The First ProjectUnderstand the problems and define the design problem considering characteristic of a structure. Perform the optimization using optimization algorithm in software such as Excel Automatic Design Laboratory Example 1 Calculate the compressive stress σc in the circular piston rod (see figure) when a force P = 10 lb is applied to the brake pedal. Assume that the line of action of the force P is parallel to the piston rod, which has diameter 0.22 in. Also, the other dimensions shown in the figure (2.0 in. and 9.0 in.) are measured perpendicular to the line of action of the force P. Automatic Design Laboratory Example 2 A long retaining wall is braced by wood shores set at an angle of part of the figure. The shores are evenly spaced, 10 ft apart. 30° and supported by concrete thrust blocks, as shown in the first For analysis purpose, the wall and shores are idealized as shown in the second part of the figure. Note that the base of the wall and both ends of the shores are assumed to be pinned. The pressure of the soil against the wall is assumed to be triangularly distributed, and the resultant force acting on a 10-ft length of the wall is F=45 k. If each shore has a 6 in. 6 in. square cross section, what is the compressive stress σc in the shores? Automatic Design Laboratory Example 3 A tie down on the deck of a sailboat consists of a bent bar bolted at both ends, as shown in the figure. The diameter dB of the bar is 6 mm, the diameter dW of the washers is 22 mm, and the thickness t of the fiberglass deck is 10 mm. If the allowable shear stress in the fiberglass is 2.1 ㎫, and the allowable bearing pressure between the washer and the fiberglass is 3.8 ㎫, what is the allowable load Pallow on the tie-down? Automatic Design Laboratory If the allowable shear stress in the bolts is 14 ksi.Example 4 A torque T0 is transmitted between two flanged shafts by means of four ¾-in. The diameter of the bolt circle is d = 6 in.) Automatic Design Laboratory . what is the maximum permissible torque? (disregard friction between the flanges. bolts (see figure). If the lifeboat weights 1500 lb. what is the maximum weight that should be carried in the lifeboat? 15° with the horizontal. A pin of diameter d = 0.Example 5 A lifeboat hangs from two ship’s davits. one on each side of the davit. as shown in the figure. and the allowable shear stress in the pins is 4000 psi. Cables attached to the lifeboat pass over the pulleys and wind around winches that raise and lower the lifeboat.80 in. The allowable tensile force in each cable is Automatic Design Laboratory . passes through each davit and supports two pulleys. The lower parts of the cables are vertical and the upper parts make and angle α = 1800 lb. Example 6 What is the maximum possible value of the clamping force in the jaws of the pliers shown in the figure if a = 90 mm. and is the maximum permissible value of the applied load P if a factor of safety of 3. b = 40 mm.5 with respect to failure of the pin is to be maintained? the ultimate shear stress in the 6-mm diameter pin is 320 ㎫? What Automatic Design Laboratory . 28L. based upon an allowable compressible stress σC in the connecting Automatic Design Laboratory . which is supported by bearings. The piston slides without friction in a cylinder and is subjected to a force P (assumed to be constant) while moving to the right in the figure. (a) Obtain a formula for the maximum permissible force Pallow rod. The connecting rod. A = 63. exerts a resisting moment M against the crank arm.62 ㎟ and R = 0. The axle at C.Example 7 The piston in an engine is attached to a connecting rod AB. (b) Calculate the force Pallow for the following data: σC = 150 ㎫ . which in turn is connected to a crank arm BC (see figure). is attached at both ends by pins. The cranks arm rotates about the axle at C with the pin at B moving in a circle of radius R. which has cross-sectional area A and length L. (Note: Disregard the rounded corners of the tube when calculating its weight. The tube hangs from a pin of diameter d that is held by the cables at points A and B.Example 8 A square steel tube of length L = 6. The cross section is a allowable shear stress in the pin is 60 ㎫.0 m and width b2 = 250 mm is hoisted by a crane (see figure). Determine the minimum diameter of the pin in order to support the weight of the tube. and the allowable bearing stress the pin and the tube is 90 ㎫.) Automatic Design Laboratory . If the wall thickness of the post is 0. what is the minimum permissible value of the outer diameter d2? Automatic Design Laboratory .Example 9 A tubular post of outer diameter d2 is guyed by two cables fitted with turnbuckles (see figure). Also. the angle between the cables and the ground is 60°. Both cables are tightened to a tensile force of 32 k. thus producing tension in the cables and compression in the post.. and the allowable compressive stress in the post is σC = 6000 psi. The cables are tightened by rotating the turnbuckles.5 in. 700 lb/ft. The span of the cable is L.800. 2 qL3 h 16 1 + δ = 8hEA 3L2 Automatic Design Laboratory . The cable consists of 27. and the origin of coordinates is at midspan. Therefore. integrate along the curve of the cable to obtain an equation for the elongation δ.572 parallel wires of diameter 0. Hint: Determine the tensile force T at any point n the cable from a free-body diagram of part of the cable: then determine the elongation of an element of the cable of length ds. which is uniform in intensity along the horizontal. let us represent the central region AOB of one of the main cables [see part (b) of the figure] as a parabolic cable supported at points A and B and carrying a uniform load of intensity q along the horizontal. (a) Derive the following formula for the elongation of cable AOB shown in part (b) of the figure: ` (b) Calculate the elongation δ of the central span of one of the main cables of the Golden Gate Bridge. finally. q = 12. the sag is h. the axial rigidity is EA. and E = 28. h = 470 ft. for which the dimensions and properties are L = 4200 ft.Example 10 The main cables of a suspension bridge [see part (a) of the figure] follow a curve that is nearly parabolic bridge deck.196 in.000 psi. respectively? (Note: See table 2-1 in Section 2. (a) What percent of the total load is now carried by the middle cable? (b) What are the stresses σM and σ0 in the middle and outer cables. The tensions in the cables are adjusted so that each cable carries one-third of the load (i.) Automatic Design Laboratory .. 20 kN).Example 11 Three steel cables jointly support a load of 60 kN (see figure).2 for properties of cables. Later. the load is increased by 40 kN to a total load of 100 kN.e. The diameter of the middle cable is 20 mm and the diameter of each outer cable is 12 mm. Example 12 A bumping post at the end of a track in a railway yard has a spring constant k = 6. The maximum possible displacement d of the end of the striking plate is 460 mm.1 MN/m (see figure). What is the maximum velocity that a railway car of weight W = 470 kN can have without damaging the bumping post when it strikes it? Automatic Design Laboratory . If the jump off point is 60 m above the water. and if it is desired to maintain a clearance of 10 m between the jumper and the water.1 kN (see figure). braking her fall with a long elastic shock cord having axial rigidity EA = 2. what length L of cord should be used? Automatic Design Laboratory .Example 13 A bungee jumper having a mass of 50 kg leaps from a bridge. Automatic Design Laboratory . Calculate the shear force V and bending moment M at the inboard end of the wing.Example 14 Under cruising conditions the distributed load acting on the wing of small airplane has the idealized variation shown in the figure. The outer diameter of the pipe is 150 mm. as shown in the figure. its thickness is 6 mm. Determine the maximum bending stress in the pipe due to its own weight. Automatic Design Laboratory . and its weight density is 18 kN/m3. The length of the pipe is L = 13 m and the distance between lifting points is s = 4 m.Example 15 A fiberglass pipe is hoisted by a crane using a sling. acting as shown in the figure.5 in. assuming the wheel gage L = 57 in.Example 16 A railroad tie (or sleeper) is subjected to two rail loads. each of magnitude P = 36 k. Automatic Design Laboratory . Calculate the maximum bending stress σmax in the tie due to the loads P. and the overhang length a = 19. The reaction q of the ballast is assumed to be uniformly distributed over the length of the tie. which has cross-sectional dimensions b = 12 in. and h = 10 in. Example 17 A stile crossing a pipeline at a chemical plant is supported by two fiberglass frames as shown in part (a) of the figure. height 150 mm. The slope of the inclined members of the frame is 2 on 3.8 kN/m acting on the horizontal part of the frame. Determine the maximum bending stress at the midsection of the frame due to a uniform load q = 2.82 m [see part (b) of the figure]. Automatic Design Laboratory . The cross section of the frame is I shaped with width 100 mm.46 m and height h = 0. Each frame has a span L = 5. and thickness 10 mm [see part (c) of the figure]. The wood beams. as shown in the figure.5 in.) Automatic Design Laboratory . (Note: the weight density γ of water equals 62.4 lb/ft3. which have thickness t = 2.. are simply supported by horizontal steel beams at A and B. Plot the stress σmax (psi) as the ordinate and the depth d (ft) as the abscissa.Example 18 A small dam of height h = 6 ft is constructed of vertical wood beams AB. Construct a graph showing the maximum bending stress σmax in the wood beams versus the depth d of the water above the lower support at B. 5 ㎫. known as balks. what is their minimum Automatic Design Laboratory .4 m long and that the balks are simply supported with a span of 3. which are called chesses. (this load includes an allowance for the weights of the chesses and balks. required width bmin? If the balks have a square cross section.Example 19 A pontoon bridge (see figure) is constructed of two wood beams. assume that a uniform floor load of 10 ㎪ acts over the chesses.) Also. assume that the chesses are 2. that span between adjacent pontoons and support the transverse floor beams. For purposes of design. The allowable bending stress in the wood is 17.6 m. Each beam has length L1 = 2. width b. with L2 = 2.5 kN/m3. determine the required dimensions b and h. Assuming that the middle cantilever supports 50 % of the load and each outer cantilever supports 25% of the load.1 m. The dimensions of the balcony floor are L1 × L2.5 m. Automatic Design Laboratory .Example 20 A small balcony constructed of wood is supported by three identical cantilever beams (see figure). The design load is 5.5 ㎪ acting over the entire floor area. (This load accounts for all loads except the weights of the cantilever beams.) The allowable bending stress in the cantilevers is 15 ㎫. which have a weight density γ = 5. and height h = 4b/3. Example 21 A horizontal shelf AD of length L = 36 in. and thickness t = 0. is supported by brackets at B and C [see part (a) of the figure].. width b = 12 in. which includes the weight of the shelf itself.75 in.. A uniform load of intensity q. Determine the maximum permissible value of the load q if the allowable bending stress in the shelf is σallow = 750 psi and the position of the supports is adjusted for maximum load-carrying capacity. Automatic Design Laboratory . The brackets are adjustable and may be placed in any desired positions between the ends of the shelf. acts on the shelf [see part (b) of the figure]. and presses against the base at A when the water level is not too much high (note that the panel will rotate about the pin at B if the depth d of the water exceeds a certain maximum depth dmax). The panel is pivoted at point B. which is height h above the base. and let γ be the weight density of water. Also. Derive the following formula for the minimum allowable thickness of the panel: t min 8γh 3 = σ allow sin 2 α ( ) (Note: To aid in deriving the formula. consider only the effects of bending in the panel. disregard the weight of the panel itself.Example 22 Water pressure acts against an inclined panel ABC that serves as a barrier (see figure). The panel has thickness t and is inclined at an angle α to horizontal.) Automatic Design Laboratory . observe that the maximum stress in the panel occurs when the depth of the water reaches the maximum depth dmax. The allowable bending stress in the panel is σallow. and consider the planks to act as simple beams between the piles. thick (actual dimension) that are supported by vertical wood piles of 12 in. as shown in the figure. calculate the maximum permissible spacing s of the piles. The lateral earth pressure is p1 = 100 lb/ft2 at the top of the wall and the top of the wall and p2 = 400 lb/ft2 at the bottom.) Automatic Design Laboratory . (Hint: Observe that the spacing of the piles may be governed by the load – carrying capacity of either the planks or the piles. Consider the piles to act as cantilever beams subjected to a trapezoidal distribution of load. diameter (actual dimension). To be on the safe side.Example 23 A retaining wall 5 ft high is constructed of horizontal wood planks 3 in. Assuming that the allowable stress in the wood is 1200 psi. assume that the pressure on the bottom plank is uniform and equal to the maximum pressure. tapered circular tubes (see figure).375 in. and dB = 10. At what distance x from the free end does the maximum bending stress occur? What is the magnitude σmax of the maximum bending stress? What is the ratio of the maximum stress to the largest stress σB at the support? Automatic Design Laboratory . respectively. Appendix D). and therefore the section modulus may be obtained from the formula S = πd2t/4.5 in. Because the thickness is small compared to the diameters. each beam may be represented as a cantilever AB of length L = 25 ft subjected to a lateral load P = 550 lb at the free end.5 in. For purposes of analysis. the moment of inertia at any cross section may be obtained from the formula I = πd3t/8 (see Case 22. The tubes have constant thickness t = 0. and average diameters dA = 3.Example 24 A tall signboard is supported by two vertical beams consisting of thin-walled. at ends A and B. Example 25 A square wood platform. The allowable bending stress for the planks is 2400 psi and the allowable shear stress is 100 psi. When analyzing the planks. noting especially that the reactions are distributed loads instead of concentrated loads.5 in. × 6 in.5 in.. see Appendix F) dimensions (actual dimensions 3. note that the maximum shear forces occur at the inside faces of the supporting beams.5 in. (a) Determine the allowable platform load w1 (lb/ft2) based upon the bending stress in the planks. The beams have 4 in.) supported on two 8-ft long beams. (c) Which of the preceding values becomes the allowable load wallow on the platform? (Hints: Use care in constructing the loading diagram for the planks. (b) Determine the allowable platform load w2 (lb/ft2) based upon the shear stress in the planks.) Automatic Design Laboratory . Also. nominal thickness tongue-and-groove planks (actual thickness 1. rests on masonry walls (see figure). disregard their weights and assume that their reactions are uniformly distributed over the top surfaces of the supporting beams. nominal The planks are designed to support a uniformly distributed load w (lb/ft2) acting over the entire top surface of the platform. 8 ft × 8 ft in area. × 5. The deck of the platform is constructed of 2 in. allowable bending stress of 7. the weight of one log and the planks it supports is equivalent to a uniform load of 850 N/m acting on the log. The logs are Douglas fir with average diameter 300 mm. Determine the maximum permissible wheel load W based upon (a) and Automatic Design Laboratory .Example 26 A simple log bridge in a remote area consists of two parallel logs with planks across them (see figure). which spans 2. Thus. Because the wheelbase of the truck is greater than 2. Assume that the weight of the truck is equally distributed between the two logs. only one set of wheels is on the bridge at a time. A truck moves slowly across the bridge.8 ㎫. the wheel load on one log is equivalent to a concentrated load W acting at any position along the span.5 ㎫.5 m. and (b) an allowable shear stress of 0. In addition.5 m. The center of gravity of the arm is 1.Example 27 An aluminum pole for a street light weighs 2300 N and supports an arm that weight 330 N (see figure).2 m from the axis of the pole. The outside diameter of the pole (at its base) is 225 mm and its thickness is 18 m. Automatic Design Laboratory . Determine the maximum tensile and compressive stresses σt and σc. respectively. in the pole (at its base) due to the weighs. Example 28 A cylindrical brick chimney of height H weights w = 825 lb/ft of height (see figure). The inner and outer diameters are d1 = 3 ft and d2 = 4 ft. The wind pressure against the side of the chimney is p = 10 lb/ft2 of projected area. respectively. Automatic Design Laboratory . Determine the maximum height H if there is to be no tension in the brickwork. (b) Determine the maximum permissible depth dmax of the water if there is to be no tension in the concrete. respectively.Example 29 A plain concrete wall (i. Assume plain concrete has weight density γc = 23 kN/m3.3 m.. at the base of the wall when the water level reaches the top (d = h). (a) Determine the maximum tensile and compressive stresses σt and σc. The height of the wall is h = 2 m and the thickness of the wall is t = 0.e. a wall with no steel reinforcement) rests on a secure foundation and serves as a small dam (see figure). Automatic Design Laboratory . . the vertical boards may be modeled as a beam AB. which provides a fixed support.Example 30 A temporary wood flume serving as a channel for irrigation water is shown in the figure. and the height h to the tie rods is 50 in. Thus.. Automatic Design Laboratory . The vertical boards forming the sides of the flume are sunk in the ground. the depth d of the water is 40 in. The top of the flume is held by tie rods that are tightened so that there is no deflection of the boards at that point. what is the maximum bending stress σ in the boards? (Hint: The numerically largest bending moment occurs at the fixed support. supported and loaded as shown in the last part of the figure. Assuming that the thickness t of the boards is 1.5 in.. 11. The moduli of elasticity for the steel and wood are Es = 30 106 psi and Ew = 1. thick steel plates on top and bottom. respectively.5 106 psi.5 in. The beam consists of a wood member ( 4 in. Automatic Design Laboratory . Calculate the maximum bending stresses σs in the steel plates and σw in the wood member due to the uniform load.25 in.Example 31 A simple beam on a 12 ft span supports a uniform load of intensity 600 lb/ft (see figure). in cross section) that is reinforced by 0. 30.) Automatic Design Laboratory . thickness 10 mm) is coated with brittle lacquer that cracks when the strain exceeds 200 10-6 (see figure).Example 32 A spherical steel pressure vessel (diameter 600 mm. What internal pressure p will cause the lacquer to develop cracks? (Assume E = 205㎬ and ν = 0. What is the minimum permissible thickness tmin of the cylinder wall? Automatic Design Laboratory .91 in and the compressive force F is 3600 lb. as shown in the figure. The diameter d of the piston is 1. The maximum allowable shear stress τallow in the wall of the cylinder is 6000 psi.Example 33 A cylinder filled with oil is under pressure from a piston. The allowable stresses in the arms are 15. what is the required diameter d of the arms? Automatic Design Laboratory .000 psi in tension and 7. Each arm is offset by the distance b = 7. as shown in the figure.Example 34 A gondola on a ski lift is supported by two bent arms. from the line of action of the weight force W.0 in. If the loaded gondola weighs 1300 lb.500 psi in shear. located on the outer surface at the base of the pipe.2 m above the base.0 m 0.Example 35 A sign is supported by a pipe (see figure) having outer diameter 100 mm and inner diameter 80 mm. The wind pressure against the sign is 1. and its lower edge is 3. Determine the maximum in-plane shear stresses due to he wind pressure on the sign at points A. The dimension of the sign are 2. Automatic Design Laboratory .75 m.8 ㎪. and C. B. and the length L = 4. A cable CBD passes through a fitting that is welded to the side of the post. If the deflection at the top of the post is limited to δ = 20 mm. The cable is Automatic Design Laboratory . respectively.Example 36 A steel post AB of hollow circular cross section is fixed at the base and free at the top (see figure). 0 m.) and the angles between the cable and the ground are α = 53. The distance between the plane of the cable (plane CBD) and the axis of the post is e = 100 mm. pretensioned by tightening the turnbuckles. The inner an outer diameters are d1 = 96 mm and d2 = 110 mm.13°. what is the maximum allowable tensile force T in the cable? (Assume E = 205 ㎬. Automatic Design Laboratory . If member BD is an A-36 steel rod of radius 2 in. determine the maximum load P that can be supported by the truss without causing the member to buckle.Example 37 The members of the truss are assumed to be pin connected. Assume that each members have a different radius. Automatic Design Laboratory . and a wall thickness of 0. and suspends himself uniformly from the center of the high bar. The pipe is made of L2 steel and has an outer diameter 1 in.125 in. Determine the maximum bending stress in the pipe (bar) and its maximum deflection.Example 38 The acrobat has a weight of 150 lb. Example 39 The A-36 steel pipe has an outer diameter of 2 in. Assume the ends of the pipe are pin connected.. Automatic Design Laboratory . If it is held in place by a guywire. so that it can support a maximum horizontal load of P=4 kip without causing the pipe to buckle. determine the pipe’s required inner diameter to the nearest 1/8 in. σ allowable = 67 ksi. determine the required diameter of the shaft to the nearest 1/8 in. Automatic Design Laboratory . If the bearings at A and B exert only vertical forces on the shaft.Example 40 The two pulleys attached to the shaft are loaded as shown. τ allowable = 12 ksi. using the maximum-shear –stress theory and maximum-distortion-energy theory. Determine the maximum load that can safely be supported.Example 41 The steel beam has an allowable bending stress σ allowable = 140 MPa and an allowable shear stress of τ allowable = 90 MPa. Automatic Design Laboratory . Example 42 The boat has a weight of 2300 lb and a center of gravity at G. Automatic Design Laboratory . determine the absolute maximum bending stress developed in the main strut of the trailer. Consider the strut to be a box-beam having the dimensions shown and pinned at C. If it rests on the trailer at the smooth contact A and can be considered pinned at B. Example 43 The chair is supported by an arm that is hinged so it rotates about the vertical axis at A. determine the maximum bending stress at section a-a. If the load on the chair is 180 lb and the arm is a hollow tube section having the dimensions shown. Automatic Design Laboratory . and the allowable bending stress is σ allowable = 22 ksi. Automatic Design Laboratory . The sleeve bearings at A and B support only vertical forces.Example 44 Determine the smallest allowable diameter of the shaft which is subjected to the concentrated forces. Due to soil friction. and a concentrated torque of 80 kN·m acts at the bit.Example 45 The A-36 steel posts are “drilled” at constant angular speed into the soil using the rotary installer. assume the torque along the post varies linearly as shown. If the post has an inner diameter of 200 mm and an outer diameter of 225 mm. determine the relative angle of twist of end A of the post with respect to end B when the post reaches the depth indicated. Automatic Design Laboratory . 5 in.Example 46 The device shown is used to mix soils in order to provide in-situ stabilization. determine the angle of twist of the shaft of A relative to B and the absolute maximum shear stress in the shaft if each mixing blade is subjected to the torques shown. If the mixer is connected to an A-36 steel tubular shaft that has an inner diameter of 3 in. Automatic Design Laboratory . and an outer diameter of 4. Determine the maximum shear stress in the shaft if the couple forces have a magnitude of F=30 lb.Example 47 The steel shaft has an diameter of 1 in. Automatic Design Laboratory . and is screwed into the wall using a wrench. What is the magnitude of stress in each rod? The diameter of each rod is given in the figure.Example 48 The 50-lb lamp is supported by three steel rods connected by a ring at A. Determine the angle of orientation O of AC such that the average normal stress in rod AC is twice the average normal stress in rod AD. Automatic Design Laboratory .