For the overhang beam shown below, (a) draw the free body diagram of the beam, (b) draw the shear and moment diagrams, (c) find moment of inertia of its cross-section, (d) determine the
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- A simple beam ACE is constructed with square cross sections and a double taper (see figure). The depth of the beam at the supports is dAand at the midpoint is dc= 2d 4. Each half of the beam has length L. Thus, the depth and moment of inertia / at distance x from the left-hand end are, respectively, in which IAis the moment of inertia at end A of the beam. (These equations are valid for .x between 0 and L, that is, for the left-hand half of the beam.) Obtain equations for the slope and deflection of the left-hand half of the beam due to the uniform load. From the equations in part (a), obtain formulas for the angle of rotation 94at support A and the deflection Scat the midpoint.A two-axle carriage that is part of an over head traveling crane in a testing laboratory moves slowly across a simple beam AB (sec figure). The load transmitted to the beam from the front axle is 2200 lb and from the rear axle is 3800 lb. The weight of the beam itself may be disregarded. Determine the minimum required section modulus S for the beam if the allowable bending stress is 17,0 ksi, the length of the beam is 18 ft, and the wheelbase of the carriage is 5 ft. Select the most economical I-beam (S shape) from Table F-2(a), Appendix F.A frame ABCD is constructed of steel wide-flange members (W8 x 21; E = 30 x ID6 psi) and subjected to triangularly distributed loads of maximum intensity q0acting along the vertical members (see figure). The distance between supports is L = 20 ft and the height of the frame is h = 4 ft. The members are rigidly connected at B and C. Calculate the intensity of load q0 required to produce a maximum bending moment of 80 kip-in. in the horizontal member BC. If the load q0 is reduced to one-half of the value calculated in part (a), what is the maximum bending moment in member BC? What is the ratio of this moment to the moment of 80 kip-in. in part (a)?
- A weight W = 4000 lb falls through a height h = 0.5 in, onto the midpoint of a simple beam of length L = 10 ft (see figure). Assuming that the allowable bending stress in the beam is = 18,000 psi and E = 30 x 10* psi, select the lightest wide-flange beam listed in Table F-l(a) in Appendix F that will be satisfactory.The 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?A C 200 x 17.1 channel section has an angle with equal legs attached as shown; the angle serves as a lintel beam. The combined steel section is subjected to a bending moment M having its vector directed along the z axis, as shown in the figure. The cent roi d C of the combined section is located at distances xtand ycfrom the centroid (C1) of the channel alone. Principal axes yl and yvare also shown in the figure and properties Ix1,Iy1and 0pare given. Find the orientation of the neutral axis and calculate the maximum tensile stress exand maximum compressive stress if the angle is an L 76 x 76 x 6.4 section and M = 3.5 kN - m. Use the following properties for principal axes for the combined section:/^, = 18.49 X 106 nrai4,/;| = 1.602 X 106 mm4, ep= 7.448*(CW),_r£ = 10.70 mm,andvf= 24.07 mm.
- The three beams shown have approximately the same cross-sectional area. Beam 1 is a W 14 X 82 with flange plates; beam 2 consists of a web plate with four angles; and beam 3 is constructed of 2 C shapes with flange plates. Which design has the largest moment capacity? Which has the largest shear capacity? Which is the most economical in bending? Which is the most economical in shear? Assume allowable stress values are: = 18 ksi and ra=11 ksi. The most economical beam is that having the largest capacity-to-weight ratio. Neglect fabrication costs in answering parts (c) and (d) above. Note: Obtain the dimensions and properties of all rolled shapes from tables in Appendix F..2 A ligmio.irc ii supported by two vorlical beams consistins: of thin-walled, tapered circular lubes (see ligure part at. for purposes of this analysis, each beam may be represented as a cantilever AB of length L = 8.0 m subjected to a lateral load P = 2.4 kN at the free end. The tubes have a constant thickness ; = 10.0 mm and average diameters dA = 90 mm and dB = 270 mm at ends A and B, re s pec lively. Because the thickness is small compared to the diameters, the moment of inerlia at any cross section may be obtained from the formula / = jrrf3;/8 (see Case 22, Appendix E); therefore, the section modulus mav be obtained from the formula S = trdhlA. (a) At what dislance A from the free end docs the maximum bending stress occur? What is the magnitude trllul of the maximum bending stress? What is the ratio of the maximum stress to the largest stress (b) Repeat part (a) if concentrated load P is applied upward at A and downward uniform load q {-x) = 2PIL is applied over the entire beam as shown in the figure part b What is the ratio of the maximum stress to the stress at the location of maximum moment?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 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 simple beam with a W 10 x 30 wide-flange cross section supports a uniform load of intensity q = 3.0 kips/ft on a span of length L = 12 ft (sec figure). The dimensions of the cross section are q = 10.5 in., b = 5.81 in., t1= 0.510 in., and fw = 0.300 in. Calculate the maximum shear stress tjuly on cross section A—A located at distance d = 2.5 ft from the end of the beam. Calculate the shear stress rat point Bon the cross section. Point B is located at a distance a = 1.5 in. from the edge of the lower flange.A rectangular beam with semicircular notches, as shown in part b of the figure, has dimensions h = 0,88 in. and h1 = 0.80 in. The maximum allowable bending stress in the metal beam is emax = 60 ksi, and the bending moment is M = 600 lb-in. Determine the minimum permissible width bminof the beam.