For the cantilever steel beam [E = 220 GPa; I = 86 x 10 mm ], use the double-integration method to determine the deflection VA at A. Assume L = 2.5 m, P = 59 kN, and w = 21 kN/m. Answer: VA = mm.
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- Beam ACB hangs from two springs, as shown in the figure. The springs have stiffnesses Jt(and k2^ and the beam has flexural rigidity EI. What is the downward displacement of point C, which is at the midpoint of the beam, when the moment MQis applied? Data for the structure are M0 = 7.5 kip-ft, L = 6 ft, EI = 520 kip-ft2, kx= 17 kip/ft, and As = 11 kip/ft. Repeat part (a), but remove Af0 and instead apply uniform load q over the entire beam.A cantilever beam of a length L and loaded by a uniform load of intensity q has a fixed support at A and spring support at B with rotational stiffness kR. A rotation B at B results in a reaction moment MB=kRxB. Find rotation B and displacement Bat end B. Use the second-order differential equation of the deflection curve to solve for displacements at end B.Solve the preceding problem for a W 250 × 89 steel column having a length L = 10 m. Let E = 200 GPa.
- 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.)Beam AB has an elastic support kR at A, pin support at B, length L, height h (see figure), and is heated in such a manner that the temperature difference T2T1 between the bottom and top of the beam is proportional to the distance from support A. Assume the temperature difference varies linearly along the beam: T2T1=T0x in which T0 is a constant having units of temperature (degrees) per unit distance. Assume the spring at A is unaffected by the temperature change. Determine the maximum deflection max of the beam, Repeat for a quadratic temperature variation along the beam, so T2T1=T0x2 What is max for parts (a) and (b) if kR goes to infinity?.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?
- The Z-section of Example D-7 is subjected to M = 5 kN · m, as shown. Determine the orientation of the neutral axis and calculate the maximum tensile stress c1and maximum compressive stress ocin the beam. Use the following numerical data: height; = 200 mm, width ft = 90 mm, constant thickness a = 15 mm, and B = 19.2e. Use = 32.6 × 106 mm4 and I2= 2.4 × 10e mm4 from Example D-7A cantilever beam of a length L = 2.5 ft has a rectangular cross section {b = 4in,, h = Sin,) and modulus E = 10,000 ksi. The beam is subjected to a linearly varying distributed load with a peak intensity qQ= 900 lb/ft. Use the method of superposition and Cases 1 and 9 in Table H-l to calculate the deflection and rotation at B.A beam ABC with simple supports at A and B and an overhang BC supports a concentrated load P at the free end C (see figure). Determine the strain energy Ustored in the beam due to the load P. From the strain energy, find the deflection Scunder the load P. Calculate the numerical values of £/and Sc if the length L is 8 ft, the overhang length a is 3 ft, the beam is a W 10 x 12 steel wide-flange section, and the load P produces a maximum stress of 12,000 psi in the beam, (Use £ = 29 X 106 psi.)
- Solve the preceding problem using the following data: beam cross section is 100 x 150 mm, length is 3 m, and point load is P = 5 kN at mid-span, Point C is located 25 mm below the top of the beam and 0.5 m to the right of support A.Copper beam AB has circular cross section with a radius of 0.25 in. and length L = 3 ft. The beam is subjected to a uniformly distributed load w = 3.5 lb/ft. Calculate the required load P at joint B so that the total deflection at joint B is zero. Assume that£ = 16,000 ksi.A sandwich beam having steel faces enclosing a plastic core is subjected to a bending moment M = 5 kN · m. The thickness of each steel face is 1 = 3 mm with modulus of elasticity E = 200 GPa, The height of the plastic core is hp= 140 mm, and its modulus of elasticity is Ep= 800 MPa. The overall dimensions of the beam are h = 146 mm and h = 175 mm. Using the transformed-section method, determine the maximum tensile and compressive stresses in the faces and the core.