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All Textbook Solutions for Mechanics of Materials (MindTap Course List)
Find support reactions at A and B and then calculate the axial force N. shear force J and bending moment M at mid-span of AB. Let L = 14 ft. q0 = 12 lb/ft. P = 50 lb. and = 300 lb-ft.Find support reactions at A and B and then calculate the axial force N, shear force V, and bending moment M at mid-span of AB. Let L = 4 m, q0= 160 N/m, P = 200 N, and M0= 380 N. m.Segments AB and BC of beam ABC are pin connected a small distance to the right of joint B (sec figure). Axial loads act at A and at the mid-span of AB. A concentrated moment is applied at joint B. (a) Find reactions at supports A, B, and C. (b) Find internal stress resultants N, V, and M at l5 ft.Segments A B and BCD of beam A BCD are pin connected at x = 4 m. The beam is supported by a sliding support at A and roller supports at C and D (see figure). A triangularly distributed load with peak intensity of SO N/m acts on EC. A concentrated moment is applied at joint D. (a) Find reactions at supports A, C, and D. (b) Find internal stress resultants N, Y, and Mat x = 5m. (c) Repeat parts (a) and (b) for die case of the roller support at C replaced by a linear spring of stiffness kr™ 200 kN/m (see figure).Segments AB and BCD of beam ABCD are pin connected at x = 10 ft. The beam is supported by a pin support at A and roller supports at C and D; the roller at D is rotated by 30* from the x axis (see figure). A trapezoidal distributed load on BC varies in intensity from 5 lb/ft at B to 2.5 lb/ft at C. A concentrated moment is applied at joint A, and a 40-lb inclined load is applied at the mid-span or CD. (a) Find reactions at supports A, C, and D. (b) Find the resultant force in the pin connection at B. (c) Repeat parts (a) and (b) if a rotational spring(kr= 50 ft-lb/radian ) is added at A and the roller at C is removed.Consider the plane truss with a pin support at joint 3 and a crtica1 roller support at joint 5 (see figure). (a) Find reactions at support joints 3 and 5. (b) Find axial forces in truss members 11 and 13.A plane truss has a pin support at A and a roller support at E(see figure). (a) Find reactions at all supports (b) Find the ati3l force in truss member FE.A plane truss has a pin support at F and a roller support at D (see figure). (a) Find reactions at both supports (b) Find the axial force in truss member FE.Find support reactions at A and B and then use the method of joints to find all member forces. Let c = 8 ft and P = 2O kips.Find support reactions at 4 and Band then use the method of joints to find all member forces. Let b = 3 m and P = 80 kN.Repeat 1.3-9 but use the method of sections go find member forces in AC and BD.Repeat 1.3-10 but use the method of sections to find member forces in AB and DC.A space truss has three-dimensional pin supports at joints 0, B, and C, Load P is applied at joint A and acts toward point Q. Coordinates of all joints arc given in feet (see figure). (a) Find reaction force components B x, B z, and Oz (b) Find the axial force in truss member AC.A space truss is restrained at joints O, A. B. and C, as shown in the figure. Load P is applied at joint A and load IP acts downward at joint C. (a) Find reaction force components Ax, By, and B. in terms of load variable P. (b) Find the axial force in truss member AB in terms of load variable P.1.3-15 A space truss is restrained at joints A, B, and C, as shown in the figure. Load 2P is applied at in the -x direction at joint A, load 3P acts in the + - direction at joint B. and load P is applied in the + r direction al joint C. Coordinates of all joints are given in terms of dimension variable L (see figure). (a) Find reaction force components Ayand Azin terms of load variable P. (b) Find the axial force in truss member AB in terms of load variable P.A space truss is restrained at joints A, B. and C, as shown in the figure. Load P acts in the +z direction at joint Band in the -z directions at joint C. Coordinates of all joints are given in terms of dimension variable L (see figure). Let P = 5 kN and L = 2 m. (a) Find the reaction force components Az and Bx. (b) Find the axial force in truss member AB.A stepped shaft ABC consisting of two solid, circular segments is subjected to torques T}and T2acting in opposite directions, as shown in the figure. The larger segment of the shaft has a diameter of dv- 2.25 in. and a length Lt= 30 in.; the smaller segment has a diameter d2— 1.75 in. and a length L, = 20 in. The torques are T, = 21,000 lb-in. and fz=10.000 lb-in. (a) Find reaction torque TAat support A. (b) Find the internal torque T(x) at two locations: x = L1/2 and x = L1+ L2/2. Show these internal torques on properly drawn free-body diagrams (FBDs).A stepped shaft ABC consisting of two solid, circular segments is subjected to uniformly distributed torque t1acting aver segment 1 and concentrated torque t2applied at C, as shown in the figure. Segment 1 of the shaft has a diameter of d1= 57 mm and length of L1= 0.75 m; segment 2 has a diameter d2— 44 mm and length L2= 0.5 m. Torque intensity /,"= 3100 N . m/m and T2= 1100 N. m. (a) Find reaction torque TAat support A. (b) Find the internal torque T(x) at two locations: .x = L1/2 and at .x = L1+ L2/2. Show these internal torques on properly drawn free-body diagrams.A plane frame is restrained al joints A and C, as shown in the figure. Members AB and BC are pin connected at B. A triangularly distributed lateral load with a peak intensity or 90 lb/ft acts on AB. A concentrated moment is applied at joint C. (a) Find reactions at supports A and C. (b) Find internal stress resultants A', V, and \f at x = 3 ft on column AB.A plane Frame is restrained at joints A and D, as shown in the figure. Members AB and BCD are pin connected at B. A triangularly distributed lateral load with peak intensity of SO N/m acts on CD. An inclined concentrated force of 200 N acts at the mid-span of BC. (a) Find reactions at supports A and D. (b) Find resultant forces in the pins at B and C.Find support reactions at A and D and then calculate the axial force N, shear force V, and bending moment M at mid-span of AB. Let L = 14 ft, q0 = 12 lb/ft, P = 50 lb. and = 300 lb-ft.Find support reactions at A and D and then calculate the axial force N. shear force 1 and bending moment 11 at mid-span of column BD. Let L = 4 m, q0 = 160N/m, P = 200N, and M0= 380 N .m.,3-23 A 200-lb trap door (AD) is supported by a strut (BC) which is pin connected to the door at B (see figure). (a) Find reactions at supports A and C. (b) Find internal stress resultants N, V, and M on the trap door at 20 in. from A.A plane frame is constructed by using a pin connection between segments ABC and CDE. The frame has pin supports at A and E and joint loads at B and D (see figure). (a) Find reactions at supports A and E. (b) Find the resultant force in the pin at C.A plane Frame with pin supports at A and E has a cable attached at C, which runs over a friction-less pulley at F(see figure). The cable Force is known to be 500 lb. (a) Find reactions at supports A and E. (b) Find internal stress resultants N, V, and M at point H.A plane frame with a pin support at A and roller supports at C and £ has a cable attached at E. which runs over Frictionless pulleys al D and B (see figure). The cable force is known to be 400 N. There is a pin connection just Lo the left of joint C. (a) Find reactions at supports^, C, and E. (b) Find internal stress, resultants N, V, and M just to the right of joint C. (c) Find resultant force in the pin near C.A 150-lb rigid bar AB. with friction less rollers al each end. is held in the position shown in the figure by a continuous cable CAD. The cable is pinned at C and D and runs over a pulley at A. (a) Find reactions at supports A and B. (b) Find the force in the cable.A plane frame has a pin support at A and roller supports at C and E (see figure). Frame segments A BD and CDEF are joined just left of joint 1) by a pin connection. (a) Find reactions at supports A. C. and E. (b) Find the resultant force in the pin just left of D.A special vehicle brake is clamped at O when the brake force P1 is applied (see figure). Force P1= 50 lb and lies in a plane that is parallel to the x-z plane and is applied at C normal to line BC. Force P2= 40 lb and is applied al B in the -y direction. (a) Find reactions at support O. (b) Find internal stress resultants N, V, T. and M at the mid-point of segment OA.Space frame A BCD is clamped at A, except it is Free to translate in the .v direction. There is also a roller support at D, which is normal to line CDE. A triangularly distributed Force with peak intensity q0 = 75 N/m acts along AB in the positive - direction. Forces Px= 60 N and Pz = = 45 N are applied at joint C, and a concentrated moment My = 120 N . m acts at the mid-span of member BC. (a) Find reactions at supports A and I). (b) Find internal stress resultants N. E’I T, and .11 at the mid-height of segment AB.Space Frame ABC is clamped at A, except it is free to rotate at A about the x and y axes. Cables DC and EC support the frame at C. Force Py= - 50 lb is applied at the mid-span of AS, and a concentrated moment Mx= -20 in-lb acts at joint B. (a) Find reactions at support A. (b) Find cable tension Forces.A soccer goal is subjected to gravity loads (in the - z direction, w = 73 N/m for DG, BG, and BC; w = 29 N/m for all other members; see figure) and a force F = 200 N applied eccentrically at the mid-height of member DG. Find reactions at sup ports C, D, and H.An elliptical exerciser machine (see figure part a) is composed of front and back rails. A simplified plane-frame model of the back rail is shown in figure part b. Analyze the plane-frame model to find reaction forces at supports A, B. and C for the position and applied loads given in figure part b. Note that there are axial and moment releases, at the base of member 2 so that member 2 can lengthen and shorten as the roller support at B moves along the 30° incline. (These releases indicate that the internal axial force N and moment M must be zero at this locationA mountain bike is moving along a flat path at constant velocity. At some instant, the rider (weight = 670 N) applies pedal and hand forces, as shown in the figure part a. (a) Find reaction forces at the front and rear hubs. (Assume that the bike is pin supported at the rear hub and roller supported at the front hub.) (b) Find internal stress resultants N, V, and M in the inclined seat post (see figure part bA hollow circular post ABC (see figure) supports a load Pt= 1700 lb acting al the top. A second load P2is uniformly distributed around the cap plate at B. The diameters and thicknesses of the upper and lower parts of the post are AB = 1.25 in.,t AB = 0.5 in., dBC= 2.25 in_ and tBC= 0.575 in., respectively. (a) Calculate the normal stress aABin the upper part of the post. (b) If you want the lower part of the post to have the same compressive stress as the upper part, what should be the required magnitude of load P2? (c) If P1remains at 1700 lb and P2is set at 2260 lb, what new thickness of BC will result in the same compressive stress in both parts?A circular nylon pipe supports a downward load PA= 10 kN. which is uniformly distributed around a cap plate at the lop of the lower pipe. A second load PB= 20 kN is applied upward at the top. The inner and outer diameters of the upper and lower parts of the pipe are d1= 50 mm, dz= 60 mm, d3= 55 mm, and d4= 65 mm. respectively. The bottom pipe has length 400 mm and the upper pipe has length 300 mm. (a) Calculate the axial normal stress in each pipe segment. (b) Calculate the strain in each pipe segment if the elongation of the upper pipe is 3.29 mm and the elongation of the bottom part is 1.25 mm.A circular tube AB is fixed at one end and free at the other end. The tube is subjected to axial force at joint B. If the outer diameter of the tube is 3 in. and the thickness is 3/4 in., calculate the maximum normal stress in the tube.A force P of 70 N is applied by a rider to the front hand brake of a bicycle ( P is the resultant of an evenly distributed pressure). As the hand brake pivots at A. a tension T develops in the 460-mm Ions; brake cable (Ae= 1.075 mm2 ), which elongates by = 0.214 mm. Find the normal stress 8 and strain e in the brake cable. Brake cable, L = 460 mm Hand brake pivot AA bicycle rider wants to compare the effectiveness of cantilever hand brakes (see figure part a) versus V brakes (figure part b). (a) Calculate the braking force RBat the wheel rims for each of the bicycle brake systems shown. Assume that all forces act in the plane of the figure and that cable tension T = 45 lb. Also. that B the average compressive normal stress ??con the brake paid (A — 0.625 in2)? (b) For each braking system. that is the stress in the brake cable if the effective cross-sectional area is 0.00167 in2? hint: Because of symmmetry. use only the right half of each figure in your analysis.A circular aluminum tube with a length of L = 420 mm is loaded in compression by forces P (see figure). The hollow segment of length LB has outside and inside diameters of 60 mm and 35 mm, respectively. The solid segment of length 2L/3 has a diameter of 60 mm. A strain gage is placed on the outside of the hollow segment of the bar to measure normal strains in the longitudinal direction. (a) If the measured strain in the hollow segment is 6jy=47010 , what is the strain s in the solid part? Hint: The strain in the solid segment is equal to that in the hollow segment multiplied by the ratio of the area of the hollow to that of the solid segment. (b) What is the overall shortening of the bar? (c) If the compressive stress in the bar cannot exceed 48 MPa, what is the maximum permissible value of load P?The cross section of a concrete corner column that is loaded uniformly in compression is shown in the figure. A circular pipe chase cut-out of 10 i n. in diameter r tins t he height of t he co \u m n (see figure). (a) Determine the average compression stress crr in the concrete if the load is equal to 3500 kips. (b) Determine the coordinates x and y{of the point where the resultant load must act in order to produce uniform normal stress in the column.A car weighing 130 kN when fully loaded is pulled slowly up a steep inclined track by a steel cable (see figure). The cable has an effective cross-sectional area of 490 mm2, and the angle ?? of the incline is 30°. (a) Calculate the tensile stress ??tin the cable. (b) IF the allowable stress in the cable is 150 MPa, what is the maximum acceptable angle of the incline for a fully loaded car?Two steel wines support a moveable overhead camera weighing W = 28 lb (see figure part a) used For close-up to viewing of field action at sporting, events. At some instant, wire I is at an angle a = 22° to the horizontal and wire 2 is at angle fi = 40°. Wires I and 2 have diameters of 30and 35 mils, respectively. (Wire diameters are often expressed in mils; one mil equals 0.001 in.) (a) Determine the tensile stresses s and s2 in the two wires. (b) If the stresses in wires 1 and 2 must be the same, what is the required diameter of wire 1 ? (c) To stabilize the camera for windy outdoor conditions, a third wire is added (see figure part b). Assume the three wires meet at a common point coordinates (0, 0. 0) above the camera at the instant shown in figure part b. Wire I is attached to a support at coordinates (75 ft, 48 ft, 70 Ft). Wire 2 is supported at (-70 ft. 55 ft, 80 Ft). Wire 3 is supported at (-10 ft. -85 Ft, 75 ft). Assume that all three wires have a diameter of 30 mils. Find the tensile stresses in all three wiresA long re Lai nine: wall is braced by wood shores set at an angle of 30° and supported by concrete thrust blocks, as shown in the first part of the figure. The shores are evenly spaced at 3 m apart. For analysis purposes, the wall and shores are idealized as shown in the second part of the figure. Note that the base of the wall and both ends of the shores are assumed to be pinned. The pressure of the soil against the wall is assumed to be triangularly distributed, and the resultant force acting on a 3-meter length of the walls is F = 190 kN. If each shore has a 150 mm X 150 mm square cross section, what is the compressive stressA pickup truck tailgate supports a crate where Wc= 150 lb. as shown in the figure. The tailgate weighs bf = 60 lb and is supported by two cables (only one is shown in the figure). Each cable has an effective cross-sectional area Ac= 0.0l7 in. (a) Find the tensile force T and normal stress a in each cable. (b) If each cable elongates 8 = 0.01 in. due to the weight of both the crate and the tailgate, what is the average strain in the cable?"Solve the preceding problem if the mass of the tailgate is MT— 11 kg and that of the crate is hic— 6S kg. Use dimensions H = 305 mm, L = 406 mm, dc= 460 mm, and dT= 350 mm. The cable cross-sectional area is At= 11.0 mm'. (a) Find the tensile Force T and normal stress T in each cable. (b) IF each cable elongatesAn L-shaped reinforced concrete slab 12 Ft X 12 ft, with a 6 Ft X 6 ft cut-out and thickness t = 9.0 in, is lifted by three cables attached at O, B, and D, as shown in the figure. The cables are are combined at point Q, which is 7.0 Ft above the top of the slab and directly above the center of mass at C. Each cable has an effective cross-sectional area of Ae= 0.12 in2. (a) Find the tensile force Tr(i = 1, 2, 3) in each cable due to the weight W of the concrete slab (ignore weight of cables). (b) Find the average stress ov in each cable. (See Table I-1 in Appendix I for the weight density of reinforced concrete.) (c) Add cable AQ so that OQA is one continuous cable, with each segment having Force T, which is connected to cables BQ and DQ at point Q. Repeat parts (a) and (b). Hini: There are now three Forced equilibrium equations and one constrain equation, T1= T4.A crane boom of mass 450 leg with its center of mass at C is stabilized by two cables AQ and BQ (Ae= 304 mm2 for each cable) as shown in the figure. A load P = 20 KN is supported at point D. The crane boom lies in the y-z plane. (a) Find the tension forces in each cable: TAQand TBQ(kN}. Neglect the mass of the cables, but include the mass of the boom in addition to load P. (b) Find the average stress (s) in each cable.Two gondolas on a ski lift are locked in the position show in the figure while repairs are being made elsewhere. The distance between support towers is L = LOO ft. The length of each cable segment under gondolas weighing WB= 450 lb and WC= 650 lb are DAB= 12 ft, DBC= 70 a, and DCB= 20 ft. The cable sag at B is AB= 3.9 ft and that at C is A- = 7.1 ft. The effective cross-sectional area of the cables is Ae= 0.12 in". (a) Find the tension force in each segment; neglect the mass of the cable. (b) Find the average stress ( ) in each cable segment.A round bar ABC of length 2L (see figure) rotates about an axis through the midpoint C with constant angular speed w (radians per second). The material of the bar has weight density y. (a) Derive a formula for the tensile stress a’ in the bar as a function of the distance x from the midpoint C. (b) What is the maximum tensile stress a max?Two separate cables AC and BC support a sign structure of weight W = 1575 lb attached to a building. The sign is also supported by a pin support at O and a lateral restraint in the '-direction at D. (a) Find the tension in each cable. Neglect the mass of the cables. (b) Find the average stress in each cable if the area of each cable is Ae= 0.471 in2.Imagine that a long steel wire hangs vertically from a high-altitude balloon. (a) What is the greatest length (feet) it can have without yielding if the steel yields at 44) ksi? (b) If the in wire hangs from a ship at sea, what is the greatest length? (Obtain the weight densities of steel and seawater from Table I-I. Appendix I.)A steel riser pipe hangs from a drill rig located offshore in deep water (see figure). (a) What is the greatest length (meters) it can have without breaking if the pipe is suspended in the air and the ultimate strength (or breaking strength) is 550 MPa? (b) If the same riser pipe hangs from a drill rig at sea, what is the greatest length? (Obtain the weight densities of steel and sea water from Table M, Appendix I. Neglect the effect of buoyant foam casings on the pipe.)Three different materials, designated A, B. and C, are tested in tension using test specimens having diameters of 0.505 in. and gage lengths of 2.0 in. (see figure}. Al failure, the distances between the gage marks are found to be 2.13, 2.43, and 2.78 in, respectively. Also, at the Failure cross sections, the diameters are found to be 0.484, 0.39S, and 0.253 in., respectively. Determine the percent elongation and percent reduction in area of each specimen. Using your own judgment, classify each material as brittle or ductile.The strength-to-weight ratio of a structural material is defined as its load-carrying capacity divided by its weight. For materials in tension, use a characteristic tensile stress obtained from a stress-strain curve as a measure of strength. For instance, either the yield stress or the ultimate stress could be used, depending upon the particular application. Thus, the strength-to-weight ratio RS/Wfor a material in tension is defined as Rs/w= in which a is the characteristic stress and 7 is the weight density. Note that the ratio has units of length. Using the ultimate stress Uas the strength parameter, calculate the strength-to-weight ratio (in units of meters) for each of the following materials: aluminum alloy 606I-T6, Douglas fir (in bending}, nylon. structural steel ASTM-A57.2, and a titanium alloy. Obtain the material properties from Tables [-1 and 1-3 of Appendix I. When a range of values is given in a table, use the average value.A symmetrical framework consisting of three pin-connected bars is loaded by a force P (see figure). The angle between the inclined bars and the horizontal is a = 52°. The axial strain in the middle bar is measured as 0.036. Determine the tensile stress in the outer bars if they are constructed of a copper alloy having the following stress-strain relationship:A specimen of a methacrylate plastic is tested in tension at room temperature (see figure}, producing the stress-strain data listed in the accompanying table (see next page). Plot the stress-strain curve and determine the proportions limit, modulus of elasticity (which is the slope of the in it initial part of the stress-strain curve), and the yield stress at 0.2% offset. Is the material ductile or brittle?The data shown in the accompanying table are From a tensile test of high-strength steel. The test specimen has a diameter of 0.505 in. and a gage length of 2.00 in. (see figure for Prob. 1.5-3). At fracture, the elongation between the gage marks is 0.12 in. and the minimum diameter is 0.42 in. Plot the conventional stress-strain curve for the steel and determine the proportional limit, modulus of elasticity (the slope of the initial part of the stress-strain curve), yield stress at 0.1% offset, ultimate stress, percent elongation in 2.00 in., and percent reduction in area. TENSILE-TEST DATA FOR PROB. L.5-7 Laid (lb) Elongation (in,) 10000.0002 20000.0006 60000.0019 10,0000.0033 12,000 0.0039 12,900 0.0041 13,400 0.0047 13,600 0.0054 13,800 0.0063 14,000 0.0090 14,4000.0102 15,200 0.0130 16,800 0.0230 18,400 O.0336 20,000 O.05O7 22,400 0.1108 22,600 FractureA bar made of structural steel having the stress-strain diagram shown in the figure has a length of 60 in. The yield stress of the steel is 50 ksi, and the slope of the initial linear part of the stress-strain curve is 29,000 ksi (a) The bar is loaded axially until it elongates 0.2 in. and then the load is removed. How does the final length of the bar compare with its original length? (b) If the bar has a circular cross section with a diameter ii = 1.5 in. and is loaded by tensile forces P = SO kips, what is the stress in the bar? What is the permanent set of the bar?A bar of length 2.0 m is made of a structural steel having the stress-strain diagram shown in the figure. The yield stress of the steel is 250 MPa, and the slope of the initial linear part of the stress-strain curve (modulus of elasticity) is 200 GPa. The bar is loaded axially until it elongates 6.5 mm. and then the load is removed. How does the final length of the bar compare with its original length? Hint: Use the concepts illustrated in Fig. 1-39b.A bar made of structural steel having the stress-strain diagram shown in the figure has a length of 4B in. The yield stress of the steel is 42 ksi, and the slope of the initial linear part or the stress-strain curve (modulus of elasticity) is 30 X 10 ksi. The bar is loaded axially until it elongates 0.20 in., and then the load is removed. How does the final length of the bar compare with its original length? Hint: Use the concepts illustrated in Fig. l-39b.A circular bar of magnesium alloy is 750 mm long. The stress-strain diagram for the material is shown in the figure. The bar is loaded in tension to an elongation of 6.0 mm. and then the load is removed. (a) That is the permanent set of the bar? (b) If the bar is reloaded, what is the proportional limit? hint. Use the concepts illustrated in Figs. 1-39b and 1-40.An aluminum bar has length L = 6 ft and diameter d = 1.375 in. The stress-strain curse for the aluminum is shown in Fig. 1.34. The initial straight, line part of the curve has a slope (modulus of elasticity) of 10.6 × 106 psi. The bar is loaded by tensile forces P = 44.6 k and then unloaded. (a) That is the permanent set of the bar? (b) If the bar is reloaded. what is the proportional limit? hint: Use the concepts illustrated in Figs. l.39b and 1.40.A continuous cable (diameter 6 mm) with tension force T is attached to a horizontal frame member at B and C to support a sign structure. The cable passes over a small friction less pulley at D. The wire is made of a copper alloy, and the stress-strain relationship for the wire is ()=124,0001+30000.03(inMPa) (a) Find the axial normal strain in the cable and its elongation due to the load W = 6.8 kN. (b) If the forces are removed, what is the permanent set of the cable? Hint: Start with constructing the stress-strain diagram and determine the modulus of elasticity, E. and the 0.2% offset yield stress.A wine of length L = 4 ft and diameter d = 0.125 in. is stretched by tensile forces P = 600 lb. The wire is made of a copper alloy having a stress-strain relationship that may be described mathematically by =18,0001+30000.03(=ksi) in which is nondimensional and has units of kips per square inch (ksi). (a) Construct a stress-strain diagram for the material. (bj Determine the elongation, of the wire due to the Forces P. (c) IF the forces are removed, what is the permanent set of the bar? (d) If the forces are applied again, what is the proportional limit?A high-strength steel bar used in a large crane has a diameter d = 2.00 in. (sec figure). The steel has a modulus of elasticity E = 29 × 10 psi and Poisson’s ratio is v = 0.29. Because of clearance requirements, the diameter of the bar is limited to 2.001 in. when it is compressed by axial forces. What is the largest compressive load Pmaxthat is permitted?A round bar of 10 mm diameter is made of aluminum alloy 7075-T6 (see figure). When the bar is stretched by axial forces P, its diameter decreases by 0.0 16 mm. Find the magnitude of the load P. Obtain the material properties from Appendix 1.A polyethylene bar with a diameter d, = 4.0 in. is placed inside a steel lube with an inner diameter d2= 4.01 in. (see figure). The polyethylene bar is then compressed by an axial Force P. At what value of the force P will the space between the polyethylene bar and the steel tube be closed? For polyethylene, assume E = 200 ksi and v = 0.4.A square plastic bar (length LP,side dimension sP=193 mm) is inserted inside a hollow. square cast iron tube (length Lc = 400 mm. side sc= 200 mm, and thickness tc= 3 mm). (a) What is the required initial length L, of the plastic bar so that. i1cn it is compressed by some force P. the final length of bar and tube are equal to length L and, at the same time, the gap beten plastic bar and cast iron tube is closed? (b) Compare initial and final volumes for the plastic bar. Assume that E = 170 GPa. E, = 2.1 GPa. v = 0.3. and = 0.4.A polyethylene bar having rectangular cross section with a width 7.35 in. and depth 7 in. is placed inside a hollow steel square section with side dimension of 8 in. The polyethylene bar is then compressed by an axial force P. At what value of the force P will the gap between the polyethylene bar and the steel tube be closed for the first time on one side? What is the remaining gap between the polyethylene bar and the steel tube on the other side? For polyethylene, assume E = 200 ksi and v = 0.4.A circular aluminum tube of length L = 600 mm is loaded in compression by forces P (see figure). The outside and inside diameters are d2= 75 mm and d1= 63 mm, respectively. A strain gage is placed on the outside of the lube to measure normal strains in the longitudinal direction. Assume that E = 73 GPa and Poissons ratio is v = 0.33. (a) IF the compressive stress in the tube is 57 MPa, what is the load P? (b) If the measured strain is e = 78 J X 10-6, what is the shorteningA bar of monel metal with a length L = 9 in. and a diameter d = 0225 in. is loaded axially by a tensile force P (see figure). If the bar elongates by 0.0)95 in., what is the decrease in diameterA tensile test is performed on a brass specimen 10 mm in diameter using a gage length of 50 mm (see figure). When the tensile load P reaches a value of 20 kN, the distance between the gage marks has increased by 0.122 mm. (a) What is the modulus of elasticity E of the brass? (b) H the diameter decreases by 0.00830 mm, what is poison’s ratio?A hollow, brass circular pipe ABC (see figure) supports a load P1= 26.5 kips acting at the top. A second load P2= 22.0 kips is uniformly distributed around the cap plate at B. The diameters and thicknesses of the upper and lower parts of the pipe are dAB= 125 in., tAB= 0.5 in., dBC= 2.25 in., and tBC= 0.375 in., respectively. The modulus of elasticity is 14,000 ksi. When both loads are fully applied, the wall thickness of pipe segment BC increases by 200 × 10-6 in. (a) Find the increase in the inner diameter of pipe segment BC. (b) Find Frisson's ratio for the brass. (c) Find the increase in the wall thickness of pipe segment AB and the increase in the inner diameter of segment A BThree round, copper alloy bars having the same length L but different shapes are shown, in the figure. The first bar has a diameter d over its entire length, the second has a diameter d over one-fifth of its length, and the third has a diameter d over one-fifteenth of its length. Elsewhere, the second and third bars have a diameter Id. All three bars are subjected to the same axial load P. Use the following numerical data: P = 1400 kN, L = 5m,d= 80 mm, E= 110 GPa. and v = 0.33. (a) Find the change in length of each bar. (b) Find the change in volume of each bar.An angle bracket having a thickness t = 0.75 in. is attached to the flange of a column by two 5/8-inch diameter bolts (see figure). A uniformly distributed load from a floor joist acts on the lop face of the bracket with a pressure p = 275 psi. The top Face of the bracket has a length L = 8 in. and width h = 3.0 in. Determine the average bearing pressure 0b between the angle bracket and the bolts and the average shear stress T aver in the bolts. Disregard friction between the bracket and the column.Truss members supporting a roof are connected to a 26-mm-thick gusset plate by a 22-mm diameter pin, as shown in the figure and photo. The two end plates on the truss members are each 14 mm thick. (a) If the load P = 80 kN, what is the largest bearing stress acting on the pin? (b) If the ultimate shear stress for the pin is 190 MPa, what force Pult is required to cause the pin to fail in shear? Disregard friction between the plates.The upper deck ala foothill stadium is supported by braces, each of which transfer a load P = 160 kips to the base of a column (see figure part a). A cap plate at the bottom of the brace distributes the load P to four flange pates (:1 = I in)t hrough a pin(d, = 2 in.) to two gusset plates t8 = l.5 in.) (see figure parts b and c). Determine the following quantities. (a) The average shear stress i in the pin. (b) The average bearing stress between the flange plates and the pin and also between the gusset plates and the pin Disregard friction between the plates. Determine the following quantities. (a) The average shear stress i in the pin. (b) The average bearing stress between the flange plates and the pin and also between the gusset plates and the pin (7j )L Disregard friction between the plates.The inclined ladder AB supports a house painter (85 kg) at C and the weight iq = 40 K/m} of the ladder itself. Each ladder rail (t5= 4 mm) is supported by a shoe (ts= 5 mm) that is attached to the ladder rail by a bolt of diameter d = 8 mmThe Force in the brake cable of the V-brake system shown in the figure is T — 45 lb. The pivot pin at A has a diameter d. = 0.25 in. and length L„ = 5/S in. Use the dimensions shown in the figure. Neglect the weight of the brake system. (a) Find the average shear stress rjm in the pivot pin where it is anchored to the bicycle frame at B. (b) Find the average bearing stress raverin the pivot pin over segment AB. (a) Find support reactions at A and B. (b) Find the resultant force in the shoe boll at A. (c) Find maximum average shear T and bearing AB stresses in the shoe bolt at A.A steel plate of dimensions 2.5 × l.5 × 0.08 m and weighing 23.1 kN is hoisted by steel cables with lengths L1= 3.2 m and L2= 3.9 m that are each attached to the plate by a clevis and pin (see figure). The pins through the clevises are 18 mm in diameter and are located 2.0 m apart. The orientation angles are measured to be s = 94.4' and a = 54.9° For these conditions, first determine the cable forces T1and T2,then find the average shear stress Taverin both pin 1 and pin 2, and then the average bearing stress ??bbetween the steel plate and each pin. Ignore the weight of the cables.A special-purpose eye boll with a shank diameter d - 0.50 in. passes through, a hole in a steel plate of thickness tp = 0.75 in. (see Figure) and is secured by a nut with thickness t = 0.25 in. The hexagonal nut bears directly against the steel plate. The radius of the circumscribed circle for the hexagon is r = 0.40 in., so each side of the hexagon has a length 0.40 in. The tensile Forces in three cables attached to the eye bolt are T1= 800 lb, T2= 500 lb. and T3= 124 lb. (a) Find the resultant force acting on the eye bolt. (b) Determine the average bearing stress crhbetween the hexagonal nut on the eye boll and the plate. (c) Determine the average shear stress T aver in the nut and also in the steel plate.An elastomeric bearing pad consisting of two steel plates bonded to a chloroprene elastomer (an artificial rubber) is subjected to a shear force V during a static loading test (see figure). The pad has dimensions a = 125 mm and b = 240 mm, and the elastomer has a thickness t = 50 mm. When the Force V equals 12 kN, the top plate is found to have displaced laterally S.O mm with respect to the bottom plate. What is the shear modulus of elasticity G of the chloroprene?A joint between iwo concrete slabs A and B is filled, with a flexible epoxy lhal bonds securely lo the concrete (see figure). The height of the joint is h = 4.0 in., its length is L = 40 in., and its thickness is t = 0.5 in. Under the action of shear forces K the slabs displace vertically through the distance d = 0.002 in. relative lo each other. (a) What is the average shear strain in the epoxy? (b) What is the magnitude of the forces V if the shear modulus of elasticity G for the epoxy is 140 ksi?A steel punch consists of two shafts: upper shaft and lower shaft. Assume that the upper shaft has a diameter d1= 24 mm and the bottom shaft has a diameter d2= 16 mm. The punch is used to insert a hole in a 4 mm plate, as shown in the figure. If a force P - 70 kN is required to create the hole, what is the average shear stress in the plate and the average compressive stress in the upper and lower shaft of the punch?A joint between two glass plates A and B is filled with a flexible epoxy that bonds securely to the glass. The height of the joint is/p = 0.5 in, its length is L = 30 in, and its thickness is/ = 0.5 in. Shear force of I' = 25 kips is applied to the joint. Calculate the displacement of the joint if the shear modulus of elasticity G of the epoxy is 100 ksi. Calculate the average shear strain in the epoxy.A punch for making a slotted hole in ID cards is shown in the figure part a. Assume that the hole produced by the punch can be described as a rectangle (12 mm X 3 mm) with two half circles (r = 1.5 mm) on the left and the right sides. If P = 10 N and the thickness of the ID card is 1 mm, what is the average shear stress in the card?A steel riser pipe hangs from a drill rig located offshore in deep water (see figure). Separate segments are joined using bolted flange plages (see figure part b and photo). Assume that there are six bolts at each pipe segment connection. Assume that the total length of the riser pipe is L = 5000 ft: outer and inner diameters are d2= l6in.and d1= 15 in.; flange plate thickness t1= 1.75 in.; and bolt and washer diameters are db= 1.125 in..and dW. = 1.875 in., respectively. (a) If the entire length of the riser pipe is suspended in air. find the average normal stress a in each bolt, the average bearing stress abbeneath each washer, and the average shear stress t through the flange plate at each bolt location for the topmost bolted connection. (b) If the same riser pipe hangs from a drill rig at sea. what are the normal, bearing, and shear stresses in the connection? Obtain the weight densities of steel and sea water from Table I-1. Appendix I. Neglect the effect of buoyant foam casings on the riser pipeA flexible connection consisting of rubber pads (thickness f = 9 mm) bonded to steel plates is shown in the figure. The pads are 160 mm long and SO mm wide. (a) Find the average shear strain yaiTi in the rubber if the force P = 16 kN and the shear modulus for the rubber is G = 1250 kPa. (b) Find the relative horizontal displacement 3 between the interior plate and the outer plates..15 A hitch-mounted bicycle rack is designed to carry up to four 30-lb bikes mounted on and strapped to two arms Gil (sec bike loads in the figure part a) The rack is attached to the vehicle at A and is assumed to be like a cant silkier beam A BCDGII (figure part b) The light of fixed segment AB is U = 10 lb. centered 9 in. from A (see figure part b) and the rest of the rack highs W2 = 40 lb. centered 19 in. from A. Segment ABCDG is a steel tube o(2 X 2 in. with a thickness I = 118 in. Segment BCDGII pivots about a bolt at B with a diameter d1 = 0.25 in. to allow access to the rear of the vehicle without removing the hitch rack. When in use, the rack is secured in an upright posit ion by a pin C(diameter o( pin d, = 5116 in.) (see phoo and figure part C). The of returning effect of the bikes on the rack is resisted by a force couple F h at BC. (a) Find the support reactions at A for the fully loaded rack. (b) Find forces in the bolt at B and the pin at C. (c) Find average shear stresses in both the bolt at Band the pin at C. (d) Find average bearing stresses o, in the bolt at B and the pin at C.The clamp shown in the figure supports a load hanging from the lower flange of a steel beam. The clamp consists of two arms (A and B) joined by a pi n at C. The pi n has a diameter d = 12 mm. Because arm B straddles arm A, the pin is in double shear. Line I in the Figure defines the line of action of the resultant horizontal force H acting between the lower flange of the beam and arm B. The vertical distance from this line to the pin is/r = 250 mm. Line 2 defines the line of action of the resultant vertical force V acting between the flange and arm B. The horizontal distance from this line to the centerline of the beam is<- = 100 mm. The force conditions between arm A and the lower flange are symmetrical with those given for arm B. Determine the average shear stress in the pin at C when the load P = 18 kNA shock mount constructed as shown iu the figure is used to support a delicate instrument. The mount consists of an outer steel tube with inside diameter b. a central steel bar of diameter d that supports the load P, and a hollow rubber cylinder (height /r) bonded to the tube and bar (a) Obtain a formula Tor the shear stress t in the rubber at a radial distance r from the center of the shock mount. (b) Obtain a formula Tor the downward displacement S of the central bar due to the load P. assuming that G is the shear modulus of elasticity of the rubber and that the steel tube and bar are rigid.1.8.18PA spray nozzle for a garden hose requires under a water pressure force fp= 30 lb at C (see figure a force F = 5 lb to open the spring-loaded spray part c). Use dimensions given in figure part a chamber AB. The nozzle hand grip pivots about a (a) Find the force in the pin at O due to applied force F pin through a flange at O. Each of the two flanges force F has a thickness t = 1/16 in., and the pin has a diam- (b) Find average shear stress taver and bearing stress eter dp = 1/8 in. (see figure part a). The spray nozzle is attached to the garden hose with a quick release fitting at B (see figure part b). Three brass balls Find the average shear stress Ta,„ in the brass (diameter db= 3/16 in.) hold the spray head in place retaining balls al C due to water pressure Force fPA single steel strut AB with a diameter (a) Find the strut force Fs and average normal stress ds= 8 mm supports the vehicle engine hood of a in the strut. mass 20 kg that pivots about hinges at C and D (see (b) Find the average shear stress t aver in the bolt at A,figure parts a and b). The strut is bent into a loop at (C) Find the average bearing stress bon the bolt at A. its end and then attached to a bolt at A with a diameter db= 10 mm. Strut AB lies in a vertical plane.The top portion of a pole saw used to trim (a) Find the force P on the cutting Made at D if tbe small branches from trees is shown in the figure tension force in the rope is T = 25 lb (see free- part a. The cutting blade BCD (see figure parts a and body diagram in figure part b). c) applies a force P at point D. Ignore the effect of (b) Find force in the pin at C. the weak return spring attached to the cutting blade (c) Find average shear stress raver and bearing below B. Use properties and dimensions given in the stress in the support pin at C (see section a-a figure. through cutting blade in figure part c).A cargo ship is tied down to marine boll arts at a number of points along its length while its cargo is unloaded by a container handling crane. Each bollard is fastened to the wharf using anchor bolts. Three cables having known tension force magnitudes F, = ll0 kN.F, = 85kN.and F, 9OkNare secured to one bollard at a point A with coordinates (0.0.45 m. 0) in the x-r-: coordinate system shown in the figure part b. Each cable force is directed at an attachment point on the ship. Force F, is directed from point A to a point on the ship having coordinates (3 m, 9 m. 0) force F, is directed at a point with coordinates (6.5 m. 8.5 m. 2 m) and force F, is directed at a point with coordinates (8 m. 9 m. S m). The diameter of each anchor bolts is 4 24 mm. (a) Find the reaction forces and reaction moments at the base of the bollard. (b) Calculate the average shear stress in the anchor bolts (in the x-: plane). Assume each bolt cart ics an equal share of the total force.A basketball player hangs on the rim after (a) Find the reactions at the support bracket a dunk. He applies equal forces P1= P2= 110 lb at (assume that the bracket-rim assembly is a both A and B (see joint coordinates in the figure). cantilever beam). Forces P1tmd P2act parallel to the y-z plane. (b) Find connection shear stresses at bolt 2.A bicycle chain consists of a series of small links, where each are 12 mm long between the centers of the pins (see figure). You might wish to examine a bicycle chain and observe its construction. Note particularly the pins, which have a diameter of 2.5 mm. To solve this problem, make two measurements on a bicycle (see figure): (1) the length L of the crank arm from main axle to pedal axle and (2) the radius R of the sprocket (the toothed wheel, sometimes called the chainring). (a) Using your measured dimensions, calculate the tensile force T in the chain due to a force F = 800 N applied to one of the pedals. (b) Calculate the average shear stress T averin the pins.A bar of solid circular cross section is loaded in tension by forces P (see figure). The bar has a length L = 16.0 in. and diameter d = 0.50 in. The material is a magnesium alloy having a modulus of elasticity E = 6.4 × 106 psi. The allowable stress in tension is ?? allow= 17,000 psi, and the elongation of the bar must not exceed 0.04 in. What is the allowable value of the forces F?.2 A torque T0is transmitted between two flanged shafts by means of ten 20-mm bolts (see figure and photo). The diameter of the bolt circle is d = 250 mm. If the allowable shear stress in the bolts is 85 MPa. what is the maximum permissible torque?(Disregard friction between the flanges.)A tie-down on the deck of a sailboat consists of a bent bar boiled at both ends, as shown in the figure. The diameter dBof the bar is 1/4 in., the diameter D Wof the washers is 7/8 in., and the thickness is of the fiberglass deck is 3/8 in. If the allowable shear stress in the fiberglass is 300 psi, and the allowable bearing pressure between the washer and the fiberglass is 550 psi, what is the allowable load P allowon the tie-down?Two steel tubes are joined at B by four pins (dp= 11 mm), as shown in the cross section a—a in the fiaure. The outer diameters of the tubes are dAB= 41 mm and dBC= 28 mm. The wall thickness are tAB= 6.5 mm and tBC= 7.5 mm. The yield stress in tension for the steel is sy = 200 MPa and the ultimate stress in tension is ??U:= 340 MPa. The corresponding yield and ultimate values in shear for the pm are 80 MPa and 140 MPa, respectively. Finally, the yield and ultimate values in bearing R between the pins and the tubes are 260 MPa, and 450 MPa, respectively. Assume that the factors safety with respect to yield stress and ultimate stress are 3.5 and 4.5. respectively. (a) Calculate the allowable tensile force P allowconsidering tension in the tube (b) Recompute P allowfor shear in the pins.(c)Finaly, recomputed Pallowfor bearing between the pm and the tubes. Which is the tubes. Which is the controlling value value of P?A steel pad supporting heavy machinery rests on Four short, hollow, cast iron piers (see figure). The ultimate strength of the cast iron in compression in 50 ksi. The outer diameter of the piers is d = 4.5 in, and the wall thickness is t = 0.40 in. Using a factor of safety of 3.5 with respect to the ultimate strength, determine the total load P that can be supported by the pad.A steel pad supporting heavy machinery rests on four short, hollow, cast iron piers (see figure). The ultimate strength of the cast iron in compression is 400 MPa. The total load P that may be supported by the pad is 900 kN. Using a factor or safely 3.0 with respect to ultimate strength, determine the outer diameter of the pier if the thickness, is of the cross section is 12 mm.A steel riser pipe hangs from a drill rig. Individual segments of equal length L = 50 ft are joined to get her using bolted flange plates (see figure part b). There are six bolts at each pipe segment connection. The outer and inner pipe diameters are t2= 14 in. and d1= 13 in.; flange plate thickness tf= 1.5 in.; and boll and washer diameters are db= 1.125 in. and dn. = 1.875 in. Find the number n of permissible segments of pipe based on following allowable stresses. (a) The allowable tensile stress in the pipe is 50 ksi. (b) The allowable tensile stress in a bolt is 120 ksi. Find number of segments n for two cases: pipe hanging in air and pipe hanging in seawater.The rear hatch of a van (BDCG in figure part a) is supported by two hinges at Bland B2and by two struts A\B\ and A2B2(diameter ds= 10 mm), as shown in figure part b. The struts are supported at A1and A2by pins, each with a diameter d = 9 mm and passing through an eyelet of thickness / = 8 mm at the end of the strut (figure part b). A closing force P = 50 N is applied at G, and the mass of the hatch A/A = 43 kg is concentrated at C. (a) What is the force F in each strut? (Use the Free-body diagram of one half of the hatch in the figure part c.) (b) What is the maximum permissible force F in the strut, F allow, if the allowable stresses are compressive stress in the strut, 70 MPa; shear stress in the pin, 45 MPa; and bearing stress between the pin and the end of the strut, 110 MPa.A lifeboat hangs from two ship's davits. as shown in the figure. A pin of diameter d = 0.80 in. passes through each davit and supports two pulleys. are on each side of the davit. Cables attached to the lifeboat pass over the pulleys and wind around winches that raise and lower the lifeboat. The lower parts of the cables are vertical and the upper parts make an angle a =15° with the horizontal. The allowable tensile force in each cable is 1800 lb, and the allowable shear stress in the pins is 4000 psi. If the lifeboat weighs 1500 lb, what is the maximum weight that can be carried in the lifeboat?A cable and pulley system in the figure part a supports a cage of a mass 300 kg at B. Assume that this includes the mass of the cables as well. The thickness or each of the three steel pulleys is t = 40 mm. The pin diameters are dPA= 25 mm, dB= 30 mm. and dc= 22 mm (see figure part a and part b). (a) Find expressions for the resultant forces acting on the pulleys at A, B. and C in terms of cubic tension T. (b) What is the maximum weight W that can be added to the cage at B based on the following allowable stresses? Shear stress in the pins is 50 MPa; bearing stress between the pin and the pulley is 110 MPa.A ship's spar is attached at the base of a mast by a pin connection (see figure). The spar is a steel tube of outer diameter d2= 3.5 in. and inner diameter d1= 2.8 in. The steel pin has a diameter d = 1 in., and the two plates connecting the spar to the pin have a thickness t = 0.5 in. The allowable stresses are compressive stress in the spar. 10 ksi: shear stress in the pi n, 6.5 ksi: and bearing stress between t he pin and t he connecting plates, 16 ksi. Determine the allowable compressive force Pallowin the spar.What is the maximum possible value of the clamping Force C in the jaws of the pliers shown in the figure if the ultimate shear stress in the 5-mm diameter pin is 340 MPa? What is the maximum permissible value of the applied load P to maintain a factor of safety of 3.0 with respect to failure of the pin?A metal bar AB of a weight Ills suspended by a system of steel wires arranged as shown in the figure. The diameter of the wires is 5/64 in., and the yield stress of the steel is 65 ksi. Determine the maximum permissible weight W max for a factor of safety of 1.9 with respect to yielding.A plane truss is subjected to loads 2P and P at joints B and C, respectively, as shown in the figure part a. The truss bars are made of two L 102 X 76 X 6.4 steel angles (see Table F-5(b): cross-sectional area or the two angles, A = 2180 mm2, and figure part b) having an ultimate stress in tension equal to 390 MPa. The angles are connected to a 12-mm-thick gusset plate at C(figure part c) with 16-mm diameter rivets; assume each rivet transfers an equal share of the member force to the gusset plate. The ultimate stresses in shear and bearing for the rivet steel are 190 MPa and 550 MPa, respectively. Determine the allowable load Pallowif a safety factor of 2.5 is desired with respect to the ultimate load that can be carried. Consider tension in the bars, shear in the rivets, bearing between the rivets and gusset plate. Disregard friction between the plates the bars, and also bearing between the rivets and the and the weight of the truss itself.A solid bar of circular cross section (diameter d) has a hole of diameter d/5 drilled laterally through the center of the bar (sec figure). The allowable average tensile stress on the net cross section of the bar is s allow. (a) Obtain a formula for the allowable load P allowthat the bar can carry in tension. (b) calculate the value of P allowif the bar is made of brass with a diameter d = 1.75 in. and s allow= 12 ksi. Hint: Use the formulas of Case 15. Appendix E.A solid steel bar of a diameter d1= 60 mm has a hole of a diameter d2= 32 mm drilled through it (see figure). A steel pin or a diameter d2passes through the hole and is attached to supports. Determine the maximum permissible tensile load P allow in the bar if the yield stress for shear in the pin is ty= 120 MPa, the yield stress for tension in the bar is ??y= 250 MPa, and a factor of safety of 2.0 with respect to yielding is required. Hint: Use the formulas of Case 15, Appendix E.A sign of weight W is supported at its base by four bolls anchored in a concrete footing. Wind pressure P acts normal to the surface of the sign; the resultant of the uniform wind pressure is force fat the center of pressure (C.P). The wind force is assumed to create equal shear forces F/4 in the y direction at each boll (see figure parts a and c). The overturning effect of the wind force also causes an uplift force R at bolts A and C and a downward force (— R) al bolts B and D (see figure part b). The resulting effects of the wind and the associated ultimate stresses for each stress condition are normal stress in each boll (h — 60 ksi); shear through the base plate (th = 17 ksi); horizontal shear and bearing on each bolt ( tfur = 25 ksi and cr^ = 75 ksi): and bearing on the bottom washer at B (or D) (abor = 50 ksi).The piston in an engine is attached to a connecting rod AB, which in turn is connected to a crank arm BC (see figure). The piston slides without friction in a cylinder and is subjected to a force P (assumed to be constant) while moving to the right in the Figure. The connecting rod. with diameter d and length L, is attached at both ends by pins. The crank arm rotates about the axle at C with the pin at B moving in a circle of radius R. The axle at C, which is supported by bearings, exerts a resisting moment M against the crank arm. (a) Obtain a formula for the maximum permissible force Pallow. based upon an allowable compressive stress acin the connecting rod. (b) Calculate the Force Pallowfor the following data:An aluminum tube is required to transmit an axial tensile force P = 33 k (sec figure part a). The thickness of the wall of the tube is 0.25 in. (a) What is the minimum required outer diameter d minif the allowable tensile stress is 12.000 psi? (b) Repeat part (a) if the tube has a hole of a diameter J/10 at mid-length (see figure parts b and C).A copper alloy pipe with a yield stress aY= 290 MPa. is to carry an axial tensile load P = 1500 kN (see figure part a). Use a factor of safety of 1.8 against yielding. (a) If the thickness t of the pipe is one-eighth of its outer diameter, what is the minimum required outer diameter dmin? (b) Repeat part (a) if the tube has a hole of diameter dt 10 drilled through the entire tube, as shown in the figure part b.A horizontal beam AB with cross-sectional dimensions (b = 0.75 in.) X (h = 8.0 in.) is supported by an inclined strut CD and carries a load P = 2700 lb at joint B (see figure part a). The strut. which consists of two bars each of thickness 5b/8, is connected to the beam by a bolt passing through the three bars meeting at joint C (see figure part b) (a) IF the allowable shear stress in the bolt is 13,000 psi, what is the minimum required diameter dminof the bolt at C? (b) If the allowable bearing stress in the bolt is 19,000 psi, what is the minimum required diameter dminof the bolt at C?Lateral bracing for an elevated pedestrian walkway is shown in the figure part a. The thickness of the clevis plate tc= 16 mm and the thickness of the gusset plate t = 20 mm (see figure part b). The maximum force in the diagonal bracing is expected to be F = 190kN. If the allowable shear stress in the pin is 90 MPa and the allowable bearing stress between the pin and both the clevis and gusset plates is 150 MPa, what is the minimum required diameter dminof the pinA plane truss has joint loads P, 2P, and 3P at joints D. C, and B. respectively (see figure) where load variable P — 5200 lb. All members have two end plates (see figure For Prob. 1.8-2) that are pin-connected to gusset plates. Each end plate has a thickness/ — 0.6.2.5 in., and all gusset plates have a thickness tg= 1.125 in. IT the allowable shear stress in each pin is 12,000 psi and the allowable bearing stress in each pin is 18.000 psi, what is the minimum required diameter dminof the pins used at either end of member BE1Cable DB supports canopy beam OABC as shown in the figure. Find the required cross-sectional area of cable BD if the allowable normal stress is 125 MPa. Determine the required diameter of the pins at 0, B, and D if the allowable stress in shear is 80 MPa. Assume that canopy beam weight is w = 8 kN. note The pins at 0, A, D and D are in double shear. Consider only the weight of the canopy; disregard the weight of cable DB.Continuous cable ADS runs over a small Frictionless pulley at D to support beam OABC that is part of an entrance canopy Tor a building (see figure}. Assume that the canopy segment has a weight it' = 1700 lb. (a) Find the required cross-sectional area of cable ADB if the allowable stress is 18 ksi. (b) Determine the required diameter of the pins at O. A, R and D if the allowable stress in shear is 12 ksi.A suspender on a suspension bridge consist of a cable that passes over the main cable (see figure) and supports the bridge deck, which is Far below. The suspender is held in position by a metal tie that is prevented from sliding downward by clamps around the suspender cable. Let P represent the load in each part of the suspender cable, and let represent the angle of the suspender cable just above the tie. represent the allowable tensile stress in the metal tie. (a) Obtain a formula for the minimum required cross-sectional area of the lie. (b) Calculate the minimum area if P = 130 kN, = 75°, and allow=80 .A square steel tube of a length L = 20 ft and width b2= 10.0 in. is hoisted by a crane (see figure). The lube hangs from a pin of diameter d that is held by the cables at points A and B. The cross section is a hollow square with an inner dimension b1= 8.5 in. and outer dimension b2= 10,0 in. The allowable shear stress in the pin is 8,700 psi. and the allowable bearing stress between the pin and the tube is 13,000 psi. Determine the minimum diameter of the pin in order to support the weight of the tube. Note: Disregard the rounded corners of the tube when calculating its weight.A cable and pulley system at D is used to bring a 230-lcg pole (ACB) to a vertical position, as shown in the Figure part a. The cable has tensile force T and is attached at C. The length L of the pole is 6.0m, the outer diameter is d = 140 mm. and the wall thickness is t = 12 mm. The pole pivots about a pin at A in figure part b. The allowable shear stress in the pin is 60 MPa and the allowable bearing stress is 90 MPa. Find the minimum diameter of the pin at A in order to support the weight of the pole in the position shown in the figure part a.A pressurized circular cylinder has a sealed cover plate fastened with steel bolts (see figure). The pressure P of the gas in the cylinder is290psi, the inside diameter D of the cylinder is 10.0 in., and the diameter dBof the bolts is 0.50 in. I f the allowable tensile stress in the bolts is 10,000 psi, find the number n of bolts needed to fasten the cover.A tubular post of outer diameter d2is guyed by two cables fitted with turnbuckles (see figure). The cables are lightened by rotating the turnbuckles, producing tension in the cables and compression in the post. Both cables are lightened to a tensile force of 110 kN. The angle between the cables and the ground is 60° and the allowable compressive stress in the post is ?? c= 35 MPa. If the wall thickness of the post is 15 mm, what is the minimum permissible value of the outer diameter d2?A large precast concrete panel for a warehouse is raised using two sets of cables at two lift lines, as shown in the figure part a. Cable 1 has a length L1 = 22 Ft, cable 2 has a length L2= 10 ft, and the distance along the panel between lift points Band D is d = 14 ft (see figure part b). The total weight of the panel is W = 85 kips. Assuming the cable lift Forces F at each lift line are about equal, use the simplified model of one half of the panel in figure part b to perform your analysis for the lift position shown. Find the required cross-sectional area AC of the cable if its breaking stress is 91 ksi and a factor of safety of 4 with respect to failure is desired.A steel column of hollow circular cross section is supported on a circular, steel base plate and a concrete pedestal (see figure). The column has an outside diameter d = 250 mm and supports a load P - 750 kN. (a) If the allowable stress in the column is 55 MPa, what is the minimum required thickness? Based upon your result, select a thickness for the column, (Select a thickness that is an even integer, such as 10, 12. 14,.. ., in units of millimeters.) (b) If the allowable bearing stress on the concrete pedestal is 11.5 MPa, what is the minimum required diameter D of the base plate if it is designed for the allowable load Pallowthat the column with the selected thickness can supportAn elevated jogging track is supported at intervals by a wood beam AB (L = 7.5 ft) that is pinned at A and supported by steel rod BC and a steel washer at B. Both the rod (dBC= 3/16 in.) and the washer (dB= 1.0 in.) were designed using a rod tension force of TBC=415 lb. The rod was sized using a factor of safely of 3 against reaching the ultimate stress tru— 60 ksi. An allowable bearing stress sba= 565 psi was used to size the washer at B. A small platform HF is suspended below a section of the elevated track to support some mechanical and electrical equipment. The equipment load is uniform load q = 50 lb/ft and concentrated load WE= 175 lb at mid-span of beam HF. The plan is to drill a hole through beam ABaX £land install the same rod (dBC) and washer) dB) at both D and F to support beam HF. (a) Use s and to check the proposed design for rod DF and washer d,: are they acceptable? (b) Re-check the normal tensile stress in rod BC and bearing stress at 8 if either is inadequate under the additional load from platform HF. Re-design them to meet the original design criteria.A flat bar of a widths b = 60 mm and thickness t = 10 mm is loaded in tension by a force p (see figure). The bar is attached to a support by a pin of a diameter d that passes through a hole of the same size in the bar. The allowable tensile stress on the net cross section of the bar is aT= 140 MPa, the allow-able shear stress in the pin is Ts = 80 MPa, and the allowable bearing stress between the pin and the bar is as= 200 MPa. (a) Determine the pin diameter dmfor which the load P is a maximum. (b) Determine the corresponding value Pmaxof the load.Continuous cable A DB runs over a small friction less pulley al D to support beam OABC, which is part of an entrance canopy for a building (see figure}. The canopy segment has a weight W = 1700 lb that acts as a concentrated load in the middle of segment AB. (a) What is the maximum permissible value of load P at C if the allowable force in the cable is 4200 lb? (b) If P = 2300 lb, what is the required diameter of pins A, B, and D? Assume that the pins are in double shear and the allowable shear stress in the pins is 10 ksi.Continuous cable ADB runs over a small friction less pulley at D to support beam OABC, which is pan of an entrance canopy for a building (see figure). A downward distributed load with peak intensity q0= 5 kNVm al O acts on the beam (see figure). Assume that canopy weight W = S kN and that the cable cross-sectional area is 100 mm". What is the required diameter of pins A, B. and D if the pins are in double shear and the allowable shear stress is SO MPa? Note that dimensions OA = AB = BC = 1.5 m.Two bars AC and BC of the same material support a vertical load P (see figure). The length L of the horizontal bar is fixed, but the angle fl can be varied by moving support A vertically and changing the length of bar AC to correspond with the new position of support A. The allowable stresses in the bars are the same in tension and compression. When the angle ft is reduced, bar AC becomes shorter, but the cross-sectional areas of both bars increase because the axial forces are larger. The opposite effects occur if the angle 0 is increased. Thus, the weight of the structure (which is proportional to the volume) depends upon the angle ft. Determine the angle ft so that the structure has minimum weight without exceeding the allowable stresses in the bars. Note: The weights of the bars are very small compared to the force P and may be disregarded.A 10-ft rigid bar AB is supported with a vertical translational spring at A and a pin at B. The bar is subjected to a linearly varying distributed load with maximum intensity q0. Calculate the vertical deform at ion of the spring if the spring constant is 4 kips/in.Rigid bar ABC is supported with a pin at A and an elastic steel rod at C. The elastic rod has a diameter of 25 mm and modulus of elasticity E = 200 GPa. The bar is subjected to a uniform load q on span AC and a point load at B. Calculate the change in length of the elastic rod. What is the vertical displacement at point B?The L-shaped arm ABCD shown in the figure lies in a vertical plane and pivots about a horizontal pin at A. The arm has a constant cross-sectional area and total weight W. A vertical spring of stiffness k supports the arm at point B. (a) Obtain a formula for the elongation of the spring due to the weight of the arm. (b) Repeat part (a) if the pin support at A is moved to D.A steel cable with a nominal diameter of 25 mm (see Table 2-1) is used in a construction yard to lift a bridge section weighing 38 kN. as shown in the figure. The cable has an effective modulus of elasticity E = 140 GPa. (a) If the cable is 14 m long, how much will it stretch when the load is picked up? (b) If the cable is rated for a maximum load of 70 kN, that is the factor of safety with respect to failure of the cable?A steel wire- and an aluminum allay wire have equal lengths, and support equal loads P (see figure). The moduli of elasticity for the steel and aluminum alloy are Ea= 30,000 ksi and Ea= 11,000 ksi, respectively. (a) IF the wires have the same diameters, what is the ratio of the elongation of the aluminum alloy wire to the elongation of the steel wire? (b) If the wires stretch the same amount, what is the ratio of the diameter of the aluminum alloy wire to the diameter of the steel wire? (c) If the wires have the same diameters and same load P, what is the ratio of the initial length of the aluminum alloy wire to that of the steel wire if the aluminum alloy wire stretches 1.5 limes that of the steel wire? (d) If the wires have the same diameters, same initial length, and same load P. what is the material of the upper wire if it elongates 1.7 times that of the steel wire?By what distance h does the cage shown in the figure move downward when the weight W is placed inside it? (See the figure.) Consider only the effects of the stretching of the cable, which has axial rigidity EA = 10,700 kN. The pulley at A has a diameter da= 300 mm and the pulley at B has a diameter dB= 150 mm. Also, the distance L1= 4.6 m, the distance L2=10.5 m, and the weight W = 22 kN. Note: When calculating the length of the cable. include the parts of the cable that go around the pulley sat A and B.Rigid bar ACB is supported by an elastic circular strut DC having an outer diameter of 15 in. and inner diameter of 14.4 in. The strut is made of steel with a modulus elasticity of E = 29,000 ksi. Point load P = 5 kips is applied at B. Calculate the change in length of the circular strut DC. What is the vertical displacement of the rigid bar at point B?A plastic cylinder is held snugly between a rigid plate and. a foundation by two steel bolts (see figure). Determine the compressive stress erFin the plastic when the nuts on the steel bolts are tightened by one complete turn. Data For the assembly are as follows: length L = 200 mm, pilch of the bolt threads p= 1.0 mm, modulus of elasticity for steel Ez= 200 GPa, modulus of elasticity for the plastic Ep = 7.5 GPa, cross-sectional area of one boll As= 36.0mm2, and cross-sectional area of the plastic cylinder Af=960 mm2.A safety valve on the top of a tank containing steam under pressure p has a discharge hole of diameter d(see figure). The valve is designed to release the steam when the pressure reaches the value Pmax If the natural length of the spring, is L and its stiffness is k, what should be the dimension ft of the valve? (Express your result as a formula for h.)The device shown in the figure consists of a prismatic rigid pointer ABC supported by a uniform translational spring of stiffness k = 950 N/m. The spring is positioned a distance P = 165 nun from the pinned end A of the pointer. The device is adjusted so that, when there is no load P, the pointer reads zero on the angular scale. (a) If the load P = 11 N, al what distance .v should the load be placed so that the pointer will read ?? = 2.5° on the scale (see figure part a)? (b) Repeal part (a) if a rotational spring E1= kb-6 is added al A (see figure part b). (c) Lel.x = 7b/8.What is P maxif 0 cannot exceed 2"? Include spring krin your analysis. (d) Now, if the weight of the pointer ABC is known to be W =3N and the weight or the spring is Ws= 2.75 N, what initial angular position (Left in degrees) of the pointer will result in a zero reading on the angular scale once the pointer is released from rest? Assume P = kr=0. (e) If the pointer is rotated lo a vertical position (see figure part c), find the required load P applied at mid-height of the pointer that will result in a pointer reading of 0 = 2.5" on the scale. Consider the weight of the pointer W. in your analysis.A small lab scale has a rigid L-shaped frame ABC consisting of a horizontal arm AB (length b = 10 in.) and a vertical arm BC (length c = 7 in.) pivoted al point B. The pivot is attached to the outer frame BCD that stands on a laboratory bench. The position of the pointer al C is controlled by a spring, Jt = 5 lb/in., that is attached to a threaded rod. The pitch of the threads is p = 1/16 in. Under application of load W, 12 revolutions of the nut are required to bring the pointer back to the mark. Calculate the weight W.A small lab scale has a rigid L-shaped frame ABC consisting of a horizontal aim AB (length b = 30 cm) and a vertical arm BC (length c = 20 cm) pivoted at point B. The pivot is attached to the outer frame BCD that stands on a laboratory bench. The position of the pointer at C is controlled by two parallel springs, each having a spring constant k = 3650 N/m. that are attached to a threaded rod. The pitch of the threads is p = 1.5 mm. If the weight is 65 N. how many revolutions of the nut are required to bring the pointer back to the mark?Two rigid bars are connected to each other by two linearly elastic springs. Before loads are applied, the lengths or the springs are such, that the bars are parallel and the springs are without stress. (a) Derive a formula for the displacement E4at point 4 when the load P is applied at joint 3 and moment PL is applied at joint 1. as shown in the figure part a. (Assume that the bars rotate through very small angles under the action of load P.) (b) Repeat part (a) if a rotational spring, kr= kL2, is now added at joint 6. What is the ratio of the deflection d4 in the figure part a to that in the figure part b ?The three-bar truss ABC shown in the figure part a has a span L = 3 m and is constructed of steel pipes having a cross-sectional area A = 3900 mm2and modulus of elasticity E = 200 GPa. Identical loads P act both vertically and horizontally at joint C, as shown. (a) If P = 475 kN, what is the horizontal displacement of joint B? (b) What is the maximum permissible load value Pmaxif the displacement of joint B is limited to 1.5 mm? (c) Repeat parts (a) and (b) if the plane truss is replaced by a space truss (see figure part b).An aluminum wire having a diameter d = 1/10 in. and length L = 12 ft is subjected to a tensile load P (see figure). The aluminum has a modulus of elasticity E = 10,600 ksi If the maximum permissible elongation of the wire is l/8 in. and the allowable stress in tension is 10 ksi, what is the allowable load Pmax?A uniform bar AB of weight W = 25 N is supported by two springs, as shown in the figure. The spring on the left has a stiffness k[= 300 N/m and natural length Lt=250 mm. The corresponding quantities for the spring on the right are k2= 400 N/m and L^ = 200 mm. The distance between the springs is L = 350 mm, and the spring on the right is suspended from a support that is a distance it = SO mm below the point of support for the spring on the left. Neglect the weight of the springs. (a) At what distance x from the left-hand spring (figure part a) should a load P = 18 N be placed in order to bring the bar to a horizontal position? (b) If P is now removed, what new value of k{is required so that the bar (figure part a) will hang in a horizontal position underweight If? (c) If P is removed and kt= 300 N/m. what distance b should spring ktbe moved to the right so that the bar (figure part a) will hang in a horizontal position under weight II"? (d) If the spring on the left is now replaced by two springs in series (kt= 300 N/m, kt) with overall natural length Lt= 250 mm (see figure part b). what value of k; is required so that the bar will hang in a horizontal position under weight IF?A hollow, circular, cast-iron pipe (Ec =12,000 ksi) supports a brass rod (Ec= 14,000 ksi} and weight W — 2 kips, as shown. The outside diameter of the pipe is dc= 6 in. (a) If the allowable compressive stress in the pipe is S00O psi and the allowable shortening of the pipe is 0.02 in., what is the minimum required wall thickness trmm? (Include the weights of the rod and steel cap in your calculations.) (b) What is the elongation of the brass rod Srdue to both load Wand its own weight? (c) What is the minimum required clearance h?The horizon Lai rigid beam A BCD is supported by vertical bars BE and CF and is loaded by vertical Forces P, = 400 KN arid P2= 360 kN acting at points A and D, respectively (see figure). Bars BE and CF are made of steel (£ = 200 GPa} and have cross-sectional areas Ag=11,100 mm" and ABE= 9280 mm-. The distances between various points on the bars are shown in the figure. Determine the vertical displacements SAand SDof points A and D, respectively.Two pipe columns (AB, FC) are pin-connected to a rigid beam (BCD), as shown in the figure. Each pipe column has a modulus of E, but heights (L1or L2) and outer diameters (d1or different for each column. Assume the inner diameter of each column is 3/4 of outer diameter. Uniformly distributed downward load q = 2PIL is applied over a distance of 3L/4 along BC, and concentrated load PIA is applied downward at D. (a) Derive a formula for the displacementA framework ABC consists of two rigid bars AB and BC. Each having a length b (see the first part of the figure part a). The bars have pin connections at A, B, and C and are joined by a spring of stiffness k. The spring is attached at the midpoints of the bars. The framework has a pin support at A and a roller support al C, and the bars are at an angle a to the horizontal. When a vertical load P is applied at joint B (see the second part of the figure part a.) the roller support C moves to the right, the spring is stretched, and the angle of the bars decreases from a to the angle ??. (a) Determine the angle 0 and the increase S in the distance between points A and C. Also find reactions at A and C. (Use the following data: b = 200 mm. ft = 3.2 kN/m. a = 45°. and P = 50 N.) (b) Repeat part (a) if a translational spring kt= kll is added at C and a rotational spring kr= kb-l2 is added at A (see figure pan b).Solve the preceding problem for the following data: b = 8.0 in., k = 16 lb/in., a = 45°, and P = 10 lb.The length of the end segments of the bar (see figure) is 20 in. and the length of the prismatic middle segment is 50 in. Also, the diameters at cross sections A. B, C, and D are 0.5, 1.0, 1.0, and 0.5 in., respectively, and the modulus of elasticity is 18 ,000 ksi. (a) Calculate the elongation of a copper bar of solid circular cross section with tapered ends when it is stretched by axial loads of magnitude 3.0 kips (see figure). (b) If the total elongation of the bar cannot exceed 0.025 in., what are the required diameters at B and C? Assume that diameters at A and D remain at 0.5 in.A long, rectangular copper bar under a tensile load P hangs from a pin that is supported by two steel posts (see figure). The copper bar has a length of 2.0 m, a cross-sectional area of4S00 mm", and a modulus of elasticity Ec= 120 GPa. Each steel post has a height of 0.5 m, a cross-sectional area of 4500 mm2, and a modulus of elasticity E = 200 GRa. (a) Determine the downward displacementAn aluminum bar AD (see figure) has a cross-sectional area of 0.40 in- and is loaded by Forces Pi= 1700 lb, Pz- 1200 lb, and P3 = 1300 lb. The lengths of the segments of the bar are ti = 60 in., b = 24 in.T and c = 36 in. (a) Assuming that the modulus of elasticity is E = 10.4 × 10o psi. calculate the change in length of the bar. Does the bar elongate or shorten? (b) By what amount ^should the load Pibe increased so that the bar does not change in length when the three loads are applied? (c) IF Pzremains at 1300 lb, what revised cross-sectional area For segment AB will result in no change of length when all three loads are applied?A vertical bar consists of three prismatic segments A\, A,, and A3with cross-sectional areas of 6000 mm", 5000 mm", and 4000 mm", respectively. The bar is made of steel with E = 200 GPa. Calculate the displacements at points B. D, and E. Ignore the weight of the bar.A vertical bar is loaded with axial loads at points B, C, and D. as shown in the figure. The bar is made of steel with a modulus of elasticity E = 29,000 ksi., The bar has a cross-sectional area of 8.24 in2. Calculate the displacements at points B, C, and D. Ignore the weight of the barRepeat Problem 2.3-4, but now include the weight of the bar. Sec Table 1.1 in Appendix I for the weight density of steel.-7 Repeat Problem 2.3-5, but n include the weight of the bar. See Table I-I in Appendix I for the weight density of steel.A rectangular bar of length L has a slot in the middle half of its length (see figure). The bar has width ft, thickness t. and modulus of elasticity E. The slot has width ft/4. (a) Obtain a formula for the elongation E of the bar due to the axial loads P. (b) Calculate the elongation of the bar if the material is high-strength steel, the axial stress in the middle region is 160 MPa. the length is 750mm, and the modulus of elasticity is 210 GPa. (c) IF the total elongation of the bar is limited lo 3^ = 0.475 mm, what is the maximum length of the slotted region? Assume that the axial stress in the middle region remains at 160 MPa.Solve the preceding problem if the axial stress in the middle region is 24,000 psi, the length is 30 in., and the modulus of elasticity is 30 × 106 psi. In part (c), assume that dmax = 0.02 inA two-story building has steel columns AB in the first floor and BC in the second floor, as shown in the figure. The roof load P:equals 400 KN, and the second-floor load P-, equals 720 kN. Each column has a length L = 3.75 m. The cross-sectional areas of the first- and second-floor columns are 11,000 mm" and 3900 mm", respectively. (a) Assuming that E = 206 GPa. determine the total shortenings aof the two columns due to the combined action of the loads Ptand P,. (b) How much additional load P0can be placed at t he top of t he column (point C) if t he total shortening: SACis not to exceed 4.0 mm?A steel bar is 8.0 Ft long and has a circular cross section of diameter d1= 0.75 in. over one-half of its length and diameter d2= 0.5 in. over the other half (see figure on following page part a}. The modulus of elasticity is E = 30 × 10° psi. (a) How much will the bar elongate under a tensile load P = 5000 lb? (b) If the same volume of material is made into a bar of constant diameter d and length 8.0 ft. what will be the elongation under the same load PI (c) If the uniform axial censorial load q = 1250 lb/ft is applied to the left over segment I (see figure part b}, find the ratio of the total elongation of the bar to that in parts (a) and (bA bar ABC of length L consists of two parts of equal lengths but different diameters. Segment AB has diameter dt= 100 mm, and segment BC has diameter d2= 60 mm. Both segments have a length B2= 0.6 m. A longitudinal hole of diameter d is drilled through segment AB for one-half of its length (distance L/4 = 0.3 m). The bar is made of plastic having a modulus of elasticity E = 4.0 GPa. Compressive loads P = 110 kN act at the ends of the bar. (a) If the shortening of the bar is limited to 8.0 mm. what is the maximum allowable diameter dmaof the hole? (See figure part a.) (b) Now. if daaxis instead set at P 1/2. at what distance h from end C should load P be applied to limit the bar shortening to 8.0 mm? (See figure part b.) (c) Finally, if loads t are applied at the ends and ''mm is the permissible length .v of the hole if shortening is to be limited to 8.0 mm? (See figure part c.)A woodpile, driven into the earth, supports a load P entirely by friction along its sides (see figure part a). The friction force/per unit length of the pile is assumed to be uniformly distributed over the surface of the pile. The pile has a length L, cross-sectional area A. and modulus of elasticity E. (a) Derive a formula for the shorteningConsider the copper lubes joined in the strength of the copper is err = 200 MPa, figure using a "sweated" joint. Use the properties and what is the maximum load P]iwathat can be dimensions given. applied to the joint if the desired factor of safety in shear is FS_ - 2 and in tension is (a) Find the total elongation of segment 2-3-4 (The nonprismalic cantilever circular bar shown has an internal cylindrical hole of diameter dtl From 0 to x so the net area of the cross section n for segment I is A. Load P is applied at x, and load Ptl is applied at x = L. Assume that E is constant. (a) Find reaction force Ry (b) Find internal axial forces Ntin segments I and 2. (c} Find .v required to obtain axial displacement at joint 3 of*16 A prismatic bar AB of length L, cross-sectional area A, modulus of elasticity E, and weight Changs vertically under its own weight (see figure). (a) Derive a formula for the downward displacement Scof point E. located at distance It from the lower end of the bar. (b) What is the elongation SBof the entire bar? (c) What is the ratio £ of the elongation, of the upper half of the bar to the elongation of the lower half of the bar? (d) If bar A B is a riser pipe hanging from a drill rig at sea. what is the total elongation of the pipe? Let L = 1500 m, A - 0.ol57 m2, and E = 210 GPa. See Appendix 1 for weight densities of steel and sea water. (See Probs. 1.4-2 and J.7-13 for additional figures.)A flat bar of rectangular cross section, length L, and constant thickness t is subjected to tension by forces P (see figure). The width of the bar varies linearly from b1at the smaller end to b2at the larger end. Assume that the angle of taper is small. (a) Derive the following formula Tor the elongation of the ban (b) Calculate the elongation, assuming L = 5 ft, t = 1.0 in., P = 25 kips, ft, = 4.0 in., ft, = 6.0 bland E = 30 × 106 psi.A flat brass bar has length L, constant thickness t, and a rectangular cross section whose width varies linearly between b2at the fixed support to b1at the free end (see figure). Assume that the taper of the bar is small. The bar has modulus of elasticity E. Calculate the displacements ??Band ??cif P = 200 kN, L = 2 m, t = 20 mm, b, = 100 mm, b, = 115 mm, and E = 96 GPa.Repeat Problem 2.3-18, but assume that the bar is made of copper alloy. Calculate the displacements SBand Scif P = 50 kips, L = 5 ft = 3/5 in., b1= 2.75 in., b2= 3 in., and E = 16,000 ksi.Repeat Problem 2.3-18, but assume that the bar is made of aluminum alloy. If P2= 200 kN, what is P1so that displacementA slightly tapered bar AB of solid circular crass section and length L is supported at end B and subjected to a tensile load P at the free end A. The diameters of the bar at ends A and B are dAand dB. respectively. Determine the length of the bar if the elongation of the bar due to the load P = 45 kips is 0.02 in. Assume that E = 10,400 ksi.A circular aluminum alloy bar of length L = 1.8 m has a slot in the middle half of its length (see figure). The bar has a radius r = 36 mm and modulus of elasticity E = 72 GPa. The slot has a height 2a = r/4. Calculate the elongation of the bar due to forces P applied at the ends if the axial stress in the middle region is known to be 180 MPa.A long, slender bar in the shape of a right circular cone with length L and base diameter d hangs vertically under the action of its own weight (see figure). The weight of the cone is W and the modulus of elasticity of the material is E. Derive a formula for the increase S in the length of the bar due to its own weight. (Assume that the angle of taper of the cone is small.)A post AB supporting equipment in a laboratory is tapered uniformly throughout its height H (see figure). The cross sections of the post are square, with dimensions b × b at the top and 1.5b × 1.5b at the base. Derive a formula For the shortening 8 of the post due to the compressive load P acting at the top. (Assume that the angle of taper is small and disregard the weight of the post itself.)The main cables of a suspension bridge (see figure part a) follow a curve that is nearly parabolic because the primary load on the cables is the weight of the bridge deck, which is uniform in intensity along the horizontal. Therefore, represent the central region AOB of one of the main cables (see part b of the figure) as a parabolic cable supported at points A and B and carrying a uniform load of intensity q along the horizontal. The span of the cable is L, the sag is /i, the axial rigidity is EA\ and the origin of coordinates is at mid span. (a) Derive the following formula for the elongation of cable AOB shown in part b or the figure: (b) Calculate the elongation 5 of the central span of one of the main cables of the Golden Gate Bridge for which the dimensions and properties are L = 4200 ft,h = 470 ft, q = 12,700 lb/ft, and E = 23,300,000 psi The cable consists of 27,572 parallel wires of diameter 0.196 in. Hint: Determine the tensile force Tal any point in the cable from a free-body diagram of part of the cable; then determine the elongation of an element of the cable of length ds: finally, integrate along the curve of the cable to obtain an equation for the elongation £.A uniformly tapered lube AB of circular cross section and length L is shown in the figure. The average diameters at the ends are dAand d£= 2d t. Assume E is constant. Find the elongation S of the tube when it is subjected to loads P acting at the ends. Use the following numerical data:^ = 35 mm, L = WO mm, E = 2.1 GPa. and P = 25 tN. Consider the following cases. (a) A hole of constant diameter dAis drilled from B toward A to form a hollow section of length x - U2. (b) A hole of variable diameter a\.x) is drilled, from B toward A to form a hollow section of length x = L/2 and constant thickness t = dA/20.A vertical steel bar ABC is pin-supported at its upper end and loaded by a force Ptat its lower end. A horizontal beam BDE is pinned lo the vertical bar al joint B and supported at point D. Load P2and moment M are applied at end E. Calculate the vertical displacement 8Cat point C if the loads are Pt= 2.5 kip. P2= 1 kip. and M = 25 kip-in. The modulus of elasticity is £ = 29,000 ksi and cross-sectional areas are At= 0.25 in" and A2= 0.15 in". Ignore the weight of the bar.A T-frame structure is torn posed of a prismatic beam ABC and a nonprismatic column DBF. The beam and the column have a pin support at .A and D, respectively. Both members are connected with a pin at B. The lengths and properties of the members are shown in the figure. Find the vertical displacement of the column at points F and B. Plot axial force (AFD) and axial displacement (ADD) diagrams For column DBF.A T-frame structure is composed of prismatic beam ABC and nonprismatic column DBF that are joined at B by a friction less pin connection. The beam has a sliding, support at A and the column is pin supported at F (see figure). Beam ABC and. column segment DB have cross-sectional area A; column segment BF has area 2A. The modulus of elasticity E is the same for both members. Load 2P is applied downward at C, and load P acts at D. Find expressions for the downward displacements of column DBF at D (5D) and also at B (Repeat Problem 2.3-29 if vertical load P at D is replaced by a horizontal load P at D (see figure).A bar ABC revolves in a horizontal plane about a vertical axis at the midpoint C (see figure). The bar, which has a length 2L and crass-sectional area A, revolves at constant angular speed at. Each half of the bar (AC and BC) has a weight W, and supports a weight W2at its end. Derive the following formula for the elongation of one-half of the bar (that is. the elongation of either AC ar BC). =L223gEA(w1+3w2) in which E is t he modulus of elasticity of the material of the bar and g is the acceleration of gravity.The assembly shown in the figure consists of a brass core (diameter d:= 0.25 in.) surrounded by a steel shell {inner diameter d2= 0.23 in., outer diameter di= 0.35 in.}. A load .P compresses the core and shell that both have a length L = 4.0 in. The module of elasticity of the brass and steel are Eb=15 X 10fi psi and Es= 30 X 10fi psi, respectively. (a) What load P will compress the assembly by 0.003 in? (b) IF the allowable stress in the steel is 22 ksi and the allowable stress in the brass is 16 ksi. what is the allowable compressive load Pallow? (Suggestion: Use the equations derived in Example 2-8.)A cylindrical assembly consisting of a brass core and an aluminum collar is compressed by a load P (see figure). The length of the aluminum collar and brass core is 350 mm. the diameter of the core is 25 mm, and the outside diameter of the collar is 40 mm. Also, the module of elasticity of the aluminum and brass are 72 GPa and 100 GPa, respectively. (a) If the length of the assembly decreases by 0.1% when the load P is applied, what is the magnitude of the load? (b) What is the maximum permissible load f^ if the allowable stresses in the aluminum and brass are SO MPa and 120 M Pa, respectively?A steel bar with a uniform cross section, is fixed at both ends. A load P = 2.5 tips is applied at point C. The bar has a cross-sectional area of 8 in2. Calculate the reactions at joints A and B and the displacement at joint C. Assume that the modulus of elasticity E = 29,000 ksi.A horizontal rigid bar ABC is pinned at end A and supported by two cables at points B and C. A vertical load P = 10 kN acts at end C of the bar. The two cables are made of steel with a modulus elasticity E = 200 GPa and have the same cross-sectional area. Calculate the minimum cross-sectional area of each cable if the yield stress of the cable is 400 MPa and the factor of safely is 2.0. Consider load P only; ignore the weight of bar ABC and the cables.A solid circular steel cylinder S is encased in a hollow circular aluminum tube A. The cylinder and tube are compressed between the rigid plates of a testing machine which applies forces P. Calculate the allowable value of the compressive force if the yield stresses of sleel and aluminum are srs = 50 Ksi and ??A= 60 ksi, respectively. Assume that As= 12 in2. AA= 6 in2. L = 20 in., Es= 29,000 tsi and EA=10,600 ksiThree prismatic bars, two of material A and one of material B. transmit a tensile load P (see figure). The two outer bars (material A) are identical. The cross-sectional area of the middle bar (material B) is 50% larger than the cross-sectional area of one of the outer bars. Also, the modulus of elasticity of material A is twice that of material B. (a) What fraction of the load P is transmitted by the middle bar? (b) What is the ratio of the stress in the middle bar to the stress in the outer bars? (c) What is the ratio of the strain in the middle bar to the strain in the outer bars?A circular bar ACB of a diameter d having a cylindrical hole of length .r and diameter till from A to C is held between rigid supports at A and B. A load P acts at U2from ends A and B. Assume E is constant. (a) Obtain formulas for the reactions R, and RBat supports A and B. respectively, due to the load P (see figure part a). (b) Obtain a formula for the displacement S at the point of load application (see figure part a). (c) For what value of x is RB= (6/5)?,? (See figure part a.) (d) Repeat part (a) if the bar is now rotated to a vertical position, load P is removed, and the bar is hanging under its own weight (assume mass density = p). (See figure part b.) Assume that x = LI2.Bar ABC is fixed at both ends (see figure) and has load P applied at B. Find reactions at A and C and displacement SBif P = 200 kN. L = 2 m, t = 20 mm, b, = 100 mm, b2= 115 mm, and E = 96 GPa.Repeat Problem 2.4-8, but assume that the bar is made of aluminum alloy and that BC is prismatic. Assume that P = 20 kim. L = 3 ft.t = 314 in., b1 2m.b 2.Sin.andElO.400ksi.A plastic rod AB of length L = 0.5 m has a diameter d1= 30 mm (see figure). A plastic sleeve CD of length c = 0.3 m and outer diameter d2= 45 mm is securely bonded to the rod so that no slippage can occur between the rod and the sleeve. The rod is made of an acrylic with a modulus of elasticity E1= 3.1 GPa, and the sleeve is made of a polyamide with E2= 2.5 GPa. (a) Calculate the elongation d of the rod when it is pulled by axial forces P = 12 kN. (b) If the sleeve is extended for the full length of the rod, what is the elongation? (c) If the sleeve is removed, what is the elongation?2.4-11 Three steel cables jointly support a load of 12 kips (see figure). The diameter of the middle cable is 3/4 in. and the diameter of each outer cable is 1/2 in. The tensions in the cables are adjusted so that each cable carries one-third of the load (ie., 4 kip). Later, the load is increased by 9 kips to a total load of 21 kips. (a) What percent of the total load is now carried by the middle cable? (b) What are the stresses crvand 0 in the middle and outer cables, respectively? Note: See Table 2-1 in Section 2.2 for properties of cables.The fixed-end bar ABCD consists of three prismatic segments, as shown in the figure. The end segments have a cross-sectional area A1= 840 mm2and length Lt= 200 mm. The middle segment has a cross-sectional area A2= 1260 mm2 and length L2= 250 mm. Loads PBand Pcare equal to 25.5 kN and 17.0 kN, respectively. (a) Determine the reactions RAand RDat the fixed supports. (b) Determine the compressive axial force FBCin the middle segment of the bar.A lube structure is acted on by loads at B and D, as shown in the figure. The tubes are joined using two flange plates at C that are boiled together using six 0.5-in. diameter bolts. (a) Derive formulas for the reactions RAand REat the ends of the bar. (b) Determine the axial displacements S£. Sc, and SDat points B, C. and D. respectively. (c) Draw an axial-displacement diagram (ADD) in which the abscissa is the distance x From support A to any point on the bar and the ordinate is the horizontal displacement Sat that point. (d) Find the maximum value of the load variable P if allowable normal stress in the bolts is 14 ksi.A hollow circular pipe (see figure} support s a load P that is uniformly distributed around a cap plate at the top of the lower pipe. The inner and outer diameters of the upper and lower parts of the pipe are d1= 50 mm, d2= 60 mm, rf3 = 57 mm, and d1= 64 mm, respectively. Pipe lengths are Lt= 2 m and L, = 3 m. Neglect the self-weight of the pipes. Assume that cap plate thickness is small compared to I, and E,. Let E = 110 MPa. (a) If the tensile stress in the upper part is d = 10.5 MPa. what is load PI Also, what are reactions ft, at the upper support and R-, at the lower support? What is the stress ar(MPa) in the lower part? (b) Find displacement S(mm) at the cap plate. Plot the axial force diagram (AFD) [Ar(.f)] and axial displacement diagram (ADD)[5(.t)]. (c) Add the uniformly distributed load q along the censorial axis of pipe segment 2. Find q (kN/m) so that It, = 0. Assume that load P from part (a) is also applied.The aluminum and steel pipes shown in the figure are Fastened to rigid supports at ends A and B and to a rigid plate C at their junction. The aluminum pipe is twice as long as the steel pipe. Two equal and symmetrically placed loads P act on the plate at C. (a) Obtain formulas for the axial stresses saand sain the aluminum and steel pipes, respectively. (b) Calculate the stresses for the following data: P = 12 kips. Aa= 8.92 in2, cross-sectional area of aluminum pipe. As = 1.03 in2, cross-sectional area of steel pipe, Ea= 10 × 106 psi, modulus of elasticity of aluminum, and Es= 29 × 106 psi, modulus of elasticity of steel.A rigid bar of weight W = SOO N hangs from three equally spaced vertical wines( length L = 150 mm, spacing a = 50 mm J: two of steel and one of aluminum. The wires also support a load P acting on the bar. The diameter of the steel wires is ds= 2 mm, and the diameter of the aluminum wire is d = A mm. a Assume £,=210 GPa and EB« 70 GPa. What load Pallowcan be supported at the mitl-point of the bar (x = a) if the allowable stress in the steel wires is 220 MPa and in the aluminum wire is 80 MPa? (See figure part (b) What is /*,Ikw» if the load is positioned at .v = all1? (See figure part a.) (c) Repeat part (b) if the second and third wires are switched as shown in the figure part b.A bimetallic bar (or composite bar) of square cross sec lion with dimensions 2b X lb is construe ted of two different metals having module of elasticity E2and E2(see figure). The two parts of the bar have the same cross-sectional dimensions. The bar is compressed by forces P acting through rigid end plates. T h e line of action of t he loads has an eccentricity e of such magnitude that each part of the bar is stressed uniformly in compression. (a) Determine the axial forces Ptand P2in the two parts of the bar. (b} Determine the eccentricity e of the loads. (c) Determine the ratio C|/tr2 of the stresses in the two parts of the bar.S Three-bar truss ABC (see figure) is constructed of steel pipes having a cross-sectional area A = 3500 mm- and a modulus of elasticity E = 210 GPa. Member BC is of length L = 2.5 m, and the angle between members AC and A B is known to be 60°. Member AC length is b = 0.71 L. Loads P = 185 kN and 2P = 370 kN act vertically and horizontally at joint C. as shown. Joints A and B are pinned supports. (Use the low of shines and law of cosines to find missing dimensions and angles in the figure.) (a) Find the support reactions at joints A and B. Use horizontal reaction B, as the redundant. (b) What is the maximum permissible value of load variable P if the allowable normal stress in each trussmemberis 150 MPa?A horizontal rigid bar of weight If' = 72001b is supported by three slender circular rods that are equally spaced (see Figure)'. The two outer rods are made of aluminum (£-, = 10 X 10 psi) with diameter f/j = 0.4 in. and length Ly= 40 in. The inner rod is magnesium {F-, = 6.5 X 10* psi) with diameter rf, and length L-,. The allowable stresses in the aluminum and magnesium are 24,000 psi and 13,000 psi, respectively. IF it is desired to have all three rods loaded to their maximum allowable values, what should be the diameter d2and length LzoF the middle rod?A rigid bar ABCD is pinned at point B and supported by springs at A and D (see figure). The springs at A and D have stiff nesses k1= 10 kN/m and k2= 25 kK/m. respectively, and the dimensions a, b, and c are 250 mm. 500 mm, and 200 mm, respectively. A load P acts at point C. If the angle of rotation of the bar due to the action of the load P is limited to 3*, what is the maximum permissible load Pmax?A rigid bar AB if of a length B = 66 in. is. hinged to a support at A and supported by two vertical wires attached at points C and D (see figure). Both wires have the same cross-sectional area (A = 0.0272 in2) and are made of the same material (modulus E = 30 X 106 psi). The wire at C has a length h = 18 in. and the wire at D has a length twice that amount. The horizontal distances are c = 20 in. and d = 50 in. (a) Determine the tensile stresses acand aDin the wires due to the load P = 340 lb acting at end B of the bar. (b) Find the downward displacement 8Bat end B of the bar.Find expressions For all support reaction forces in the plane frame with load 2P applied at C, as shown in the figure. Joint A is a sliding support, joint D is pinned, and joint F is a roller support. Assume that member AC is a flat prismatic bar of length L, width b, and thickness t. Beam ABC is pinned to column DBF at mid-height (point B). Column DBF has constant thickness t and width h for DB but width 2b for BF. Consider load 2P at C only; neglect the weights of all members. The modulus of elasticity E is the same for both members. Select reaction RFas the redundant.A trimetallic bar is uniformly compressed by an axial force P = 9 kips applied through a rigid end plate (see figure}. The bar consists of a circular steel core surrounded by brass and copper tubes. The steel core has a diameter of L.25 i n., the brass tube has an outer diameter of 1.75 in., and the copper tube has an outer diameter of 2.25 in. The corresponding modulus of elasticity are f, = 30, 000 ksi, Eb= 16,000 ksi, and E = 18,000 ksi Calculate the compressive stresses ers, ab, and cin the steel, brass, and copper, respectively, due to the force P.Find expressions for all support reaction Forces in the plane frame with load 3P applied at C as shown in the figure. Joints A and D are pin supported, and there is a roller support at joint F. The lengths and the properties of the members are shown in the figure. Neglect the weights of all members. Select Rfas the redundant.The rails of a railroad track are welded together at their ends (to form continuous rails and thus eliminate the clacking sound of the wheels) when the temperature is 60°F. What compressive stress ?? =6.5×10-6 /? is produced in the rails when they are heated by the sun to 120"F if the coefficient of thermal expansion a = the modulus of elasticity E = 30 × 106 psi?A circular steel rod of diameter d is subjected to a tensile force P = 3.5 kN (see figure). The allow able stresses in tension and shear are 118 MPa and 48 MPa, respectively. What is the minimum permissible diameter dminof the rod?A rigid bar of weight W = 750 lb hangs from three equally spaced wires: two of steel and one of aluminum (see figure). The diameter of the wires is 1/8 in. Before they were loaded, all three wires had the same length. What temperature increase T in all three wires will result in the entire load being carried by the steel wires? (Assume Es= 30 × 106 psi, as= 6.5 × 10-6 /'F, and aa= 12 × 10-6F.)A steel rod. of 15-mm diameter is held snugly (but without am1 initial stresses) between rigid walls by the arrangement shown in the figure part a. (For the steel rod, use a = 12 X KT6fX and E = 200 GPa.) (a) Calculate the temperature drop AT (degrees Celsius) al which the average shear stress in the 12-mm diameter bolt becomes 45 MPa. Also, what is the normal stress in the rod? (b) What are the average bearing stresses in the bolt and clevis al A and between the washer {dw= 20 mm) and wall (r = IS mm) at £? (c) If the connection to the wall at B is changed to an end plate with two bolts (see figure part b), what is the required diameter dhof each bolt if the temperature drop is A J" = 38°C and the allowable bolt stress is 90 MPa?A bar AB of length L is held between rigid supports and heated no uniformly in such a manner that the temperature increase A T at a distance. from end A is given by the expression ?T = ?TBH/P?. where ATsis the increase in temperature at end of the bar (see figure part a). (a) Derive a formula for the compressive stress ar in the bar. (Assume that the material has modulus of elasticity E and coefficient of thermal expansion a). (b) Now modify the formula in part (a) if the rigid support at A is replaced by an elastic support at A having a spring constant Jt (see figure part b)_ Assume that only bar AB is subject to the temperature increase,A beam is constructed using two angle sections (L 102 × 76 × 6.4) arranged back to back, as shown in the figure. The beam is fixed al joint A and attached to an elastic support having a spring constant k = l750 kN/m al joint B. Assume only the beam is subjected to temperature increase AT = 45°C. Calculate the thermal stress developed in the beam and the displacement at point B. Assume that a = 12 X 10-6/?. Let E = 205 GPaA W 8 × 28 beam of a length 10 ft is held between immoveable supports. The beam has a modulus of elasticity E = 29,000 ksi and coefficient of thermal expansion a = 6.5 ×10-6 /?. If the temperature of the beam is raised uniformly by an amount AT = 20°F, calculate the thermal stress aTin the beam.A plastic bar ACB having two different solid circular cross sections is held between rigid supports, as shown in the figure. The diameters in the left-and right-hand parts are 50 mm and 75 mm. respectively. The corresponding lengths are 225 mm and 300 mm. Also, the modulus of elasticity E is 6.0 GPa. and the coefficient of thermal expansion a is 100 × 10-6 loC. The bar is subjected to a uniform temperature increase of 30°C. (a) Calculate the following quantities: (I) the compressive force N in the bar. (2) the maximum compressive stress a/, and (3) the displacement Scof point C. (b) Repeat part (a) if the rigid support at A is replaced by an elastic support having spring constant k = 50 MN/m (see figure part b; assume that only the bar ACB is subject to the temperature increase).,5-9 A flat aluminum alloy bar is fixed at both ends. Segment AB has a slight taper. If the temperature of the bar is raised uniformly by an amount AT = 20PF, find reactions al A and C. What is the displacement at B? Assume that L = 3 ft, t = 1/4 in., b1= 2 in., b2= 2.5 in., E = 10,400 ksi, and the coefficient of thermal expansion a = 13 X l0-6/'F.Repeat Problem 2.5-9 for the flat bar shown in the figure but assume that and thatA circular steel rod AB? (diameter d, = 1.0 in., length L1= 3.0 Ft) has a bronze sleeve (outer diameter d2= 1-25 in., length L2= 1.0 ft) shrunk onto it so that the two parts are securely bonded (see figure). Calculate the total elongation 6 of the steel bar ¦due to a temperature rise AT = 500°F. (Material properties are as follows: For steel, Es= 30 ×106 psi and as= 6.5 × lO6/?; for bronze, Eb= 15 × 106psi and ab= 11 × l0-6/?.)A circular, aluminum alloy bar of a length L = 1.8 m has a slot in the middle half of its length (see figure). The bar has a radius r = 36 mm and modulus of elasticity E = 72 GPa. The slot has a height 2a = r/4. If the temperature of the beam is raised uniformly by an amount AT = 15°C, calculate the thermal stress aTdeveloped in the bar. Assume that ?? = 23 × 10-6/?.Rectangular bars of copper and aluminum are held by pins at their ends, as shown in the figure. Thin spacers provide a separation between the bars. The copper bars have cross-sectional dimensions 0.5 in. × 2.0 in., and the aluminum bar has dimensions 1.0 in. × 2.0 in. Determine the shear stress in the 7/16-in. diameter pins if the temperature is raised by 100°F. (For copper, Et= 18,000 ksi and ac = 9.5 × 10-6/?; for aluminum, Ea= 10,000 ksi and aa= 13 × 10-6/?.) Suggestion: Use the results of Example 2-10A brass sleeve S is fitted over a steel bolt B (see figure), and the nut is lightened until it is just snug. The bolt has a diameter dB= 25 mm, and the sleeve has inside and outside diameters d1= 26 mm and d2= 36 mm. respectively. Calculate the temperature rise .?T that is required to produce a. compressive stress of 25 MPa in the sleeve. (Use material properties as follows: for the sleeve, as= 21 × 10-6 /? and Es= 100 GPa; for the boll, aB= 10 × 10-6 /? and EB= 200PaA rigid triangular frame is pivoted at C and held by two identical horizontal wires at points A and B (see figure). Each wire has an axial rigidity EA = 120 kips and coefficient of thermal expansion a = 12.5 X 10-6/°F. (a) If a vertical load P = 500 lb acts at point D, what are the tensile forces TAand TBin the wires at A and B, respectively? (b) If both wires have their temperatures raised by 180°F while the load P is acting, what are the forces TAand TB (c) What further increase in temperature will cause the wire at B to become slack?,5-16 A rigid bar ABCD is pinned at end A and supported by two cables at points Band C (see figure). The cable at B has a nominal diameter dA copper bar AB with a length 25 in. and diameter 2 in. is placed in position at room temperature with a gap of 0.008 in. between end A and a rigid restraint (see figure). The bar is supported at end B by an elastic spring with a spring constant k= 1.2 × 106 lb/in. (a) Calculate the axial compressive stress crcin the bar if the temperature of the bar only rises 50 F. (For copper, use a = 9.6 × 10-6/ and E = 16 × 106 psi.) (b) What is the force in the spring? (Neglect gravity effects.) (c) Repeat part (a) if k ? 8.A steel wire AB is stretched between rigid supports (see figure). The initial priestess in the wire is 42 MPa when the temperature is 20°C. (a) What is the stress cr in the wire when the temperature drops to 0°C? (b) At what temperature Twill the stress in the wire become zero? (Assume d = 14 X 10 FC and E = 200 GPa)-19 The mechanical assembly shown in the figure consists of an aluminum tube, a rigid end plate, and two steel cables. The slack is removed from the cables by rotating the turnbuckles until the assembly is snug but with no initial stresses. Afterward, the turnbuckles are tightened by 1.5 turns. Calculate the forces in the tube and the cables and determine the shortening of the tube. As= 0.85 in2 for each cable, AA= 4.5 in2, L = 20 in., Es= 29,000 ksi, EA= 10,600 ksi, and p = 1/16 inA bar AB having a length L and axial rigidity EA is fixed at end A (see figure). At the other end, a small gap of dimension sexists between the end of the bar and a rigid surface. A load P acts on the bar at point C. which is two-thirds of the length from the fixed end. IF the support reactions produced by load P are to be equal in magnitude, what should be the size of the gap?Pipe 2 has been inserted snugly into Pipe I. but the holes Tor a connecting pin do not line up; there is a gap s. The user decides to apply either force P:lo Pipe I or force P-, to Pipe 2, whichever is smaller. Determine the following using the numerical properties in the box. (a) If only P{is applied, find Pt{tips} required to close gap s; if a pin is then inserted and Ptremoved, what are reaction forces RAand RBfor this load case? (b) If only P2is applied, find P2{kips) required to close gap a; if a pin is inserted and P2removed, what are reaction forces R^ and RBfor this load case? (c) What is the maximum shear stress in the pipes, for the loads in parts (a) and (b)? (d) If a temperature increase IT is to be applied to the entire structure to close gaps{instead of applying forces Ptand P2), find the AT required to close the gap. If a pin is inserted after the gaphas closed, what are reaction forces .''.', and RBfor this case? (e) Finally, if the structure (with pin inserted) then cools to the original ambient temperature, what are reaction forces Rtand PA non prism elk- bar ABC made up of segments AB(length £,, cross-sectional area .Inland BC (length i-,, cross-sectional area A2) is fixed at end A and free al end C (see figure). The modulus of elasticity of the bar is E. A small gap of d intension s exists between the end of the bar and an elastic spring of length Lj and spring constant k3. If bar ABC only (not tin? spring} is subjected to temperature increase A3", determine the following. (a) Write an expression for reaction forces R^ and RDif the elongation of /I BC exceeds gap length s. (b) Find expressions for the displacements of points B and C if the elongation of ABC exceeds gap length s.Wires B and C are attached to a support at the left-hand end and to a pin-supported rigid bar at the right-hand end (see figure). Each wire has cross-sectional area A =0.03 in2 and modulus of elasticity E = 30 X 106 psi. When the bar is in a vertical position, the length of each wire is L = 80 in. However, before being attached to the bar, the length of wire B was 79.98 in. and wire C was 79.95 in. Find the tensile forces TBand Tc in the wires under the action of a force P = 700 lb acting at the upper end of the bar.A rigid steel plate is supported by three posts of high-strength concrete each having an effective cross-sectional area A = 40,000 mm2 and length L = 2 m (see figure). Before the load P is applied, the middle post is shorter than the others by an amount s = 1.0 mm. Determine the maximum allowable load Pallowif the allowable compressive stress in the concrete is sallow = 20 MPa. (Use E = 30 GPa for concrete.)A capped cast-iron pipe is compressed by a brass rod, as shown. The mil is turned until it is just snug, then add an additional quarter turn to pre-compress the cast-iron pipe. The pitch of the threads of the bolt ap = 52 mils (a mil is one-thousandth of an inch). Use the numerical properties provided. (a) What stresses a and arwill be produced in the cast-iron pipe and brass rod. respectively, by the additional quarter turn of the nut? (b) Find the bearing stress ahbeneath the washer and the shear stress t(in the steel cap.A plastic cylinder is held snugly between a rigid plate and. a foundation by two steel bolts (see figure). Determine the compressive stress erFin the plastic when the nuts on the steel bolts are tightened by one complete turn. Data For the assembly are as follows: length L = 200 mm, pilch of the bolt threads p= 1.0 mm, modulus of elasticity for steel Ez= 200 GPa, modulus of elasticity for the plastic Ep = 7.5 GPa, cross-sectional area of one boll As= 36.0mm2, and cross-sectional area of the plastic cylinder Af=960 mm2.2.5.27PConsider the sleeve made From two copper tubes joined by tin-lead solder over distance s. The sleeve has brass caps at both ends that are held in place by a steel bolt and washer with the nut turned just snug at the outset. Then, two "loadings" are applied: a = 1/2 turn applied to the nut; at the same time, the internal temperature is raised by ?T = 30°C. (a) Find the forces in the sleeve and boll, Psand PB, due to both the priestess in the bolt and the temperature increase. For copper, use EI= 120 GPa and ac= 17 × W+C; for steel, use E, = 200 GPa and a, = 12 × 10-6/°C. The pitch of the boll threads is p = 1.0 mm. Assume s = 26 mm and bolt diameter dB= 5 mm. (b) Find the required length of the solder joint, s, if shear stress in the sweated joint cannot exceed the allowable shear stress t ™ 18.5 MPa. (c) What is the final elongation of the entire assemblage due to both temperature change A T and the initial prestress in the bolt?A polyethylene tube (length L) has a cap that when installed compresses a spring (with under-formed length L1) by an amount ?? = (L1 = L). Ignore deformations of the cap and base. Use the force at the base of the spring as the redundant. Use numerical properties given in the boxes. (a) What is the resulting Force-in the spring, Fk? (b) What is the resulting Force in the tube, Ftl (c) What is the filial length of the tube, Lf? (d) What temperature change ?T inside the tube will result in zero force in the springPrestressed concrete beams are sometimes manufactured in the following manner. High-strength steel wires are stretched by a jacking mechanism that applies a force Q, as represented schematically in part a of the figure. Concrete is then poured around the wires to form a beam, as shown in the figure part b. After the concrete sets properly, the jacks are released, and the force Q is removed (see part c of the figure). Thus, the beam is left in a prestressed condition with the wires in tension and the concrete in compression. Assume that the prestressing force Q produces in the steel wires an initial stress a0= 620 MPa. If the module of elasticity of the steel and concrete are in the ratio 12:1 and the cross-sectional areas are in the ratio 1:50. what are the final stresses a5and a5 in the two materials?2.5.31PA steel bar of rectangular cross section (1.5 in. × 2.0 in.) Carries a tensile load P (see figure). The allowable stresses in tension and shear are 14,500 psi and 7,100 psi, respectively. Determine the maximum permissible load Pmax.A circular steel rod of diameter d is subjected to a tensile force P = 3.5 kN (see figure). The allow able stresses in tension and shear are 118 MPa and 4S MPa, respectively. What is the minimum permissible diameter dminof the rod?A standard brick (dimensions 8 in. × 4 in. × 2.5 in ) is compressed lengthwise by a force P. as shown in the figure, If the ultimate shear stress for brick is 1200 psi and the ultimate compressive stress is 3600 psi. what force Pmax is required to break the brick?A brass wire of diameter d = 2.42 mm is stretched tightly between rigid supports so that the tensile force is T = 98 N (see figure). The coefficient of thermal expansion for the wire is 19.5 × 10-6/°C. and the modulus of elasticity is E = 110 GPa. (a) What is the maximum permissible temperature drop AT if the allowable shear stress in the wire is 60 MPa? (b) At what temperature change does the wire go slack?2.6.5PA steel bar with a diameter d = 12 mm is subjected to a tensile load P = 9.5 kN (see figure). (a) What is the maximum normal stresser smax in the bar? (b) What is the maximum shear stress tmax? (c) Draw a stress element oriented at 45° to the axis of the bar and show all stresses acting on the faces of this element. (d) Repeat part (c) for a stress element oriented at 22.5° to the axis of the bar.During a tension lest of a mild-steel specimen (see figure), the extensometer shows an elongation of 0.00120 in. with a gage length of 2 in. Assume that the steel is stressed below the proportional limit and that the modulus of elasticity E = 30 × 10 psi. (a) What is the maximum normal stress (j^, in the specimen? (b) What is the maximum shear stress tmax? (c) Draw a stress element oriented at an angle of 45° to the axis of the bar, and show all stresses acting on the faces of this element.A copper bar with a rectangular cross section is held without stress between rigid supports (see figure). Subsequently, the temperature of the bar is raised 50°C (a) Determine the stresses on all faces of the elements A and B, and show these stresses on sketches of the elements. (Assume = 17.5 × 10-6/? and E = 120 GPa ) (b) If the shear stress at B is known to be 48 MPa at some inclination 8, find angleA prismatic bar with a length L = 3 ft and cross-sectional area A = 8 in2 is compressed by an axial centroidal load P = 10 kips. Determine the complete state of stress acting on an inclined section pq that is cut through the bar at an angle W = 35", and show the stresses on a properly oriented stress element.A prismatic bar with a length L = 1 m and cross-sectional area A = 1200 mm2 is supported at the ends. The bar is then subjected to a temperature increase ?T = 25°C. Calculate the complete stale of stress acting on an inclined section .rs that is cut through the bar at an angle ?? = 45*. Use E = 200 GPa and the coefficient of thermal expansion a = 12×10-6/?.The plane truss in the figure is assembled From steel C 10 X 20 shapes (see Table 3(a) in Appendix F). Assume that L = 10 ft and b = 0 71 L. (a) If load variable P = 49 kips, what is the maximum shear stress Tmaxin each truss member? (b) What is the maximum permissible value of load variable P if the allowable normal stress is 14 ksi and the allowable shear stress is 7.5 ksi?Plastic bar of diameter d = 32 mm is compressed in a testing device by a Force P = 190 N that is applied as shown in the figure. (a) Determine the normal and shear stresses acting: on all faces of stress elements oriented at (1 ) an angle 8 = 00, (2) an angle ?? = 22.5s, and (3) an angle ?? = 45°. In each case, show the stresses on a sketch of a properly oriented element. What are smaxtmax (b) Find smax and tmax in the plastic bar if a re-cantering spring of stiffness k is inserted into the testing device, as shown in the figure. The spring stillness is 1/6 of the axial stiffness of the plastic bar.A plastic bar of rectangular cross section (ft = 1.5 in. and h = 3 in.) fits snugly between rigid supporls at room temperature (68oF) but with no initial stress (see Figure). When the temperature of the bar is raised to 160oF, the compressive stress on an inclined plane pq at mid-span becomes 1700 psi. (a) What is the shear stress on plane pq? (Assume a = 60 × 10-6/*t and E = 450 × 103psi.) (b) Draw a stress element oriented to plane pq and show the stresses acting on all laces of this element. (c) If the allowable normal stress is 3400 psi and the allowable shear stress is 1650 psi. what is the maximum load P (in the positive x direction), which can be added at the quarter point (in addition to thermal effects given) without exceeding allowable stress values in the bar?A copper bar of rectangular cross section (b = 18 mm and k = 40 mm) is held snugly (but without any initial stress) between rigid supports (see figure). The allowable stresses on the inclined plane pq at mid-span, for which ?? = 55°. are specified as 60 MPa in compression and 30 MPa in shear. (a) What is the maximum permissible temperature rise AT if the allowable stresses on plane pq are not to be exceeded? (Assume a = 17 × 10-6/oC and E = 120 GPa.) (b) If the temperature increases by the maximum permissible amount, what are the stresses on plane pq? (c) If the temperature rises AT = 2SoC how far to the right of end A (distance BL, which is expressed as a fraction of length L) can load P = 15 kN be applied without exceeding allowable stress values in the bar? Assume that a =75 MPa and t = 35 MPa.A circular brass bar with a diameter J is member AC in truss ABC thai has load P = 5000 lb applied at joint C. Bar AC is composed of two segments brazed together on a plane pq. making an angle a 36 with the axis of the bar (see figure). The allowable stresses in the brass are 13.500 psi in tension and 6500 psi in shear. On the brazed joint, the allow. able stresses arc 6000 psi in tension and 3000 psi in shear. What is the tensile force NAC in bar AC? What is the minimum required diameter d of bar AC?Two boards are joined by gluing along a scarf joint, as shown in the figure. For purposes of cutting and gluing, the angle a between the plane of the joint and the faces of the boards must be between 10° and 40f. Under a tensile load P, the normal stress in the boards is 4.9 MPa. (a) What axe the normal and shear stresses acting on the glued joint if a = 20°? (b) If the allowable shear stress on the joint is 2.25 MPa. what is the largest permissible value of the angle ct? (c) For what angle a with the shear stress on the glued joint be numerically equal to twice the normal stress on the joint?Acting on the sides of a stress element cut from a bar in uniaxial stress are tensile stresses of 10,000 psi and 5000 psi, as shown in the figure. (a) Determine the angle 0 and the shear stress T and show all stresses on a sketch of the element. (b) Determine the maximum normal stress amaxand the maximum shear stress Tmax in the material.A prismatic bar is subjected to an axial force that produces a tensile stress ????=65 MPa and a shear stress T??= 23 MPa on a certain inclined plane (see figure). Determine the stresses acting oil all laces of a stress element oriented at ?? = 30°, and show the stresses on a sketch of the element.The normal stress on plane pq of a prismatic bar in tension (see figure) is found to be 7500 psi. On plane rs, which makes an angle B = 30° with plane pq. the stress is found to be 2500 psi. Determine the maximum normal stress a and maximum shear stress tmax in the bar.A tension member is to be constructed of two pieces or plastic glued along plane pq (see figure). For purposes of cutting and gluing, the angle ?? must be between 25 and 45. The allowable stresses on the glued joint in tension and shear are 5.0 MPa and 3.0 MPa, respectively (a) Determine the angle it so that the bar will carry the largest load P. (Assume that the strength of the slued joint controls the design.) (b) Determine the maximum allowable load ?? if the cross-sectional area of the bar is 225-21 Plastic bar AB of rectangular cross section (6 = 0.75 in. and h = 1.5 in.) and length L = 2 Ft is Fixed at A and has a spring support (Ar = 18 kips/in.) at C (see figure). Initially, the bar and spring have no stress. When the temperature of the bar is raised hy foot. the compressive stress on an inclined plane pq at Lq = 1.5 Ft becomes 950 psi. Assume the spring is massless and is unaffected by the temperature change. Let a = 55 × l0-6p and E = 400 ksi. (a) What is the shear stresst9 on plane pq? What is angle 07 =1 Draw a stress element oriented to plane pq, and show the stresses acting on all laces of this element. (c) If the allowable normal stress is ± 1000 psi and the allowable shear stress is ±560 psi, what is the maximum permissible value of spring constant k if the allowable stress values in the bar are not to be exceeded? (d) What is the maximum permissible length L of the bar if the allowable stress values in the bar are not be exceeded? (Assume £ = IB kips/in.) (e) What is the maximum permissible temperature increase (A7") in the bar if the allowable stress values in the bar are not to be exceeded? (Assume L = 2 ft and k = L& kips/inA compression bar having a square cross section with sides b = 50 mm is subjected to load P. The bar is constructed Tram two pieces of wood that are connected by a glued joint along plane pq that is inclined at angle a = 35°. The allowable stress in the wood in compression is 11.5 MPa and in shear is 4.5 MPa. Also, the allowable stress in the glued joint in compression is 3.5 MPa and in shear is 1.25 MPa. Determine the maximum load P that can be applied to the barA prismatic bar AD of length L, cross-sectional area A. and modulus of elasticity E is subjected to loads 5P, 3P, and P acting at points B, C, and D, respectively (see figure). Segments AB, BC, and CD have lengths L/6, L/2, and L/3, respectively. (a) Obtain a formula for the strain energy U of the bar. (b) Calculate the strain energy if P = 6 kips, L = 52 in., A = 2.76 in2, and the material is aluminum with E = 10.4 × 106 psi.A bar with a circular cross section having two different diameters d and 2d is shown in the figure. The length of each segment of the bar is L/2T and the modulus of elasticity of the material is E. (a) Obtain a formula for the strain energy U of the bar due to the load P. (b) Calculate the strain energy if the load P = 27 kN, the length L = 600 mm, the diameter d = 40 mm, and the material is brass with E = 105 GPa.A three-story steel column in a building supports roof and floor loads as shown in the figure. The story height H is 10.5 ft. the cross-sectional area A of the column is 15.5 in2, and the modulus of elasticity E of the steel is 30 × 106 psi. Calculate the strain energy U of the column assuming P1= 40 kips and P2= P3= 60 kips.The bar ABC shown in the figure is loaded by a force P acting at end C and by a force Q acting at the midpoint B. The bar has a constant axial rigidity EA. (a) Determine the strain energy U1of the bar when the Force P acts alone (Q = 0). (b) Determine the strain energy U2when the force Q acts alone (P = 0). (c) Determine the strain energy U3when the Forces P and Q act simultaneously upon the barDetermine the strain energy per unit volume (units of psi) and the strain energy per unit weight (units of in ) that can be stored in each or the materials listed in the accompanying table, assuming that the material is stressed to the proportional limit. DATA FOR PROBLEM 2.7-5 Material Weight Density (lb/in3) Modulus of Elasticity (ksi) Proportional Limit (psi) Mild sleel 0.284 30,000 36,000 Tool steel 0.284 30,000 75,000 Aluminum 0.0984 10,500 60,000 Rubber (soft) 0.0405 0.300 300The truss ABC shown in the Figure is subjected to a horizontal load P at joint B. The two bars are identical with cross-sectional area A and modulus of elasticity E. (a) Determine the strain energy U of the truss if the angle ß = 60°. (b) Determine the horizontal displacement dBof joint B by equating the strain energy of the truss to the work done by the load.-7 The truss A BC Shawn in the figure supports a horizontal load P1= 300 lb and a vertical load P2= 9001b. Both bars have a cross-sectional area A = 2.4 in2 and are made of steel with E = 30 X 106 psi. (a) Determine the strain energy U1of the truss when the load P1acts alone (P2= 0). (b) Determine the strain energy U2when the load P2acts alone (P1= 0). (c) Determine the strain energy U3when both loads act simultaneously.The statically indeterminate structure shown in the figure consists of a horizontal rigid bar AB supported by five equally spaced springs. Springs l, 2, and 3 have stiff nesses 3k, 5k. and k, respectively. When unstressed, the lower ends of all Five springs lie along: a horizontal line. Bar AB. which has weight W, causes the springs to elongate by an amount S. (a) Obtain a formula For the total strain energy of the springs in terms of the downward displacement d of the bar. (b) Obtain a formula for the displacement S by equating the strain energy of the springs to the work done by the weight W.A slightly tapered bar AB of rectangular cross section and length L is acted upon by a force P (see figure). The width of the bar varies uniformly From b2at end A to b1at end B. The thickness t is constant. (a) Determine the strain energy U of the bar. (b) Determine the elongation ?? of the bar by equating the strain energy to the work done by the force P.A compressive load P is transmitted through a rigid plate to three magnesium-alloy bars that are identical except that initially the middle bar is slightly shorter than the other bars (see figure). The dimensions and properties of the assembly are as follows: length L = 1.0 m, cross-sectional area of each bar A = 3000 mm", modulus of elasticity E = AS GPa, and the gap s = 1.0 mm. (a) Calculate the load Ptrequired to close the gap. (b) Calculate the downward displacement 5 of the rigid plate when P = 400 kN. (c) Calculate the total strain energy V of the three bars when P = 400 kN (d) Explain why the strain -energy V is not equal to PS/2. Hint: Draw a load-displacement diagram.A block B is pushed against three springs by a force P (see figure). The middle spring has a stillness K1and the outer springs each have stiffness k^. Initially, the springs are unstressed, and the middle spring is longer than the outer springs (the difference in length is denoted s). (a) Draw a force-displacement diagram with the force P as ordinate and the displacement x of the block as abscissa. (b) From the diagram, determine the strain energy U1 of the springs when x = 2s. (c) Explain why the strain energy E, is not equal to PA bungee cord that behaves linearly elastically has an unstressed length L0= 760 mm and a stiffness k = 140 N/m. The cord is attached to two pegs, distance/? = 380 mm apart, and is pulled at its midpoint by a Force P = 80 N (see figure). (a) How much strain energy U is stored in the cord? (b) What is the displacement Scof the point where the load is applied? (c) Compare the strain energy (with the quantity PSC12. Note: The elongation of the cord is not small compared lo its original length.A sliding collar of weight W = 150 lb falls From a height h = 2.0 in. onto a flange at the bottom of a slender vertical rod (see figure). The rod has a length L = 4.0 ft, cross-sectional area A = 0.75 in2, and modulus of elasticity E = 30 X 106 psi. Calculate the following quantities: (a) the maximum downward displacement of the flange, (b) the maximum tensile stress in the rod, and (c) the impact factor.Solve the preceding problem if the collar has mass M = 80 kg, the height h = 0.5 m, the length L = 3.0 m, the cross-sectional area A = 350mm2. and the modulus of elasticity E = 170 GPa.2.8.3PA block weighing W = 5.0 N drops inside a cylinder from a height h = 200 mm onto a spring having stiffness k = 90 N/m (see figure), (a) Determine the maximum shortening of the spring due to the impact and (b) determine the impact factor.Solve the preceding problem for W = 1.0 lb. h = 12 in.,and k =0.511,/in.2.8.6PA weight W = 4500 lb falls from a height h onto a vertical wood pole having length L = 15 ft, diameter d = 12 in., and modulus of elasticity E = 1.6 × 106 psi (see figure). If the allowable stress in the wood under an impact load is 2500 psi. what is the maximum permissible height h?2.8.8P2.8.9PA bumping post at the end of a track in a railway yard has a spring constant k = 8.0 MN/m (see figure). The maximum possible displacement d or the end of the striking plate is 450 mm. What is the maximum velocity vmaxthat a railway car of weight W = 545 kN can have without damaging the bumping post when it strikes it?A bumper for a mine car is constructed with a spring of stiffness k = 1120 lb/in. (see figure). If a car weighing 3450 lb is traveling at velocity v = 7 mph when it strikes the spring, what is the maximum shortening of the spring?A bungee jumper having a mass of 55 kg leaps from a bridge, braking her fall with a long elastic shock cord having axial rigidity EA = 2.3 kN (see figure). If the jumpoff point is 60 m above the water, and if it is desired to maintain a clearance of 10 m between the jumper and the water, what length L of cord should be used?2.8.13PA rigid bar AB having a mass M = 1.0 kg and length L = 0.5 m is hinged at end A and supported at end B by a nylon cord BC (see figure). The record has cross-sectional area A = 30 mm2. length b = 0.25 m. and modulus of elasticity E = 2.1 GPa. If the bar is raised to its maximum height and then released, what is the maximum stress in the cord?The flat bars shown in parts a and b of the figure are subjected to tensile forces P = 3.0 kips. Each bar a has thickness t = 0.25 in. (a) For the bar with a circular hole, determine the maximum stresses for hole diameters d = 1 in. and d = 2 in. if the width b = 6.0 in. (b) For the stepped bar with shoulder fillets, determine the maximum stresses for fillet radii R = 0.25 in. and R = 0.5 in. if the bar widths are b = 4.0 in. and c = 2.5 in.The flat bars shown in parts a and b of the figure are subjected to tensile forces P = 2.5 kN. Each bar has thickness t = 5.0 mm. (a) For the bar with a circular hole, determine the maximum stresses for hole diameters d = 12 mm and d = 20 mm il" the width h = 60 mm. (b) For the stepped bar with shoulder fillets, determine the maximum stresses Tor fillet radii R = 6 mm and R = 10 mm if the bar widths are h = 60 mm and c = 40 mm.A flat bar of width b and thickness t has a hole of diameter d drilled through it (see figure). The hole may have any diameter that will fit within the bar. What is the maximum permissible tensile load Pmaxif the allowable tensile stress in the material is st?Around brass bar of a diameter d1= 20mm has upset ends each with a diameter d2= 26 mm (see figure). The lengths of the segments of the bar are L1= 0.3 m and L2= 0.1 m. Quarter-circular fillets are used at the shoulders of the bar, and the modulus of elasticity of the brass is E = 100 GPa. If the bar lengthens by 0.12 mm under a tensile load P, what is the maximum stress ??maxin the bar?2.10.5P,10-6 A prismatic bar with a diameter d0= 20 mm is being compared with a stepped bar of the same diameter(d| = 20 mm) that is enlarged in the middle region lo a diameter d2= 25 mm (see figure). The radius of the fillets in the stepped bar is 2.0 mm. (a) Does enlarging the bar in the middle region make it stronger than the prismatic bar? Demonstrate your answer by determining the maximum permissible load for the prismatic bar and the maximum permissible load F, for the enlarged bar, assuming that the allowable stress for the material is 80 MPa. (b) What should be the diameter d0of the prismatic bar if it is to have the same maximum permissible load as does the stepped bar?A stepped bar with a hole (see figure) has widths h = 2.4 in. and c = 1.6 in. The fillets have radii equal to 0.2 in. What is the diameter d max of the largest hole that can be drilled through the bar without reducing the load-carrying capacity?A bar AB of length L and weight density y hangs vertically under its awn weight (see figure). The stress-strain relation forth? Material is given by the Romberg-Osgood equation [Eq. (2-71)]: Derive the formula For the elongation of the bar.A prismatic bar of length L = 1.8 m and cross-sectional area A = 480 mm" is loaded by forces P{= 30 kN and A = 60 kN (see figure) The bar is constructed of magnesium alloy having a stress-strain curve described by the Ram berg-Osgood equation: 45.000 618 UW id which u has units of mega pascals (MPa). (a) Calculate the displacement 8t- of the end of the bar when the load P:acts alone. (b) Calculate the displacement when the load P, acts alone. (c) Calculate the displacement when both loads act simultaneously.