5. The rigid beam rests on two short posts AC and 80 as shown Post AC is made from steel with E-200 GP and 80 from aluminum with E-70 GP. The diameters of posts AC and 80 are 25 mm and 40 mm, respectively. Determine the vertical displacement of point on AB 100 AN
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- An aluminum bar having a rectangular cross section (2.0 in. × 1.0 in.) and length L = 30 in. is compressed by axial loads that have a resultant P = 2800 lb acting at the midpoint of the long side of the cross section (sec figure). Assuming that the modulus of elasticity E is equal to 10 × 106 psi and that the ends of the bar are pinned, calculate the maximum deflection and the maximum bending moment Mmax.An S6 × 12.5 steel cantilever beam AB is supported by a steel tic rod at B as shown. The tie rod is just taut when a roller support is added at Cat a distance s to the left of £, then the distributed load q is applied to beam segment AC, Assume E = 30 × 106 psi and neglect the self-weight of the beam and tie rod. Sec Table F-2(a) in Appendix F for the properties of the S-shape beam. (a) What value of uniform load q will, if exceeded, result in buckling of the tie rod if L1, =6 ft, s = 2 ft, H = 3 ft, and d = 0.25 in.? (b) What minimum beam moment of inertia ibis required to prevent buckling of the tie rod if q = 200 lb/ft, L1, = 6 ft, H = 3 ft, d = 0.25 in., and s = 2 ft? (c) For what distance s will the tic rod be just on the verge of buckling if q = 200 lb/ft, L1= 6 ft, M = 3 ft, and d = 0.25 in.?A fixed-end b earn is subjected to a point load at mid-span. The beam has a rectangular cross section (assume that the h/b ratio is 2) and is made of wood (E = 11GPa). Find height h of the cross section if the maximum displacement of the beam is 2 mm. Calculate the displacement of the beam at the inflection points.
- The tapered cantilever beam AB shown in the figure has a solid circular cross section. The diameters at the ends A and B are dAand dB= 2dA, respectively. Thus, the diameter d and moment of inertia / at distance v from the free end are, respectively, in which IAis the moment of inertia at end A of the beam. Determine the equation of the deflection curve and the deflection SAat the free end of the beam due to the load P.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)?.20 Determine the plastic moment Mpfor beam having the cross section shown in the figure ey=210 MPa.
- 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.The tapered cantilever beam AB shown in the figure has a thin-walled, hollow circular cross sections of constant thickness t. The diameters at the ends A and B are dAand dB= 2dA, respectively. Thus, the diameter d and moment of inertia / at distance x from the free end are, respectively, in which IAis the moment of inertia at end A of the beam. Determine the equation of the deflection curve and the deflection 8 Aat the free end of the beam due to the load P..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 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 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?A tapered cantilever beam A B of length L has square cross sections and supports a concentrated load P at the free end (sec figure part a). The width and height of the beam vary linearly from kAat the free end to hBat the fixed end. Determine the distance.y from the free end A to the cross section of maximum bending stress if hE= 3h4, What is the magnitude ffœai of the maximum bending stress? What is the ratio of the maximum stress to the largest stress B at the support? Repeat part (a) if load P is now applied as a uniform load of intensity q = P/L over the entire beam, A is restrained by a roller support, and B is a sliding support (see figure part b).