PROB. 5.5-12 5.5-13 Determine the maximum tensile stress o, (due to pure bending about a horizontal axis through C by positive bending moments M) for beams having cross sections as follows (see figure). (a) A semicircle of diameter d (b) An isosceles trapezoid with bases b¡ = b and b, = 4b/3, and altitude h (c) A circular sector with a = m/3 and r = d/2 C- a (a) (b) (c)
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- A canti lever beam A B of a n isosceles t rapezoi-dal cross section has a length L = 0.8 m, dimensions bx= 80 mm and b2= 90 mm, and height h = 110 mm (see figure). The beam is made of brass weighing 85 kN/m3. Determine the maximum tensile stress asand maximum compressive stressA cantilever beam AB having rectangular cross sections with varying width bxand varying height hxis subjected to a uniform load of intensity q (sec figure). If the width varies linearly with x according to the equation hx= bBxiL^ how should the height hxvary as a function of v in order to have a fully stressed beam? (Express hxin terms of the height hBat the fixed end of the beam.)The cross section of a steel beam is shown in the figure. This beam is subjected to a bending moment M having its vector at an angle 8 to the - axis. Determine the orientation of the neutral axis and calculate the maximum tensile stress tiand maximum compressive stress tcin the beam. Assume that e = 22.5° and M = 4.5 kN · m. Use cross-sectional properties Ix=93.14 × 106 mm4, Iy= 152.7 X 10e mm4, and 9 = 27.3º.
- The beams shown in the figure are subjected to bending moments M = 2100 lb-in. Each beam has a rectangular cross section with height h = 1.5 in. and width b = 0,375 in, (perpendicular to the plane of the figure). For the beam with a hole at m id height, determine the maximum stresses for hole diameters d = 0.25,0.50, 0.75, and 1,00 in. For the beam with two identical notches (inside height /j| = 1.25 in.), determine the maximum stresses for notch radii R = 0.05, 0.10, 0.15 and 0.20 in.The cross section of a composite beam made of aluminum and steel is shown in the figure. The moduli of elasticity are TA= 75 GPa and Es= 200 GPa. Under the action of a bending moment that produces a maximum stress of 50 M Pa in the aluminum, what is the maximum stress xs in the steel? If the height of the beam remains at 120 mm and allowable stresses in steel and aluminum are defined as 94 M Pa and 40 M Pa, respectively, what heights h and h. arc required for aluminum and steel, respectively, so that both steel and aluminum reach their allowable stress values under the maximum moment?A beam with a T-section is supported and loaded as shown in the figure. The cross section has width b = 2 1/2 in., height c = 3 in., and thickness t = 3/8 in. Determine the maximum tensile and compressive stresses in the beam. If the allowable stresses in tension and compression are 18 ksi and 12 ksi, respectively, what is the required depth h of the beam? Assume that thickness t remains at 3/8 in. and that flange width/) = 2.5 in. Find the new values of loads P and q so that the allowable tension (18 ksi) and compression (12 ksi) stresses are reached simultaneously for the beam. Use the beam cross section in part (a) (see figure) and assume that Lh and L3are unchanged.
- The cross section of a rectangular beam having a width b and height h is shown in part a of the figure. For reasons unknown to the beam designer, it is planned to add structural projections of width b/9 and height d/9 the top and bottom of the beam (see part b of the figure). For what values of d is the bending-moment capacity of the beam increased? For what values is it decreased?. A cantilever beam (width b = 3 in. and depth h = 6 in,) has a length L = 5 ft and is subjected to a point load P and a concentrated moment M = 20 kip-ft at end B. If normal stress trx= 0 at point C, located 0.5 in. below the top of the beam and 1 ft to the right of point Atfind point load P. Also show the complete state of plane stress on the element at point C.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.
- A beam with a channel section is subjected to a bending moment M having its vector at an angle 0 to the 2 axis (see figure). Determine the orientation of the neutral axis and calculate the maximum tensile stress et and maximum compressive stress ecin the beam. Use the following data: C 8 × 11.5 section, M = 20 kip-in., tan0=l/3. See Table F-3(a) of Appendix F for the dimensions and properties of the channel section.The beams shown in the figure are subjected to bending moments M = 250 N · m. Each beam has a rectangular cross section with height f1= 44 mm and width d = 10 mm (perpendicular to the plane of the figure). For the beam with a hole at m id height, determine the maximum stresses for hole diameters d = 10, 16,22, and 28 mm. For the beam with two identical notches (inside height d= 40 mm), determine the maximum stresses for notch radii R = 2,4, 6, and 8 mm.A cantilever beam AB with rectangular cross sections of a constant width b and varying height /iT is subjected to a uniform load of intensity q (see figure). How should the height hxvary as a function of .y (measured from the free end of the beam) in order to have a fully stressed beam? (Express hxin terms of the height hBat the fixed end of the beam.)