Mechanics of Materials (MindTap Course List)
Mechanics of Materials (MindTap Course List)
9th Edition
ISBN: 9781337093347
Author: Barry J. Goodno, James M. Gere
Publisher: Cengage Learning
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Textbook Question
Chapter 5, Problem 5.6.25P

A beam of square cross section (a = length of each side) is bent in the plane of a diagonal (see figure). By removing a small amount of material at the top and bottom corners, as shown by the shaded triangles in the figure, you can increase the section modulus and obtain a stronger beam, even though the area of the cross section is reduced.

  1. Determine the ratio ß defining the areas that should be removed in order to obtain the strongest cross section in bending.

  • By what percent is the section modulus increased when the areas arc removed?
  •   Chapter 5, Problem 5.6.25P, A beam of square cross section (a = length of each side) is bent in the plane of a diagonal (see

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    Chapter 5 Solutions

    Mechanics of Materials (MindTap Course List)

    Ch. 5 - A thin, high-strength steel rule (E = 30 x 10ft...Ch. 5 - A simply supported wood beam AB with a span length...Ch. 5 - Beam ABC has simple supports at A and B and an...Ch. 5 - A simply supported beam is subjected to a in early...Ch. 5 - Each girder of the lift bridge (sec figure) is 180...Ch. 5 - A freight-car axle AS is loaded approximately as...Ch. 5 - A seesaw weighing 3 lb/ft of length is occupied by...Ch. 5 - During construction of a highway bridge, the main...Ch. 5 - The horizontal beam ABC of an oil-well pump has...Ch. 5 - A railroad tie (or sleeper) is subjected to two...Ch. 5 - A fiberglass pipe is lifted by a sling, as shown...Ch. 5 - A small dam of height h = 2.0 m is constructed of...Ch. 5 - Determine the maximum tensile stress (7, (due to...Ch. 5 - Determine the maximum bending stress emaxdue to...Ch. 5 - A simple beam A B of a span length L = 24 ft is...Ch. 5 - Determine the maximum tensile stress erand maximum...Ch. 5 - A cantilever beam A3, loaded by a uniform load and...Ch. 5 - A canti lever beam A B of a n isosceles t...Ch. 5 - A cantilever beam, a C12 x 30 section, is...Ch. 5 - A frame ABC travels horizontally with an...Ch. 5 - A beam ABC with an overhang from B to C supports a...Ch. 5 - A cantilever beam AB with a rectangular cross...Ch. 5 - A beam with a T-section is supported and loaded as...Ch. 5 - Consider the compound beam with segments AB and...Ch. 5 - A small dam of a height h = 6 ft is constructed of...Ch. 5 - A foot bridge on a hiking trail is constructed...Ch. 5 - A steel post (E=30×106) having thickness t = 1/8...Ch. 5 - Beam ABCDE has a moment release just right of...Ch. 5 - A simply supported wood beam having a span length...Ch. 5 - A simply supported beam (L = 4.5 m) must support...Ch. 5 - The cross section of a narrow-gage railway bridge...Ch. 5 - A fiberglass bracket A BCD with a solid circular...Ch. 5 - A cantilever beanie B is loaded by a uniform load...Ch. 5 - A simple beam of length L = 5 m carries a uniform...Ch. 5 - A simple beam AB is loaded as shown in the figure....Ch. 5 - A pontoon bridge (see figure) is constructed of...Ch. 5 - A floor system in a small building consists of...Ch. 5 - The wood joists supporting a plank Floor (see...Ch. 5 - A beam ABC with an overhang from B to C is...Ch. 5 - -12 A "trapeze bar" in a hospital room provides a...Ch. 5 - A two-axle carriage that is part of an over head...Ch. 5 - A cantilever beam AB with a circular cross section...Ch. 5 - A propped cantilever beam A BC (see figure) has a...Ch. 5 - A small balcony constructed of wood is supported...Ch. 5 - A beam having a cross section in the form of an un...Ch. 5 - A beam having a cross section in the form of a...Ch. 5 - Determine the ratios of the weights of four beams...Ch. 5 - Prob. 5.6.20PCh. 5 - A steel plate (called a cover ploie) having...Ch. 5 - A steel beam ABC is simply supported at A and...Ch. 5 - A retaining wall 6 ft high is constructed of...Ch. 5 - A retaining wall (Fig. a) is constructed using...Ch. 5 - A beam of square cross section (a = length of each...Ch. 5 - The cross section of a rectangular beam having a...Ch. 5 - A tapered cantilever beam A B of length L has...Ch. 5 - .2 A ligmio.irc ii supported by two vorlical beams...Ch. 5 - Prob. 5.7.3PCh. 5 - Prob. 5.7.4PCh. 5 - Prob. 5.7.5PCh. 5 - A cantilever beam AB with rectangular cross...Ch. 5 - A simple beam ABC having rectangular cross...Ch. 5 - A cantilever beam AB having rectangular cross...Ch. 5 - The shear stresses t in a rectangular beam arc...Ch. 5 - .2 Calculate the maximum shear stress tmaxand the...Ch. 5 - A simply supported wood beam is subjected to...Ch. 5 - A simply supported wood beam with overhang is...Ch. 5 - Two wood beams, each of rectangular cross section...Ch. 5 - A cantilever beam of length L = 2 m supports a...Ch. 5 - A steel beam of length L = 16 in. and...Ch. 5 - A beam of rectangular cross section (width/) and...Ch. 5 - A laminated wood beam on simple supports (figure...Ch. 5 - A laminated plastic beam of square cross section...Ch. 5 - A wood beam AB on simple supports with span length...Ch. 5 - A simply supported wood beam of rectangular cross...Ch. 5 - A square wood platform is 8 ft × 8 ft in area and...Ch. 5 - A wood beam ABC with simple supports at A and B...Ch. 5 - A wood pole with a solid circular cross section (d...Ch. 5 - A simple log bridge in a remote area consists of...Ch. 5 - A vertical pole consisting of a circular tube of...Ch. 5 - A circular pole is subjected to linearly varying...Ch. 5 - A sign for an automobile service station is...Ch. 5 - A steel pipe is subjected to a quadratic...Ch. 5 - -1 through 5.10-6 A wide-flange beam (see figure)...Ch. 5 - -1 through 5.10-6 A wide-flange beam (see figure)...Ch. 5 - -1 through 5.10-6 A wide-flange beam (see figure)...Ch. 5 - -1 through 5.10-6 A wide-flange beam (see figure)...Ch. 5 - -1 through 5.10-6 A wide-flange beam (see figure)...Ch. 5 - -1 through 5.10-6 A wide-flange beam (see figure)...Ch. 5 - A cantilever beam AB of length L = 6.5 ft supports...Ch. 5 - A bridge girder A B on a simple span of length L =...Ch. 5 - A simple beam with an overhang supports a uniform...Ch. 5 - A hollow steel box beam has the rectangular cross...Ch. 5 - A hollow aluminum box beam has the square cross...Ch. 5 - The T-beam shown in the figure has cross-sectional...Ch. 5 - Calculate the maximum shear stress tmax. in the...Ch. 5 - A prefabricated wood I-beam serving as a floor...Ch. 5 - A welded steel gird crhaving the erass section...Ch. 5 - A welded steel girder having the cross section...Ch. 5 - A wood box beam is constructed of two 260 mm × 50...Ch. 5 - A box beam is constructed of four wood boards as...Ch. 5 - Two wood box beams (beams A and B) have the same...Ch. 5 - A hollow wood beam with plywood webs has the...Ch. 5 - A beam of a T cross section is formed by nailing...Ch. 5 - The T-beam shown in the figure is fabricated by...Ch. 5 - A steel beam is built up from a W 410 × 85 wide...Ch. 5 - The three beams shown have approximately the same...Ch. 5 - Two W 310 × 74 Steel wide-flange beams are bolted...Ch. 5 - A pole is fixed at the base and is subjected to a...Ch. 5 - A solid circular pole is subjected to linearly...Ch. 5 - While drilling a hole with a brace and bit, you...Ch. 5 - An aluminum pole for a street light weighs 4600 N...Ch. 5 - A curved bar ABC having a circular axis (radius r...Ch. 5 - A rigid Trame ABC is formed by welding two steel...Ch. 5 - A palm tree weighing 1000 lb is inclined at an...Ch. 5 - A vertical pole of aluminum is fixed at the base...Ch. 5 - Because of foundation settlement, a circular tower...Ch. 5 - A steel bracket of solid circular cross section is...Ch. 5 - A cylindrical brick chimney of height H weighs w =...Ch. 5 - A flying but tress transmit s a load P = 25 kN,...Ch. 5 - A plain concrete wall (i.e., a wall with no steel...Ch. 5 - A circular post, a rectangular post, and a post of...Ch. 5 - Two cables, each carrying a tensile force P = 1200...Ch. 5 - Prob. 5.12.16PCh. 5 - A short column constructed of a W 12 × 35...Ch. 5 - A short column with a wide-flange shape is...Ch. 5 - A tension member constructed of an L inch angle...Ch. 5 - A short length of a C 200 × 17.1 channel is...Ch. 5 - The beams shown in the figure are subjected to...Ch. 5 - The beams shown in the figure are subjected to...Ch. 5 - A rectangular beam with semicircular notches, as...Ch. 5 - A rectangular beam with semicircular notches, as...Ch. 5 - A rectangular beam with notches and a hole (see...
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  • A wood beam reinforced by an aluminum channel section is shown in the figure. The beam has a cross section of dimensions 150 mm x 250 mm, and the channel has a uniform thickness of 6.5 mm. If the allowable stresses in the wood and aluminum are 8 M Pa and 38 M Pa, respectively, and if their moduli of elasticity are in the ratio 1 to 6, what is the maximum allowable bending moment for the beam?
    -1 through 5.10-6 A wide-flange beam (see figure) is subjected to a shear force V. Using the dimensions of the cross section, calculate the moment of inertia and then determine the following quantities: The maximum shear stress tinixin the web. The minimum shear stress rmin in the web. The average shear stress raver (obtained by dividing the shear force by the area of the web) and the ratio i^/t^ The shear force carried in the web and the ratio V^tV. Noie: Disregard the fillets at the junctions of the web and flanges and determine all quantities, including the moment of inertia, by considering the cross section to consist of three rectangles. 5.10-3 Wide-flange shape, W 8 x 28 (see Table F-L Appendix F); V = 10 k
    -1 through 5.10-6 A wide-flange beam (see figure) is subjected to a shear force V. Using the dimensions of the cross section, calculate the moment of inertia and then determine the following quantities: The maximum shear stress tinixin the web. The minimum shear stress rmin in the web. The average shear stress raver (obtained by dividing the shear force by the area of the web) and the ratio i^/t^ The shear force carried in the web and the ratio V^tV. Note: Disregard the fillets at the junctions of the web and flanges and determine all quantities, including the moment of inertia, by considering the cross section to consist of three rectangles. 5.10-4 Dimensions of cross section: b = 220 mm, f = 12 mm, h = 600 mm, hx= 570 mm, and V = 200 kN.
  • The beam ABC shown in the figure is simply supported at A and B and has an overhang from B to C. The loads consist of a horizontal force P1= 4,0 kN acting at the end of a vertical arm and a vertical force P2= 8.0 kN acting at the end of the overhang, Determine the shear force Fand bending moment M at a cross section located 3,0 m from the left-hand support. Note: Disregard the widths of the beam and vertical arm and use centerline dimensions when making calculations, Find the value of load A that results in V = 0 at a cross section located 2.0 m from the left-hand support. If P2= 8.0 kN, find the value of load P1that results in M = 0 at a cross section located 2,0 m from the left-hand support.
    -1 through 5.10-6 A wide-flange beam (see figure) is subjected to a shear force V. Using the dimensions of the cross section, calculate the moment of inertia and then determine the following quantities: The maximum shear stress tinixin the web. The minimum shear stress rmin in the web. The average shear stress raver (obtained by dividing the shear force by the area of the web) and the ratio i^/t^. The shear force i^/t^ carried in the web and the ratio V^tV. Note: Disregard the fillets at the junctions of the web and flanges and determine all quantities, including the moment of inertia, by considering the cross section to consist of three rectangles. 5.10-6 Dimensions of cross section: b = 120 mm, a = 7 mm, h = 350 mm, hx= 330 mm, and K=60kN.
    -1 through 5.10-6 A wide-flange beam (see figure) is subjected to a shear force V. Using the dimensions of the cross section, calculate the moment of inertia and then determine the following quantities: The maximum shear stress tinixin the web. The minimum shear stress rmin in the web. The average shear stress raver (obtained by dividing the shear force by the area of the web) and the ratio i^/t^ The shear force carried in the web and the ratio K b/K. Note: Disregard the fillets at the junctions of the web and flanges and determine all quantities, including the moment of inertia, by considering the cross section to consist of three rectangles. 5.10-2 Dimensions of cross section: b = 180 mm, v = 12 mm, h = 420 mm, i = 380 mm, and V = 125 kN.
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