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P8.26 A small aluminum alloy [E = 70 GPa] tee shape is used as a simply supported beam as shown in Figure P8.26a. For this beam, a = 160 mm. The cross-sectional dimensions (Figure P8.26b) of the tee shape are b = 20 mm, d = 30 mm, and t = 6 mm. After loads are applied to the beam at B, C, and D, a compressive normal strain of 440 is measured from a strain gage located at c = 8 mm below the topmost edge of the tee stem. What is the applied load P?
Solution
Centroid location in y direction: (reference axis at bottom of shape)
Moment of inertia about the z axis:
(measured upward from bottom edge of section)
The y coordinate at the strain gage is:
The bending strain at H is thus:
FIGURE P8.26a
FIGURE P8.26b
The bending stress at H is:
From the flexure moment, determine the bending moment that produces this bending stress at H:
Shear-force and bending-moment diagrams
From the shear-force and bending-moment diagrams, we find that the bending moment at section 1–1 is:
Thus, the load P can be calculated as:
P8.27 A solid steel shaft supports loads PA = 250 N and PC = 620 N as shown in Figure P8.27. Assume a = 500 mm, b = 700 mm, and c = 600 mm. The bearing at B can be idealized as a roller support and the bearing at D can be idealized as a pin support. If the allowable bending stress is 105 MPa, determine the minimum diameter that can be used for the shaft.
Solution
Shear-force and bending-moment diagrams
Maximum bending moment magnitude M = 142,615.4 N-mm
Minimum required section modulus
Section modulus for solid circular section
FIGURE P8.27
P8.28 A simply supported wood beam (Figure P8.28a/29a) with a span of L = 6 m supports a uniformly distributed load of w0. The beam width is b = 120 mm and the beam height is h = 300 mm (Figure P8.28b/29b). The allowable bending stress of the wood is 8.0 MPa. Calculate the magnitude of the maximum load w0 that may be carried by the beam.
FIGURE P8.28a/29a
Maximum allowable bending moment M:
P8.29 A simply supported wood beam (Figure P8.28a/29a) with a span of L = 24 ft supports a uniformly distributed load of w0 = 450 lb/ft. The allowable bending stress of the wood is 1,200 psi. If the aspect ratio of the solid rectangular wood beam is specified as h/b = 3.0 (Figure P8.28b/29b), calculate the minimum width b that can be used for the beam.
Solution
Shear-force and bending-moment diagrams
Maximum bending moment magnitude M = 25,600 lb ft = 307,200 lb in.
Minimum required section modulus
2 307,200lbin. 256in. 1,200lb/in.
Section modulus for solid rectangular section
Theaspectratioofthesolidrectangularwoodbeam isspecifiedas h/b =3.0;therefore,thesectionmodulus can be expressed as:
Minimum allowable beam width
The corresponding beam height h is
FIGURE P8.28a/29a
FIGURE P8.28b/29b
P8.30 A cantilever timber beam (Figure P8.30a/31a) with a span of L = 4.25 m supports a linearly distributed load with maximum intensity of w0 = 5.5 kN/m. The allowable bending stress of the wood is 7.0 MPa. If the aspect ratio of the solid rectangular timber is specified as h/b = 0.67 (Figure P8.30b/31b), determine the minimum width b that can be used for the beam.
Solution
Maximum moment magnitude: The maximum bending moment magnitude in the cantilever beam occurs at support A:
The aspect ratio of the solid rectangular wood beam is specified as h/b = 0.67; therefore, the section modulus can be expressed as:
FIGURE P8.30a/31a
FIGURE P8.30b/31b
P8.31 A cantilever timber beam (Figure P8.30a/31a) with a span of L = 14 ft supports a linearly distributed load with maximum intensity of w0 The beam width is b = 15 in. and the beam height is h = 10 in. (Figure P8.30b/31b). The allowable bending stress of the wood is 1,000 psi. Calculate the magnitude of the maximum load w0 that may be carried by the beam.
Solution
Section modulus for rectangular cross section about horizontal centroidal axis
Maximum allowable moment
Maximum moment magnitude: The maximum bending moment magnitude in the cantilever beam occurs at support A:
FIGURE P8.30a/31a
FIGURE P8.30b/31b
P8.32 The lever shown in Figure P8.32 must exert a force of P = 2,700 lbs at B as shown. The lever lengths are a = 10.5 in. and b = 46.0 in. The allowable bending stress for the lever is 12,000 psi. If the lever height h is to be three times the lever thickness t (i.e., h/t = 3), what is the minimum thickness t that can be used for the lever?
Solution Equilibrium
Maximum bending moment: The maximum bending moment magnitude in the lever occurs at B: ( ) ( )( ) max 616.304 lb46.0 in.10.5 in.21,878.8 lbin.
Minimum required section modulus
21,878.8 lbin.1.8232 in. 12,000 lb/in.
Section modulus for solid rectangular section 32 /12 /26 Ithth
The aspect ratio of the solid rectangular lever is specified as h/t = 3; therefore, the section modulus can be expressed as:
Minimum allowable lever thickness 331.51.8232
The corresponding lever height h is
P8.33 The beam shown in Figure P8.33 will be constructed from a standard steel W-shape using an allowable bending stress of 30 ksi. The beam span is L = 24 ft, and the beam loads are w = 1.5 kips/ft and P = 20 kips.
(a) Develop a list of five acceptable shapes that could be used for this beam. Include the most economical W14, W16, W18, W21, and W24 shapes on the list of possibilities.
(b) Select the most economical W shape for this beam.
Shear-force and bending-moment diagrams
Maximum bending moment magnitude M = 228 kip·ft
Minimum required section modulus
228 kipft12 in./ft 91.2 in. 30 kips/in.
(a) Acceptable steel W-shapes
68,103in.
57,92.2in. W18 55,98.3in.
W21 50,94.5in. W24 55,114in.
(b) Most economical W-shape W21 50 Ans.
P8.34 The beam shown in Figure P8.34 will be constructed from a standard steel W-shape using an allowable bending stress of 30 ksi. The beam span is L = 24 ft, and the beam loads are w = 1,100 lb/ft and 2w = 2,200 lb/ft.
(a) Develop a list of four acceptable shapes that could be used for this beam. Include the most economical W10, W12, W14, and W16 shapes on the list of possibilities.
(b) Select the most economical W shape for this beam.
Shear-force and bending-moment diagrams
Maximum bending moment magnitude M = 121.275 kip·ft
Minimum required section modulus
(a) Acceptable steel W-shapes
(b) Most economical W-shape
Ans.
P8.35 The beam shown in Figure P8.35 will be constructed from a standard steel W-shape using an allowable bending stress of 165 MPa.
(a) Develop a list of four acceptable shapes that could be used for this beam. Include the most economical W310, W360, W410, and W460 shapes on the list of possibilities.
(b) Select the most economical W shape for this beam.
Maximum moment magnitude: The maximum bending moment magnitude occurs at the base of the cantilever beam:
P8.36 The beam shown in Figure P8.36 will be constructed from a standard steel HSS-shape using an allowable bending stress of 30 ksi.
(a) Develop a list of three acceptable shapes that could be used for this beam. On this list, include the most economical HSS8, HSS10, and HSS12 shapes.
(b) Select the most economical HSS-shape for this beam.
Shear-force and bending-moment diagrams
Maximum bending moment magnitude M = 45.56 kip ft
Minimum required section modulus
(a) Acceptable steel HSS shapes
are acceptable
(b) Most economical HSS shape