김경종
15 KiB
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y 3 m 3 45° 2 x 3 m 1 10 kN z 10 kN
Figure P5–50
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y 4 m 2 x 4 m 20 kN · m 3 25 kN · m 1 15 kN z
Figure P5–51
5.52–5.57 Solve the grid structures shown in Figures P5–52—P5–57 using a computer program.
For grids P5–52—P5–54, let E = 3 0 \times 1 0 ^ { 6 } psi, G = 1 2 \times 1 0 ^ { 6 } psi, I = 2 0 0 ~ \mathrm { i n } ^ { 4 } , and J = 1 0 0 \mathrm { i n } ^ { 4 } , except as noted in the figures. In Figure P5–54, let the cross elements
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y 4 3 2000 lb 10 ft 10 ft 1 2 1' z x
Figure P5–52
have I = 5 0 \mathrm { i n } ^ { 4 } and J = 2 0 \mathrm { i n } ^ { 4 } , with dimensions and loads as in Figure P5–53. For grids \mathrm { P 5 } { - } 5 5 { - } \mathrm { P 5 } { - } 5 7 , let E = 2 1 0 \mathrm { \ G P a } , G = 8 4 \mathrm { \ G P a } , I = 2 \times 1 0 ^ { - 4 } \mathrm { m } ^ { 4 } , J = 1 \times 1 0 ^ { - 4 } \mathrm { m } ^ { 4 } , and A = 1 \times 1 0 ^ { - 2 } \mathrm { m } ^ { 2 } .
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y 1 kip 1 kip 1 kip 1 kip 1 kip 1 kip 1 kip 1 kip 1 kip 1 kip 1 kip x 10 ft 6 @ 6 ft = 36 ft z
Figure P5–53
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(all loads 1 kip each) y x 10 ft 6 @ 6 ft = 36 ft z
Figure P5–54
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y 1 x 40 kN 60 kN·m 5 50 kN·m 3 4 m 4 m 40 kN 50 kN·m 4 60 kN·m 2 6 4 m 4 m z
Figure P5–55
5.58–5.59 Determine the displacements and reactions for the space frames shown in Figures P5–58 and P5–59. Let I _ { x } = 1 0 0 ~ \mathrm { i n } ^ { 4 } , I _ { y } = 2 0 0 ~ \mathrm { i n } ^ { 4 } , I _ { z } = 1 0 0 0 ~ \mathrm { i n } ^ { 4 } , E = 3 0 { , } 0 0 0 ksi, G = 1 0 { , } 0 0 0 ksi, and A = 1 0 0 \mathrm { i n } ^ { 2 } for both frames.
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Fy = -5 kip 2 mx = -100 k-ft Fz = 40 kip 3 20 ft 4 10 ft 10 ft y x z
Figure P5–58
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20 ft y 2 5 20 ft Mz = -40 k-ft 8 Mz = -50 k-ft 1 x z 20 ft 7 30 kip 3 10 ft 4 6 10 ft
Figure P5–59
Use a computer program to assist in the design problems in Problems 5.60–5.72.
5.60 Design a jib crane as shown in Figure P5–60 that will support a downward load of 6000 lb. Choose a common structural steel shape for all members. Use allowable stresses of 0 . 6 6 S _ { y } \ : ( S _ { y } is the yield strength of the material) in bending, and 0:60Sy in
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c = 29 in. b = 58 in. a = 100 in. A d = 58 in. e = 86 in. F 6000 lb B E C D
Figure P5–60
tension on gross areas. The maximum deflection should not exceed 1 / 3 6 0 of the length of the horizontal beam. Buckling should be checked using Euler’s or Johnson’s method as applicable.
5.61 Design the support members, AB and C D , for the platform lift shown in Figure P5–61. Select a mild steel and choose suitable cross-sectional shapes with no more than a 4 : 1 ratio of moments of inertia between the two principal directions of the cross section. You may choose two different cross sections to make up each arm to reduce weight. The actual structure has four support arms, but the loads shown are for one side of the platform with the two arms shown. The loads shown are under operating conditions. Use a factor of safety of 2 for human safety. In developing the finite element model, remove the platform and replace it with statically equivalent loads at the joints at B and D. Use truss elements or beam elements with low bending stiffness to model the arms from B to D , the intermediate connection, E to F , and the hydraulic actuator. The allowable stresses are 0:66Sy in bending and 0 . 6 0 S _ { y } in tension. Check buckling using either Euler’s method or Johnson’s method as appropriate. Also check maximum deflections. Any deflection greater than 1=360 of the length of member AB is considered too large.
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72 30 30 32 D 24 F B 600 lb 800 lb 600 lb 30 C 24 E 30 A 45° Dimensions are in inches
Figure P5–61
5.62 A two-story building frame is to be designed as shown in Figure P5–62. The members are all to be I-beams with rigid connections. We would like the floor joists beams to have a 15-in. depth and the columns to have a 10 in. width. The material is to be A36 structural steel. Two horizontal loads and vertical loads are shown. Select members such that the allowable bending in the beams is 24,000 psi. Check buckling in the columns using Euler’s or Johnson’s method as appropriate. The allowable deflection in the beams should not exceed 1=360 of each beam span. The overall sway of the frame should not exceed 0.5 in.
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w = 150 lb/ft w = 300 lb/ft 5' 5000 lb 8' 10,000 lb 10' 10' 15'
Figure P5–62
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3.0' 9.0' A B C 0.9' 3' E D 3' 0.9' 7.5' F 2.3' G 30° 2.5 kip
Figure P5–63
5.63 A pulpwood loader as shown in Figure P5–63 is to be designed to lift 2.5 kip. Select a steel and determine a suitable tubular cross section for the main upright member BF that has attachments for the hydraulic cylinder actuators AE and DG. Select a steel and determine a suitable box section for the horizontal load arm AC. The horizontal load arm may have two different cross sections AB and BC to reduce weight. The finite element model should use beam elements for all members except the hydraulic cylinders, which should be truss elements. The pinned joint at B between the upright and the horizontal beam is best modeled with end release of the end node of the top element on the upright member. The allowable bending stress is 0:66Sy in members AB and BC. Member BF should be checked for buckling. The allowable deflection at C should be less than 1=360 of the length of BC. As a bonus, the client would like you to select the size of the hydraulic cylinders AE and DG.
5.64 A piston ring (with a split as shown in Figure P5–64) is to be expanded by a tool to facilitate its installation. The ring is sufficiently thin (0.2 in. depth) to justify using conventional straight-beam bending formulas. The ring requires a displacement of 0.1 in. at its separation for installation. Determine the force required to produce this separation. In addition, determine the largest stress in the ring. Let E = 1 8 \times 1 0 ^ { 6 } psi, G = 7 \times 1 0 ^ { 6 } psi, cross-sectional area A = 0 . 0 6 \ \mathrm { i n } . ^ { 2 } , and principal moment of inertia I = 4 . 5 \times 1 0 ^ { - 4 } \ \mathrm { i n } . ^ { 4 } . The inner radius is 1.85 in., and the outer radius is 2.15 in. Use models with 4, 6, 8, 10, and 20 elements in a symmetric model until convergence to the same results occurs. Plot the displacement versus the number of elements for a constant force F predicted by the conventional beam theory equation of Reference [8].
\delta = \frac {3 \pi F R ^ {3}}{E I} + \frac {\pi F R}{E A} + \frac {6 \pi F R}{5 G A} \quad \text { where } R = 2. 0 \text { in. and } \delta = 0. 1 \text { in. }
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δ=0.1 in. required due to F
Figure P5–64
5.65 A small hydraulic floor crane as shown in Figure P5–65 carries a 5000-lb load. Determine the size of the beam and column needed. Select either a standard box section or a wide-flange section. Assume a rigid connection between the beam and column. The column is rigidly connected to the floor. The allowable bending stress in the beam is 0:60Sy. The allowable deflection is 1=360 of the beam length. Check the column for buckling.
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8 in. 72 in. A B 60 in. 5000 lb C 20° D
Figure P5–65
5.66 Determine the size of a solid round shaft such that the maximum angle of twist between C and B is 0.26 degrees per meter of length and the deflection of the beam is less than 0.005 inches under the pulley C for the loads shown. Assume simple supports at bearings A and B. Assume the shaft is made from cold-rolled AISI 1020 steel. (Recommended angles of twist in driven shafts can be found in Machinery’s Handbook, Oberg, E., et. al., 26th ed., Industrial Press, N.Y., 2000.)
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0.4 m 0.5 m 0.15 m A C B x y z 2 kN D T 5 kN
Figure P5–66
5.67 The shaft shown supports a winch load of 780 lb and a torsional moment of 7800 lbin. at F (26 inches from the center of the bearing at A). In addition, a radial load of 500 lb and an axial load of 400 lb act at point E from a worm gearset. Assume the maximum stress in the shaft cannot be larger than that obtained from the maximum distortional energy theory with a factor of safety of 2.5. Also make sure the angle of twist is less than 1.5 deg between A and D. In your model, assume the bearing at A to be frozen when calculating the angle of twist. Bearings at B; C, and D can be assumed as simple supports. Determine the required shaft diameter.
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A B C D E F Shaft Winch drum 10" 10" 12"
Figure P5–67
5.68 Design the gabled frame subjected to the external wind load shown (comparable to an 80 mph wind speed) for an industrial building. Assume this is one of a typical frame spaced every 20 feet. Select a wide flange section based on allowable bending stress of 20 ksi and an allowable compressive stress of 10 ksi in any member. Neglect the possibility of buckling in any members. Use ASTM A36 steel.
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Wind h 16 ft 11 ft L = 40 ft
(a)
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3.00 psf 7.50 psf
(b)
Figure P5–68
5.69 Design the gabled frame shown for a balanced snow load shown (typical of the Midwest) for an apartment building. Select a wide flange section for the frame. Assume the allowable bending stress not to exceed 140 MPa. Use ASTM A36 steel.
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740 MPa 3 m (4 m spacing of frames) 4 m 6 m
Figure P5–69
5.70 Design a gantry crane that must be able to lift 10 tons as it must lift compressors, motors, heat exchangers, and controls. This load should be placed at the center of one of the main 12-foot-long beams as shown in Figure P5–70 by the hoisting device location. Note that this beam is on one side of the crane. Assume you are using
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8 ft 12 ft 2 ft 3 ft 15 ft 10 T
Figure P5–70
ASTM A36 structural steel. The crane must be 12 feet long, 8 feet wide, and 15 feet high. The beams should all be the same size, the columns all the same size, and the bracing all the same size. The corner bracing can be w ide flange sections or some other common shape. You must verify that the structure is safe by checking the beam’s bending strength and allowable deflection, the column’s buckling strength, and the bracing’s buckling strength. Use a factor of safety against material yielding of the beams of 5. Verify that the beam deflection is less than L/360, where L is the span of the beam. Check Euler buckling of the long columns and the bracing. Use a factor of safety against buckling of 5. Assume the column-to-beam joints to be rigid while the bracing (a total of eight braces) is pinned to the column and beam at each of the four corners. Also assume the gantry crane is on rollers with one roller locked down to behave as a pin support as shown.
5.71
Design the rigid highway bridge frame structure shown in Figure P5–71 for a moving truck load (shown below) simulating a truck moving across the bridge. Use the load shown and place it along the top girder at various locations. Use the allowable stresses in bending and compression and allowable deflection given in the Standard Specifications for Highway Bridges, American Association of State Highway and Transportation Officials (AASHTO), Washington, D.C. or use some other reasonable values.