steel has a 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 rigid horizontal arm AB (see figure part a). The distance from the outside of the pole to the line of action of each force is c = 5.0 in. (see figure part b) and the pole height is L = 14in. (a) If the allowable shear stress in the pole is 4500 psi, what is the minimum required diameter dminof the pole? Find the torsional stiffness of the pole (kip-in./rad). Assume that G = 10,800 ksi. If two translational springs, each with stiffness k = 33 kips/in., are added at 2(75 from A and B (see figure part c), repeat part (a) to find dmin. Hint: Consider the pole and pair of springs as "springs in parallel."A vertical pole of solid, circular cross section is twisted by horizontal forces P = 5kN acting at the ends of a rigid horizontal arm AB (see figure part a). The distance from the outside of the pole to the line of action of each force is c = 125 mm (sec figure part b) and the pole height L = 350 mm. (a) If the allowable shear stress in the pole is 30 MPa, what is the minimum required diameter dminof the pole? (b) What is the torsional stiffness of the pole (kN · m/rad)? Assume that G = 28 GPa. (c) If two translation al springs, each with stiffness k =2550 kN/m, are added at 2c/5 from A and B (see figure part c), repeat part (a) to find dmin. Hint: Consider the pole and pair of springs as "springs in parallel."A solid brass bar of diameter d = 1.25 in. is subjected to torques T1as shown in part a of the figure. The allowable shear stress in the brass is 12 ksi. What is the maximum permissible value of the torques T1? If a hole of diameter 0.625 in. is drilled longitudinally through the bar, as shown in part b of the figure, what is the maximum permissible value of the torques T2? What is the percent decrease in torque and the percent decrease in weight due to the hole?A hollow aluminum tube used in a roof structure has an outside diameter d2= 104mm and an inside d1= 82 mm (see figure). The tube is 2.75 m long, and the aluminum has shear modulus G7 = 28GPa. If the tube is twisted in pure torsion by torques acting at the ends, what is the angle of twist (in degrees) when the maximum shear stress is 48 MPa? What diameter d is required for a solid shaft (see figure) to resist the same torque with the same maximum stress? What is the ratio of the weight of the hollow tube to the weight of the solid shaft?A circular tube of inner radius r1and outer radius r2is subjected to a torque produced by forces P = 900 lb (see figure part a). The forces have their lines of action at a distance b = 5.5 in. from the outside of the tube. (a) If the allowable shear stress in the tube is 6300 psi and the inner radius r1= 1.2 in., what is the minimum permissible outer radius r2? (b) If a torsional spring of stiffness = 450 kip-in./rad is added at the end of the tube (see figure part b), what is the maximum value of forces P lithe allowable shear stress is not to be exceeded? Assume that the tube has a length of L 18 in., outer radius of r2= 1.45 in. and shear modulus G = 10.800 ksi. Hint: Consider the tube and torsional spring as “springs in parallel.”.1 A stepped shaft ABC consisting of two solid circular segments is subjected to torques ‘1 and T, acting in opposite directions, as shown in the figure. The larger segment of the shaft has a diameter of d1= 2.25 in. and length L1= 30 in.; the smaller segment has a diameter of d2 1.75 in. and length of L1= 20 in. The material is steel with shear modulus G = 11 × 106 psi, and the torques are T1 = 20.000 lb-in, and T2= 8000 lb-in. (a) Calculate the maximum shear stress tmaxin the shaft and the angle of twist (in degrees) at end C. (b) If the maximum shear stress in BC must be the same as that in AB. what is the required diameter of segment BC? What is the resulting twist at end C?A circular tube of outer diameter d3= 70 mm and inner diameter d2= 60 mm is welded at the right-hand end to a fixed plate and at the left-hand end to a rigid end plate (see figure). A solid, circular bar with a diameter of d1= 40 mm is inside of, and concentric with, the tube. The bar passes through a hole in the fixed plate and is welded to the rigid end plate. The bar is 1.0 m long and the tube is half as long as the bar. A torque T = 1000 N · m acts at end A of the bar. Also, both the bar and tube arc made of an aluminum alloy with a shear modulus of elasticity G = 27 G Pa. Determine the maximum shear stresses in both the bar and tube. Determine the angle of twist (in degrees) at end A of the bar.A stepped shaft ABCD consisting of solid circular segments is subjected to three torques, as shown in the figure. The torques have magnitudes of 12.5 kip-in., 9.8 kip-in., and 9.2 kip-in. The length of each segment is 25 in. and the diameters of the segments arc 3.5 in., 2.75 in., and 2,5 in. The material is steel with shear modulus of elasticity G = 11.6 × 10-3ksi. Calculate the maximum shear stress in the shaft and the angle of twist (in degrees) at end D? If each segment must have the same shear stress, find the required diameter of each segment in part (a) so that all three segments have shear stress from part (a). What is the resulting angle of twist at D?A solid, circular bar ABC consists of two segments, as shown in the figure. One segment has a diameter of d1= 56 mm and length of L1= 1.45 m; the other segment has a diameter of d2= 48 mm and length of L2= 1.2 m. What is the allowable torque Tallowif the shear stress is not to exceed 30 M Pa and the angle of twist between the ends of the bar is not to exceed 1.25°? (Assume G = 80GPa.)A hollow tube ABCDE constructed of monel metal is subjected to five torques acting in the directions shown in the figure. The magnitudes of the torques are T1= 1000 lb-in., T2= T4= 500 lb-in., and T3= T5= 800 lb-in. The tube has an outside diameter of d2= 1.0 in. The allowable shear stress is 12,000 psi and the allowable rate of twist is 2.0°/ft. Determine the maximum permissible inside diameter d1, of the tube.A shaft with a solid, circular cross section consisting of two segments is shown in part a of the figure. The left-hand segment has a diameter of 80 mm and length of 1.2 m; the right-hand segment has a diameter of 60 mm and length of 0,9 m. Shown in part b of the figure is a hollow shaft made of the same material and having the same length. The thickness t of the hollow shaft is d/10, where d is the outer diameter. Both shafts are subjected to the same torque. If the hollow shaft is to have the same torsional stiffness as the solid shaft, what should be its outer diameter d? If torque T is applied at either end of both shafts and the hollow shaft is to have the same maximum shear stress as the solid shaft, what should be its outer diameter d?3.4.7P Two sections of steel drill pipe, joined by bolted flange plates at Ä are being tested to assess the adequacy of both the pipes. In the test, the pipe structure is fixed at A, a concentrated torque of 500 kN - m is applied at x = 0.5 m, and uniformly distributed torque intensity t1= 250 kN m/m is applied on pipe BC. Both pipes have the same inner diameter = 200 mm. Pipe AB has thickness tAB=15 mm, while pipe BC has thickness TBC= 12 mm. Find the maximum shear stress and maximum twist of the pipe and their locations along the pipe. Assume G = 75 GPa.3.4.9P -10. A tapered bar AB with a solid circular cross section is twisted by torques T(see figure). The diameter of the bar varies linearly from dAat the left-hand end to dBat the right-hand end. (a) Confirm that the angle of twist of the tapered bar is (b) For what ratio dBld4will the angle of twist of the tapered bar be one-half the angle of twist of a prismatic bar of diameter d41 (The prismatic bar is made of the same material- has the same length, and is subjected to the same torque as the tapered bar.)A tapered bar AB with a solid circular cross section is twisted by torques T = 36,000 lb-in. (sec figure). The diameter of the bar varies linearly from dAat the left-hand end to dBat the right-hand end. The bar has length L = 4,0 ft and is made of an aluminum alloy having shear modulus of elasticity G = 3.9 × 106 psi. The allowable shear stress in the bar is 15,000 psi and the allowable angle of twist is 3.0°. If the diameter at end B is 1.5 times the diameter at end A, what is the minimum required diameter dAat end A?The bar shown in the figure is tapered linearly from end A to end B and has a solid circular cross section. T lie 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 = 82GPa. If the torque T = 180 N · m and the allowable angle of twist is 0.3°, what is the minimum allowable diameter dBat the larger end of the bar?The non prismatic, cantilever circular bar shown has an internal cylindrical hole from 0 to y, so the net polar moment of inertia of the cross section for segment 1 is (7/8 )Ip. Torque Tis applied at _y and torque 772 is applied at .v = L. Assume that G is constant. Find the reaction moment Ry. Find internal torsional moments Tiin segments 1 and 2. Find x required to obtain twist at joint 3 of tf3 = TLtGIp. What is the rotation at joint 2, Draw the torsional moment (TMD:7(.,0 _v L) and displacement (TDD: M_y),0 x L) diagrams.A uniformly tapered tube AB with a 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 dAand dB= 2dA. The polar moment of inertia may be represented by the approximate formula Ipttd3t4[see Eq. (3-21)]. Derive a formula for the angle of twist e of the tube when it is subjected to torques T acting at the ends.A uniformly tapered aluminum-alloy tube AB with a circular cross section and length L is shown in the figure. The outside diameters at the ends arc dAand dB= 2dA. A hollow section of length LB and constant thickness t = dA/10 is cast into the tube and extends from B halfway toward A. (a) Find the angle of twist é of the tube when it is subjected to torques T acting at the ends. Use numerical values: dA= 2.5 in., L = 48 in., G = 3.9 × 106 psi, and T =40,000 in.-lb. (b) Repeat part (a) if the hollow section has con stant diameter rfj (see figure part b)For the thin nonprismatic steel pipe of constant thickness t and variable diameter d shown with applied torques at joints 2 and 3, determine the following. Find the reaction moment Ry. Find an expression for twist rotation O2at joint 3. Assume that G is constant. Draw the torsional moment diagram (TMD:T .17 A mountain-bike rider going uphill applies torque T = Fd(F = l5lb, d = 4 in.) to the end of the handlebars ABCD by pulling on the handlebar extenders DE. Consider the right half of the handlebar assembly only (assume the bars are fixed at the fork at A). Segments AB and CD are prismatic with lengths L, = 2 in.andL3 = 8.5 in, and with outer diameters and thicknesses d01 = 1.25 in. 101 = 0.125 in. and d03 = O.87in.,i03 = 0.ll5in, respectively as shown. Segment BC’ of length L, = 1.2 in. however. is tapered, and outer diameter and thickness vary linearly between dimensions at B and C. Consider torsion effects only. Assume G = 4000 ksi is constant. Derive an integral expression for the angle of twist of half of the handlebar tube when it is subjected to torque T = Fd acting at the end. Evaluate ‘b1-, for the given numerical1ues.A prismatic bar AB of length L and solid circular cross section (diameter d) is loaded by a distributed torque of constant intensity t per unit distance (sec figure). Determine the maximum shear stress tmaxin the bar. Determine the angle of twist between t the ends of the bar.A prismatic bar AB with a solid circular cross section (diameter d) is loaded by a distributed torque (see figure). The intensity of the torque, that is, the torque per unit distance, is denoted i(x) and varies linearly from a maximum value iAat end A to zero at end B. Also, the length of the bar is L and the shear modulus of elasticity of the material is G. Determine the maximum shear stress in the bar. Determine the angle of twist between the ends of the bar.A magnesium-alloy wire of diameter d = 4mm and length L rotates inside a flexible tube in order to open or close a switch from a remote location (see figure). A torque Tis applied manually (either clockwise or counterclockwise) at end 5, thus twisting the wire inside the tube. At the other end A, the rotation of the wire operates a handle that opens or closes the switch. A torque T0 = 0.2 N · m is required to operate the switch. The torsional stiffness of the tube, combined with friction between the tube and the wire, induces a distributed torque of constant intensity t = 0.04N m/m (torque per unit distance) acting along the entire length of the wire. (a) If the allowable shear stress in the wire is T allow = 30 MPa, what is the longest permissible length Lmaxof the wire?A nonprismatic bar ABC with a solid circular cross section is loaded by distributed torques (sec figure). The intensity of the torques, that is, the torque per unit distance, is denoted t(x) and varies linearly from zero at A to a maximum value T0/L at B. Segment BC has linearly distributed torque of intensity r(x) = T0/3L of opposite sign to that applied along AB. Also, the polar moment of inertia of AB is twice that of BC and the shear modulus of elasticity of the material is G. Find the reaction torque RA. Find internal torsional moments T(x) in segments AB and BC. Find the rotation t0 Find the maximum shear stress tmaxand its location along the bar, Draw the torsional moment diagram (TMD:T(x),0 < x < L).-22 Two tubes (AB, BC) of the same material arc connected by three pins (pin diameter = d ) just left of B as shown in the figure. Properties and dimensions for each tube are given in the figure. Torque 2ris applied at x = 2L/5 and uniformly distributed torque intensity tQ= 37/L is applied on tube BC. (a) Find the maximum value of load variable T(N m) based on allowable shear (tx) and bearing(cha ) stresses in the three pins which connect the two tubes at B. Use the following numerical properties: L = 1.5m, E = 74GPa, v = 0.33, dp= 18mm, ta=45MPa, =90 MPa, di=85 mm, di = T$ mm, and d3— 60 mm. (b) What is the maximum shear stress in the tubes for the applied torque in part (a)?A circular copper bar with diameter d = 3 in. is subjected to torques T = 30 kip-in. at its ends. Find i lie maximum shear, ion si le. and compressive stresses in the tube and their corresponding strains. Assume that G = 6000 ksi.A circular steel tube with an outer diameter of 75 mm and inner diameter of 65 mm is subjected to torques 7"at its ends. Calculate the maximum permissible torque 7jliajl if the allowable normal strain is ea= 5 X 10"4 Assume that G = 15 GPa.A hollow aluminum shaft (see figure) has an outside diameter d2= 4.0 in. and inside diameter d1= 2.0 in. When twisted by torques T, the shaft has an angle of twist per unit distance equal to 0,54°/ft. The shear modulus of elasticity of the aluminum is G = 4.0 x 106 psi. Determine the maximum tensile stress emax in the shaft. Determine the magnitude of the applied torques T.A hollow steel bar (G = 80 GPa ) is twisted by torques T (see figure). The twisting of the bar produces a maximum shear strain ymax= 640 x 10-6 rad. The bar has outside and inside diameters of 150 mm and 120 mm, respectively. Determine the maximum tensile strain in the bar. Determine the maximum tensile stress in the bar. What is the magnitude of the applied torques T?A tubular bar with outside diameterd2= 4.0 in, is twisted by torques T = 70,0 kip-in. (see figure). Under the action of these torques, the maximum tensile stress in the bar is found to be 6400 psi. Determine the inside diameter rtf of the bar. If the bar has length L = 48.0 in. and is made of aluminum with shear modulus G = 4,0 × 106 psi, what is the angle of twist d (in degrees) between the ends of the bar? (c) Determine the maximum shear strain y (in radians)?A solid circular bar of diameter d = 50 mm (see figure) is twisted in a testing maching until the applied torque reaches the value T = 500 N ·. At this value of torque, a strain gage oriented at 45º to the axis of the bar gives a reading e = 339 × 10-6. What is the shear modulus G of the material?-7 A steel tube (G = 11.5 x 106 psi) has an outer diameter d2= 2.0 in. and an inner diameter dt=1,5 in. When twisted by a torque 7", the tube develops a maximum normal strain of 170 x 10-6. What is the magnitude of the applied torque T?A solid circular bar of steel (G = 78 GPa) transmits a torque T = 360 N - m. The allowable stresses in tension, compression, and shear arc 90 MPa, 70 MPa, and 40 MPa, respectively. Also, the allowable tensile strain is 220 x 10-6, Determine the minimum required diameter d of the bar, If the bar diameter d = 40 mm, what is Tmax?The normal strain in the 45n direction on the surface of a circular tube (sec figure) is 880 × 10 when the torque T = 750 lb-in. The tube is made of copper alloy with G = 6.2 × 106 psi and y = 0.35. If the outside diameter d2of the tube is 0.8 in., what is the inside diameter dt? If the allowable normal stress in the tube is 14 ksi, what is the maximum permissible inside diameter d?An aluminum tube has inside diameter dx= 50 mm, shear modulus of elasticity G = 27 GPa, v = 0.33, and torque T = 4.0 kN · m. The allowable shear stress in the aluminum is 50 MPa, and the allowable normal strain is 900 X 10-6. Determine the required outside diameter d2 Re-compute the required outside diameter d2, if allowable normal stress is 62 MPa and allowable shear strain is 1.7 X 10-3.-11 A solid steel bar (G = 11.8 X 106 psi ) of diameter d = 2,0 in. is subjected to torques T = 8.0 kip-in. acting in the directions shown in the figure. Determine the maximum shear, tensile, and compressive stresses in the bar and show these stresses on sketches of properly oriented stress elements. Determine the corresponding maximum strains (shear, tensile, and compressive) in the bar and show these strains on sketches of the deformed elements.A solid aluminum bar (G = 27 GPa ) of diameter d = 40 mm is subjected to torques T = 300 N - m acting in the directions shown in the figure, Determine the maximum shear, tensile, and compressive stresses in the bar and show these stresses on sketches of properly oriented stress elements. Determine the corresponding maximum strains (shear, tensile, and compressive) in the bar and show these strains on sketches of the deformed elements.Two circular aluminum pipes of equal length L = 24 in. arc loaded by torsional moments T (sec figure). Pipe I has outside and inside diameters d2= 3 in. and L2, = 2.5 in., respectively. Pipe 2 has a constant outer diameter of d2along its entire length L and an inner diameter of d1but has an increased inner diameter of d3= 2.65 in. over the middle third. Assume that E = 10,400 ksi, u = 0.33, and allowable shear stress ra= 6500 psi. Find the maximum acceptable torques that can be applied to Pipe 1; repeat for Pipe 2. If the maximum twist e of Pipe 2 cannot exceed 5/4 of that of Pipe 1, what is the maximum acceptable length of the middle segment? Assume both pipes have total length L and the same applied torque T. Find the new value of inner diameter d3of Pipe 2 if the maximum torque carried by Pipe 2 is to be 7/8 of that for Pipe L If the maximum normal strain in each pipe is known to bemax = 811 x 10-6, what is the applied torque on each pipe? Also, what is the maximum twist of each pipe? Use the original properties and dimensions.A generator shaft in a small hydroelectric plant turns at 120 rpm and delivers 50 hp (see figure). (a) If the diameter of the shaft is d = 3,0 in., what is the maximum shear tmaxïin the shaft? (b) If the shear stress is limited to 4000 psi, what is the minimum permissible diameter dmax of the shaft?A motor drives a shaft at 12 Hz and delivers 20 kW of power (sec figure), If the shaft has a diameter of 30 mm, what is the maximum shear stress tmaxin the shaft? If the maximum allowable shear stress is 40 M Pa, what is the maximum permissible diameter tmaxof the shaft? c A motor driving a solid circular steel shaft with diameter d = 1.5 in, transmits 50 hp to a gear at B, The allowable shear stress in the steel is 6000 psi. Calculate the required speed of rotation (number of revolutions per minute) so that the shear stress in the shaft does not exceed the allowable limit.3.7.4P The propeller shaft of a large ship has an outside diameter 18 in. and inside diameter 12 in,, as shown in the figure. The shaft is rated for a maximum shear stress of 4500 psi. If the shaft is turning at 100 rpm, what is the maximum horsepower that can be transmitted without exceeding the allowable stress? If the rotational speed of the shaft is doubled but the power requirements remain unchanged, what happens to the shear stress in the shaft?The drive shaft for a truck (outer diameter 60 mm and inner diameter 40 mm) is running at 2500 rpm (see figure). (a If the shaft transmit 115 kW. what is the maximum shear stress in the shaft? (b) If the allowable shear stress is 30 MPa, what is the maximum power that can be transmitted?A hollow circular shaft for use in a pumping station is being designed with an inside diameter equal to 0.75 times the outside diameter. The shaft must transmit 400 hp at 400 rpm without exceeding the allowable shear stress of 6000 psi. Determine the minimum required outside diameter d.A tubular shaft being designed for use on a construction site must transmit 120 kW at 1,75 Hz, The inside diameter of the shaft is to be one-half of the outside diameter. If the allowable shear stress in the shaft is 45 MPa, what is the minimum required outside diameter d?A propeller shaft of solid circular cross section and diameter d is spliced by a collar of the same material (see figure). The collar is securely bonded to both parts of the shaft. What should be the minimum outer diameter dyof the collar in order that the splice can transmit the same power as the solid shaft?What is the maximum power that can be delivered by a hollow propeller shaft (outside diameter 50 mm, inside diameter 40 mm, and shear modulus of elasticity 80 GPa) turning at 600 rpm if the allowable shear stress is 100 MPa and the allowable rate of twist is 3.0°/m?A motor delivers 275 hp at 1000 rpm to the end of a shaft (see figure). The gears at B and Ctake out 125 and 150 hp, respectively. Determine the required diameter d of the shaft if the allowable shear stress is 7500 psi and the angle of twist between the motor and gear C is limited to 1.5°. (Assume G = 11.5 × 106 psi, L1= 6 ft, and L2= 4ft)3.7.12P A solid circular bar ABCD with fixed supports is acted upon by torques T0and 2T0at the locations shown in the figure. (a) Obtain a formula for the maximum angle of twist 0maxof the bar. (b) What is 0max if the applied torque T0at B is reversed in direction?A solid circular bar A BCD with fixed supports at ends A and D is acted upon by two equal and oppositely directed torques T0, as shown in the figure. The torques arc applied at points B and C each of which is located at distance x from one end of the bar. (The distance x may vary from zero to L/2.) For what distance x will the angle of twist at points B and C be a maximum? What is the corresponding angle of twist 0max?A solid circular shaft AB of diameter d is fixed against rotation at both ends (sec figure), A circular disk is attached to the shaft at the location shown. What is the largest permissible angle of rotation 0max of the disk if the allowable shear stress in the shaft is Tallow? [Assume that a >b. Also, use Eqs. (3-50a and b) of Example 3-9 to obtain the reactive torques.]A ho 1 low st e el shaft ACB of outside diameter 50 mm and inside diameter 40 mm is held against rotation at ends A and B (see figure). Horizontal forces Pare applied at the ends of a vertical arm that is welded to the shaft at point C. Determine the allowable value of the forces P if the maximum permissible shear stress in the shaft is 45 MPa.A stepped shaft ACB having solid circular cross sections with two different diameters is held against rotation at the ends (sec figure). If the allowable shear stress in the shaft is 6000 psi, what is the maximum torque (T0) that may be applied at section C? Find (T0)max if the maximum angle of twist is limited to 0.55º. Let G = 10,600 ksi.A stepped shaft ACB having solid circular cross sections with two different diameters is held against rotation at the ends (see figure), (a) If the allowable shear stress in the shaft is 43 MPa, what is the maximum torque (T0)max that may be applied at section C? (b) Find (ï^),^ if the maximum angle of twist is limited to 1.85', Let G = 28 GPa.A stepped shaft ACE is held against rotation at ends A and B and subjected to a torque T0acting at section C(see figure). The two segments of the shaft (AC and CB) have diameters dAand dg, respectively, and polar moments of inertia IpAand IpBrespectively. The shaft has length L and segment AC has length a. For what ratio a/L will the maximum shear stresses be the same in both segments of the shaft? For what ratio a/L will the internal torques be the same in both segments of the shaft?A solid circulai' aluminum bar AB is fixed at both ends and loaded by a uniformly distributed torque 150N·n/m. The bar has diameter d = 30 mm. Calculate the reactive torques at the supports and the angle of twist at midspan. Assume that G = 28 GPa.Two sections of steel drill pipe, joined by bolted flange plates at B, arc subjected to a concentrated torque 4000 kip-in. at x = 3 ft, and a uniformly distributed torque t0= 50 kip-ft/ft is applied on pipe BC. Let G = 11,800 ksi and assume that pipes AB and BC have the same inner diameter, d = 12 in. Pipe AB has a thickness tAB= 3/4 in., and pipe BC has a thickness tBC= 5/8 in. Find the reactive torques at A and C and the maximum shear stresses in each segment.A circular bar AB of length L is fixed against rotation at the ends and loaded by a distributed torque t(x) that varies linearly in intensity from zero at end A to t0at end B (sec figure). Obtain formulas for the fixed-end torques TAand TB. Find an expression for the angle of twist 0(x). What is 0max, and where does it occur alone the bar?A circular bar AB with ends fixed against rotation has a hole extending for half of its length (sec figure). The outer diameter of the bar is d2= 3.0 in., and the diameter of the hole is d1= 2.4 in. The total length of the bar is L = 50 in. At what distance x from the left-hand end of the bar should a torque T0be applied so that the reactive torques at the supports will be equal? Based on the solution for x in part (a), what is 0max and where does it occur? Assume that T0= 87.4 kip-in. and G = 10,600 ksi.A solid steel bar of diameter d1= 25.0 mm is enclosed by a steel tube of outer diameter d3= 37.5 mm and inner diameter d2= 30.0 mm (see figure). Both bar and tube arc held rigidly by a support at end A and joined securely to a rigid plate at end B. The composite bar, which has a length L = 550 mm, is twisted by a torque T = 400 N ·m acting on the end plate. Determine the maximum shear stresses T1and T2in the bar and tube, respectively. Determine the angle of rotation 0(in degrees) of the end plate, assuming that the shear modulus of the steel is G = 80 GPa. Determine the torsional stiffness kTof the composite bar.A solid steel bar of diameter d1= 1.50 in. is enclosed by a steel tube of outer diameter d3= 2.25 in, and inner diameter d2= 1,75 in. (see figure). Both bar and tube arc held rigidly by a support at end A and joined securely to a rigid plate at end B. The composite bar, which has length L = 30.0 in., is twisted by a torque T = 5000 lb-in, acting on the end plate. Determine the maximum shear stresses r, and r2in the bar and tube, respectively. Determine the angle of rotation 0 (in degrees) of the end plate, assuming that the shear modulus of the steel is G = 116 × 106 psi. Determine the torsional stiffness kTof the composite bar.The composite shaft shown in the figure is manufactured by shrink-Fitting a steel sleeve over a brass core so that the two parts act as a single solid bar in torsion. The outer diameters of the two parts are dY= 40 mm for the brass core and d2= 50 mm for the steel sleeve. The shear moduli of elasticity are Gb= 36 GPa for the brass and Gs= 80 GPa for the steel. (a) Assuming that the allowable shear stresses in the brass and steel are rb= 48 MPa and ts= 80 MPa, respectively, determine the maximum permissible torque Tmax that may be applied to the shaft. (b) If the applied torque T = 2500 kN · m, find the required diameter d2so that allowable shear stress t3is reached in the steel.The composite shaft shown in the figure is manufactured by shrink-fitting a steel sleeve over a brass core so that the two parts act as a single solid bar in torsion. The outer diameters of the two parts arc d1= 1.6 in. for the brass core and d2= 2.0 in. for the steel sleeve. The shear moduli of elasticity are Gb= 5400 ksi for the brass and G1 = 12,000 ksi for the steel (a) Assuming that the allowable shear stresses in the brass and steel are th= 4500 psi and ts= 7500 psi, respectively, determine the maximum permissible torque 7*mM that may be applied to the shaft. (b) If the applied torque T = 15 kip-in., find the required diameter d2so that allowable shear stress rsis reached in the steel.A steel shaft (Gs= 80 GPa) of total length L = 3.0 m is encased for one-third of its length by a brass sleeve (Gb= 40 GPa) that is securely bonded to the steel (see figure). The outer diameters of the shaft and sleeve are d1= 70 mm and d2= 90 mm, respectively. Determine the allowable torque 71 that may be applied to the ends of the shaft if the angle of twist between the ends is limited to 8,0e. Determine the allowable torque T2if the shear stress in the brass is limited to rh= 70 MPa. Determine the allowable torque 7. if the shear stress in the steel is limited to ts= 110 MPa. What is the maximum allowable torque Tmaxif all three of the preceding conditions must be satisfied?A uniformly tapered aluminum-ally tube AB of circular cross section and length L is fixed against rotation at A and B, as shown in the figure. The outside diameters at the ends are dAand dA.A hollow section of lenth L/2 and constant thickness t = dA/10 is cast into the tube and extends from B half-way toward A. Torque T0is applied at L/2. (a) Find the reactive torques at the supports, TA and TB. Use numerical values as follows: dA = 2.5 in., L = 48., G = 309 × 106 psi, and T0= 40,000 in.-lb. (b) Repeat part (a) if the hollow sections has constant diameter dA.Two pipes {L, = 2.5 m and L, = 1.5 m) are joined al B by flange plales (thickness (, = 14 mm) with five bolts [dlt, = 13 mm] arranged in a circular pal tor n (see figure). Also, each pipe segment is atlaehed to a wall (at .1 and ( '. see figure! using a base plate Uh = 15 mm) and four bolts (dM, = 16 mm). All bolts are tightened until just snug. Assume £, = 110 GPa,E2 = 73 GPa,», = 0.33,andv, = 0.25. Neglect the self-weight of the pipes, and assume the pipes are in a stress-free stale initially. The cross-sectional areas of the pipes are At = 1500 mm: and A2 = (3/5)4. The ollter diameter of Pipe 1 is 60 mm. The outer diameter of Pipe 2 is equal to the inner diameter of Pipe 1. The bolt radius r = 64 mm for both base and flange plates. (a) If torque '/'is applied at .v = Lt. find an expression for reactive torques Iit and IL in terms of T. (b) Find the maximum load variable /'(i.e., Tmal) if allowable torsional stress in the two pipes is Tall0* = 65 MPa-id Draw torsional moment iTMD i and torsional displacement (TDD) diagrams. Label all key ordinales. What is '/>.ll('.' (d) Find mail, if allowable shear and bearing stresses in the base plate and flange bolts cannot be exceeded. Assume allowable stresses in shear an.: :vari:'.g I all bolls are r |Nill, = 45 MPa andtr MaK =90 MPa. (e) Remove torque Tat x — L,. Now assume the flange-plate bolt holes are misaligned by some angle ß (see figure). Find the expressions for reactive torques Rx and R2 if the pipes are twisted to align the flange-plate bolt holes, bolts are then inserted, and the pipes released. (f) What is the maximum permissible misalignment angle ß mix if allowable stresses in shear and bearing for all bolts [from part (d)] are not to be exceeded?A solid circular bar of steel (G = 1L4 × 106 psi) with length L = 30 in, and diameter d = 1.75 in, is subjected to pure torsion by torques T acting at the ends (see figure). Calculate the amount of strain energy V stored in the bar when the maximum shear stress is 4500 psi. From the strain energy, calculate the angle of twist 0 (in degrees).A solid circular bar of copper (G = 45 GPa) with length L = 315n m and diameter d = 40 mm is subjected to pure torsion by torques T acting at the ends (see figure). Calculate the amount of strain energy U stored in the bar when the maximum shear stress is 32 MPa. From the strain energy, calculate the angle of twist (in degrees).A stepped shaft of solid circular cross sections (see figure) has length L = 45 in., diameter d2=1.2 in., and diameter d1= 1.0 in. The material is brass with G = 5.6 × 106 psi. Determine the strain energy U of the shaft if the angle of twist is 3.0°.A stepped shaft of solid circular cross sections (see figure) has length L = 0.80 m, diameter d2= 40 mm, and diameter d2= 30 mm. The material is steel with G = 80 GPa. Determine the strain energy U of the shaft if the angle of twist is 1.0º.A circular tube AB is fixed at one end and free at the other. The tube is subjected to concentrated torques as shown in the figure. If the outer radius of the tube is 1.5 in, and the thickness is 3/4 in., calculate the strain energy stored in the tube. Let G= 11,800 ksi.A cantilever bar of circular cross section and length L is fixed at one end and free at the other (see figure). The bar is loaded by a torque Tat the free end and by a distributed torque of constant intensity r per unit distance along the length of the bar. What is the strain energy U1 of the bar when the load Tracts alone? What is the strain energy U2when the load r acts alone? What is the strain energy b when both loads act simultaneously?Obtain a formula for the strain energy U of the statically indeterminate circular bar shown in the figure. The bar has fixed supports at ends A and B and is loaded by torques 2T0and T0at points C and D, respectively. Hint: Use Eqs. (3-50a and b) of Example 3-9 to obtain the reactive torques.A statically indeterminate stepped shaft ACE is fixed at ends A and B and loaded by a torque TQat point C (see figure). The two segments of the bar are made of the same material, have lengths L4and LB, and have polar moments of inertia IAand Ipb. Determine the angle of rotation 4>of the cross section at Cby using strain energy. Hint: Use Eq, (3-55b) to determine the strain energy Urn terms of the angle d?. Then equate the strain energy to the work done by the torque to. Compare your result with Eq. (3-52) of Example 3-9.Derive a formula for the strain energy U of the cantilever bar shown in the figure. The bar has circular cross sections and length L. It is subjected to a distributed torque of intensity t per unit distance. The intensity varies linearly from r = 0 at the free end to a maximum value t = /0 at the support.A thin-walled hollow tube AB of conical shape has constant thickness I and average diameters dAand dBat the ends (see figure). Determine the strain energy U oT the tube when it is subjected to pure torsion by torques T, Determine the angle of twist of the tube. Note: Use the approximate formula / ird^tlA for a thin circular ring; see Case 22 of Appendix E.A hollow circular tube A fits over the end of a solid circular bar B, as shown in the figure. The far ends of both bars are fixed. Initially, a hole through bar B makes an angle ß with a line through two holes in tube A. Then bar B is twisted until the holes are aligned, and a pin is placed through the holes. When bar B is released and the system returns to equilibrium, what is the total strain energy U of the two bars? (Let lAand lBrepresent the polar moments of inertia of bars A and B, respectively. The length L and shear modulus of elasticity G are the same for both bars.)A heavy flywheel rotating at n revolutions per minute is rigidly attached to the end of a shaft of diameter d (see figure). If the bearing at A suddenly freezes, what will be the maximum angle of twist <£of the shaft? What is the corresponding maximum shear stress in the shaft? (Let L = length of the shaft, G = shear modulus of elasticity, and / = mass moment of inertia of the flywheel about the axis of the shaft. Also, disregard friction in the bearings at Sand Cand disregard the mass of the shaft.) Hint: Equate the kinetic energy of the rotating flywheel to the strain energy of the shaft.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 kip-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 approximate theory give conservate or nonconservative results?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 tminof the tube so that the maximum shear stress in the tube will not exceed the maximum shear stress in the solid bar.A thin-walled aluminum tube of rectangular cross section (sec fig me) has a centerline dimensions b = 6.0 in. and b = 4.0 in. The wall thickness t is constant and equal to 0.25 in. Determine the shear stress in the tube due to a torque T = 15 kip-in. Determine the angle of twist (in degrees) if the length L of the tube is 50 in. and the shear modulus G is 4.0 x 106 psi.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. Determine the shear stress in the tube due to a torque T = 1650 N · m. Determine the angle of twist (in degrees) if the length L of the tube is 1.2 m and the shear modulus G is 75 GPa.A square tube section has side dimension of 20 in. arid thickness of 0.5 in. If the section is used for a 10-ft-long beam subjected to 1250 kip-in, torque at both ends, calculate the maximum shear stress and the angle of twist between the ends. Use G = 11,600 ksi.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 U1in the tube to the strain energy U2in the solid bar if the maximum shear stresses are the same in both cases? (For the tube, use the approximate theory for thin-walled bars.)A thin-walled steel tube having an elliptical cross section with constant thickness t (see figure) is subjected to a torque T = 18 kip-in. Determine the shear stress 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. Note: See Appendix E, Case 16, for the properties of an ellipse.Calculate the shear stress 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 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 rand 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, Express the ratio 12terms of the non-dimensional ratio ß = r/t. Calculate the ratio of angles of twist for ß = 5, 10, and 20. What conclusion about the accuracy of the approximate theory do you draw from these results?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 1b-in. Calculate the minimum required wall thickness Tminif the allowable shear stress is 4500 psi and the allowable rate of twist is 0.01 rad/ft.A thin tubular shaft with a circular cross section (see figure) and 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 theory for a circular bar.A thin-walled rectangular tube has uniform thickness t and dimensions a x 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 Lmof the median line of the cross section and the torque T remain constant? From your results, show that the shear stress is smallest when the tube is square (ß = 1).A long, thin-walled tapered tube AB with a 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 dAand dB, respectively. Derive the following formula for the angle of twist of the tube: Hint: If the angle of taper is small, you may obtain approximate results by applying the formulas for a thin-walled prismatic tube to a differential element of the tapered tube and then integrating along the axis of the tube.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 Tmax?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 Rminthat 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 at the stress concentration for values as follows: D1= 0.7,0.8, and 0.9 in. Plot a graph showing versus D?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 will this stress be reached? Is this diameter an upper or a lower limit on the value of D2?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 1b-in. What is the smallest permissible diameter D1?Calculate the shear force V and bending moment Mata cross section just to the right of the 800 lb load acting on the simple beam AB shown in the figure.Determine the shear force V and bending moment M just right of the 6 kN load on the simple beam AB shown in the figure.Determine the shear force V and bending moment M at the midpoint of the beam with overhangs (see figure). Note that one load acts downward and the other upward, and clockwise moments Pb are applied at each support. Repeat if moments Ph are moved to the ends of the beam (Fig, b).Calculate the shear force V and bending moment M at a cross section located just right of the 4 kN load on the cantilever beam AB shown in the figure.Consider the beam with an overhang shown in the figure. Determine the shear force V and bending moment M at a cross section located 18 ft from the left-hand end A. Find the required magnitude of load intensity q acting on the right half of member SC that will result in a zero shear force on the cross section IS ft from A.The beam ABC shown in the figure is simply supported at A and B and has an overhang from B to C. The loads consist of a horizontal force P1= 4,0 kN acting at the end of a vertical arm and a vertical force P2= 8.0 kN acting at the end of the overhang, Determine the shear force Fand bending moment M at a cross section located 3,0 m from the left-hand support. Note: Disregard the widths of the beam and vertical arm and use centerline dimensions when making calculations, Find the value of load A that results in V = 0 at a cross section located 2.0 m from the left-hand support. If P2= 8.0 kN, find the value of load P1that results in M = 0 at a cross section located 2,0 m from the left-hand support.The beam ABCD shown in the figure has overhangs at each end and carries a uniform load of intensity q (Fig. a). For what ratio b/L will the bending moment at the midpoint of the beam be zero? Repeat for a triangular load with peak intensity q0at L/2 (Fig. b).At a full d raw, an archer applies a pull of 130 N to the bowstring of the bow shown in the figure. Determine the bending moment at the midpoint of the bow.A curved bar ABC is subjected to loads in the form of two equal and opposite forces P, as shown in the figure. The axis of the bar forms a semicircle of radius r. Determine the axial force N, shear force K and bending moment M acting at a cross section defined by the angle Under cruising conditions, the distributed load acting on the wing of a small airplane has the idealized variation shown in the figure. Calculate the shear force V and bending moment M at 4 m from the tip of the wing.A beam ABCD with a vertical arm CE is supported as a simple beam al A and D (see figure part a). A cable passes over a small pulley that is attached to the arm at E. One end of the cable is attached to the beam at point B. (a) What is the force P in the cable if the bending moment in the beam just lo the left of point C is equal numerically to 640 lb-ft? Note: Disregard the widths of the beam and vertical arm and use centerline dimensions when making calculations. (b) Repeat part (a) if a roller support is added at C and a shear release is inserted just left of C (see figure part b).A simply supported beam AB supports a trapezoid ally distributed load (see figure). The intensity of the load varies linearly from 50 kN/m at support A to 25 kN/m at support B, Calculate the shear force V and bending moment M at the midpoint of the beam.Beam ABCD represents a reinforced-concrete foundation beam that supports a uniform load of intensity q1= 3500 lb/ft (see figure). Assume that the soil pressure on the underside of the beam is uniformly distributed with intensity q2 Find the shear force VBand bending moment MBat point B. Find the shear force Vmand bending moment M at the midpoint of the beam.Find shear (V) and moment (M) at x = 3L/4 for the beam shown in Fig. a. Let MA= 24 kN m,P = 48 kN, L = 6 m, and q0= 8 kN/m. Repeat for the beam in Fig, b (first solve for the reaction moment at fixed support A).Find expressions for shear force V and moment M at mid-span of beam AB in terms of peak load intensity q0and beam length variables a and L Let a = 5L/b.Find expressions for shear force V and moment Mat x = 2L/3 of beam (a) in terms of peak load intensity q0and beam length variable L. Repeat for beam (b) but at x = L/2.Find expressions for shear force V and moment Mat x = 2L/3 of beam (a) in terms of peak load intensity q0 and beam length variable L. Repeat for beam (b).Find expressions for shear force V and moment M at x = x0of beam AB in terms of peak load intensity q0and beam length variable L. Let x0= L/2.Find expressions for shear force V and moment M at x = x0of beam AB in terms of peak load intensity q0and beam length variable L. Let x0= 2L/3.Find expressions for shear force V and moment M at x = L/2 of beam BC. Express V and M in term s of peak load intensity q0and be a m length variable L.A cable with force P is attached to a frame at A and runs over a frictionless pulley at D. Find expressions for shear force V and moment M at x = LI2 of beam BC.Find expressions for shear force V and moment M at v = L/2 of beam AB in structure (a). Express V and M in terms of peak load intensity q0and beam length variable L. Repeat for structure (b) but find Fand M at m id-span of member BC.A cable with force P is attached to a frame at D and runs over a frictionless pulley at A Find expressions for shear force V and moment M at x = L/3 of beam AB.Frame ABCD carries two concentrated loads (2P at T and P at ZX see figure) and also a linearly varying distributed load on AB, Find expressions for shear force Fand moment A/at x = L/3 of beam AB in terms of peak load intensity q0, force P, and beam length variable L. Let q0= P/L.Frame ABC has a moment release just left of joint B. Find axial force N, shear force V, and moment M at the top of column AB. Write variables N, V, and M in terms of variables P and L.The simply supported beam ABCD is loaded by a weight W = 27 kN through the arrangement shown in the figure part a. The cable passes over a small frictionless pulley at B and is attached at E to the end of the vertical arm. Calculate the axial force JV, shear force V, and bending moment M at section C, which is just to the left of the vertical arm. Note: Disregard the widths of the beam and vertical arm and use centerline dimensions when making calculations. Repeat part (a) if a roller support is added at C and a moment release is inserted just left of C (see figure part b).The centrifuge shown in the figure rotates in a horizontal plane (the x-y plane) on a smooth surface about the z axis (which is vertical) with an angular acceleration a. Each of the two arms has a weight w per unit length and supports a weight W = 2B/L at its end. Derive formulas for the maximum shear force and maximum bending moment in the arms, assuming b = L/9 and c = L/10.Draw the shear-Force and bending-moment diagrams for a simple beam AB supporting two equal concentrated loads P (see figure). Repeat if the left-hand load is upward and the right-hand load is downward.A simple beam AB is subjected to a counter clockwise couple of moment M1acting at distance a from the left-hand support (see figure). Draw the shear-force and bending-moment diagrams for this beam. Also draw the shear-force and bending-moment diagrams if a second moment M0 is added at distance a from support B Draw the shear-force and bending-moment diagrams for a cantilever beam AB carrying a uniform load of intensity q over one-half of its length (see figure).The cantilever beam AB shown in the figure is subjected to a concentrated load P at the midpoint and a counterclockwise couple of moment M1= PL/A at the free end. Draw the shear-force and bending-moment diagrams for this beam.Cantilever beam AB carries an upward uniform load of intensity q1from x = 0 to L/2 (see Fig. a) and a downward uniform load of intensity q from x = L/2 to L. Find q1in terms of q if the resulting moment at A is zero. Draw V and M diagrams for the case of both q and qtas applied loadings. Repeat part (a) for the case of an upward triangularly distributed load with peak intensity q0(see Fig. b). For part (b), find q0, instead of q1 The simple beam AB shown in the figure is subjected to a concentrated load P and a clockwise couple M1= PL/3 acting at the third points. Draw the shear-force and bending-moment diagrams for this beam.A simple beam AB subjected to couples M1and 3M2 acting at the third points is shown in the figure. Draw the shear-force and bending-moment diagrams for this beam.A simply supported beam ABC is loaded by a vertical load P acting at the end of a bracket BDE (see figure). Draw the shear-force and bending-moment diagrams for beam abc. Now assume that load Pat £ is directed to the right. The vertical dimension BD is Li 5. Draw axial-force, shear-force, and bending-moment diagrams for ABC.A simply supported beam ABC is loaded at the end of a bracket BDE (see figure). Draw axial-force, shear-force, and bending-moment diagrams for ABC.A beam ABC is simply supported at A and B and has an overhang BC (see figure). The beam is loaded by two forces P and a clockwise couple of moment Pa at D that act through the arrangement shown. Draw the s hear-force and bending-mo ment diagrams for beam ABC. If moment Pa at D is replaced by moment M, find an expression for M in terms of variables P and a so that the reaction at B goes to zero. Plot the associated shear-force and bending-moment diagrams for beam ABC.Beam ABCD is simply supported at B and C and has overhangs at each end (see Fig, a). The span length is L and each overhang has length L/3. A uniform load of intensity q acts along the entire length of the beam. Draw the shear-force and bending-moment diagrams for this beam. Repeat part (a) if the uniform load is replaced with a triangularly distributed load with peak intensity q0= q at mid-span (see Fig. b).Draw the shear-force and bending-moment diagrams for a cantilever beam AB acted upon by two different load cases. A distributed load with linear variation and maximum intensity q0(see figure part a). A distributed load with parabolic variation and maximum intensity q0(see figure part b).The simple beam AB supports a triangular load of maximum intensity q0= 10 lb/in. acting over one-half of the span and a concentrated load P = 80 lb acting at m id span (see figure). Draw the shear-force and bending-moment diagrams for this beam.The beam AB shown in the figure supports a uniform load of intensity 3000 N/m acting over half the length of the beam. The beam rests on a foundation that produces a uniformly distributed load over the entire length. Draw the shear-force and bending-moment diagrams for this beam. Repeat part (a) for the distributed load variation shown in Fig. b.A cantilever beam AB supports a couple and a concentrated load, as shown in the figure. Draw the shear-force and bending-moment diagrams for this beam.The cantilever beam A B shown in the figure is subjected to a triangular load acting over one-half of its length and a concentrated load acting at the free end. Draw the shear-force and bending-moment diagrams for this beam Beam ABC has simple supports at .A and B. an overhang BC and the distributed loading shown in the figure. Draw the shear-force and bending-moment diagrams for this beam.Beam ABC with an overhang at one end supports a partial uniform load of intensity 12 kN/m and a concentrated moment of magnitude 4 kN · m at C (see figure). Draw the shear-force and bending-moment diagrams for this beam.Consider the two beams shown in the figures. Which beam has the larger maximum moment? First, find support reactions: then plot axial force (N), shear (V), and moment (M) diagrams for both beams. Label all critical N, V, and M values and also the distance to points where N, V. and/or M are zero.The three beams in the figure have the same loading. However, one has a moment release just to the left of C, the second has a shear release just to the right of C, and the third has an axial release just to the left of C. Which beam has the largest maximum moment? First, find support reactions; then plot axial force (N), shear ( V) and moment (M) diagrams for all three beams. Label all critical N, K, and M values and also the distance to points where N, K and loi M are zero.The beam ABC shown in the figure is simply supported at A and B and has an overhang from B to C Draw the shear-force and bending-moment diagrams for beam ABC. Note: Disregard the widths of the beam and vertical arm and use centerline dimensions when making calculations.A simple beam AB is loaded by two segments of uniform load and two horizontal and vertical forces acting at the ends of a vertical arm (see figure). Draw the shear-force and bending-moment diagrams for this beam.Two beams (see figure) are loaded the same and have the same support conditions. However, the location of internal axial, shear, and moment releases is different for each beam (see figures). Which beam has the larger maximum moment?The beam A BCD shown in the figure has overhangs that extend in both directions for a distance of 4.2 m from the supports at B and C which are 1.2 m apart. Draw the shear-force and bending-moment diagrams for this overhanging beam.A beam ABCD with a vertical arm CE is supported as a simple beam at .1 and D (see figure). A cable passes over a small pulley that is attached to the arm at E. One end of the cable is attached to the beam at point B. The tensile force in the cable is 1800 lb. Draw the shear-Force and bending-moment diagrams for beam A BCD. Note: Disregard the widths of the beam and vertical arm and use centerline dimensions when making calculations. Repeat part (a) if a roller support is added at C and a shear release is inserted just left of C (see figure part b).Beams ABC and CD are supported at A,C, and D and are joined by a hinge (or moment release) just to the left of C. The support at A is a sliding support (hence reaction A = 0 for the loading shown below). Find all support reactions; then plot shear (F) and moment (M) diagrams. Label all critical Fand M values and also the distance to points where either F and/or M are zero.The simple beam ACE shown in the figure is subjected to a triangular load of maximum intensity q0= 200 lb/ft at a = 8 ft and a concentrated moment M = 400 Ib-ft at A. Draw the shear-force and bending-moment diagrams for this beam, Find the value of distanced that results in the maximum moment occurring at L/2. Draw the shear-force and bending-moment diagrams for this case. Find the value of distance a for which Mmaxis the largest possible value.A beam with simple supports is subjected to a trapezoidally distributed load (see figure). The intensity of the load varies from 1.0 kN/m at support A to 2.5 kN/m at support B. Draw the shear-force and bending-moment diagrams for this beam. Assume that Mfl at B is zero. Find the required moment MQat B so that the maximum moment in the beam does not exceed 1.0 kN · m.A beam of length L is designed to support a uniform load of intensity q (see figure). If the supports of the beam are placed at the ends, creating a simple beam, the maximum bending moment in the beam is qL2/8. However, if the supports of the beam are moved symmetrically toward the middle of the beam (as shown), the maximum bending moment is reduced. Determine the distance a between the supports so that the maximum bending moment in the beam has the smallest possible numerical value. Draw the shear-force and bending-moment diagrams for this condition. Repeat part (a) if the uniform load is replaced with a triangularly distributed load with peak intensity q0= q at mid-span (see Fig. b).The compound beam ABCDE shown in the figure consists of two beams (AD and DE) joined by a hinged connection (or moment release) at D, The moment release can transmit a shear force but not a bending moment. Draw the shear-force and bending-moment diagrams for this compound beam.Draw the shear-force and bending-moment diagrams for beam AB with a sliding support at A and an elastic support with spring constant K at B acted upon by two different load cases: A distributed load with linear variation and maximum intensity q0(see figure part a). A distributed load with parabolic variation with maximum intensity q0(see figure part b).The shear-force diagram for a simple beam is shown in the figure. Determine the loading on the beam and draw the bending-moment diagram, assuming that no couples act as loads on the beam.The shear-force diagram for a beam is shown in the figure. Assuming that no couples act as loads on the beam, determine the forces acting on the beam and draw the bending-moment diagram.A compound beam (see figure) has an internal moment release just to the left of B and a shear release just to the right of C Reactions have been computed at A, C, and D and are shown in the figure. First, confirm the reaction expressions using statics; then plot shear (V) and moment (W) diagrams. Label all critical Fand M values and also the distance to points where either V and/or M are zero.A compound beam (see figure) has an shear release just to the left of C and a moment release just to the right of C. A plot of the moment diagram is provided below the beam for applied load P at B and triangular distributed loads v(x) on segments Z/C and CD. First, solve for reactions using statics; then plot axial force (A) and shear force (K) diagrams. Confirm that the moment diagram is that shown below. Label all critical N, V, and M values and also the distance to points where N, V, and/or M are zero.A simple beam AB supports two connected wheel loads 3P and 2P that are a distance d apart (see figure). The wheels may be placed at any distance x from the left-hand support of the beam. (Assume P = 12 kN, d = 2 m, and 1 = 15 m.) (a) Determine the distance .y that will produce the maximum shear force in the beam, and also determine the maximum shear force Vmax. (b) Determine the distance v that will produce the maximum bending moment in the beam, and also draw the corresponding bending-moment diagram.The inclined beam represents a ladder with the Following applied loads: the weight (W) of the house painter and the distributed weight (u) of the ladder itself. Find support reactions at A and B: then plot axial force (N), shear (V), and moment (M) diagrams. Label all critical N, V, and M values and also the distance to points where any critical ordmates are zero. Plot N, V, and M diagrams normal to the inclined ladder. Repeat part (a) for the case of the ladder suspended from a pin at B and traveling on a roller support perpendicular to the floor at A.Beam ABC is supported by a tie rod CD as shown. Two configurations are possible: pin support at A and downward triangular load on AB or pin at B and upward load on AB. Which has the larger maximum moment? First, find all support reactions; then plot axial force (N), shear (V), and moment (M) diagrams for ABC only and label all critical N, V, and M values. Label the distance to points where any critical ordinates are zero.A plane frame (see figure) consists of column AB and beam BC that carries a triangular distributed load (see figure part a). Support A is fixed, and there is a roller support at C. Beam BC has a shear release just right of joint B. Find the support reactions at A and C then plot axial-force (N), shear-force (V), and bending-moment (M) diagrams for both members. Label all critical N,K and M values and also the distance to points where any critical ordinates are zero. Repeat part (a) if a parabolic lateral load acting to the right is now added on column AB (figure part b).The plane frame shown in the figure is part of an elevated freeway system. Supports at A and D arc fixed, but there are moment releases at the base of both columns (AB and DE) as well as in column BC and at the end of beam BE. Find all support reactions; then plot axial-force (N), shear (F), and moment (M) diagrams for all beam and column members. Label all critical N, V, and M values and also the distance to points where any critical ordinatcs are zero.A steel wire with a diameter of d = 1/16 in. is bent around a cylindrical drum with a radius of R = 36 in. (see figure). Determine the maximum normal strain emax What is the minimum acceptable radius of the drum if the maximum normal strain must remain below yield? Assume E = 30,000 ksi and ey= 100 ksi. If R = 36 in., what is the maximum acceptable diameter of the wire if the maximum normal strain must remain below yield?A copper wire having a diameter ofd = 4 mm is bent into a circle and held with the ends just touching (see figure), If the maximum permissible strain in the copper is = 0.0024, what is the shortest length L of wire that can be used? If L = 5.5 m, what is the maximum acceptable diameter of the wire if the maximum normal strain must remain below yield? Assume E = 120 GPa and(7K= 300 MPa.A 4.75-in, outside diameter polyethylene pipe designed to carry chemical waste is placed in a trench and bent around a quarter-circular 90° bend (see figure). The bent section of the pipe is 52 ft long. Determine the maximum compressive strain If the normal strain cannot exceed 6.1 × 10-3 , what is the maximum diameter of the pipe? If d = 4.75 in., what is the minimum acceptable length of the bent section of the pipe?A cantilever beam AB is loaded by a couple M0at its free end (see figure). The length of the beam is L = 2.0 m, and the longitudinal normal strain at the top surface is E = 0.0010. The distance from the top surface of the beam to the neutral surface is c = 85 mm. Calculate the radius of curvature p, the curvature and the vertical deflection S at the end of the beam. If allowable strain £:i = 0.0008, what is the maximum acceptable depth of the beam? [Assume that the curvature is unchanged from part(a)]. If allowable strain Ea= 0.0008, c = 85 mm, and L = 4 m, what is deflection S?A thin strip of steel with a length of L =19 in. and thickness of t = 0,275 in. is bent by couples M0(see figure). The deflection at the midpoint of the strip (measured from a line joining its end points) is found to be 0.30 in. Determine the longitudinal normal strain ë at the top surface of the strip. If allowable strain £a= 0-0008, what is the maximum acceptable thickness of the strip? If allowable strain £a= 0.0008, t = 0.275 in., and L = 32 in., what is deflection S? If allowable strain sa= 0.0008, t = 0.275 in., and the deflection cannot exceed 1,0 in., what is the maximum permissible length of the strip?A bar of rectangular cross section is loaded and supported as shown in the figure. The distance between supports is L = 1.75 m, and the height of the bar is h = 140 mm. The deflection at the midpoint is measured as 2.5 mm. (a) What is the maximum normal strain £ at the top and bottom of the bar? (b) If allowable strain ea= 0.0006 and the deflection cannot exceed 4.3 mm, what is the maximum permissible length of the bar?A simply supported beam with a length L = 10 ft and height 7 in. is bent by couples Müinto a circular arc with downward deflection S at the midpoint. If the curvature of the beam is 0,003 ft-1, calculate the deflection, 5, at the mid-span of the beam and the longitudinal strain at the bottom fiber given that the distance between the neutral surface and the bottom surface is 3.5 in.A cantilever beam is subjected to a concentrated moment at B, The length of the beam L = 3 m and the height h = 600 mm. The longitudinal strain at the top of the beam is 0,0005 and the distance from the neutral surface to the bottom surface of the Iva m is 300 nun. Find the radius of curvature, the curvature, and the deflection of the beam at B.A thin strip of hard copper (E = 16,000 ksi) having length L = 90 in, and thickness; E= 3/32 in. is bent into a circle and held with the ends just touching (sec figure). Calculate the maximum bending stress emaxin the strip, By what percent docs the stress increase or decrease if the thickness of the strip is increased by 1/32 in,? (c) Find the new length of the strip so that the stress in part (b)(t = 1/8 in. and L = 90 in.) is equal to that in part (a) (t = 3/32 in. and L = 90 in.).A steel wire (E = 200 GPa) of a diameter d = L25 mm is bent around a pulley of a radius i/o = 500 mm (see figure), What is the maximum stress true in the wire? By what percent docs the stress increase or decrease if the radius of the pulley is increased by 25%? By what percent docs the stress increase or decrease if the diameter of the wire is increased by 25% while the pulley radius remains at Rq = 500 mm?A thin, high-strength steel rule (E = 30 x 10ft psi) having a thickness t = 0.175 in. and length L = 48 in. is bent by couples Mcinto a circular arc subtending a central angle a = 40° (sec figure), What is the maximum bending stress emax. in the rule? By what percent docs the stress increase or decrease if the central angle is increased by 10%? What percent increase or decrease in rule thickness will result in the maximum stress reaching the allowable value of 42 ksi?A simply supported wood beam AB with a span length L = 4 m carries a uniform load of intensity 4 = 5.8 kN/m (see figure). Calculate the maximum bending stress rmax„ due to the load q if the beam has a rectangular cross section with width h = 140 mm and height h = 240 mm. Repeat part (a) but use the trapezoidal distributed load shown in the figure part b.Beam ABC has simple supports at A and B and an overhang from B to C. The beam is constructed from a steel W 16 × 3L The beam must carry its own weight in addition to uniform load q = 150 lb/ft. Determine the maxima m tensile and compressive stresses in the beam.A simply supported beam is subjected to a in early varying distributed load q(x)=xL with maximum intensity q0at B. The beam has a length L = 4 m and rectangular cross section with a width of 200 mm and height of 300 mm. Determine the maximum permissible value for the maximum intensity, q0, if the allowable normal stresses in tension and compression are 120 M Pa.Each girder of the lift bridge (sec figure) is 180 ft long and simply supported at the ends. The design load for each girder is a uniform load of intensity 1,6 kips/ft. The girders are fabricated by welding three steel plates to form an I-shaped cross section (see figure) having section modulus S = 3600 in3. What is the maximum bending stress rmaxin a girder due to the uniform load?A freight-car axle AS is loaded approximately as shown in the figure, with the forces P representing the car loads (transmitted to the axle through the axle boxes) and the forces R representing the rail loads (transmitted to the axle through the wheels). The diameter of the axle is d = 82 mm, the distance between centers of the rails is Z., and the distance between the forces P and R is A = 220 mm. Calculate the maximum bending stress vmaxin the axle if P = 50 kN.A seesaw weighing 3 lb/ft of length is occupied by two children, each weighing 90 lb (see figure). The center of gravity of each child is 8 ft from the fulcrum. The board is 19 ft long, 8 in. wide, and 1.5 in. thick. What is the maximum bending stress in the board?During construction of a highway bridge, the main girders are cantilevered outward from one pier toward the next (see figure). Each girder has a cantilever length of 48 m and an I-shaped cross section with dimensions shown in the figure. The load on each girder (during construction) is assumed to be 9,5 kN/m, which includes the weight of the girder. Determine the maximum bending stress in a girder due to this load.The horizontal beam ABC of an oil-well pump has the cross section shown in the figure. If the vertical pumping force acting at end C is 9 kips and if the distance from the line of action ofthat force to point B is 16 ft, what is the maximum bending stress in the beam due to the pumping force?A railroad tie (or sleeper) is subjected to two rail loads, each of magnitude P = 175 kN, acting as shown in the figure. The reaction q of the ballast is assumed to be uniformly distributed over the length of the tie, which has cross-sectional dimensions b = 300 mm and h = 250 mm. Calculate the maximum bending stress mmaxin the tie due to the loads P, assuming the distance L = 1500 mm and the overhang length a = 500 mm.A fiberglass pipe is lifted by a sling, as shown in the figure. The outer diameter of the pipe is 6,0 in., its thickness is 0.25 in,, and its weight density is 0,053 lb/in3 The length of the pipe is L = 36 ft and the distance between lifting points is s = 11 ft. Determine the maximum bending stress in the pipe due to its own weight, Find the spacing s between lift points which minimizes the bending stress. What is the minimum bending stress? What spacing s leads to maximum bending stress? What is that stress?A small dam of height h = 2.0 m is constructed of vertical wood beams AB of thickness t = 120 mm, as shown in the figure. Consider the beams to be simply supported at the top and bottom. Determine the maximum bending stress emaxin the beams, assuming that the weight density of water is y = 9.81kN/m3.Determine the maximum tensile stress (7, (due to pure bending about a horizontal axis through C by positive bending moments (B) for beams having cross sections as follows (see figure). A semicircle of diameter d. An isosceles trapezoid with bases (b) = b and f2 = 4W3 and altitude h. A circular sector with a = ît/3 and r = d/2.Determine the maximum bending stress emaxdue to pure bending by a moment (M) for a beam having a cross section in the form of a circular core (see figure). The circle has diameter d and the angle ß = 60°. Hint: Use the formulas given in Appendix E, Cases 9 and 15.A simple beam A B of a span length L = 24 ft is subjected to two wheel loads acting at a distance d = 5 ft apart (see figure). Each wheel transmits a load P = 3.0 kips, and the carriage may occupy any position on the beam. Determine the maximum bending stress Gmaxdue to the wheel loads if the beam is an I-beam having section modulus S = 16.2 in3. If d = 5 ft. Find the required span length L to reduce the maximum stress in part (a) to 18 ksi. If L = 24 ft, Find the required wheel spacing s to reduce the maximum stress in part (a) to 18 ksi.Determine the maximum tensile stress erand maximum compressive stress ecdue to the load P acting on the simple beam, 40 (see figure). (a) Data are P = 6.2 kN, L = 3.2 m, d = L25 m, b = SO mm, t = 25 mm, h = 120 mm, and A, = 90 mm. (b) Find the value of d for which tensile and compressive stresses arc the largest. What are these stresses?A cantilever beam A3, loaded by a uniform load and a concentrated load (sec figure), is constructed of a channel section. (a) Find the maximum tensile stresser, and maxi-mum compressive stress trcif the cross section has the dimensions indicated and the moment of inertia about the - axis (the neutral axis) is t = 3.36 in4. Note: The uniform load represents the weight of the beam. Find the maximum value of the concentrated load if the maximum tensile stress cannot exceed 4 ksi and the maximum compressive stress is limited to 14.5 ksi. How far from A can load P = 250 lb be positioned if the maximum tensile stress cannot exceed 4 ksi and the maximum compressive stress is limited to 14.5 ksi?A canti lever beam A B of a n isosceles t rapezoi-dal cross section has a length L = 0.8 m, dimensions bx= 80 mm and b2= 90 mm, and height h = 110 mm (see figure). The beam is made of brass weighing 85 kN/m3. Determine the maximum tensile stress asand maximum compressive stress A cantilever beam, a C12 x 30 section, is subjected to its own weight and a point load at B. Find the maximum permissible value of load f at B (kips) if the allowable stress in tension and compression is o\. =18 ksi.A frame ABC travels horizontally with an acceleration a0(see figure). Obtain a formula for the maximum stress emax in the vertical arm AB, which has length thickness t, and mass density p.A beam ABC with an overhang from B to C supports a uniform load of 200 lb/ft throughout its length (sec figure). The beam is a channel section with dimensions as shown in the figure. The moment of inertia about the - axis (the neutral axis) equals 8.13 in4. Calculate the maximum tensile stress ofand maximum compressive stress trt due to the uniform load. Find the required span length a that results in the ratio of larger to smaller compressive stress being equal to the ratio of larger to smaller tensile stress for the beam. Assume that the total length L = a + h = 18 ft remains unchanged.A cantilever beam AB with a rectangular cross section has a longitudinal hole drilled throughout its length (see figure). The beam supports a load P = 600 N. The cross section is 25 mm wide and 50 mm high, and the hole has a diameter of 10 mm. Find the bending stresses at the top of the beam, at the top of the hole, and at the bottom of the beam.A beam with a T-section is supported and loaded as shown in the figure. The cross section has width b = 2 1/2 in., height c = 3 in., and thickness t = 3/8 in. Determine the maximum tensile and compressive stresses in the beam. If the allowable stresses in tension and compression are 18 ksi and 12 ksi, respectively, what is the required depth h of the beam? Assume that thickness t remains at 3/8 in. and that flange width/) = 2.5 in. Find the new values of loads P and q so that the allowable tension (18 ksi) and compression (12 ksi) stresses are reached simultaneously for the beam. Use the beam cross section in part (a) (see figure) and assume that Lh and L3are unchanged.Consider the compound beam with segments AB and BCD joined by a pin connection (moment release) just right of B (see figure part a). The beam cross section is a double-T made up from three 50 mm × 150 mm wood members (actual dimensions, see figure part b), (a) Find the cent raid C of the double-T cross section (c1:c2): then compute the moment of inertia, [I2 (mm4 )]. (b) Find the maximum tensile normal stress ifand maximum compressive normal stress tt. (kPa) for the loading shown. (Ignore the weight of the beam.)A small dam of a height h = 6 ft is constructed of vertical wood beams AB, as shown in the figure. The wood beams, which have a thickness I = 2.5 in., are simply supported by horizontal steel beams at A and Ä Construct a graph showing the maximum bending stress tram in the wood beams versus the depth d of the water above the lower support at B. Plot the stress0mas(psi) as the ordinate and the depth d(ft) as the abscissa. Note: The weight density y of water equals 62.4 lb/ft3.A foot bridge on a hiking trail is constructed using two timber logs each having a diameter d = 0.5 m (see figure a). The bridge is simply supported and has a length L = 4 m. The top of each log is trimmed to form the walking surface (see Fig, b)LA simplified model of the bridge is shown in Fig. g. Each log must carry its own weight w = 1.2 kN/m and the weight (P = 850 N) of a person at mid-span, (see Fig. b). Determine the maximum tensile and compressive stresses in the beam (Fig, b) due to bending. If load h is unchanged, find the maximum permissible value of load ... if the allowable normal stress in tension and compression is 2.5 M Pa.A steel post (E=30×106) having thickness t = 1/8 in. and height L = 72 in. support a stop sign (see figure), where s = 12.5 in. The height of the post L is measured from the base to the centroid of the sign. The stop sign is subjected to wind pressure p = 20 lb/ft2 normal to its surface. Assume that the post is fixed at its base. What is the resultant load on the sign? (Sec Appendix E, Case 25, for properties of an octagon, n =8.) What is the maximum bending stress in the post? Repeat part (b) if the circular cut-outs arc eliminated over the height of the post.Beam ABCDE has a moment release just right of joint B and has concentrated moment loads at D and E. In addition, a cable with tension P is attached at fand runs over a pulley at C (Fig, a). The beam is constructed using two steel plates, which arc welded to form a T cross section (see Fig. b). Consider ßexuralstresses only Find the maximum permissible value of load variable P if the allowable bending stress is 130 M Pa. Ignore the self-weight of the frame members and let length variable L = 0.75 m.A simply supported wood beam having a span length L = 12 ft is subjected to unsymmetrical point loads, as shown in the figure. Select a suitable size for the beam from the table in Appendix G. The allowable bending stress is 1800 psi and the wood weighs 35 Lb/ft3.A simply supported beam (L = 4.5 m) must support mechanical equipment represented as a distributed load with intensity q = 30 kN/m acting over the middle segment of the beam (see figure). Select the most economical W-shape steel beam from Table F-l(b) to support the loads. Consider both the distributed force q and the weight of the beam. Use an allowable bending stress of 140 MPa.The cross section of a narrow-gage railway bridge is shown in part a of the figure. The bridge is constructed with longitudinal steel grinders that support the wood cross ties. The bridge is constructed with longitudinal steel girders that support the wood cross ties. The girders are restrained against lateral buckling by diagonal bracing, as indicated by the dashed lines. The spacing of the girders is S1= 50 in. and the spacing of the rails is s2= 30 in. The load transmitted by each rail to a single tie is P = 1500 1b. The cross section of a tie, shown in part b of the figure, has a width b =5.0 in. and depth d. Determine the minimum value of d based upon an allowable bending stress of 1125 psi in the wood tie. (Disregard the weight of the tie itself.)A fiberglass bracket A BCD with a solid circular cross section has the shape and dimensions shown in the figure, A vertical load P = 40 N acts at the free end D. Determine the minimum permissible diameter ^nwi °f tnc bracket if the allowable bending stress in the material is 30 MPa and/? = 37 mm. Note: Disregard the weight of the bracket itself. If d = 10 nun, b = 37 mm, and= 30 MPa, what is the maximum value of load P if vertical load P at D is replaced with horizontal loads P at B and D (see figure part b)?A cantilever beanie B is loaded by a uniform load q and a concentrated load P, as shown in the figure. Select the most economical steel C shape from Table F-3(a) in Appendix F; use q = 20 lb/ft and P = 300 lb (assume allowable normal stress is cra= IS ksi). Select the most economical steel S shape from Table F-2(a) in Appendix F; use q = 45 lb/ft and P = 2000 lb (assume allowable normal stress is tra= 20 ksi), Select the most economical steel W shape from Table F-1(a) in Appendix F; use q = 45 lb/ft and P = 2000 lb (assume allowable normal stress is (T = 20 ksi). However, assume that the design requires that the W shape must be used in weak axis bending, i.e., it must bend about the 2-2 (or v) axis of the cross section. Note: For parts (a), (b), and (c), revise your initial beam selection as needed to include the distributed weight of the beam in addition to uniform load q.A simple beam of length L = 5 m carries a uniform load of intensity q = 5,8 kN/m and a concentrated load 22.5 kN (see figure). (a) Assuming tra]]ow = 110 MPa, calculate the required section modulus S. Then select the most economical wide-flange beam (W shape) from Table F-l(b) in Appendix F, and recalculate S, taking into account the weight of the beam. Select a new beam if necessary. (b) Repeat part (a), but now assume that the design requires that the W shape must be used in weak axis bending (i.e., it must bend about the 2-2 (or y) axis of the cross section).A simple beam AB is loaded as shown in the figure. Calculate the required section modulus S if ^aibw = IS,000 psi, L = 32 ft, P = 2900 lb, and g = 450 lb/ft. Then select a suitable I-beam (S shape) from Table F-2(a), Appendix F, and recalculate 5 taking into account the weight of the beam. Select a new beam size if necessary. What is the maximum load P that can be applied to your final beam selection in part (a)?A pontoon bridge (see figure) is constructed of two longitudinal wood beams, known as bulks, that span between adjacent pontoons and support the transverse floor beams, which arc called chesses. For purposes of design, assume that a uniform floor load of 7.5 kPa acts over the chesses. (This load includes an allowance for the weights of the chesses and balks.) Also, assume that the chesses are 2.5 m long and that the balks are simply supported with a span of 3.0 m. The allowable bending stress in the wood is 15 MPa. If the balks have a square cross section, what is their minimum required width b^l Repeat part (a) if the balk width is 1.5 b and the balk depth is b; compare the cross-sectional areas of the two designs.A floor system in a small building consists of wood planks supported by 2-in. (nominal width) joists spaced at distance s and measured from center to center (see figure). The span length L of each joist is 12 ft, the spacing s of the joists is 16 in., and the allowable bending stress in the wood is 1250 psi. The uniform floor load is 120 lb/ft", which includes an allowance for the weight of the floor system itself. Calculate the required section modulus S for the joists, and then select a suitable joist size (surfaced lumber) from Appendix G, assuming that each joist may be represented as a simple beam carrying a uniform load. What is the maximum floor load that can be applied to your final beam selection in part (a)?The wood joists supporting a plank Floor (see figure) are 38 mm × 220 mm in cross section (actual dimensions) and have a span length of L = 4.0 m. The floor load is 5.0 kPa, which includes the weight of the joists and the floor. (a) Calculate the maximum permissible spacing s of the joists if the allowable bending stress is 14 M Pa. (Assume that each joist may be represented as a simple beam carrying a uniform load.) (b) If spacing s = 406 mm, what is the required depth ft of the joist? Assume all other variables remain unchanged.A beam ABC with an overhang from B to C is constructed of a C 10 × 30 channel section with flanges facing upward (sec figure). The beam supports its own weight (30 lb/ft) plus a triangular load of maximum intensity g0 acting on the overhang. The allowable stresses in tension and compression arc IS ksi and 12 ksi, respectively. Determine the allowable triangular load intensity allow if tne distance L equals 4 ft. What is the allowable triangular load intensity fallow ^tnc Dcam is rotated 180e about its longitudinal centroidal axis so that the flanges are downward?-12 A "trapeze bar" in a hospital room provides a means for patients to exercise while in bed (see figure). The bar is 2.1 m long and has a cross section in the shape of a regular octagon. The design load is 1.2 kN applied at the midpoint of the bar. and the allowable bending stress is 200 M Pa. Determine the minimum height h of the bar. (Assume that the ends of the bar are simply supported and that the weight of the bar is negligible.)A two-axle carriage that is part of an over head traveling crane in a testing laboratory moves slowly across a simple beam AB (sec figure). The load transmitted to the beam from the front axle is 2200 lb and from the rear axle is 3800 lb. The weight of the beam itself may be disregarded. Determine the minimum required section modulus S for the beam if the allowable bending stress is 17,0 ksi, the length of the beam is 18 ft, and the wheelbase of the carriage is 5 ft. Select the most economical I-beam (S shape) from Table F-2(a), Appendix F.A cantilever beam AB with a circular cross section and length L = 750 mm supports a load P = 800 N acting at the free end (see figure). The beam is made of steel with an allowable bending stress of 120 MPa. Determine the required diameter dmm(figure part a) of the beam, considering the effect of the beam's own weight. Repeat part (a) if the beam is hollow with wall thickness t = df$ (figure part b); compare the cross-sectional areas of the two designs.A propped cantilever beam A BC (see figure) has a shear release just right of the mid-span. (a) Select the most economical wood beam from the table in Appendix G; assume4 = 55 lb/ft, L = 16 ft, o~aw= 1750 psi, and raw= 375 psi. Include the self-weight of the beam in your design. (b) If a C 10 x 25 steel beam is now used for beam ABC, what is the maximum permissible value of load variable q? Assume üi = 16 ksi and L = 10 ft. Include the self-weight of the beam in your analysis.A small balcony constructed of wood is supported by three identical cantilever beams (see figure). Each beam has length Lt = 2.1 m, width r>, and height h = Abi3. The dimensions of the balcony floor are L\ x Z.^ where L\ = 2.5 im The design load is 5.5 kPa acting over the entire floor area. (This load accounts for all loads except the weights of the cantilever beams, which have a weight density y = 5.5 kN/mJ.) The allowable bending stress in the cantilevers is 15 MPa. Assuming that the middle cantilever supports 50% of the load and each outer cantilever supports 25% of the load, determine the required dimensions b and ft.A beam having a cross section in the form of an un symmetric wide-flange shape (sec figure) is subjected to a negative bending moment acting about the 2 axis. Determine the width b of the top flange in order that the stresses at the top and bottom of the beam will be in the ratio 4:3, respectively.A beam having a cross section in the form of a channel (sec figure) is subjected to a bending moment acting about the z axis. Calculate the thickness t of the channel in order that the bending stresses at the top and bottom of the beam will be in the ratio 7:3, respectively.Determine the ratios of the weights of four beams that have the same length, are made of the same material, are subjected to the same maximum bending moment, and have the same maximum bending stress if their cross sections are (I) a rectangle with height equal to twice the width, (2) a square, (3) a circle, and (4) a pipe with outer diameter d and wall thickness f = di% (sec figures).5.6.20P A steel plate (called a cover ploie) having cross-sectional dimensions 6,0 in. × 0.5 in. is welded along the full length of the bottom flange of a W 12 × 50 wide-flange beam (sec figure, which shows the beam cross section). What is the percent increase in the smaller section modulus (as compared to the wide-flange beam alone)?A steel beam ABC is simply supported at A and fiand has an overhang BC of length L = 150 mm (see figure). The beam supports a uniform load of intensity q = 4,0 kN/m over its entire span AB and l.5g over BC. The cross section of the beam is rectangular with width h and height 2b. The allowable bending stress in the steel iso"a|]ûW = 60 MPa, and its weight density is y = 77.0 kN/m . Disregarding the weight of the beam, calculate the required width b of the rectangular cross section. Taking into account the weight of the beam, calculate the required width b.A retaining wall 6 ft high is constructed of horizontal wood planks 2.5 in. thick (actual dimension) that are supported by vertical wood piles of a 12 in, diameter (actual dimension), as shown in the figure. The lateral earth pressure is pt=125 lb/ft2 at the top of the wall and p2= 425 lb/ft2 at the bottom. Assuming that the allowable stress in the wood is 1175 psi, calculate the maximum permissible spacing s of the piles. Find the required diameter of the wood piles so that piles and planks (f = 2.5 in.) reach the allowable stress at the same time. Hint: Observe that the spacing of the piles may be governed by the load-carrying capacity of either the planks or the piles. Consider the piles to act as cantilever beams subjected to a trapezoidal distribution of load, and consider the planks to act as simple beams between the piles. To be on the safe side, assume that the pressure on the bottom plank is uniform and equal to the maximum pressure.A retaining wall (Fig. a) is constructed using steel W-shape columns and concrete panel infill (Fig, b). Each column is subjected to lateral soil pressure with peak intensity q0(Figs, b and c). The tensile and compressive strength of the beam is 600 MPa. Select the most economical W 360 section from Table F-l(b) based on safety factor of 3.0.A beam of square cross section (a = length of each side) is bent in the plane of a diagonal (see figure). By removing a small amount of material at the top and bottom corners, as shown by the shaded triangles in the figure, you can increase the section modulus and obtain a stronger beam, even though the area of the cross section is reduced. Determine the ratio ß defining the areas that should be removed in order to obtain the strongest cross section in bending. By what percent is the section modulus increased when the areas arc removed?The cross section of a rectangular beam having a width b and height h is shown in part a of the figure. For reasons unknown to the beam designer, it is planned to add structural projections of width b/9 and height d/9 the top and bottom of the beam (see part b of the figure). For what values of d is the bending-moment capacity of the beam increased? For what values is it decreased?A tapered cantilever beam A B of length L has square cross sections and supports a concentrated load P at the free end (sec figure part a). The width and height of the beam vary linearly from kAat the free end to hBat the fixed end. Determine the distance.y from the free end A to the cross section of maximum bending stress if hE= 3h4, What is the magnitude ffœai of the maximum bending stress? What is the ratio of the maximum stress to the largest stress B at the support? Repeat part (a) if load P is now applied as a uniform load of intensity q = P/L over the entire beam, A is restrained by a roller support, and B is a sliding support (see figure part b)..2 A ligmio.irc ii supported by two vorlical beams consistins: of thin-walled, tapered circular lubes (see ligure part at. for purposes of this analysis, each beam may be represented as a cantilever AB of length L = 8.0 m subjected to a lateral load P = 2.4 kN at the free end. The tubes have a constant thickness ; = 10.0 mm and average diameters dA = 90 mm and dB = 270 mm at ends A and B, re s pec lively. Because the thickness is small compared to the diameters, the moment of inerlia at any cross section may be obtained from the formula / = jrrf3;/8 (see Case 22, Appendix E); therefore, the section modulus mav be obtained from the formula S = trdhlA. (a) At what dislance A from the free end docs the maximum bending stress occur? What is the magnitude trllul of the maximum bending stress? What is the ratio of the maximum stress to the largest stress (b) Repeat part (a) if concentrated load P is applied upward at A and downward uniform load q {-x) = 2PIL is applied over the entire beam as shown in the figure part b What is the ratio of the maximum stress to the stress at the location of maximum moment?5.7.3P 5.7.4P 5.7.5P A cantilever beam AB with rectangular cross sections of a constant width b and varying height /iT is subjected to a uniform load of intensity q (see figure). How should the height hxvary as a function of .y (measured from the free end of the beam) in order to have a fully stressed beam? (Express hxin terms of the height hBat the fixed end of the beam.)A simple beam ABC having rectangular cross sections with constant height A and varying width bxsupports a concentrated load P acting at the midpoint (see figure). How should the width bxvary as a function of x in order to have a fully stressed beam? (Express bxin terms of the width bgat the midpoint of the beam.)A cantilever beam AB having rectangular cross sections with varying width bxand varying height hxis subjected to a uniform load of intensity q (sec figure). If the width varies linearly with x according to the equation hx= bBxiL^ how should the height hxvary as a function of v in order to have a fully stressed beam? (Express hxin terms of the height hBat the fixed end of the beam.)The shear stresses t in a rectangular beam arc given by Eq. (5-43): in which Fis the shear force, / is the moment of inertia of the cross-sectional area, /lis the height of the beam, and i] is the distance from the neutral axis to the point where the shear stress is being determined (Fig. 5-32). By integrating over the cross-sectional area, show that the resultant of the shear stresses is equal to the shear force V..2 Calculate the maximum shear stress tmaxand the maximum bending stress emaxin a wood beam (see figure) carrying a uniform load of 22.5 kN/m (which includes the weight of the beam) if the length is 1.95 m and the cross section is rectangular with width 150 mm and height 300 mm, and the beam is either (a) simply supported as in the figure part a, or b has a sliding support at right as in the figure part b.A simply supported wood beam is subjected to uniformly distributed load q. The width of the beam is 6 in, and the height is 8 in. Determine the normal stress and the shear stress at point C. Show these stresses on a sketch of a stress element at point C.A simply supported wood beam with overhang is subjected to uniformly distributed load q. The beam has a rectangular cross section with width b = 200 mm and height h = 250 mm. Determine the maximum permissible value q if the allowable bending stress is eall= 11 MPa, and the allowable shear stress is Tall= 1.2 MPa.Two wood beams, each of rectangular cross section (3.0 in. x 4.0 in., actual dimensions), are glued together to form a solid beam with dimensions 6.0 in. x 4.0 in. (sec figure). The beam is simply supported with a span of S ft. What is the maximum moment Mmaxthat may be applied at the left support if the allowable shear stress in the glued joint is 200 psi? (Include the effects of the beams own weight, assuming that the wood weighs 35 lb/ft3.) Repeat part (a) if Mmaxis based on allowable bending stress of 2500 psi.A cantilever beam of length L = 2 m supports a load P = 8,0 kN (sec figure). The beam is made of wood with cross-sectional dimensions 120 mm x 200 mm. Calculate the shear stresses due to the load/"at points located 25 mm, 50 mm, 75 mm, and 100 mm from the top surface of the beam. From these results, plot a graph showing the distribution of shear stresses from top to bottom of the beam.A steel beam of length L = 16 in. and cross-sectional dimensions h = 0.6 in. and h = 2 in. (see figure) supports a uniform load of intensity if = 240 lb/in., which includes the weight of the beam. Calculate the shear stresses in the beam (at the cross section of maximum shear force) at points located 1/4 in., 1/2 in., 3/4 in., and I in, from the top surface of the beam. From these calculations, plot a graph showing the distribution of shear stresses from top to bottom of the beam.A beam of rectangular cross section (width/) and height supports a uniformly distributed load along its entire length L. The allowable stresses in bending and shear are and TaUow, respectively. If the beam is simply supported, what is the span length Lübelow which the shear stress governs the allowable load and above which the bending stress governs? If the beam is supported as a cantilever, what is the length L() below which the shear stress governs the allowable load and above which the bending stress governs?A laminated wood beam on simple supports (figure part a) is built up by gluing together four 2 in. X 4 in. boards (actual dimensions) to form a solid beam 4 in. x 8 in. in cross section, as shown in the figure part b. The allowable shear stress in the glued joints is 62psi, the allowable shear stress in the wood is 175 psi, and the allowable bending stress in the wood is 1650 psi. (a) If the beam is 12 ft long, what is the allowable load P acting at the one-third point along the beam, as shown? (Include the effects of the beam s own weight, assuming that the wood weighs 35 lb/ft3.) (b) Repeat part (a) if the beam is assembled by gluing together two 3 in. x 4 in. boards and a 2 in. X 4 in. board (sec figure part c).A laminated plastic beam of square cross section is built up by gluing together three strips, each 10 mm x 30 mm in cross section (see figure). The beam has a total weight of 3.6 N and is simply supported with span length L = 360 mm. Considering the weight of the beam (q), calculate the maximum permissible CCW moment M that may be placed at the right support. The allowable shear stress in the glued joints is 0.3 MPa. The allowable bending stress in the plastic is 8 MPa.A wood beam AB on simple supports with span length equal to 10 ft is subjected to a uniform load of intensity 125 lb/ft acting along the entire length of the beam, a concentrated load of magnitude 7500 lb acting at a point 3 ft from the right-hand support, and a moment at A of 18,500 ft-lb (sec figure). The allowable stresses in bending and shear, respectively, are 2250 psi and 160 psi. From the table in Appendix G, select the lightest beam that will support the loads (disregard the weight of the beam). Taking into account the weight of the beam (weight density = 35 lb/ft3), verify that the selected beam is satisfactory, or if it is not, select a new beam.A simply supported wood beam of rectangular cross section and span length 1.2 m carries a concentrated load P at midspan in addition to its own weight (see figure). The cross section has width 140 mm and height 240 mm. The weight density of the wood is 5,4 kN/m3. Calculate the maximum permissible value of the load P if (a) the allowable bending stress is MPa and (b) the allowable shear stress is 0,8 MPa.A square wood platform is 8 ft × 8 ft in area and rests on masonry walls (see figure). The deck of the platform is constructed of 2-in. nominal thickness tongue-and-groove planks (actual thickness 1.5 in.; sec Appendix CL) supported on two S-ft long beams. The beams have 4 in. × (i in. nominal dimensions (actual dimensions 3.5 in. × 5.5 in.). The planks arc designed to support a uniformly distributed load n ( lb/ft" i acting over the entire top surface of the platform. I be allowable bending stress for the planks is 2400 psi and the allowable shear stress is 100 psi. W ben analyzing the planks, disregard their weights and assume that their reactions are uniformly distributed over the top surfaces of the supporting beams. (a) Determine the allowable platform load Mr. (lb/ft2) based upon the bending stress in the planks. (b) Determine the allowable platform load if-. (lb/ft-) based upon the shear stress in the planks. (c) Which of the preceding values becomes the allowable load alolow on the platform? Hints: Use care in constructing the loading diagram for the planks, noting especially that the reactions are distributed loads instead of concentrated loads. Also, note that the maximum shear forces occur at the inside faces of the supporting beams.A wood beam ABC with simple supports at A and B and an overhang ÄChas height h = 300 mm (sec figure). The length of the main span of the beam is L = 3.6 m and the length of the overhang is L/3 = 1,2 m. The beam supports a concentrated load 3P = 18 kN at the midpoint of the main span and a moment PLI2 = 10,8 kN - m at the free end of the overhang. The wood has a weight density y = 5,5 kN/m3. Determine the required width h of the beam based upon an allowable bending stress of 8,2 MPa, Determine the required width based upon an allowable shear stress of 0.7 MPa.A wood pole with a solid circular cross section (d = diameter) is subjected to a triangular distributed horizontal force of peak intensity q0= 20 lb/in. (see figure). The length of the pole is L = 6 ft, and the allowable stresses in the wood arc 1900 psi in bending and 120 psi in shear. Determine the minimum required diameter of the pole based upon (a) the allowable bending stress, and (b) the allowable shear stress.A simple log bridge in a remote area consists of two parallel logs with planks across them (see figure). The logs arc Douglas fir with an average diameter 300 mm. A truck moves slowly across the bridge, which spans 2.5 m. Assume that the weight of the truck is equally distributed between the two logs. Because the wheelbase of the truck is greater than 2,5 m, only one set of wheels is on the bridge at a time. Thus, the wheel load on one log is equivalent to a concentrated load W acting at any position along the span. In addition, the weight of one log and the planks it supports is equivalent to a uniform load of 850 N/m acting on the log. Determine the maxi mum permissible wheel load W based upon (a) an allowable bending stress of 7.0 MPa and (b) an allowable shear stress of 0.75 MPa.A vertical pole consisting of a circular tube of outer diameter 5 in. and inner diameter 4.5 in. is loaded by a linearly varying distributed force with maximum intensity of q0, Find the maximum shear stress in the pole.A circular pole is subjected to linearly varying distributed force with maximum intensity t0. Calculate the diameter daof the pole if the maximum allowable shear stress for the pole is 75 M Pa.A sign for an automobile service station is supported by two aluminum poles of hollow circular cross section, as shown in the figure. The poles are being designed to resist a wind pressure of 75 lb/ft" against the full area of the sign. The dimensions of the poles and sign are hx= 20 ft, /r =5 ft, and h = 10 ft. To prevent buckling of the walls of the poles, the thickness e is specified as one-tenth the outside diameter d. (a) Determine the minimum required diameter of the poles based upon an allowable bending stress of 7500 psi in the aluminum. (b) Determine the minimum required diameter based upon an allowable shear stress of 300 psi.A steel pipe is subjected to a quadratic distributed load over its height with the peak intensity q0at the base (see figure). Assume the following pipe properties and dimensions: height L, outside diameter d = 200 mm, and wall thickness f = 10 mm. Allowable stresses for flexure and shear are o~a=125 MPa and Ta= 30 MPa, If L = 2.6 m, Fmd^0ayM (kN/m), assuming that allowable flexure and shear stresses in the pipe are not to be exceeded. If q0= 60 kN/m, find the maximum height Lraajl(m) of the pipe if the allowable flexure and shear stresses in the pipe arc not to be exceeded.-1 through 5.10-6 A wide-flange beam (see figure) is subjected to a shear force V. Using the dimensions of the cross section, calculate the moment of inertia and then determine the following quantities: The maximum shear stress tinixin the web. The minimum shear stress rmin in the web. The average shear stress t (obtained by dividing the shear force by the area of the web) and the ratio tmax/taver. The shear force Vweb/V carried in the web and the Vweb/V. Note: Disregard the fillets at the junctions of the web and flanges and determine all quantities, including the moment of inertia, by considering the cross section to consist of three rectangles. 5.10-1 Dimensions of cross section: b = 6 in,, ï = 0.5 in., h = 12 in,, A, = 10.5 in., and V = 30 k.-1 through 5.10-6 A wide-flange beam (see figure) is subjected to a shear force V. Using the dimensions of the cross section, calculate the moment of inertia and then determine the following quantities: The maximum shear stress tinixin the web. The minimum shear stress rmin in the web. The average shear stress raver (obtained by dividing the shear force by the area of the web) and the ratio i^/t^ The shear force carried in the web and the ratio K b/K. Note: Disregard the fillets at the junctions of the web and flanges and determine all quantities, including the moment of inertia, by considering the cross section to consist of three rectangles. 5.10-2 Dimensions of cross section: b = 180 mm, v = 12 mm, h = 420 mm, i = 380 mm, and V = 125 kN.-1 through 5.10-6 A wide-flange beam (see figure) is subjected to a shear force V. Using the dimensions of the cross section, calculate the moment of inertia and then determine the following quantities: The maximum shear stress tinixin the web. The minimum shear stress rmin in the web. The average shear stress raver (obtained by dividing the shear force by the area of the web) and the ratio i^/t^ The shear force carried in the web and the ratio V^tV. Noie: Disregard the fillets at the junctions of the web and flanges and determine all quantities, including the moment of inertia, by considering the cross section to consist of three rectangles. 5.10-3 Wide-flange shape, W 8 x 28 (see Table F-L Appendix F); V = 10 k

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