NAME: Given Formulae: Law of Cosines: Law of Sines:

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1 NME: Given Formulae: Law of Cosines: EXM 3 PST PROBLEMS (LESSONS 21 TO 28) 100 points Thursday, November 16, 2017, 7pm to 9:30, Room 200 You are allowed to use a calculator and drawing equipment, only. Formulae provided 2.5 hour time limit This massive compilation of problems represents about 7 years of past exams C b a Law of Sines: c B Formulae: Normal stress = P/, where P is the normal force on the cut and is the area of the cut. Shear stress avg = V/, where V is the shear force on the cut and is the area of the cut. Normal strain = L/L, where L is the change in length and L is the original length (K, gauge length) Shear strain xy = the change in angle of x-y, where x-y are initially perpendicular or xy = /2, where is the deformed angle between x-y, measured in radians and is obviously. Hooke s Law = E, where E = modulus of elasticity (K Elastic Modulus or Young s Modulus) Poisson s Ratio = - ( lat / long ), where the specimen is loaded in the long direction, resulting in long strain, as well as lat strain. Hooke s Law for Shear: = G, where G is the shear modulus of elasticity (K, the modulus of rigidity) The elastic properties E, G, and are related by: Some Moments of Inertia Formulae Elastic Deformation of an xially Loaded bar: Deformation due to thermal expansion: Δ Bending M/EI = 1/ = -y/ =My/I = E Torsion = Tc/J = TL/GJ Power P = T where T is torque and is angular frequency (rad/s) 1hp = 550 ft-lb/s 1Watt = 1 N-m/s J = ( /2)c 4 for a solid, circular shape of radius c. Page 1 of 19

2 1. (15 points). The motor is spinning at 3600 rpm. sensor in shaft (2) indicates that the maximum (extreme fiber) shear stress in the shaft is 5 ksi. Determine the power output of the motor in units of Horsepower. Given: shafts (1) and (2) have diameters of 1 inch. 2. (18 points). The 12 x 12 member shown is made of plain, unreinforced concrete. It will fail if either ). the compressive stress reaches 4.0 ksi, or B). the tensile stress reaches 0.40 ksi. Determine the maximum force P that may be applied, if it is applied 4 off-center, as shown. P (18 points). Determine the angle of twist between and C, C. Given: Shaft (1) has a diameter of 40mm Shaft (2) has a diameter of 80mm Shaft (3) has a diameter of 40mm ll shafts are solid steel (G = 80 GPa) pplied torques, as shown. Page 2 of 19

3 1. (25 points). Determine the maximum power that the gear train can supply if the allowable shear stress allow = 50 MPa, then determine the rotation angle of gear D relative to gear C when the gear train is operating at this maximum power if G=80GPa. Given: Motor at spins at 3600 rpm Shaft (1) Diameter = 10mm, Length L 1 = 500mm Shaft (2) Diameter = 15mm, Length L 2 = 400mm 2. (25 points). Determine the maximum force P that may be applied to the C-clamp, shown. Given: The clamp is made from an alloy with an allowable stress of 100 MPa in compression, but only 50 MPa in tension. Page 3 of 19

4 1. (25 points). Determine the maximum force P that may be applied to the C-clamp, shown. Given: The clamp is made from an alloy with an allowable stress of 70 MPa in compression, but only 50 MPa in tension. 1. (27 points). Determine the maximum uniformly-distributed load w (report the answer in units of lbs/inch) that can safely be applied to the wooden beam if the applied stresses must be limited to: Maximum shear stress anywhere in the beam must not exceed 200 psi. Maximum shear stress on any glue joint must not exceed 100 psi. Maximum normal stress due to bending must not exceed 1500 psi. The wooden beam is made by gluing together the boards, shown. w (lbs/inch) 100 B 1 x8 10 Side View (not to scale) 1 x10 10 Cross-Section View I = 535 in 4 Page 4 of 19

5 2. (17 points). The motor shown supplies 15 hp at 1,800 rpm at. Both shafts are made of steel (allowable shear stress tallow = 12 ksi). Determine: The torque T (report your answer in units of kip-in) in shaft (1) and (2). The rotational speed of Gear D (report your answer in units of rpm). The required shaft diameter if they will both be the same size and must not have shear stress exceeding allow = 12 ksi. 3. (5 points). Determine the shear stress between the top flange and the web of the I-beam shown. The beam is supported by a pin at and a roller at B. It is subjected to 10 kip-in end-moments, as shown. There are no other applied loads. M = 10 kip-in M = 10 kip-in 4 x1 Flange 100 B 4 x1 Flange 6 x1 Web Your job: find the shear stress here. Side View Cross-Section View Page 5 of 19

6 1. (17 points) The timber beam is to be notched at its ends as shown. If it is to support the loading shown, determine the smallest depth d of the beam at the notch if the allowable shear stress is allow=450psi. The beam has a width of 8 inches. d 2. (15 points) The motor shown supplies 10 hp at 1500 rpm at. The bearings shown permit free rotation of the shafts. If both shafts (1) and (2) are to be solid shafts of the same diameter, determine the minimum the torque in each shaft and diameter that may be used if the shear stress is limited to no more than psi. 3. (2 points). For the previous problem: at what rpm does shaft (2) spin? Page 6 of 19

7 4. (12 points) n 18-inch-long segment of a beam is shown. It is known to have a negative bending moment of 42 kip-in at point and a negative bending moment of 36 kip-in at point B. Determine the resultant forces on each end of board number (1), then determine the shear stress that is in the glue that joins board (1) to board (3). (3) 5. (2 points) TRUE or FLSE. For the previous problem, the resultant normal force on board (1) is equal to the resultant normal force on board (2), at point B. 6. (2 points) TRUE or FLSE. For Problem 4, the resultant normal force on board (3) is zero at point B. Page 7 of 19

8 1. (20 points) Determine the maximum uniformly distributed load w (units: kips/inch) that may be applied to the beam without failing the glue that holds blocks (1) to block (2). Given: Blocks (1) are glued to Block (2). The maximum allowable shear stress in this glue is glue = 1 ksi. = in (measured from the bottom), I=61.27 in 4 Blocks (1) are 1 x2, Block (2) is 6 x1, Block (3) is 6 x1 w =? (1) (2) (1) (3) 100 inches SIDE VIEW CROSS SECTION 2. (15 points) The motor shown supplies 12 hp at 1800 rpm at. The bearings shown permit free rotation of the shafts. If the shear stress in shaft (2) must be limited to 6000 psi, determine the minimum acceptable diameter for shaft (2) if a solid shaft is used. Given: Gear B has 48 teeth and is 4.8-in in diameter, while Gear C has 30 teeth and is 3.0-in in diameter. Page 8 of 19

9 3. (12 points) The 12-in.-long beam segment shown is subjected to internal bending moments of M=700 ft-lbs and MB=400 ft-lbs, as shown. It is subjected to a constant shear force. If the beam was built by gluing together the top flange and the web, determine the shear stress in the glue. Given: = 4.5 in (measured from the bottom)., I=49.88 in 4, The top board is 1 x 4.5, while the bottom board is 1 x 6 4. (3 points) Explain and illustrate why a hollow shaft makes more efficient use of material than a solid shaft, when used as a torsional member. Page 9 of 19

10 1. (2 points). The circular rod shown below is subjected to torsion T, only. The shear stress acting on the cross-section is: a. Constant over the cross-section b. Equal to zero c. Maximum at a distance c from the x-axis d. Maximum at the x-axis 2. (3 points). For the beam below, label the exact location at which the shear stress is maximum and briefly explain how you know this is the max point. w = 1 kip/ft 10 feet Cross-Section 3. (12 points) Determine the maximum shear stress for a rectangular cross-section in terms of the applied shear force V and the cross-sectional area. Given: Beam with concentrated force V, on the end. Cross section has a width of b and a depth of h. V h V Beam Side View b Cross-Section Page 10 of 19

11 4. (15 points) Determine the maximum shear stress that occurs in a weld. Given: Box beam with 12 x12 outside dimensions, and 10 x10 inside dimensions. The box beam is made by welding 4 plates together, at the locations shown. w = 1 kip/ft Welds 10 feet Cross-Section 12 x12 outside dimensions 10 x10 inside dimensions 1 wall thickness 1. (10 points) Determine the maximum deflection of the beam below. The beam shown has E=1000 ksi, I = 10 in 4, L=100 inches, M = 1 kip-inch. M = 1 kip-in 1 kip-in = M (3 points). Two plastic C-shaped members are to be glued together for form a hollow square cantilever box beam that will be subjected to a horizontal force. Which configuration ( or B) will minimize the shear stress on the glue? Force Force Glue Joint Glue Joint Fixed Support () Fixed Support (B) Page 11 of 19

12 3. (12 points). Given: The shear force on the beam cross-section is constant over the 16 interval from to B. It is known that the extreme-fiber normal stresses are = 20 ksi at and B = 15 ksi at B. The I-shaped cross-section is composed of 1 x8 flanges and a 1 x8 web. Determine the shear stress between the flange and the web, between and B. = 20 ksi B = 15 ksi 1 x8 Flange B Determine here 1 x8 W b 1 x8 Flange SIDE VIEW Showing Normal Stresses CROSS-SECTION M M B B 3D VIEW M causes maximum normal stress of 20 ksi MB causes maximum normal stress of 15 ksi Page 12 of 19

13 4. (17 points). 1.5m diameter boiler (pressure vessel) is constructed out of curved steel plates that are fastened together at their ends using a butt joint consisting of two cover plates and rivets having a diameter of 10-mm and spaced 50-mm apart, as shown. If the steam pressure in the boiler is 1.00 MPa, determine the average shear stress in the rivets. (2) cover plates p = 1.00 MPa 1.5m Diameter Boiler Cross Section 10mm diam rivet 10mm diam rivet 5. (17 points). plastic beam is constructed by gluing a C-shape to a rectangular shape. If the beam is loaded, as shown, determine the maximum shear stress in the glue. w = 500 lbs/ft ft 2 ft 10 1 Cross-Section I = 241 in 4 Page 13 of 19

14 1. (5 points). Two 2 x 4 boards are glued together and act as a cantilever beam over a span of 20 inches, subjected to an unknown point load P. It is known that the normal bending stresses at the extreme fibers have magnitudes of 1000psi. Determine the shear stress in the glue. Fixed End P max = 1000psi (20 points). The steel I-beam is considered to have failed if either: a. The normal bending stress exceeds 30 ksi, or b. The shear stress exceeds 15 ksi Problem: Determine the maximum uniform load w that may be applied without failure occurring. Given: The beam spans 50 inches over pin and roller supports. The cross-section is composed of 1 x 10 flanges and a ½ x 18 web, as shown. NOTE: This is NOT a Mohr s Circle problem. w =? CROSS SECTION 1 x 10 Flange ½ x 18 Web 50 inches 1 x 10 Flange Page 14 of 19

15 3. (20 points). The steel I-beam is considered to have failed if either: c. The normal bending stress exceeds 25 ksi, or d. The shear stress exceeds 15 ksi Problem: Determine the maximum uniform load w that may be applied without failure occurring. Given: The beam spans 80 inches over pin and roller supports. The cross-section is composed of ½ x 10 flanges and a ½ x 19 web, as shown. NOTE: This is NOT a Mohr s Circle problem. w =? ½ x 10 Flange ½ x 19 Web 80 inches ½ x 10 Flange Page 15 of 19

16 1. (3 points). For which location would the shear forces on the nails be greatest (circle the correct answer)? Given: The left fixed support for the beam is denoted (1) and the right free end is denoted as (2). The nails on the top of the beam are denoted B and the nails on the bottom of the beam are denoted. a. (1) b. (1)B c. (2) d. (2)B e. (1) and (1)B (they are the same) w f. (2) and (2)B (they are the same) (1) (2) 2. (22 points). Draw the stresses acting on element shown, labeling their magnitudes and directions. Do not draw any other orientation (this is NOT a Mohr s Circle Problem): Given: The horizontal force of 5000 lbs is applied to the top of the cantilever beam at a distance that is 5 from the centerline of the beam, as shown. The beam has a 10 x10 cross-section 5000 lbs x10 Cross Section Page 16 of 19

17 1. (25 points) box beam is fabricated from two plywood webs that are secured to the solid lumber boards at its top and bottom flanges. The beam supports a concentrated load of P=5000 lbs at the center of a 16-foot span. 3/8 diameter bolts connect the plywood webs and the solid lumber flanges at a spacing of 12 inches along the beam. Supports and C are considered as a pin and roller. Determine: The maximum shear stress in the plywood webs. The average shear stress in the bolts ft 8 ft 1. (3 points) hydraulic jack is shown below, consisting of a 3 diameter piston that is inside a hollow cylinder. If the piston has 3000 psi oil pressure behind it, determine the force P that this jack is currently lifting. 3000psi oil pressure behind the piston 3 diameter piston 2. (7 points). rural water supply system consists of a water tank that is up on a hill, supplying water to a community in the valley. If the water level is at an elevation that is 100-ft above the community it supplies, determine the minimum wall thickness needed for the supply pipe if it has an inside diameter of 4 so that the hoop stresses do not exceed 1000 psi. Given: a cubic foot of water weighs 62.4 lbs. Water Level 100-ft Water Tank 4 diameter water supply pipe Page 17 of 19

18 1. (20 points) Write the function for the deflection of beam B, v(x) and the function for the slope of beam B, (x), where the origin for x is at fixed support, directed toward free-end B, then determine the maximum deflection and maximum slope of the beam. Given: Distributed load that increases linearly from 0 at to 1 kip/inch at B. EI = 1,000,000 kip-in 2. w = 1 kip/inch B 60 inches x 2. (20 points) Determine the maximum force P that the gantry crane can safely support if the maximum allowable normal stress allow = 30 ksi. Given: the gantry crane BC is made out of hollow square tubing with outside dimensions of 3 inches and a wall thickness of ¼. ssume the weight of the crane, itself, is negligible. B 50 inches C 3 inches 100 inches P 2.5 inches 3 inches Side View of Gantry Crane 2.5 inches Cross Section of hollow square tubing 1. (3 points) What is Pascal s Law? 2. (2 points). TRUE or FLSE. The member below is a two-force member which implies that the forces on each end are collinear and that there is no internal bending or shear forces present in the member Page 18 of 19

19 3. (20 points) n S-Hook is used for heavy lifting. It is made out of 2 diameter circular steel rod and is bent into a 6 radius, where the radius is measured to the centroid of the rod. Determine the maximum load P that the hook can sustain without exceeding the allowable normal stress all = 20 ksi, ssume that the most severe stresses occur at section -. P r = 6 r = 6 P 6. (20 points): Write the v(x) function of the deflected elastic shape for the uniformly-loaded beam in terms of w, L, and EI. Then report the maximum deflection in units of feet if w=1 kip/ft, L=20-ft, and EI=20,833 kip-ft 2. Given: w=constant EI=constant L 13. (12 points) Use double-integration to determine the deflection at midspan. EI is constant. 1. (20 points) Write the function for the deflection v(x) of beam B due to the concentrated moment at B, and the function for the slope (x), where the origin for x is at pinned support, directed toward roller support B. Write these functions in terms of M, L, x, and EI. Given: Concentrated moment M at position B. L B M Page 19 of 19 x