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- The compound beam shown in the figure consists of a cantilever beam AB (length L) that is pin-connected to a simple beam BD (length 2L). After the beam is constructed, a clearance c exists between the beam and a support at C, midway between points B and ZX Subsequently, a uniform load is placed along the entire length of the beam. What intensity q of the load is needed to close the gap at C and bring the beam into contact with the support?A fixed-end beam AB carries point load P acting at point C. The beam has a rectangular cross section (b = 75 mm, h = 150 mm). Calculate the reactions of the beam and the displacement at point C. Assume that E = 190 GPa.A fixed-end beam AB of a length L is subjected to a uniform load of intensity q acting over the middle region of the beam (sec figure). Obtain a formula for the fixed-end moments MAand MBin terms of the load q, the length L, and the length h of the loaded part of the beam. Plot a graph of the fixed-end moment MAversus the length b of the loaded part of the beam. For convenience, plot the graph in the following nondimensional form: MAqL2/l2versusbL with the ratio b/L varying between its extreme values of 0 and 1. (c) For the special case in which ù = h = L/3, draw the shear-force and bending-moment diagrams for the beam, labeling all critical ordinates.
- A thin steel beam AB used in conjunction with an electromagnet in a high-energy physics experiment is securely bolted to rigid supports (see figure), A magnetic field produced by coils C results in a force acting on the beam. The force is trapezoidally distributed with maximum intensity q0= 18 kN/m. The length of the beam between supports is L = 200 mm, and the dimension c of the trapezoidal load is 50 mm. The beam has a rectangular cross section with width b = 60 and height h = 20 mm. Determine the maximum bending stress max and the maximum deflection for the beam. (Disregard any effects of axial deformations and consider only the effects of bending. Use E = 200 GPa.)A cantilever beam model is often used to represent micro-clectrical-mechanical systems (MEMS) (sec figure}. The cantilever beam is made of polysilicon (E = 150 GPa) and is subjected to an electrostatic moment M applied at the end of the cantilever beam. 1 f dimensions arc h — 2 [im, h — 4 ^m, and L = 520 ^mt find expressions for the tip deflection and rotation of the cantilever beam in terms of moment M.A r o lukI f/frm f «m t ub e of ou t sid e d ia met er ^ and a copper core of diameter dxare bonded to form a composite beam, as shown in the figure, (a) Derive formulas for the allowable bending moment M that can be carried by the beam based upon an allowable stress <7Ti in the titanium and an allowable stress (u in the copper (Assume that the moduli of elasticity for the titanium and copper are Er- and £Cu, respectively.) (b) If d1= 40 mm, d{= 36 mm, ETl= 120 GPa, ECu= 110 GPa, o-Ti = 840 MPa, and ctqj = 700 MPa, what is the maximum bending moment Ml (c) What new value of copper diameter dtwill result in a balanced design? (i.e., a balanced design is that in which titanium and copper reach allow- able stress values at the same time).
- An L-shaped reinforced concrete slab 12 Ft X 12 ft, with a 6 Ft X 6 ft cut-out and thickness t = 9.0 in, is lifted by three cables attached at O, B, and D, as shown in the figure. The cables are are combined at point Q, which is 7.0 Ft above the top of the slab and directly above the center of mass at C. Each cable has an effective cross-sectional area of Ae= 0.12 in2. (a) Find the tensile force Tr(i = 1, 2, 3) in each cable due to the weight W of the concrete slab (ignore weight of cables). (b) Find the average stress ov in each cable. (See Table I-1 in Appendix I for the weight density of reinforced concrete.) (c) Add cable AQ so that OQA is one continuous cable, with each segment having Force T, which is connected to cables BQ and DQ at point Q. Repeat parts (a) and (b). Hini: There are now three Forced equilibrium equations and one constrain equation, T1= T4.A beam is constructed using two angle sections (L 102 × 76 × 6.4) arranged back to back, as shown in the figure. The beam is fixed al joint A and attached to an elastic support having a spring constant k = l750 kN/m al joint B. Assume only the beam is subjected to temperature increase AT = 45°C. Calculate the thermal stress developed in the beam and the displacement at point B. Assume that a = 12 X 10-6/?. Let E = 205 GPaA weight W = 4500 lb falls from a height h onto a vertical wood pole having length L = 15 ft, diameter d = 12 in., and modulus of elasticity E = 1.6 × 106 psi (see figure). If the allowable stress in the wood under an impact load is 2500 psi. what is the maximum permissible height h?
- A fixed-end beam is loaded by a uniform load q = 15 kN/m and a point load P = 30 kN at mid-span. The beam has a length of 4 m and modulus of elasticity of 205 GPa. Find reactions at A and B. Calculate the height of the beam if the displacement at mid-span is known to be 3 mm. Assume that the beam has rectangular cross section with h/b = 2.A column ABC is supported at ends A and C and compressed by an axial load P (figure a). Lateral support is provided at point B but only in the plane of the figure; lateral support perpendicular to the plane of the figure is provided only at A and C. The column is constructed of two channel sections (C 6 × 8.2) back to back (see figure b). The modulus of elasticity of the column is E = 29,500 ksi and the proportional limit is 50 ksi. The height of the column is L = 15 ft. Find the allowable value of load P using a factor of safety of 2.5.A gold-alloy microbeam attached to a silicon wafer behaves like a cantilever beam subjected to a uniform load (see figure). The beam has a length L = 27.5 m and rectangular cross section of a width b = 4.0 m and thickness t = 0.88 m. The total load on the beam is 17.2 N. If the deflection at the end of the beam is 2.46 m is what is the modulus of elasticity Egof the gold alloy? (Use the formulas of Example 9-2.)