A-7.7: A cantilever beam with rectangular cross section (b = 95 mm, h = 300 mm) supports load P = 160 kN at its free end. The ratio of the magnitudes of the principal stresses (01/02) at point A (at distance c = 0.8 m from the free end and distance d = 200 mm up from the bottom) is approximately: (A) 5 (B) 12 (C) 18 (D) 25
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- Two identical, simply supported beams AB and CD are placed so that they cross each other at their midpoints (sec figure). Before the uniform load is applied, the beams just touch each other at the crossing point. Determine the maximum bending moments (mab)max* and (MCD)max beams AB and CD, respectively, due to the uniform load if the intensity of the load is q = 6.4 kN/m and the length of each beam is L = 4 m..10 A built-up bourn supporting a condominium balcony is made up of a structural T (one half of a W 200 x 31.3) for the top flange and web and two angles (2 L 2 / b / 6.4. long legal back-lo-backl lot the bottom flange and web. as shown. The beam is subjected to a bending moment .1/ having its vector at an angle ft lo the z axis (see figure). Determine the or ion ta I ion of the neutral axis and calculate the maximum tensile stress ir, and maximum compressive stress tr. in ".he beam. .Assume that 9 = 30°andM = 15 kN · m. Use the numerical properties: c =4.111mm, c2 =4.169 mm, of = 134 mm, I, = 76 mm, A = 4144 mm 3 =3.88 X 106 mm 4, and = 34.18 X 10 mm 4.The cross section of an unbalanced wide-flange beam is shown in the figure. Derive the following formula for the distance /h from the centerline of one flange to the shear center S: h1t2b23ht1b13+t2b23 Also, check the formula for the special cases of a T-beam (b2= t2=0) and a balanced wide-flange beam (t2= ttand b2= ty).
- A 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 cross-sectional dimensions of a beam are shown. Assume b-2.125 in., tt-0.3125 in. and d=2.125 in. (a) If the bending stress at point K is 3360 psi (T), determine the internal bending moment M₂ acting about the z centroidal axis of the beam. (b) Determine the bending stress o, at point H (positive if tension, negative if compression). H N K Answers: bf (a) M₂ = i (b) σH = i d lb-ft. psi.
- Determine the internal forces at point C (i.e. Vc. Nc, Mc) of the cantilever beam shown. Point A is fixed support 30 kN 60 kN/m 2 m A C 50 kNm 3 m 3 m Vc = +75 kN, Nc = 0, Nc = 0, Mc = +80 kN-m o Vc = -120 kN, Mc = +75 kN-m o Vc = -75 kN, Nc = 0, Mc = -90 kN-m o Vc = +75 kN, Nc = 0, Mc = -75 kN-m o None of the above o BThe cross-sectional dimensions of a beam are shown. Assume b=3.625 in., tetw=0.1875 in. and d=4.125 in. (a) If the bending stress at point K is 2470 psi (T), determine the internal bending moment M, acting about the z centroidal axis of the beam. (b) Determine the bending stress oy at point H (positive if tension, negative if compression). K d H Answers: (a) M2 = i Ib-ft. (b) oH = i psi.A beam is subjected to equal bending moments of M, = 3000 N-m. The cross-sectional dimensions areb = 190 mm, c = 30 mm, d = 73 mm, and t = 7 mm. Determine: (a) the centroid location (measured with respect to the bottom of the cross-section), the moment of inertia about the z axis, and the controlling section modulus about the z axis. (b) the bending stress at point H. Tensile stress is positive, while compressive stress is negative. (c) the bending stress at point K. Tensile stress is positive, while compressive stress is negative. (d) the maximum bending stress produced in the cross section. Tensile stress is positive, while compressive stress is negative. M, M2 H d (typ) K b Answer: (a) y= mm 1z = i mm4 S= i mm3 (b) ƠH = MPа ( c) σκ-1 MPа (d) Omax= MPa
- The composite beam in Fig. a is made of wood and reinforced with a steel strap located on its bottom side. If the beam is subjected to a bending moment of M = 2 kN # m, determine the normal stress at points B and C. Take Ew = 12 GPa and Est = 200 GPa.The cross-sectional dimensions of the beam are a = 4.8 in, b = 6.5 in, d=4.5 in, and t= 0.30 in. The internal bending moment about the z centroidal axis is Mz= -3.90 kip-ft. Answers both in psiThe beam cross section shown below has been proposed for a short pedestrian bridge. The cross section will consist of two pipes that are welded to a rectangular web plate. Dimensions of the cross section are: h = 430 mm tw = 16 mm d = 110 mm t = 4.8 mm Additionally: • The area of each pipe is A = 1586 mm2. • The moment of inertia of the entire beam cross section about the z centroidal axis is IZ = 194710000 mm4. If the beam will be subjected to a shear force of V = 170 kN, determine the shear stress at point K, located at yK = 70 mm below the z centroidal axis.