Draw the shear-force and bending-moment diagrams for a simple beam as shown in Figure Q2 with a uniform load of intensity q acting over part of the span if q = 20 kN, a = b= 1 m, and c= 0.5 m. A В Figure Q2
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- Find expressions for shear force V and moment M at v = L/2 of beam AB in structure (a). Express V and M in terms of peak load intensity q0and beam length variable L. Repeat for structure (b) but find Fand M at m id-span of member BC.Cantilever beam AB carries an upward uniform load of intensity q1from x = 0 to L/2 (see Fig. a) and a downward uniform load of intensity q from x = L/2 to L. Find q1in terms of q if the resulting moment at A is zero. Draw V and M diagrams for the case of both q and qtas applied loadings. Repeat part (a) for the case of an upward triangularly distributed load with peak intensity q0(see Fig. b). For part (b), find q0, instead of q1The hollow box beam shown in the figure is subjected to a bending moment M of such magnitude that the flanges yield but the webs remain linearly elastic. (a) Calculate the magnitude of the moment M if the dimensions of the cross section are A = 15 in., A] = 12.75 in., h = 9 in., and ey =7.5 in. Also, the yield stress is eY = 33 ksi. (b) What percent of the moment M is produced by the elastic core?
- 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 beam supporting a uniform load of intensity q throughout its length rests on pistons at points A, C and B (sec figure). The cylinders are filled with oil and are connected by a tube so that the oil pressure on each piston is the same. The pistons at A and B have diameter d1and the piston at C has diameter D2. (a) Determine the ratio of d2to d1so that the largest bending moment in the beam is as small as possible. Under these optimum conditions, what is the largest bending moment Mmaxin the beam? What is the difference in elevation between point C and the end supports?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.
- Find expressions for shear force V and moment M at mid-span of beam AB in terms of peak load intensity q0and beam length variables a and L Let a = 5L/b.Draw the shear-force and bending-moment diagrams for a cantilever beam AB acted upon by two different load cases. A distributed load with linear variation and maximum intensity q0(see figure part a). A distributed load with parabolic variation and maximum intensity q0(see figure part b).-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 t (obtained by dividing the shear force by the area of the web) and the ratio tmax/taver. The shear force Vweb/V carried in the web and the Vweb/V. 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. 5.10-1 Dimensions of cross section: b = 6 in,, ï = 0.5 in., h = 12 in,, A, = 10.5 in., and V = 30 k.
- (a) A simple beam AB with length L and height h supports a uniform load of intensity q (see the figure part a). Obtain a formula for the curvature shortening A of this beam. Also, obtain a formula for the maximum bending stress b in the beam due to the load q. Now assume that the ends of the beam are pinned so that curvature shortening is prevented and a horizontal force H develops at the supports (see the figure part b). Obtain a formula for the corresponding axial tensile stress t . Using the formulas obtained in parts (a) and (b), calculate the curvature shortening , the maximum bending stress b, and the tensile stress t for the following steel beam: length L = 3m, height h = 300 mm, modulus of elasticity E = 200 GPa, and moment of inertia I = 36 x 106 mm4. Also, the load on the beam has intensity q = 25 kN/m. Compare the tensile stress tproduced by the axial forces with the maximum bending stress bproduced by the uniform load.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.)A propped cantilever beam is subjected to two triangularly distributed loads, each with a peak load intensity equal to q0(see figure), lind the expressions for reactions at A and C using superposition. Plot shear and moment diagrams.