Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Mechanics of Materials (MindTap Course List)
Solve the preceding problem for the following data: diameter LO m, thickness 48 mm, pressure 22 MPa, modulus 210 GPa. and Poisson's ratio 0.29A spherical stainless-steel tank having a diameter of 26 in. is used to store propane gas at a pressure of 2075 psi. The properties of the steel are as follows: yield stress in tension, 140,000 psi; yield stress in shear, 65,000 psi; modulus of elasticity, 30 X 106 psi; and Poisson's ratio, 0.28. The desired factor of safety with respect to yielding is 2.8. Also, the normal strain must not exceed 1250 X 10-6. (a) Determine the minimum permissible thickness fin of the tank. (b) If the tank thickness is 0.30 in. and normal strain is measured at 990 x 10~ , what is the internal pressure in the tank at that point?Solve the preceding problem if the diameter is 480 mm, the pressure is 20 MPa, the yield stress in tension is 975 MPa, the yield stress in shear is 460 MPa, the factor of safety is 2,75, the modulus of elasticity is 210 GPa, Poissorfs ratio is 0.28, and the normal strain must not exceed 1190 x 10" . For part (b), assume that the tank thickness is 8 mm and the measured normal strain is 990 x 10~ .: A hollow, pressurized sphere having a radius r = 4.8 in, and wall thickness t = 0.4 in. is lowered into a lake (see figure). The compressed air in the tank is at a pressure of 24 psi (gage pressure when the tank: is out of the water). At what depth D0will the wall of the tank be subjected to a compressive stress of 90 psi?A fire extinguisher tank is designed for an internal pressure of 825 psi. The tank has an outer diameter of 4.5 in. and thickness of O.OS in. Calculate the longitudinal stress, the circumferential stress, and the maximum shear stresses (out-of-plane and in-plane) at the outer surface of the tank.8.3.2PA scuba t a n k (see fig ure) i s bci ng d e signed fo r an internal pressure of 2640 psi with a factor of safety of 2,0 with respect to yielding. The yield stress of the steel is 65,000 psi in tension and 32,000 psi in shear, (a) If the diameter of the tank is 7,0 in., what is the minimum required wall thickness? (b) If the wall thickness is 0.25 in., what is the maximum acceptable internal pressure?A tall standpipc with an open top (see figure) has diameter d = 2,2 m and wall thickness t = 20 mm, (a) What height h of water will produce a circumferential stress of 12 MPa in the wall of the standpipe? (b) What is the axial stress in the wall of the tank due to the water pressure?An inflatable structure used by a traveling circus has the shape of a half-circular cylinder with closed ends (see figure). The fabric and plastic structure is inflated by a small blower and has a radius of 40 ft when fully inflated. A longitudinal scam runs the entire length of the "ridge" of the structure. If the longitudinal scam along the ridge tears open when it is subjected to a tensile load of 540 pounds per inch of seam, what is the factor of safety n against tearing when the internal pressure is 0,5 psi and the structure is fully inflated?A thin-walled cylindrical pressure vessel of a radius r is subjected simultaneously to internal gas pressure p and a compressive force F acting at the ends (see figure), (a) What should be the magnitude of the force F in order to produce pure shear in the wall of the cylinder? (b) If force F = 190 kN, internal pressure p = 12 MPa, inner diameter = 200 mm, and allowable normal and shear stresses are 110 MPa and 60 MPa, respectively, what is the required thickness of the vessel?A strain gage is installed in the longitudinal direction on the surface of an aluminum beverage can (see figure). The radius-to-thickness ratio of the can is 200. When the lid of the can is popped open, the strain changes by e0 = 187 x 10-6. (a) What was the internal pressure/* in the can? (Assume E = 10 x 106 psi and v = 0J3.) (b) What is the change in strain in the radial direction when the lid is opened?A circular cylindrical steel tank (see figure) contains a volatile fuel under pressure, A strain gage at point A records the longitudinal strain in the tank and transmits this information to a control room. The ultimate shear stress in the wall of the tank is 98 MPa, and a factor of safety of 2,8 is required. (a) At what value of the strain should the operators take action to reduce the pressure in the tank? (Data for the steel are modulus of elasticity E = 210 GPa and Poisson's ratio v = 0.30.) (b) What is the associated strain in the radial directionA cylinder filled with oil is under pressure from a piston, as shown in the figure. The diameter d of the piston is 1,80 in. and the compressive force F is 3500 lb. The maximum allowable shear stress t^^, in the wall of the cylinder is 5500 psi. What is the minimum permissible thickness t„± of the cylinder wall? (Sec figureSolve the preceding problem if F =90 mm, F = 42 kN, and t = 40°MPaA standpipe in a water-supply system (see figure) is 12 ft in diameter and 6 in. thick. Two horizontal pipes carry water out of the standpipe; each is 2 ft in diameter and 1 in. thick. When the system is shut down and water fills the pipes but is not moving, the hoop stress at the bottom of the standpipe is 130 psi, (a) What is the height h of the water in the standpipe? (b) If the bottoms of the pipes are at the same elevation as the bottom of the stand pipe. what is the hoop stress in the pipes?A cylindrical tank with hemispherical heads is constructed of steel sections that are welded circumferentially (see figure). The tank diameter is 1.25 m, the wall thickness is 22 mm, and the internal pressure is 1750 kPa, (a) Determine the maximum tensile stress h in the heads of the tank. (b) Determine the maximum tensile stress c in the cylindrical part of the tank. (c) Determine the tensile stress <7W acting perpendicular to the welded joints, (d) Determine the maximum shear stress rtiin the heads of the tank. (e) Determine the maximum shear stress tcin the cylindrical part of the tank: A cylindrical tank with diameter d = 18 in, is subjected to internal gas pressure p = 450 psi. The tank is constructed of steel sections that arc welded circum fereiitially (sec figure). The heads of the tank are hemispherical. The allowable tensile and shear stresses are 8200 psi and 3000 psi, respectively. Also, the allowable tensile stress perpendicular to a weld is 6250 psi. Determine the minimum required thickness /minof (a) the cylindrical part of the tank and (b) the hemispherical heads.A pressurized steel tank is constructed with a helical weld that makes an angle at = 55" with the longitudinal axis (see figure). The tank has radius r = 0.6 m, wall thickness t = 18 mm, and internal pressure p = 2.8 MPa. Also, the steel has modulus of elasticity E = 200 GPa and Poisson's ratio v = 0.30. Determine the following quantities for the cylindrical part of the tank. (a) The circumferential and longitudinal stresses. (b) The maximum in-plane and out-of-plane shear stresses. (c) The circumferential and longitudinal strains. (d) The normal and shear stresses acting on planes parallel and perpendicular to the weld (show these stresses on a properly oriented stress elementSolve the preceding problem for a welded Tank with a = 62°,r = 19 irM = 0.65 in.,p = 240 psu E = 30 X 106 psi, and v = 0.30A wood beam with a cross section 4 x 6 in. is simply supported at A and B. The beam has a length of 9 ft and is subjected to point load P = 5 kips at mid-span. Calculate the state of stress at point C located 4 in. below the top of the beam and 4.5 ft to the right of support A. Neglect the weight of the beam8.4.2PA simply supported beam is subjected to two point Loads, each P = 500 lb, as shown in the figure. The beam has a cross section of 6 X 12 in. Find the state of plane stress at point C located S in. below the top of the beam and 0,5 ft to the right of support A. Also find the principal stresses and the maximum shear stress at CA cantilever beam with a width h = 100 mm and depth h = 150 mm has a length L = 2 m and is subjected to a point load P = 500 N at B. Calculate the state of plane stress at point C located 50 mm below the top of the beam and 0,5 m to the right of point A, Also find the principal stresses and the maximum shear stress at C. Neglect the weight of the beam.A beam with a width h = 6 in. and depth h = 8 in. is simply supported at A and B. The beam has a length L = 10 ft and is subjected to a linearly varying distributed load with peak intensity q0= 1500 lb/ft. Calculate the state of plane stress at point C located 3 in. below the top of the beam and 0.1 ft to the left of point B. Also find the principal stresses on element C. Neglect the weight of the beam.Beam ABC with an overhang BC is subjected to a linearly varying distributed load on span AB with peak: intensity q0= 2500 N/m and a point load P = 1250 N applied at C. The beam has a width ft = 100 mm and depth h = 200 mm. Find the state of plane stress at point D located 150 mm below the top of the beam and 0.2 m to the left of point B. Also find the principal stresses at D>Neglect the weight of the beam.A cantilever beam(Z, = 6 ft) with a rectangular cross section (/> = 3.5 in., h = 12 in.) supports an upward load P = 35 kips at its free end. (a) Find the state of stress ((7T, o^., and r in ksi) on a plane-stress element at L/2 that is i/ = 8 in. up from the bottom of the beam. Find the principal normal stresses and maximum shear stress. Show these stresses on sketches of properly oriented elements. (b) Repeat part (a) if an axial compressive centroidal load N = 40 kips is added at BSolve the preceding problem for the following data:P = 160 kN,JV = 200 tN,L = 2 m,b = 95 mm, h = 300 mm, and d = 200 mmA simple beam with a rectangular cross section (width, 3,5 inL; height, 12 in,) carries a trapczoi-dally distributed load of 1400 lb/ft at A and 1000 lb/ft at B on a span of 14 ft (sec figure). Find the principal stresses 2 and the maximum shear stress r__ at a cross section 2 ft from the left-hand support at each of the locations: (a) the neutral axis, (b) 2 in. above the neutral axis, and (c) the top of the beam. (Disregard the direct compressive stresses produced by the uniform load bearing against the top of the beam.)An overhanging beam ABC has a guided support at A, a rectangular cross section, and supports an upward uniform load q = PtL over AB and a downward concentrated load P at the free end C {see figure). The span length from A to B is L, and the length of the overhang is L12. The cross section has a width of A and a height A. Point D is located midway between the supports at a distance d from the top face of the beam. Knowing that the maximum tensile stress (principal stress) at point Z> is tr, = 38 MPa, determine the magnitude of the load P. Data for the beam are L = 1.75 m, b = 50 mm, // = 220 mm, and d = 55 mm.Solve the preceding problem if the stress and dimensions aallow = 2450 pai, L = 80 in., b = 2.5 in,, h = 10 in., and d = 2.5 inA cantilever wood beam with a width b = 100 mm and depth h = 150 mm has a length L = 2 m and is subjected to point load P at mid-span and uniform load q = 15 N/m. (a) If the normal stress trx= 0 at point C, located 120 mm below the top of the beam at the fixed support A, calculate the point load P, Also show the complete state of plane stress on the element at point C (b) Repeat Part a if er = 220 kPa. Assume that element C is a sufficient distance from support A so that stress concentration effects are negligible.. A cantilever beam (width b = 3 in. and depth h = 6 in,) has a length L = 5 ft and is subjected to a point load P and a concentrated moment M = 20 kip-ft at end B. If normal stress trx= 0 at point C, located 0.5 in. below the top of the beam and 1 ft to the right of point Atfind point load P. Also show the complete state of plane stress on the element at point C.A beam with a wide-flange cross section (see figure) has the following dimensions: h = 120 mm, r = 10 mm, h = 300 mm, and /ij = 260 mm. The beam is simply supported with span length L = 3,0 im A concentrated load P = 120 kN acts at the midpoint of the span. At across section located 1.0 m from the left-hand support, determine the principal stresses tr, and tr2and the maximum shear stress Tmax at each of the following locations: (a) the top of the beam, (b) the top of the web, and (c) the neutral axisA beam with a wide-flange cross section (see figure) has the following dimensions: b = 5 in., t = 0.5 in,, ft = 12 in., and /?, = 10.5 in. The beam is simply supported with span length L = 10 ft and supports a uniform load q = 6 kips/fL Calculate the principal stresses *rl and and the maximum shear stress t__ at a cross section located [|] JA 3 ft from the left-hand support at each of the following locations: (a) the bottom of the beam, (b) the bottom of the web, and (c) the neutral axisA W 200 x 41.7 wide-flange beam (see Table F-l(b), Appendix F) is simply supported with a span length of 2.5 m (see figure). The beam supports a concentrated load of 100 kN at 0.9 m from support B. At a cross section located 0,7 m from the left-hand support, determine the principal stresses tr, and 2and the maximum shear stress rnMJt at each of the following locations: (a) the top of the beam, (b) the top of the web, and (c) the neutral axis,A W 12 x 35 steel beam is fixed at A. The beam has length L = 6 ft and is subjected to a linearly varying distributed load with peak intensity q0=830 lb/ft. Calculate the state of plane stress at point C located 3 in, below the top of the beam at mid-span. Also find the principal normal stresses and the maximum shear stress at C. Include the weight of the beam. Sec Table F-l(a), Appendix F, for beam propertiesA W 360 x 79 steel beam is fixed at A. The beam has a length of 2.5 m and is subjected to a linearly varying distributed load with maximum intensity q0= 500 N/m on segment AB and a uniformly distributed load of intensity qQon segment EC. Calculate the state of plane stress at point D located 220 mm below the top of the beam and 0\3 m to the left of point B. Find the principal normal stresses and the maximum shear stress at D. Include the weight of the beam, See Table F-l(b), Appendix F, for beam properties.A W 12 X 14 wide-flange beam (see Table F-l(a), Appendix F) is simply supported with a span length of 120 in. (see figure). The beam supports two anti-symmetrically placed concentrated loads of 7,5 kips each. At a cross section located 20 in. from the right-hand support, determine the principal stresses (7]and (7\ and the maximum shear stress Tmaw at each of the following locations: (a) the top of the beam, (b) the top of the web, and (c) the neutral axis,A cantilever beam with a T-section is loaded by an inclined force of magnitude 6.5 kN (see figure), The line of action of the force is inclined at an angle of 60" to the horizontal and intersects the top of the beam at the end cross section. The beam is 2.5 m long and the cross section has the dimensions shown. Determine the principal stresses irtand 2 and the maximum shear stress rmax at points A and B in the web of the beam near the support.Beam A BCD has a sliding support at A, roller supports at C and A and a pin connection at B (see figure). Assume that the beam has a rectangular cross section (b = 4 in., h = 12 in.). Uniform load q acts on ABC and a concentrated moment is applied at D. Let load variable q = 1750 lb/ft, and assume that dimension variable L = 4 ft. First, use statics to confirm the reaction moment at A and the reaction forces at Cand A as given in the figure. Then find the ratio of the magnitudes of the principal stresses (crj/os) just left of support Cat a distance d = 8 in. up from the bottom,, Solve the preceding problem using the numerical data: /) = 90mm, h = 280 mm, d = 210 mm, q = 14 kN/m, and L = L2 m.A W 12 x 35 steel cantilever beam is subjected to an axial load P = 10 kips and a transverse load V = 15 kips. The beam has length L = 6 ft, (a) Calculate the principal normal stresses and the maximum shear stress for an clement located at C near the fixed support. Neglect the weight of the beam, (b) Repeat Part a for point D which is 4 in. above point C (see figure). See Table F-l(a), Appendix F, for beam properties.A W 310 x 52 steel beam is subjected to a point load P = 45 kN and a transverse load V = 20 kN at B. The beam has length L =2m.(a) Calculate the principal normal stresses and the maximum shear stress on element D located on the web right below the top flange and near the fixed support. Neglect the weight of the beam, (b) Repeat Part a atcentroid C (sec figure). See Table F-l(b), Appendix F, for beam propertiesA solid circular bar is fixed at point A. The bar is subjected to transverse load V = 70 lb and torque T = 300 lb-in. at point B. The bar has a length L = 60 in. and diameter d = 3 in. Calculate the principal normal stresses and the maximum shear stress at clement 1 located on the bottom surface of the bar at fixed end A (see figure), Assume that element 1 is a sufficient distance from support A so that stress concentration effects are negligibleA cantilever beam with a width h = 100 mm and depth h = 150 mm has a length L = 2 m and is subjected to a point load P = 500 N at B. Calculate the state of plane stress at point C located 50 mm below the top of the beam and 0,5 m to the right of point A, Also find the principal stresses and the maximum shear stress at C. Neglect the weight of the beam.Solve the preceding problem using the following data: beam cross section is 100 x 150 mm, length is 3 m, and point load is P = 5 kN at mid-span, Point C is located 25 mm below the top of the beam and 0.5 m to the right of support A.A cylindrical tank subjected to internal pressure/? is simultaneously compressed by an axial force F = 72 kN (see figure). The cylinder has diameter d = 100 mm and wall thickness t = 4 mm. Calculate the maximum allowable internal pressure /?max based upon an allowable shear stress in the wall of the tank of 60 MPa.A cylindrical pressure vessel having a radius r = 14 in. and wall thickness t = 0,5 in, is subjected to internal pressure p = 375 psi, In addition, a torque T = 90 kip-ft acts at each end of the cylinder (see figure), (a) Determine the maximum tensile stress ctniXand the maximum in-plane shear stress Tmjv in the wall of the cylinder. (b) If the allowable in-plane shear stress is 4.5 ksi, what is the maximum allowable torque T\ (c) If 7 = 150 kip-ft and allowable in-plane shear and allowable normal stresses are 4.5 ksi and 11.5 ksi, respectively, what is the minimum required wall thicknessA pressurized cylindrical tank with flat ends is loaded by torques T and tensile forces P (sec figure), The tank has a radius of r = 125 mm and wall thickness t = 6.5 mm. The internal pressure p = 7.25 MPa and the torque T = 850 N m. (a) What is the maximum permissible value of the forces P if the allowable tensile stress in the wall of the cylinder is 160 MPa? (b) If forces P = 400 kN, what is the maximum acceptable internal pressure in the tank?A cylindrical pressure vessel with flat ends is subjected to a torque T and a bending moment M (see figure), The outer radius is 12.0 in. and the wall thickness is 1.0 in. The loads are T = 800 kip-in,, M = 1000 kip-in., and the internal pressure p = 900 psi. Determine the maximum tensile stress r maximum compressive stress c, and maximum shear stress rmas in the wall of the cylinder.The tensional pendulum shown in the figure consists of a horizontal circular disk of a mass M = 60 kg suspended by a vertical steel wire (G = SOGPa) of a length L = 2 m and diameter d = 4 mm. Calculate the maximum permissible angle of rotation <£m:ls of the disk (that is, the maximum amplitude of torsional vibrations) so that the stresses in the wire do not exceed 100 MPa in tension or 50 MPa in shear.The hollow drill pipe for an oil well (sec figure) is 6,2 in. in outer diameter and 0.75 in. in thickness. Just above the bit, the compressive force in the pipe (due to the weight of the pipe) is 62 kips and the torque (due to drilling) is 185 kip-in. Determine the maximum tensile, compressive, and shear stresses in the drill pipe.Solve the preceding problem if the diameter is 480 mm, the pressure is 20 MPa, the yield stress in tension is 975 MPa, the yield stress in shear is 460 MPa, the factor of safety is 2,75, the modulus of elasticity is 210 GPa, Poissorfs ratio is 0.28, and the normal strain must not exceed 1190 x 10" . For part (b), assume that the tank thickness is 8 mm and the measured normal strain is 990 x 10~. A segment of a generator shaft with a hollow circular cross section is subjected to a torque T = 240 kip-in, (see figure). The outer and inner diameters of the shaft arc 8.0 in, and 6.25 in., respectively. What is the maximum permissible compressive load /'that can be applied to the shaft if the allowable in-plane shear stress is rAi(m= 6250 psiA post having a hollow, circular cross section supports a P = 3.2 kN load acting at the end of an arm that is h = 1.5 m long (see figure). The height of the post is L = 9 m, and its section modulus isS = 2.65 x 10 mmJ. Assume that the outer radius of the post is r2= 123 mm, and the inner radius is r}=117 mm. (a) Calculate the maximum tensile stress and \ maximum in-plane shear stress Tm:ls at point A on the outer surface of the post along the x axis due to the load P. Load P acts at B along line BC. (b) If the maximum tensile stress and maximum in-plane shear stress at point A arc limited to 90 MPa and 38 MPa, respectively, what is the largest permissible value of the load PIA sign is supported by a pole of hollow circular cross section, as shown in the figure. The outer and inner diameters of the pole are 10.5 in. and 8.5 in,, respectively. The pole is 42 ft high and weighs 4 kips, The sign has dimensions 8 ft x 3 ft and weighs 500 lb, Note that its center of gravity is 53.25 in. from the axis of the pole. The wind pressure against the sign is 35 lb/ft-, (a) Determine the stresses acting on a stress element at point A, which is on the outer surface of the pole at the "front" of the pole, that is, the part of the pole nearest to the viewer, (b) Determine the maximum tensile, compressive, and shear stresses at point AA sign is supported by a pipe (see figure) having an outer diameter 110 mm and inner diameter 90 mm. The dimensions of the sign are 2.0 m X 1.0 m, and its lower edge is 3.0 m above the base. Note that the center of gravity of the sign is 1.05 m from the axis of the pipe. The wind pressure against the sign is 1.5 kPa. Determine the maximum in-plane shear stresses due to the wind pressure on the sign at points /I, B, and C, located on the outer surface at the base of the pipe.A traffic light and signal pole is subjected to the weight of each traffic signal We = 45 lb and the weight of the road lamp WL= 55 lb. The pole is fixed at the base. Find the principal normal stresses and the maximum shear stress on element B located 19 ft above the base (see figure). Assume that the weight of the pole and lateral arms is included in the signal and lamp weightsRepeat the preceding problem but now find the stress state on Element A at the base. Let W%= 240 N, WL= 250 N, / = 5 mm, d = 360 mm. See the figure for the locations of clement A and all loadsA bracket ABCD having a hollow circular cross section consists of a vertical arm AB{L = 6 ft), a horizontal arm BC parallel to the v0 axis, and a horizontal arm CD parallel to the -0 axis (see figure). The arms BC and CD have lengths b}= 3.6 ft and b2= 2,2 ft, respectively. The outer and inner diameters of the bracket are d-, = 7,5 in. and dx= 6,8 in. An inclined load P = 2200 lb acts at point D along line DH. Determine the maximum tensile, compressive, and shear stresses in the vertical armA gondola on a ski lift is supported by two bent arms, as shown in the figure. Each arm is offset by the distance b = ISO mm from the line of action of the weight force W. The allowable stresses in the arms are 100 MPa in tension and 50 MPa in shear. If the loaded gondola weighs 12 kN, what is the minimum diameter roof the arms?Beam A BCD has a sliding support at A, roller supports at C and A and a pin connection at B (see figure). Assume that the beam has a rectangular cross section (b = 4 in., h = 12 in.). Uniform load q acts on ABC and a concentrated moment is applied at D. Let load variable q = 1750 lb/ft, and assume that dimension variable L = 4 ft. First, use statics to confirm the reaction moment at A and the reaction forces at C and A as given in the figure. Then find the ratio of the magnitudes of the principal stresses (crj/os) just left of support Cat a distance d = 8 in. up from the bottom, The pedal and crank are in a horizontal plane and points A and B are located on the top of the crank. The load P = 160 lb acts in the vertical direction and the distances (in the horizontal plane) between the line of action of the load and points A and B are b\ = 5.0 in., h-, = 2.5 in., and/>3 = 1.0 in. Assume that the crank has a solid circular cross section with diameter d = 0.6 inA double-decker bicycle rack made up of square steel tubing is fixed at A (figure a). The weight of a bicycle is represented as a point load applied at B on a plane frame model of the rack (figure b). (a)Find the state of plane stress on an element C located on the surface at the left side of the vertical tube at the base A. Include the weight of the framing system. (Assume weight density y = 77 kN/m .) (b) Find the maximum shear stresses on an element at C and show them on a sketch of a properly oriented element. Assume that element C is a sufficient distance from support A so that stress concentration effects are negligibleA semicircular bar AB lying in a horizontal plane is supported at B (sec figure part a). The bar has a centerline radius R and weight q per unit of length (total weight of the bar equals TiqR). The cross section of the bar is circular with diameter d. (a) Obtain formulas for the maximum tensile stress v maximum compressive stress c, and maximum in-plane shear stress rmdX at the top of the bar at the support due to the weight of the bar. (b) Repeat part (a) if the bar is a quarter-circular segment (see figure part b) but has the same total weight as the semicircular bar.Repeat Problem 8.5-22 but replace the square tube column with a circular tube having a wall thickness r = 5 mm and the same cross-sectional area (3900 mm2) as that of the square tube in figure b in Problem 8.5-22. Also, add force P. = 120 N at B (a) Find the state of plane stress at C. (b) Find maximum normal stresses and show them on a sketch of a properly oriented element. (c) Find maximum shear stresses and show them on a sketch of a properly oriented element.An L-shaped bracket lying in a horizontal plane supports a load P = 1501b (see figure). The bracket has a hollow rectangular cross section with thickness t = 0,125 in. and outer dimensions b = 2,0 in. and h = 3,5 in. The centerline lengths of the arms are fy = 20 in, and b2= 30 in. Considering only the load P, calculate the maximum tensile stress ut, maximum compressive stress crc, and maximum shear stress TmaK at point A, which is located on the top of the bracket at the support,A horizontal bracket ABC consists of two perpendicular arms AB of a length 0.75 m and BC of a length 0,5 m. The bracket has a solid, circular cross section with a diameter equal to 65 mm. The bracket is inserted in a friction less sleeve at A (which is slightly larger in diameter), so it is free to rotate about the r0 axis at A and is supported by a pin at C Moments are applied at point C M{=1.5 kN -m in the x direction and A/3 = L0 kN-m acts in the — z direction. Considering only the moments Mxand M2, calculate the maximum tensile stress t, the maximum compressive stress crc* and the maximum in-plane shear stress Tmax at point />, which is located at support A on the side of the bracket at mid-height, An arm A BC lying in a horizontal plane and supported at A (see figure) is made of two identical solid steel bars AB and BC welded together at a right angle. Each bar is 22 in. long. (a) Knowing that the maximum tensile stress (principal stress) at the top of the bar at support A due solely to the weights of the bars is 1025 psi, determine the diameter d of the bars. (b) If the allowable tensile stress is 1475 psi and each bar has a diameter d = 2.0 in,, what is the maximum downward load P that can be applied at C(in addition to self-weightA crank arm consists of a solid segment of length bxand diameter rf, a segment of length bltand a segment of length byas shown in the figure. Two loads P act as shown: one parallel to — vand another parallel to —y. Each load P equals 1.2 kN. The crankshaft dimensions are A] = 75 mm, fr> = 125 mm, and b3= 35 mm. The diameter of the upper shaft isd = 22 mm, (a) Determine the maximum tensile, compressive, and shear stresses at point A, which is located on the surface of the shaft at the z axis. (b) Determine the maximum tensile, compressive, and shear stresses at point B, which is located on the surface of the shaft at the y axisA moveable steel stand supports an automobile engine weighing W = 750 lb, as shown in the figure part a. The stand is constructed of 2.5 in. x 2.5 in. x 1/8 in.-thick steel tubing. Once in position, the stand is restrained by pin supports at B and C. Of interest are stresses at point A at the base of the vertical post; point A has coordinates (x = 1.25, y = 0, z = 1.25) in inches. Neglect the weight of the stand. (a) Initially, the engine weight acts in the — z direction through point Q, which has coordinates (24, 0, 1.25) inches. Find the maximum tensile, compressive, and shear stresses at point A. (b) Repeat part (a) assuming now that, during repair, the engine is rotated about its own longitudinal axis (which is parallel to the x axis) so that Warts through Q [with coordinates (24, 6, 1.25) in inches] and force F = 200 lb is applied parallel to the y axis at distance d = 30 inA mountain bike rider going uphill applies a force P = 65 N to each end of the handlebars AB CD, made of aluminum alloy 7075-T6, by pulling on the handlebar extenders (DF on right handlebar segment). Consider the right half of the handlebar assembly only (assume the bars are fixed at the fork at A), Segments AB and CD are prismatic with lengths Lvand L3 and with outer diameters and thicknesses J01, /01 and d03, /03, respectively, as shown. Segment BC of length L2, however, is tapered, and outer diameter and thickness vary linearly between dimensions at B and C Consider shear, torsion, and bending effects only for segment AD; assume DFis rigid. Find the maximum tensile, compressive, and shear stresses adjacent to support A. Show where each maximum stress value occursDetermine the maximum tensile, compressive, and shear stresses acting on the cross section of the tube at point A of the hitch bicycle rack shown in the figure. The rack is made up of 2 in. x 2 in. steel tubing which is 1/8 in. thick. Assume that the weight of each of four bicycles is distributed evenly between the two support arms so that the rack can be represented as a cantilever beam (ABCDEF) in the xy plane. The overall weight of the rack alone is W = 60 lb directed through C, and the weight of each bicycle is P = 30 lb.8.5.32PA plumber's valve wrench is used to replace valves in plumbing fixtures. A simplified model of the wrench (see figure part a) consists of pipe AB (length L. outer diameter D inner diameter dy), which is fixed at A and has holes of a diameter dhon either side of the pipe at B. A solid, cylindrical bar CBD (lengths, diameter^) is inserted into the holes at B and only one force F = 55 lb is applied in the -Z direction at C to loosen the fixture valve at A (see figure part c). Let G = 11,800 ksi, v = 0.30, L = 4 in., a = 4.5 in., d2= 1.25 in., dx= 1 in., and dh= 0.25 in. Find the state of plane stress on the top of the pipe near A (at coordinates A" = 0,Y = Q,Z = d->12), and show all stresses on a plane stress element (see figure part b). Compute the principal stresses and maximum shear stress, and show them on properly rotated stress elementsA compound beam ABCD has a cable with force P anchored at C The cable passes over a pulley at D, and force P acts in the —x direction, There is a moment release just left of B. Neglect the self-weight of the beam and cable. Cable force P = 450 N and dimension variable L = 0.25 m. The beam has a rectangular cross section (b = 20 mm, it = 50 mm). (a) Calculate the maximum normal stresses and maximum in-plane shear stress on the bottom surface of the beam at support A. (b) Repeat part (a) for a plane stress element located at mid-height of the beam at A. (c) If the maximum tensile stress and maximum in-plane shear stress at point A are limited to 90 MPa and 42 MPa, respectively, what is the largest permissible value of the cable force P?A steel hanger bracket ABCD has a solid, circular cross section with a diameter of d = 2 in. The dimension variable is b = 6 in. (see figure). Load P = 1200 lb is applied at D along a line DIh the coordinates of point H arc (8/>, — 5b, 2b), Find normal and shear stresses on a plane stress element on the surface of the bracket at A. Then find the principal stresses and maximum shear stress. Show each stress state on properly rotated elementsThe equation of the deflection curve for a cantilever beam is v(x)=m0x22EI Describe the loading acting on the beam. Draw the moment diagram for the beam.The equation of the deflection curve for a simply supported beam is v(x)=q024EI(2Lx3x4L3x) Derive the slope equation of the beam. Derive the bending-moment equation of the beam. Derive the shear-force equation of the beam. Describe the loading acting on the beam.-3 The deflection curve for a simple beam AB (see figure) is given by v=q0x360LEI(7L410L2x2+3x4) Describe the load acting on the beam.The deflection curve for a simple beam AB (sec figure) is given by v=q0L44EIsinxL Describe the load acting on the beam. Deter mine the reactions RAand RBat the supports, Determine the maximum bending moment Mmax.The deflection curve for a cantilever beam AB (sec figure) is given by v=q0x2120LEI(10L210L2x+5Lx2x3) Describe the load acting on the beam.The deflection curve for a cantilever beam AB (see figure) is given by v=q0x2360L2EI(45L440L3x+15L2x2x4) Describe the load acting on the beam. Determine the reactions RAand M 4at the support.A simply supported beam is loaded with a point load, as shown in the figure. The beam is a steel wide flange shape (W 12 ×35) in strong axis bending. Calculate the maximum deflection of the beam and the rotation at joint A if L = 10 ft, a = 7 ft, b = 3 ft, and P = 10 kips. Neglect the weight of the beam.A I-meter-long, simply supported copper beam (E = 117 GPa) carries uniformly distributed load q. The maximum deflection is measured as 1.5 mm. Calculate the magnitude of the distributed load q if the beam has a rectangular cross section (width b = 20 mm, height h = 40 mm). If instead the beam has circular cross section and q = 500 N/m, calculate the radius r of the cross section. Neglect the weight of the beam.A wide-flange beam (W 12 x 35) supports a uniform load on a simple span of length L = 14 ft (see figure). Calculate the maximum deflection max at the midpoint and the angles of rotation Sat the supports if q = 1.8 kips/ft and E = 30 × 106 psi. (Use the formulas of Example 9-1.)A uniformly loaded, steel wide-flange beam with simple supports (see figure) has a downward deflection of 10 mm at the midpoint and angles of rotation equal to 0.01 radians at the ends. Calculate the height h of the beam if the maximum bending stress is 90 MPa and the modulus of elasticity is 200 GPa, (Use the formulas of Example 9-L)What is the span length L of a uniformly loaded, simple beam of wide-flange cross section (see figure) if the maximum bending stress is 12,000 psi, the maximum deflection is 0.1 in., the height of the beam is 12 in., and the modulus of elasticity is 30 × 106psi? (Use the formulas of Example 9-1.)-6 Calculate the maximum deflection of a uniformly loaded simple beam if the span length L = 2.0 m, the intensity of the uniform load q = 2.0 kN/m, and the maximum bending stress = 60 MPa, The cross section of the beam is square, and the material is aluminum having modulus of elasticity E = 70 GPa. (Use the formulas of Example 9-1.)A cantilever beam with a uniform load (see figure) has a height h equal to 1/8 of the length L. The beam is a steel wide-flange section with E = 28 X 106 psi and an allowable bending stress of 17,500 psi in both tension and compression. Calculate the ratio S/L of the deflection at the free end to the length, assuming that the beam carries the maximum allowable load. (Use the formulas of Example 9-2.)A gold-alloy microbeam attached to a silicon wafer behaves like a cantilever beam subjected to a uniform load (see figure). The beam has a length L = 27.5 m and rectangular cross section of a width b = 4.0 m and thickness t = 0.88 m. The total load on the beam is 17.2 N. If the deflection at the end of the beam is 2.46 m is what is the modulus of elasticity Egof the gold alloy? (Use the formulas of Example 9-2.)Obtain a formula for the ratio c/maxof the deflection at the midpoint to the maximum deflection for a simple beam supporting a concentrated load P (see figure). From the formula, plot a graph of c/max versus the ratio a/L that defines the position of the load (0.5 < a/L < ) What conclusion do you draw from the graph? (Use the formulas of Example 9-3.)A cantilever beam model is often used to represent micro-clectrical-mechanical systems (MEMS) (sec figure}. The cantilever beam is made of polysilicon (E = 150 GPa) and is subjected to an electrostatic moment M applied at the end of the cantilever beam. 1 f dimensions arc h — 2 [im, h — 4 ^m, and L = 520 ^mt find expressions for the tip deflection and rotation of the cantilever beam in terms of moment M.B cams AB and CDE are connected using rigid link DB with hinges (or moment releases) at ends D and B (see figure a). Beam AB is fixed at joint A and beam CDE is pin-supported at joint E. Load P = 150 lb is applied at C\ Calculate the deflections of joints B and joint C Assume L = 9 ft and EI = 127,000 kip-in2. Repeat part (a) if rigid link DB is replaced by a linear spring with k = 20 kips/in (see figure b).-12 Derive the equation of the deflection curve for a cantilever beam AB supporting a load P at the free end (see figure). Also, determine the deflection Band angle of rotation bat the free end. Use the second-order differential equation of the deflection curve.-13 Derive the equation of the deflection curve for a simple beam AB loaded by a couple M0at the left-hand support (see figure). Also, determine the maximum deflection max Use the second-order differential equation of the deflection curve.-14 A cantilever beam AB supporting a triangularly distributed load of maximum intensity q0is shown in the figure. Derive the equation of the deflection curve and then obtain formulas for the deflection Band angle of rotation Bat the free end. Use the second-order differential equation of the deflection curve.A cantilever beam has a length L = 12 ft and a rectangular cross section (b = 16 in., h = 24 in.), A linearly varying distributed load with peak intensity q0acts on the beam, (a) Find peak intensity q0if the deflection at joint B is known to be 0.18 in. Assume that modulus E = 30,000 ksi. (b) Find the location and magnitude of the maximum rotation of the beam.A simple beam with an overhang is subjected to d point load P = 6kN. If the maximum allowable deflect ion at point C is 0.5 mm, select the lightest W360 section from Table F-l{b) that can be used for the beam. Assume that L = 3 m and ignore the distributed weight of the beam.-17 A cantilever beam AB is acted upon by a uniformly distributed moment (bending moment, not torque) of intensity m per unit distance along the axis of the beam (see figure). Derive the equation of the deflection curve and then obtain formulas for the deflection Band angle of rotation Bat the free end. Use the second-order differential equation of the deflection curve.-18 The beam shown in the figure has a sliding support at A and a spring support at B, The sliding support permits vertical movement but no rotation. Derive the equation of the deflection curve and determine the deflection Bat end B due to the uniform load of intensity q. Use the second-order differential equation of the deflection curve.-19 Derive the equations of the deflect ion curve for a simple beam AB loaded by a couple M0acting at distance a from the left-hand support (see figure). Also, determine the deflection B at the point where the load is applied. Use the second-order differential equation of the deflection curve.-20 Derive the equations of the deflection curve for a cantilever beam AB carrying a uniform load of intensity q over part of the span (see figure). Also, determine the deflection Bat the end of the beam. Use the second-order differential equation of the deflection curve.-21 Derive the equations of the deflection curve for a cantilever beam AB supporting a distributed load of peak intensity q0acting over one-half of the length (see figure). Also, obtain formulas for the deflections Band cat points B and C, respectively Use the second-order differentia] equation of the deflection curve.-22 Derive the equations of the deflection curve for a simple beam AB with a distributed load of peak intensity q0acting over the left-hand half of the span (see figure). Also, determine the deflection cat the midpoint of the beam. Use the second-order differential equation of the deflection curve.-23 The beam shown in the figure has a sliding support at A and a roller support at B. The sliding support permits vertical movement but no rotation. Derive the equation of the deflection curve and determine the deflection Aat end A and also cat point C due to the uniform load of intensity q = P/ L applied over segment CB and load P at x = L / 3. Use the second-order differential equation of the deflection curve.-1 Derive the equation of the deflection curve for a cantilever beam AB when a couple M0acts counterclockwise at the free end (see figure). Also, determine the deflection Band slope Bat the free end. Use the third-order differential equation of the deflection curve (the shear-force equation).-2 A simple beam AB is subjected to a distrib uted load of intensity q(x) = q0sin x/L, where q0is the maximum intensity of the load (see figure). Derive the equation of the deflection curve, and then determine the deflection max at the midpoint of the beam. Use the fourth-order differential equation of the deflection curve (the load equation).-3 The simple beam AB shown in the figure has moments 2M0and A/0 acting at the ends. Derive the equation of the deflection curve, and then determine the maximum deflection max Use the third-order differential equation of the deflection curve (the shear-force equation).-4 A beam with a uniform load has a sliding support at one end and spring support at the other. The spring has a stiffness k = 48IE/ L2. Derive the equation of the deflection curve by starting with the third-order differential equation (the shear-force equation). Also, determine the angle of rotation Bat support B.-5 The distributed load acting on a cantilever beam AB has an intensity q(x) given by the expression q0cos x/2L, whereby is the maximum intensity of the load (see figure). Derive the equation of the deflection curve, and then determine the deflection Bat the free end. Use the fourth-order differential equation of the deflection curve (the load equation).-6 A cantilever beam .4B is subjected to a parabolically valying load of intensity q(x)=q0(L2x2)/L2 where q0is the maximum intensity of the load (see figure). Derive the equation of the deflection curve, and then determine the deflection Band angle of rotation Bat the free end. Use the fourth-order dif ferential equation of the deflection curve (the load equation).-7 A beam on simple supports is subjected to a parabolically distributed load of intensity q(x) = 7q0x(L-x)/L2, where q0is the maximum intensity of the load (see figure). A Derive the equation of the deflection curve, and then determine the maximum deflection Use the fourth-order differential equation of the deflection curve (the load equation).Derive the equation of the deflection curve for beam AB with sliding support at A and roller at B* carrying a triangularly distributed load of maximum intensity q0(see figure). Also, determine the maximum deflection ôniill of the beam. Lsc the fourth-order differential equation of the deflection curve (the load equation).-9 Derive the equations of the deflection curve for beam ABC with sliding support at A and roller support at B, supporting a uniform load of intensity q acting on the overhang portion of the beam (see figure). Also, determine deflection cand angle of rotation c. Use the fourth-order differential equation of the deflection curve (the load equation).-10 Derive the equations of the deflection curve for beam AB with sliding support at A and roller support at B, supporting a distributed load of maximum intensity q0acting on the right-hand half of the beam (see figure). Also, determine deflection A, angle of rotation B , and deflection cat the midpoint. Use the fourth-order differential equation of the deflection curve (the load equation).A simply supported beam (E = 1600 ksi) is loaded by a triangular distributed load from A to C(see figure). The load has a peak intensity q0= 10 lb/ ft, and the deflection is known to be 0.01 in, at point C. The length of the beam is 12 ft, and the ratio of the height to the width of the cross section is (h:b) 2:1, Find the height h; and width h of the cross section of the beam.A simply supported beam (E = 12 GPa) carries a uniformly distributed load q = 125 N/m, and a point load P = 200 N at mid-span. The beam has a rectangular cross section (b = 75 mm, h = 200 mm) and a length of 3.6 m. Calculate the maximum deflection of the beam.Copper beam AB has circular cross section with a radius of 0.25 in. and length L = 3 ft. The beam is subjected to a uniformly distributed load w = 3.5 lb/ft. Calculate the required load P at joint B so that the total deflection at joint B is zero. Assume that£ = 16,000 ksi.Beam ABC is loaded by a uniform load q and point load P at joint C. Using the method of superposition, calculate the deflection at joint C. Assume that L = 4 m, a =2ra, q = 15 kN/m, P = 7.5 kN, £ = 200 GPa, and / = 70.8 X 106 mm4.A cantilever beam of a length L = 2.5 ft has a rectangular cross section {b = 4in,, h = Sin,) and modulus E = 10,000 ksi. The beam is subjected to a linearly varying distributed load with a peak intensity qQ= 900 lb/ft. Use the method of superposition and Cases 1 and 9 in Table H-l to calculate the deflection and rotation at B.A cantilever beam carries a trapezoidal distributed load (see figure). Let wB= 2.5 kN/m, wA= 5.0kN/m, and L = 2.5 m. The beam has a modulus E = 45GPa and a rectangular cross section with widths = 200 mm and depth/; = 300 mm. Use the method of superposition and Cases 1 and 8 in Table H-l to calculate the deflection and rotation at B.-5-7 A cantilever beam AB carries three equalaly spaced concentrated loads, as shown in the figure. Obtain formulas for the angle of rotation B and deflaction B at the free end of the beam.A simple beam AB supports five equally spaced loads P (see figure). Determine the deflection Slat the midpoint of the beam, If the same total load {5P) is distributed as a uniform load on the beam, what is the deflection èy at the midpoint? Calculate the ratio of d, t0The cantilever beam AB shown in the figure has an extension BCD attached to its free end. A force P acts at the end of the extension. Find the ratio aiL so that the vertical deflection of point B will be zero. Find the ratio aiL so that the angle of rotation at point B will be zero.Beam ACE hangs from two springs, as shown in the figure. The springs have stiffnesses kxand k2and the beam has flex lira I rigidity EL (a) What is the downward displacement of point C, which is at the midpoint of the beam, when the moment M0 is applied? Data for the structure are as follows: M0= 10,0 kN m, L = 1.8 m, EI = 216 kN m2, Jt, = 250 kN/m, and k2= 160 kN/m, (b) Repeat part (a), but remove A/() and apply a uniform load q — 3.5 kN/m to the entire beam.What must be the equation y =f(x) of the axis of the slightly curved beam AB (see figure) before the load is applied in order that the load moving along the bar, always stays at the same level?-12 Determine the angle of rotation Band deflection Bat the free end of a cantilever beam AB having a uniform load of intensity q acting over the middle third of its length (see figure).The cantilever beam ACE shown in the figure has FlexuraI rigidity EI = 2,1 x 106kip-in". Calculate the downward deflections Scand 8Sat points C and B, respectively, due to the simultaneous action of the moment of 35 kip-in. applied at point C and the concentrated load of 2,5 kips applied at the free end B.A cantilever beam is subjected to load P at mid-span and counterclockwise moment Mat B (see figure). Find an expression for moment M in terms of the load P so that the reaction moment MAat A is equal to zero. Find an expression for moment M in terms of the load P so that the deflection isêp = 0; also, what is rotation Find an expression for moment M in terms of the load P so that the rotation also, what is deflectionUse the method of superposition to find the angles of rotation 9Aand SBat the supports, and the maximum deflection for a simply supported beam subjected to symmetric loads P at distance a from each support. Assume that EI is constant, total beam length is L and a = U3. Hint: Use the formulas of Example 9-3.Repeat Problem 9,5-15 for the anti-symmetric loading shown in the figure.A cantilever beam is subjected to a quadratic distributed load q{x) over the length of the beam (see figure). Find an expression for moment M in terms of the peak distributed load intensity qQso that the deflection is = 0.A beam ABCD consisting of a simple span BD and an overhang AB\s loaded by a force P acting at the end of the bracket CEF (see figure), Determine the deflection at the end of the over h a tig. Under what conditions is this deflection upward? Under what conditions is it downward?A horizontal load P acts at end C of the bracket ABC shown in the figure. Determine the deflection 6Cof point C. Determine the maximum upward deflection 8 of member AB. Note: Assume that the flexural rigidity EI is constant throughout the frame. Also, disregard the effects of axial deformations and consider only the effects of bending due to the load P.A beam ABC having flexural rigidity EI = 75 kN irT is loaded by a force P = 800 N at end C and tied down at end A by a wire having axial rigidity EA = 900 kN (see figure). What is the deflection at point C when the load P is applied?Determine the angle of rotation 0Band deflection-22 A simple beam AB supports a uniform load of intensity q acting over the middle region of the span (see figure). Determine the angle of rotation A at the left-hand support and the deflection max at the midpoint.The overhanging beam A BCD supports two concentrated loads P and Q (see figure), For what ratio PIQ will the deflection at point B be zero? For what ratio will the deflection at point D be zero? If Q is replaced by a uniform load with intensity q (on the overhang), repeat parts (a) and (b), but find ratio Pl(qa).A thin metal strip of total weight W and length L is placed across the top of a flat table of width L2as shown in the figure. What is the clearance S between the strip and the middle of the table? (The strip of metal has flexural rigidity EI.)An overhanging beam ABC with flexural rigidity EI = 15 kip-in" is supported by a sliding support at A and by a spring of stiffness k at point fi(see figure). Span AB has a length L = 30 in. and carries a u ni form load. The overhang BC has a length b = 15 in. For what stiffness k of the spring will the uniform load produce no deflection at the free end C?A beam A BCD rests on simple supports at B and C (see figure). The beam has a slight initial curvature so that end A is 18 mm above the elevation of the supports and end D is 12 mm above. What moments Mtand M^, acting at points A and Dtrespectively, will move points A and D downward to the level of the supports? (The flexural rigidity EI of the beam is 2.5 X 106 N m2 and L = 2.5m).The compound beam ABC shown in the figure has a sliding support at A and a fixed support at C. The beam consists of two members joined by a pin connection (i.e., moment release) at B. Find the deflectionA compound beam ABC DE (see figure) consists of two parts (ABC and CDE) connected by a hinge (i.e., moment release) at C. The elastic support at B has stiffness k = EI I b3. Determine the deflection SL- at the free end E due to the load P acting at that point.A steel beam ABC is simply supported at A and held by a high-strength steel wire at B (see figure). A load P = 240 lb acts at the free end C. The wire has axial rigidity EA = 1500 x 103 lb, and the beam has flexural rigidity EI = 36 X 106 lb-in". What is the deflection-30. Calculate the deflection at point C of a beam subjected to uniformly distributed load w = 275 N/m on span AB and point load P = 10 kN at C. Assume that that L = 5 m and EI = 1.50 × 107N · m2.Compound beam ABC is loaded by point load P = 1.5 kips at distance 2aB from point A and a triangularly distributed load on segment BC with peak intensity qü= 0.5 kips/ft. If length a = 5 ft and length/) = 10 ft, find the deflection at B and rotation at A. Assume that £ = 29,000 ksi and / = 53.8 in4.The compound beam shown in the figure consists of a cantilever beam AB (length L) that is pin-connected to a simple beam BD (length 2L). After the beam is constructed, a clearance c exists between the beam and a support at C, midway between points B and ZX Subsequently, a uniform load is placed along the entire length of the beam. What intensity q of the load is needed to close the gap at C and bring the beam into contact with the support?-33 Find the horizontal deflection hand verti cal deflection vat the free end C of the frame ABC shown in the figure. (The flexural rigidity EI is con stant throughout the frame.) Note: Disregard the effects of axial deformations and consider only the effects of bending due to the load P.The fr a me A BCD shown in the heure is squeezed by two collinear forces P acting at points A and D. What is the decrease ê in the distance between points A and D when the loads P are applied? (The flexural rigidity EI is constant throughout the frame,) Note: Disregard the effects of axial deformations and consider only the effects of bending due to the loads P.A framework A BCD is acted on by counterclockwise moment M at A (see figure). Assume that Elis constant. Find expressions for reactions at supports B and C Find expressions for angles of rotation at A, 5, C, and Z). Find expressions for horizontal deflections SÂand SD, If length LA3= L12, find length LCDin terms of L for the absolute value of the ratio |sysj=i.A framework A BCD is acted on by force P at 2L/3 from 8(see figure). Assume that 7f/is constant. Find expressions for reactions at supports B and C. Find expressions for angles of rotation at A, B* C, and D. Find expressions for horizontal deflections èAand ôD. If length LAB= L i 2, find length LCDin terms of L for the absolute value of the ratioA beam ABCDE has simple supports at B and D and symmetrical overhangs at each end (see figure). The center span has length L and each overhang has length h. A uniform load of intensity q acts on the beam. Determine the ratio biL so that the deflection Bcat the midpoint of the beam is equal to the deflections SAand SEat the ends. For this value of b/L, what is the deflectionA frame ABC is loaded at point C by a force P acting at an angle öf to the horizontal (see figure). Both members of the frame have the same length and the same flexural rigidity. Determine the angle a so that the deflection of point C is in the same direction as the load. (Disregard the effects of axial deformations and consider only the effects of bending due to the load P.) Note: A direction of loading such that the resulting deflection is in the same direction as the load is called a principal direction. For a given load on a planar structure, there are two principal directions that are perpendicular to each other.The wing of a large commercial jet is represented by a simplified prismatic cantilever beam model with uniform load \v and concentrated loads P at the two engine locations (see figure). Find expressions for the tip deflection and rotation at D in terms of \\\ P, L, and EL.The wing of a small plane is represented by a simplified prismatic cantilever beam model acted on by the distributed loads shown in the figure. Assume constant El = 1200kN-m~, Find the tip deflection and rotation at B.Find an expression for required moment MA(in terms of q and L) that will result in rotation 9B= 0 due to MAand q loadings applied at the same time. Also, what is the resulting net rotation at support A?Find an expression for required moment MA(in terms of q and L) that will result in rotation êB= 0 due to MAand q loadings applied at the same time. Also, what is the resulting net rotation at support A?Find required distance d (in terms of L) so that rotation Ss= 0 is due to M and q loadings applied at the same time. Also, what is the resulting net rotationA cantilever beam has two triangular loads as shown in the figure. Find an expression for beam deflection Scusing Superposition. Find the required magnitude of load intensity q2in terms of q0so that the deflection at C is zero. Find an expression for the deflection at C if both load intensities, qxand q2, are equal to q0.-1 A cantilever beam AB is subjected to a uniform load of intensity q acting throughout its length (see figure). Determine the angle of rotation Band the deflection Bat the free end.The load on a cantilever beam AB has a triangular distribution with maximum intensity^ (see figure). Determine the angle of rotation BBand the deflection SBat the free end.A cantilever beam AB is subjected to a concentrated load P and a couple MQacting at the free end (see figure). Obtain formulas for the angle of rotation fiBand the deflection SBat end B.Determine the angle of rotation BBand the deflection SBat the free end of a cantilever beam AB with a uniform load of intensity q acting over the middle third of the length (see figure).-5 Calen1ate the deflections S 3a ndA cantileverbeam^Cßsupportstwo concentrated loads Ptand A, as shown in the figure. Calculate the deflections SBand 8Cat points B and C, respectively. Assume Px= 10 kN, P\ = 5 kN, L = 2.6 m, E = 200 GPa, and / = 20.1 x I0ft mm4.Obtain formulas for the angle of rotation 0Aat support A and the deflection SajAXat the midpoint for a simple beam AB with a uniform load of intensity q (see figure).A simple beam AB supports two concentrated loads P at the positions shown in the figure. A support C at the midpoint of the beam is positioned at distance d below the beam before the loads are applied. Assuming that d = 10 mm, L = 6m, E = 200 G Pa, and I = 198 x 106 mm4, calculate the magnitude of the loads P so that the beam just touches the support at C.A simple beam AB is subjected to a load in the form of a couple M0 acting at end B (see figure). Determine the angles of rotation A and B at the supports and the deflection at the midpoint.-10 The simple beam AB shown in the figure supports two equal concentrated loads P: one acting downward and the other upward. Determine the angle of rotation A at the left-hand end, the deflection 1under the downward load, and the deflection 2 at the midpoint of the beam.A simple beam AB is subjected to couples M0and 2A0 acting as shown in the figure. Determine the angles of rotation 04and BBat the ends of the beam and the deflection S at point D where the load M0is applied.The cantilever beam ACB shown in the figure has moments of inertia /, and I{in parts AC and CB, respectively. Using the method of superposition, determine the deflection 8Bat the free end due to the load P. Determine the ratio r of the deflection 8Bto the deflection S:at the free end of a prismatic cantilever with moment of inertia /] carrying the same load. Plot a graph of the deflection ratio r versus the ratio 12 //L of the moments of inertia. (Let /, II- vary from I to 5.)The cantilever beam ACB shown in the figure supports a uniform load of intensity q throughout its length. The beam has moments of inertia I2and IYin parts AC and CB, respectively. Using the method of superposition, determine the deflection SBat the free end due to the uniform load. Determine the ratio r of the deflection 6Bto the deflection 3Xat the free end of a prismatic cantilever with moment of inertia /] carrying the same load. Plot a graph of the deflection ratio r versus the ratio 12 //t of the moments of inertia. (Let 7, tlxvary from I to 5.)Beam ACB hangs from two springs, as shown in the figure. The springs have stiffnesses Jt(and k2^ and the beam has flexural rigidity EI. What is the downward displacement of point C, which is at the midpoint of the beam, when the moment MQis applied? Data for the structure are M0 = 7.5 kip-ft, L = 6 ft, EI = 520 kip-ft2, kx= 17 kip/ft, and As = 11 kip/ft. Repeat part (a), but remove Af0 and instead apply uniform load q over the entire beam.-4 A simple beam ABCD has moment of inertia I near the supports and moment of iertia 2I in the middle region, as shown in the figure. A uniform load of intensity q acts over the entire length of the beam. Determine the quations of the deflection curve for the left-hand half of the beam. Also, find the angle of rotation A at the left-hand support and the deflection max at the midpoint.A beam ABC has a rigid segment from A to B and a flexible segment with moment of inertia / from B to C(see figure). A concentrated load P acts at point B. Determine the angle of rotation SAof the rigidsegment, the deflection 8Bat point ß, and the maximum deflection 8.A simple beam ABC has a moment of inertia 1,5 from A to B and A from B to C (see figure). A concentrated load P acts at point B. Obtain the equations of the deflection curves for both parts of the beam. From the equations, determine the angles of rotation 0Aand Bcat the supports and the deflection 6Bat point B.The tapered cantilever beam AB shown in the figure has a thin-walled, hollow circular cross sections of constant thickness t. The diameters at the ends A and B are dAand dB= 2dA, respectively. Thus, the diameter d and moment of inertia / at distance x from the free end are, respectively, in which IAis the moment of inertia at end A of the beam. Determine the equation of the deflection curve and the deflection 8 Aat the free end of the beam due to the load P.The tapered cantilever beam AB shown in the figure has a solid circular cross section. The diameters at the ends A and B are dAand dB= 2dA, respectively. Thus, the diameter d and moment of inertia / at distance v from the free end are, respectively, in which IAis the moment of inertia at end A of the beam. Determine the equation of the deflection curve and the deflection SAat the free end of the beam due to the load P.A tapered cantilever beam A B supports a concentrated load P at the free end (see figure). The cross sections of the beam are rectangular with constant width A, depth d Aat support A, and depth ds= ^dJ2 at the support. Thus, the depth d and moment of inertia / at distance x from the free end are, respectively, in which / 4 is the moment of inertia at end A of the beam. Determine the equation of the deflection curve and the deflection S 4at the free end of the beam due to the load P.A tapered cantilever beam AB supports a concentrated load P at the free end (see figure). The cross sections of the beam are rectangular tubes with constant width b, outer Tube depth dAat A, and outer tube depth dB— ldA/2 at support B. The tube thickness is constant, as t = dA/20. IAis the moment of inertia of the outer tube at end A of the beam. If the moment of inertia of the tube is approximated as la{x) as defined, find the equation of the deflection curve and the deflection 5^ at the free end of the beam due to the load P.Repeat Problem 97-10, but now use the tapered propped cantilever tube A B with sliding support at B (see figure) that supports a concentrated load P at the sliding end. Find the equation of the deflection curve and the deflection 8Bat the sliding end of the beam due to the load P.A simple beam ACE is constructed with square cross sections and a double taper (see figure). The depth of the beam at the supports is dAand at the midpoint is dc= 2d 4. Each half of the beam has length L. Thus, the depth and moment of inertia / at distance x from the left-hand end are, respectively, in which IAis the moment of inertia at end A of the beam. (These equations are valid for .x between 0 and L, that is, for the left-hand half of the beam.) Obtain equations for the slope and deflection of the left-hand half of the beam due to the uniform load. From the equations in part (a), obtain formulas for the angle of rotation 94at support A and the deflection Scat the midpoint.A uniformly loaded simple beam AB (see figure) of a span length L and a rectangular cross section (b = width, h = height ) has a maximum bending stress m^ due to the uniform load Determine the strain energy Ustored in the beam.A simple beam AB of length L supports a concentrated load P at the midpoint (see figure). Evaluate the strain energy of the beam from the bending moment in the beam. Evaluate the strain energy of the beam from the equation of the deflection curve. From the strain energy determine the deflection S under the loadA propped cantilever beam AB of length L and with a sliding support at A supports a uniform load of intensity q (see figure). Evaluate the strain energy of the beam from the bending moment in the beam. Evaluate the strain energy of the beam from the equation of the deflection curve.A simple beam AB of length L is subjected to loads that produce a symmetric deflection curve with maximum deflection S at the midpoint of the span (see figure). How much strain energy U is stored in the beam if the deflection curve is (a) a parabola and (b) a half wave of a sine curve?A beam ABC with simple supports at A and B and an overhang BC supports a concentrated load P at the free end C (see figure). Determine the strain energy Ustored in the beam due to the load P. From the strain energy, find the deflection Scunder the load P. Calculate the numerical values of £/and Sc if the length L is 8 ft, the overhang length a is 3 ft, the beam is a W 10 x 12 steel wide-flange section, and the load P produces a maximum stress of 12,000 psi in the beam, (Use £ = 29 X 106 psi.)A simple beam ACB supporting a uniform load q over the first half of the beam and a couple of moment MQat end B is shown in the figure. Determine the strain energy U stored in the beam due to the load q and the couple M0acting simultaneously.The frame shown in the figure consists of a beam ACB supported by a strut CD. The beam has length IL and is continuous through joint C A concentrated load P acts at the free end B. Determine the vertical deflection 8Bat point B due to the load P. Note: Let El denote the flexural rigidity of the beam, and let EA denote the axial rigidity of the strut. Disregard axial and shearing effects in the beam, and disregard any bending effects in the strut.A simple beam AB of length L is loaded at the left-hand end by a couple of moment MQ(see figure). Determine the angle of rotation 0 , at support A. (Obtain the solution by determining the strain energy of the beam and then using Castigliano's theorem.)The simple beam shown in the figure supports a concentrated load P acting at distance a from the left-hand support and distance h from the right-hand support. Determine the deflection SDat point D where the load is applied. (Obtain the solution by determining the strain energy of the beam and then using Castigliano's theorem.)An overhanging beam ABC supports a concentrated load P at the end of the overhang (see figure). Span AB has length L, and the overhang has length o. Determine the deflection Scat the end of the overhang. (Obtain the solution by determining the strain energy of the beam and then using Castiglia-nos theorem.)The cantilever beam shown in the figure supports a triangularly distributed load of maximum intensity qü. Determine the deflection SBat the free end B. (Obtain the solution by determining the strain energy of the beam and then using Castigliano's theorem.)A simple beam ACB supports a uniform load of intensity q on the left-hand half of the span (see figure). Determine the angle of rotation ÔBat support B>(Obtain the solution by using the modified form of Castigliano's theorem.)A cantilever beam ACB supports two concentrated loads P\ and P-,, as shown in the figure. Determine the deflections Scand SBat points C and B, respectively. (Obtain the solution by using the modified form of Castigliano's theorem.)The cantilever beam A CB shown in the hgure is subjected to a uniform load of intensity q acting between points A and C. Determine the angle of rotation ÔAat the free end A>(Obtain the solution by using the modified form of Castigliano's theorem.)The frame A BC support s a concentrated load P at point C (see figure). Members AB and BC have lengths h and fh respectively. Determine the vertical deflection Scand angle of rotation $c at end C of the frame, (Obtain the solution by using the modified form of Ca s tig] i a no s theorem.)A simple beam ABC DE supports a uniform load of intensity iy (see figure). The moment of inertia in the central part of the beam (BCD) is twice the moment of inertia in the end parts (AB and DE). Find the deflection Scat the midpoint C of the beam. (Obtain the solution by using the modified form of Castigliano's theorem.)An overhanging beam ABC is subjected to a couple MAat the free end (see figure). The lengths of the overhang and the main span are a and L, respectively. Determine the angle of rotation 04and deflection S4at end A. (Obtain the solution by using the modified form of Castigliano's theorem.)An overhanging beam ABC rests on a simple support at A and a spring support at B (see figure). A concentrated load P acts at the end of the overhang. Span AB has length /_, the overhang has length a, and the spring has stiffness k. Determine the downward displacement 6Cof the end of the overhang. (Obtain the solution by using the modified form of Castigliano's theorem.)A symmetric beam A BCD with overhangs at both ends supports a uniform load of intensity q (see figure). Determine the deflection SDat the end of the overhang. (Obtain the solution by using the modified form of Castiglianos theorem.)A heavy object of weight W is dropped onto the midpoint of a simple beam AB from a height h (see figure). Obtain a formula for the maximum bending stress ^ma* due to tne filing weight in terms of h, st, and 5st, where it is the maximum bending stress and Sstis the deflection at the midpoint when the weight W acts on the beam as a statically applied load. Plot a graph of the ratio o"max/ö"it (that is, the ratio of the dynamic stress to the static stress) versus the ratio iifS^r(Let h/S^ vary from 0 to 10.)An object of weight Wis dropped onto the midpoint of a simple beam AB from a height h (see figure). The beam has a rectangular cross section of area A. Assuming that h is very large compared to the deflection of the beam when the weight PFis applied statically, obtain a formula for the maximum bending stress crniilx in the beam due to the falling weight.A cantilever beam AB of length L = 6 It is constructed of a W 8 x 21 wide-flange section (see figure), A weight W = 1500 lb falls through a height h = 0.25 in. onto the end of the beam. Calculate the maximum deflection £m.iy of the end of the beam and the maximum bendini* stress *rm,vdue to the falling weight, (Assume E = 30 X 10 psi,)A weight W = 20 kN falls through a height h = 1,0 mm onto the midpoint or a simple beam of length L = 3 m (see figure). The beam is made of wood with square cross section (dimension don each side) and E = 12 GPa. If the allowable bending stress in the wood is °aLLow =10MPa, what is the minimum required dimensionA weight W = 4000 lb falls through a height h = 0.5 in, onto the midpoint of a simple beam of length L = 10 ft (see figure). Assuming that the allowable bending stress in the beam is = 18,000 psi and E = 30 x 10* psi, select the lightest wide-flange beam listed in Table F-l(a) in Appendix F that will be satisfactory.An overhanging beam ABC with a rectangular cross section has the dimensions shown in the figure. A weight W = 750 N drops onto end C of the beam. If the allowable normal stress in bending is 45 MPa, what is the maximum height h from which the weight may be dropped? (Assume E = 12 G Pa,)A heavy flywheel rotates at an angular speed m about its axis of rotation. If the flywheel suddenly freezes to the axle, what will be the reaction R at support A of the beam?A simple beam AB of length L and height /; undergoes a temperature change such that the bottom of the beam is at temperature 7™, and the top of the beam is at temperature Tx(see figure). Determine the equation of the deflection curve of the beam, the angle of rotation 9Aat the left-hand support, and the deflection 8mjLXat the midpoint.A cantilever beam JA of length Land height/; (see figure) is subjected to a temperature change such that the temperature at the top is 7[ and at the bottom is 7. Determine the equation of the deflection curve of the beam, the angle of rotation BBat end and the deflection 8Bat end B,An overhanging beam ABC of height h has a sliding support at A and a roller at B, The beam is heated to a temperature Tton the top and T2on the bottom (see figure). Determine the equation of the deflection curve of the beam, the angle of rotation 6Cat end C, and the deflection Bfat end C.A simple beam AB of length L and height h (see figure) is heated in such a manner that the temperature difference 7= T{between the bottom and top of the beam is proportional to the distance from support A: that is, assume the temperature difference varies linearly along the beam: T2- Tt= Tax in which 7"0 is a constant having units of temperature (degrees) per unit distance. Determine the maximum deflection SW9Xof the beam, Repeat for a quadratic temperature variation along the beam, so T2+T1= TaxBeam AB has an elastic support kR at A, pin support at B, length L, height h (see figure), and is heated in such a manner that the temperature difference T2T1 between the bottom and top of the beam is proportional to the distance from support A. Assume the temperature difference varies linearly along the beam: T2T1=T0x in which T0 is a constant having units of temperature (degrees) per unit distance. Assume the spring at A is unaffected by the temperature change. Determine the maximum deflection max of the beam, Repeat for a quadratic temperature variation along the beam, so T2T1=T0x2 What is max for parts (a) and (b) if kR goes to infinity?A propped cantilever steel beam is constructed from a W12 × 35 section. The beam is loaded by its self-weight with intensity q. The length of the beam is 1L5 ft. Let E = 30,000 ksi. Calculate the reactions at joints A and B. Find the location of zero moment within span AB. Calculate the maximum deflection of the beam and the rotation at joint B.A fixed-end b earn is subjected to a point load at mid-span. The beam has a rectangular cross section (assume that the h/b ratio is 2) and is made of wood (E = 11GPa). Find height h of the cross section if the maximum displacement of the beam is 2 mm. Calculate the displacement of the beam at the inflection points.A propped cantilever beam AB of a length L is loaded by a counterclockwise moment M0acting at support B (see figure). Beginning with the second-order differential equation of the deflection curve (the bending-moment equation), obtain the reactions, shear forces, bending moments, slopes, and deflections of the beam. Construct the shear-force and bending-moment diagrams, labeling all critical ordinates.A fixed-end beam AB of a length L supports a uniform load of intensity q (see figure). Beginning with the second-order differential equation of the deflection curve (the bending-moment equation), obtain the reactions, shear forces, bending moments, slopes, and deflections of the beam. Construct the shear-force and bending-moment diagrams, Labeling all critical ordinales.A cantilever beam AB of a length L has a fixed support at A and a roller support at B (see figure). The support at B is moved downward through a distance B . Using the fourth-order differential equation of the deflection curve (the load equation), determine the reactions of the beam and the equation of the deflection curve. Note: Express all results in terms of the imposed displacement B.A cantilever beam of a length L and loaded by a uniform load of intensity q has a fixed support at A and spring support at B with rotational stiffness kR. A rotation B at B results in a reaction moment MB=kRxB. Find rotation B and displacement Bat end B. Use the second-order differential equation of the deflection curve to solve for displacements at end B.A cantilever beam has a length L and is loaded by a triangularly distributed load of maximum intensity q0at B. Use the fourth-order differential equation of the deflection curve to solve for reactions at A and B and also the equation of the deflection curve.A propped cantilever beam of a length L is loaded by a parabolically distributed load with a maximum intensity q0at B. Use the fourth-order differential equation of the deflection curve to solve for reactions at A and B and also the equation of the deflection curve. Repeat part (a) if the parabolic load is replaced by q0sin(x/2L).A propped cantilever beam of a length L is loaded by a parabolically distributed load with a maximum intensity q0 at A. Use the fourth-order differential equation of the deflection curve to solve for reactions at A and B and also the equation of the deflection curve. Repeat part (a) if the parabolic load is replaced by q0cos(x/2L).A fixed-end beam of a length L is loaded by a distributed load in the form of a cosine curve with a maximum intensity q0 at A. Use the fourth-order differential equation of the deflection curve to solve for reactions at A and B and also the equation of the deflection curve, Repeat part (a) using the distributed load q0sin(x/2L).A fixed-end b earn of a length L is loaded by a distributed load in the form of a cosine curve with a maximum intensity q0at A. Use the fourth-order differential equation of the deflection curve to solve for reactions at A and B and also the equation of the deflection curve. Repeat part (a) if the distributed load is now q0(1x2/L2) .A fixed-end beam of a length L is loaded by triangularly distributed load of a maximum intensity q0at B. Use the fourth-order differential equation of the deflection curve to solve for reactions at A and B and also the equation of the deflection curve.A counterclockwise moment M0acts at the midpoint of a fixed-end beam ACB of length L (see figure). Beginning with the second-order differential equation of the deflection curve (the bendingmoment equation), determine all reactions of the beam and obtain the equation of the deflection curve for the left-hand half of the beam. Then construct the shear-force and bending-moment diagrams for the entire beam, labeling all critical ordinales. Also, draw the deflection curve for the entire beam.A propped cantilever beam of a length L is loaded by a concentrated moment M0at midpoint C Use the second-order differential equation of the deflection curve to solve for reactions at A and B. Draw shear-force and bending-moment diagrams for the entire beam. Also find the equations of the deflection curves for both halves of the beam, and draw the deflection curve for the entire beam.A propped cantilever beam is subjected to uniform load q. The beam has flexural rigidity EI = 2000 kip-ft2 and the length of the beam is 10 ft. Find the intensity q of the distributed load if the maximum displacement of the beam is max = 0.125 in.Repeat Problem 10.3-15 using L = 3.5 m, max = 3 mm, and EI = 800 kN·m2.A two-span, continuous wood girder (E = 1700 ksi) supports a roof patio structure (figure part a). A uniform load of intensity q acts on the girder, and each span is of length 8 ft. The girder is made up using two 2×8 wood members (see figure part b). Ignore the weight of the beam. Use the nominal dimensions of the beam in your calculations. Find the reactions at A, B, and C. Use the method of superposition to calculate the displacement of the beam at the mid-sapn of segment AB. Hind: See Figs. 10-14c and 10-14d in Example 10-3.A fixed-end beam AB carries point load P acting at point C. The beam has a rectangular cross section (b = 75 mm, h = 150 mm). Calculate the reactions of the beam and the displacement at point C. Assume that E = 190 GPa.A fixed-end beam AB supports a uniform load of intensity q = 75 lb/ft acting over part of the span. Assume that EI = 300kip-ft2. Calculate the reactions at A and B. Find the maximum displacement and its location. Repeat part (a) if the distributed load is applied from A to B.-4-4 A cantilever beam is supported at B by cable BC. The beam carries a uniform load q = 200 N/M. If the length of the beam is L = 3 m, find the force in the cable and the reactions at A. Ignore the axial flexibility of the cable.A propped cantilever beam AB of a length L carries a concentrated load P acting at the position shown in the figure. Determine the reactions RA, RB,and MAfor this beam. Also, draw the shear-force and bendingmoment diagrams, labeling all critical ordinates.A beam with a sliding support at B is loaded by a uniformly distributed load with intensity q. Use the method of superposition to solve for all reactions. Also draw shear-force and bending-moment diagrams, labeling all critical ordinales.A propped cantilever beam of a length 2L with a support at B is loaded by a uniformly distributed load with intensity q. Use the method of superposition to solve for all reactions. Also draw shear-force and bending-moment diagrams, labeling all critical ordinates.The continuous frame ABC has a pin support at /l, roller supports at B and C, and a rigid corner connection at B (see figure). Members AB and BC each have flexural rigidity EI. A moment M0acts counterclockwise at B, Note: Disregard axial deformations in member AB and consider only the effects of bending. Find all reactions of the frame. Find joint rotations B at A, B, and C. Find the required new length of member BC in terms of L., so that B in part (b) is doubled in size.The continuous frame ABC has a pin support at A, roller supports at B and C and a rigid corner connection at B (see figure). Members AB and BC each have flexural rigidity EI. A moment Müacts counterclockwise at A. Note: Disregard axial deformations in member AB and consider only the effects of bending. Find all reactions of the frame. Find joint rotations 6 at A, B, and C. Find the required new length of member AB in terms of L, so that A in part (b) is doubled in size.Beam AB has a pin support at A and a roller support at B Joint B is also restrained by a linearly elastic rotational spring with stiffness kR, which provides a resisting moment MBdue to rotation at B. Member AB has flexural rigidity EI. A moment M0acts counterclockwise at B. Use the method of superposition to solve for all reactions. Find an expression for joint rotation Ain terms of spring stiffness kR. What is Awhen kR 0? What is Awhen kR— ? What is Awhen kR= 6EI/L?The continuous frame ABCD has a pin support at B: roller supports at A,C, and D; and rigid corner connections at B and C (see figure). Members AB, BC, and CD each have flexural rigidity EL Moment M0acts counterclockwise at B and clockwise at C. Note: Disregard axial deformations in member A Band consider only the effects of bending. Find all reactions of the frame. Find joint rotations al A, B. C, and D. Repeat parts (a) and (b) if both moments M0are counter clockwise.Two flat beams AB and CD, lying in horizontal planes, cross at right angles and jointly support a vertical load P at their midpoints (see figure). Before the load P is applied, the beams just touch each other. Both beams are made of the same material and have the same widths. Also, the ends of both beams are simply supported. The lengths of beams AB and CD are LABand LCD, respectively. What should be the ratio tABltCDof the thicknesses of the beams if all four reactions arc to be the same?-4-13 A propped cantilever beam of a length 2L is loaded by a uniformly distributed load with intensity q. The beam is supported at B by a linearly elastic spring with stiffness k. Use the method of superposition to solve for all reactions. Also draw shear-force and bending-moment diagrams, labeling all critical ordinates. Let k = 6EI/L3.A propped cantilever beam of a length 2L is loaded by a uniformly distributed load with intensity q. The beam is supported at B by a linearly elastic rotational spring with stiffness kR,which provides a resisting moment MBdue to rotation B . Use the method of superposition to solve for all reactions. Also draw shear-force and bending-moment diagrams, labeling all critical ordinates. Let kR= El/L.Determine the fixed-end moments (MAand MB) and fixed-end forces (R4and Rs) for a beam of length L supporting a triangular load of maximum intensity q0(see figure). Then draw the shear-force and bending-moment diagrams, labeling all critical ordinates.A continuous beam ABC wit h two unequal spans, one of length L and one of length 2L, supports a uniform load of intensity q (sec figure). Determine the reactions RA, RB, and RCfor this beam. Also, draw the shear-force and bending-moment diagrams, labeling all critical ordinates.Beam ABC is fixed at support A and rests (at point B) upon the midpoint of beam DE (see part a of the figure). Thus, beam, ABC may be represented as a propped cantilever beam with an overhang BC and a linearly elastic support of stiffness k at point B (see part b of the figure). The distance from A to B is L = 10 ft, the distance from B to C is L/2 = 5 ft, and the length of beam DE is L = 10 ft. Both beams have the same flexural rigidity EI. A concentrated load P = 1700 lb acts at t lie free end of beam ABC. Determine the reactions RA, RB+ and MAfor beam ABC. Also, draw the shear-force and bending-moment diagrams for beam ABC, labeling all critical ordinates.A propped cantilever beam has flexural rigidity EI =4.5MN·m2. When the loads shown are applied to the beam, it settles at joint B by 5 mm. Find the reaction at joint B.A triangularly distributed 1oad with a maximum intensity of q0= 10 lb/ft acts on propped cantilever beam AB. If the length L of the beam is 10 ft, find the reactions at A and B.A fixed-end beam is loaded by a uniform load q = 15 kN/m and a point load P = 30 kN at mid-span. The beam has a length of 4 m and modulus of elasticity of 205 GPa. Find reactions at A and B. Calculate the height of the beam if the displacement at mid-span is known to be 3 mm. Assume that the beam has rectangular cross section with h/b = 2.Uniform load q = 10 lb/ft acts over part of the span of fixed-end beam AB (see figure). Upward load P = 250 lb is applied 9 ft to the right of joint A. Find the reactions at A and B.A propped cantilever beam with a length L = 4 m is subjected to a trapezoidal load with intensities q0= 10 kN/m and q1 = 15 kN/m. Find the reactions at A and B. Hint: The loading is the sum of uniform and triangular loads.A cant i levé r b ea m i s supported by a tie rod at B as shown. Both the tie rod and the beam are steel with E = 30 x 106 psi. The tie rod is just taut before the distributed load q = 200 lb/ft is applied. Find the tension force in the tie rod. Draw shear-force and bending-moment diagrams for the beam, labeling all critical ordinates.The figure shows a nonprismatic, propped cantilever beam AB with flexural rigidity 2EI from A to C and EI from C to B. Determine all reactions of the beam due to the uniform load of intensity q. Hint: Use the results of Problems 9.7-1 and 9.7-2.A beam ABC is fixed at end A and supported by beam DE at point B (sec figure). Both beams have the same cross section and are made of the same material. Determine all reactions due to the load P. What is the numerically largest bending moment in cither beam?A three-span continuous beam A BCD with three equal spans supports a uniform load of intensity q (see figure). Determine all reactions of this beam and draw the shear-force and bending-moment diagrams, labeling all critical ordinales.A beam rests on supports at A and B and is loaded by a distributed load with intensity q as shown. A small gap exists between the unloaded beam and the support at C. Assume that span length L = 40 in. and flexural rigidity of the beam EI = 04 x 109lb-in2. Plot a graph of the bending moment at B as a function of the load intensity q. Hint: See Example 9-9 for guidance on computing the deflection at C.A propped cantilever beam is subjected to two triangularly distributed loads, each with a peak load intensity equal to q0(see figure), lind the expressions for reactions at A and C using superposition. Plot shear and moment diagrams.A propped cantilever beam is loaded by a triangular distributed load from A to C (sec figure). The load has a peak intensity q0= 10 lb/ft. The length of the beam is 12 ft. Find support reactions at A and B.A fixed-end beam AB of a length L is subjected to a moment M0acting at the position shown in the figure. Determine all reactions for this beam. Draw shear-force and bending-moment diagrams for the special case in which a = b = L/2.A temporary wood flume serving as a channel for irrigation water is shown in the figure. The vertical boards forming the sides of the flume are sunk in the ground, which provides a fixed support. The top of the flume is held by tic rods that are tightened so that there is no deflection of the boards at that point. Thus, the vertical boards may be modeled as a beam AB, supported and loaded as shown in the last part of the figure. Assuming that the thickness t of the boards is 1,5 in., the depth d of the water is 40 in., and the height h to the tie rods is 50 in., what is the maximum bending stress in the boards? Hint: The numerically largest bending moment occurs at the fixed support.Two identical, simply supported beams AB and CD are placed so that they cross each other at their midpoints (sec figure). Before the uniform load is applied, the beams just touch each other at the crossing point. Determine the maximum bending moments (mab)max* and (MCD)max beams AB and CD, respectively, due to the uniform load if the intensity of the load is q = 6.4 kN/m and the length of each beam is L = 4 m.The cantilever beam AB shown in the figure is an S6 × 12.5 steel I-beam with E = 30 × 106 psi. The simple beam DE is a wood beam 4 in. x 12 in. (nominal dimensions) in cross section with E = 1.5 x 106 psi. A steel rod AC of diameter 0.25 in., length 10 ft, and E = 30 x 106 psi serves as a hanger joining the two beams. The hanger fits snugly between the beams before the uniform load is applied to beam DE. Determine the tensile force Fin the hanger and the maximum bending moments MABand MDEin the two beams due to the uniform load, which has an intensity q = 400 lb/ft. Hint: To aid in obtaining the maximum bending moment in beam DE, draw the shear-force and bending-moment diagrams.The beam AB shown in the figure is simply supported at A and B and supported on a spring of stiffness k at its midpoint C. The beam has flexural rigidity EI and length IL. What should be the stiffness k of the spring in order that the maximum bending moment in the beam (due to the uniform load) will have the smallest possible value?The continuous frame ABC has a fixed support at A, a roller support at C, and a rigid corner connection at B (see figure). Members AB and BC each have length L and flexural rigidity EL. A horizontal force P acts at mid-height of member AB. Find all reactions of the frame. What is the largest bending moment Mmaxin the frame? Note: Disregard axial deformations in member AB and consider only the effects of bending.The continuous frame ABC has a pinned support at A, a sliding support at C, and a rigid corner connection at B (see figure). Members AB and BC each have length L and flexural rigidity EI. A horizontal force P acts at mid-height of member AB. Find all reactions of the frame. What is the largest bending moment Mmaxin the frame? Note: Disregard axial deformations in members AB and BC and consider only the effects of bending.A wide-flange beam ABC rests on three identical spring supports at points A, B, and C (see figure). The flexural rigidity of the beam is EI =6912 x 106 lb-in, and each spring has stiffness k = 62,500 lb/in. The length of the beam is L = 16 ft. If the load P is 6000 lb, what are the reactions RA, RB,and RC? Also, draw the shear-force and bending-moment diagrams for the beam, labeling all critical ordinates.A fixed-end beam AB of a length L is subjected to a uniform load of intensity q acting over the middle region of the beam (sec figure). Obtain a formula for the fixed-end moments MAand MBin terms of the load q, the length L, and the length h of the loaded part of the beam. Plot a graph of the fixed-end moment MAversus the length b of the loaded part of the beam. For convenience, plot the graph in the following nondimensional form: MAqL2/l2versusbL with the ratio b/L varying between its extreme values of 0 and 1. (c) For the special case in which ù = h = L/3, draw the shear-force and bending-moment diagrams for the beam, labeling all critical ordinates.A beam supporting a uniform load of intensity q throughout its length rests on pistons at points A, C and B (sec figure). The cylinders are filled with oil and are connected by a tube so that the oil pressure on each piston is the same. The pistons at A and B have diameter d1and the piston at C has diameter D2. (a) Determine the ratio of d2to d1so that the largest bending moment in the beam is as small as possible. Under these optimum conditions, what is the largest bending moment Mmaxin the beam? What is the difference in elevation between point C and the end supports?A thin steel beam AB used in conjunction with an electromagnet in a high-energy physics experiment is securely bolted to rigid supports (see figure), A magnetic field produced by coils C results in a force acting on the beam. The force is trapezoidally distributed with maximum intensity q0= 18 kN/m. The length of the beam between supports is L = 200 mm, and the dimension c of the trapezoidal load is 50 mm. The beam has a rectangular cross section with width b = 60 and height h = 20 mm. Determine the maximum bending stress max and the maximum deflection for the beam. (Disregard any effects of axial deformations and consider only the effects of bending. Use E = 200 GPa.)Find an expression for required moment MA(in terms of q and L) that will result in rotation A= 0 due to MAand q loadings applied at the same time.Repeat Problem 10.4-41 for the loading shown in the figure.A propped cantilever beam is loaded by two different load patterns (see figures a and b). Assume that El is constant and the total beam length is L. Find expressions for reactions at  and B for each beam. Plot shear and moment diagrams. Assume that a = L/B.A cable CD of a length H is attached to the third point of a simple beam AB of a length L (see figure). The moment of inertia of the beam is I, and the effective cross-sectional area of the cable is A. The cable is initially taut but without any initial tension, (a) Obtain a formula for the tensile force S in the cable when the temperature drops uniformly by T degrees, assuming that the beam and cable are made of the same material (modulus of elasticity E and coefficient of thermal expansion . Use the method of superposition in the solution, (b) Repeat part (a), assuming a wood beam and steel cable.A propped cantilever beam, fixed at the left-hand end A and simply supported at the right-hand end B, is subjected to a temperature differentia] with temperature T1on its upper surface and T2on its lower surface (see figure).Solve t he preceding problem by integrating the differential equation of the deflection curve.A two-span beam with spans of lengths L and L/3 is subjected to a temperature differential with temperature T1on its upper surface and T2on its lower surface (see figure). Determine all reactions for this beam. Use the method of superposition in the solution. Assume the spring support is unaffected by temperature. What are the reactions when k ?Solve the preceding problem by integrating the differential equation of the deflection curve.Assume that the deflected shape of a beam AB with immovable pinned supports (see figure) is given by the equation v = - sinx/L, where is the deflection at the midpoint of the beam and L is the length. Also, assume that the beam has constant axial rigidity EA. Obtain formulas for the longitudinal force H at the ends of the beam and the corresponding axial tensile stress t. For an aluminum-alloy beam with E = 10 × 106 psi, calculate the tensile stress twhen the ratio of the deflection S to the length L equals 1/200, 1/400, and 1/600.(a) A simple beam AB with length L and height h supports a uniform load of intensity q (see the figure part a). Obtain a formula for the curvature shortening A of this beam. Also, obtain a formula for the maximum bending stress b in the beam due to the load q. Now assume that the ends of the beam are pinned so that curvature shortening is prevented and a horizontal force H develops at the supports (see the figure part b). Obtain a formula for the corresponding axial tensile stress t . Using the formulas obtained in parts (a) and (b), calculate the curvature shortening , the maximum bending stress b, and the tensile stress t for the following steel beam: length L = 3m, height h = 300 mm, modulus of elasticity E = 200 GPa, and moment of inertia I = 36 x 106 mm4. Also, the load on the beam has intensity q = 25 kN/m. Compare the tensile stress tproduced by the axial forces with the maximum bending stress bproduced by the uniform load.A rigid bar of length L is supported by a linear elastic rotational spring with rotational stiffness ßRat A. Determine the critical load Pcr for the structure.The figure shows an idealized structure consisting of a rigid bar with pinned connections and linearly elastic springs. Rotational stiffness is denoted R, and translational stiffness is denoted . Determine the critical load PCTfor the structure from the figure part a. Find PCIif another rotational spring is added at B from the figure part b.-2-3. Two rigid bars are connected with a rotational spring, as shown in the figure. Assume that the elastic rotational spring constant is R= 75 kip-in./rad. Calculate the critical load Pcrof the system. Assume that L = 6 ft.Repeat Problem 11.2-3 assuming that R= 10 kN · m/rad and L = 2 m.The figure shows an idealized structure consisting of two rigid bars with pinned connections and linearly elastic rotational springs. Rotational stiffness is denoted ßR. Determine the critical load Pcrfor the structure.An idealized column consists of rigid bar ABCD with a roller support at B and a roller and spring support at D. The spring constant at D. is ß = 750 N/m. Find the critical load Pcrof the column.An idealized column is made up of rigid segments ABC and CD that are joined by an elastic connection at C with rotational stiffness ßR= 100 kip-in./rad. The column has a roller support at B and a sliding support at D. Calculate the critical load Pcrof the column.The figure shows an idealized structure consisting of bars AB and BC that are connected using a hinge at B and linearly elastic springs at A and B. Rotational stiffness is denoted ßRand translational stiffness is denoted ß. Determine the critical load Pcrfor the structure from the figure part a. Find PCTif an elastic connection is now used to connect bar segments AB and BC from the figure part b.The figure shows an idealized structure consisting of two rigid bars joined by an elastic connection with rotational stiffness ßR. Determine the critical load PCTfor the structure.The figure shows an idealized structure consisting of rigid bars ABC And DEF joined by a linearly elastic spring ß between C and D. The structure is also supported by translational elastic support ß at B and rotational elastic support ßRat E. Determine the critical load Pcrfor the structure.The figure shows an idealized structure consisting of an L-shaped rigid bar structure supported by linearly elastic springs at A and C. Rotational stiffness is denoted ßRand translational stiffness is denoted ß. Determine the critical load Pcrfor the structure.Rigid column ABCD has an elastic support at B with translational stiffness ß. Find an expression for the critical load Pcrof the column.An idealized column is made up of rigid bars ABC and CD that are joined by a rotational elastic connection at C with stiffness ßR. The column has a roller support at B and a pin support at D. Find an expression for the critical load Pcr of the column.An idealized column is composed of rigid bars ABC and CD joined by an elastic connection with rotational stiffness ßRIat C. There is a roller support at B and an elastic support at D with translationa1 spring stiffness ß and rotational stiffness ßR2. Find the critical buckling loads for each of the two buckling modes of the column. Assume that L = 3 m, ß = 9 kN/m, and ßR]= ßR1= ßL2. Sketch the buckled mode shapes.Repeat Problem 11.2-14 using L = 12 ft, ß = 0.25 kips/in., ßRl= 1.5ßL2, and ßR2= 2 ßR1.An idealized column is composed of rigid bars ABC and CD joined by an elastic connection with rotational stiffness ßRat C. There is an elastic support at B with translational spring stiffness ß and a pin support at D. Find the critical buckling loads for each of the two buckling modes of the column in terms of ßL. Assume that ßR= ßL2. Sketch the buckled mode shapes.Column AB has a pin support at A,a roller support at B, and is compressed by an axial load P (see figure). The column is a steel W12 × 35 with modulus of elasticity E = 29,000 ksi and proportional limit pl = 50 ksi. The height of the column is L = 12 ft. Find the allowable value of load P assuming a factor of safety n = 2,5.Slender column ABC is supported at A and C and is subjected to axial load P. Lateral support is provided at mid-height if but only in the plane of the figure; lateral support perpendicular to the plane of the figure is provided only at ends A and C. The column is a steel W shape with modulus of elasticity E = 200 GPa and proportional limit pl= 400 MPa. The total length of the column L = 9 m. If the al low-able load is 150 kN and the factor of safety is 2.5, determine the lightest W 200 section that can be used for the column. (See Table F-l(b), Appendix F).Calculate the critical load PCTfor a W 8 × 35 steel column (see figure) having a length L = 24 ft and E = 30 × 106 psi under the following conditions: The column buckles by bending about its strong axis (axis 1-1). the column buckles by bending about its weak axis (axis 2-2). In both cases, assume that the column has pinned ends.Solve the preceding problem for a W 250 × 89 steel column having a length L = 10 m. Let E = 200 GPa.Solve Problem 11.3-3 for a W 10 × 45 steel column having a length L = 28 ft.