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Prismatic cantilever beam
Taking section at a distance x from the free end.
By similar triangle,
since,
by double integration method,
where,
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- 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.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.Compound beam ABC is loaded by point load P = 1.5 kips at distance 2aB from point A and a triangularly distributed load on segment BC with peak intensity qü= 0.5 kips/ft. If length a = 5 ft and length/) = 10 ft, find the deflection at B and rotation at A. Assume that £ = 29,000 ksi and / = 53.8 in4.
- 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.The wing of a small plane is represented by a simplified prismatic cantilever beam model acted on by the distributed loads shown in the figure. Assume constant El = 1200kN-m~, Find the tip deflection and rotation at B.The cantilever beam ACB shown in the figure has moments of inertia /, and I{in parts AC and CB, respectively. Using the method of superposition, determine the deflection 8Bat the free end due to the load P. Determine the ratio r of the deflection 8Bto the deflection S:at the free end of a prismatic cantilever with moment of inertia /] carrying the same load. Plot a graph of the deflection ratio r versus the ratio 12 //L of the moments of inertia. (Let /, II- vary from I to 5.)
- The wing of a large commercial jet is represented by a simplified prismatic cantilever beam model with uniform load \v and concentrated loads P at the two engine locations (see figure). Find expressions for the tip deflection and rotation at D in terms of \\\ P, L, and EL.An overhanging beam ABC with flexural rigidity EI = 15 kip-in" is supported by a sliding support at A and by a spring of stiffness k at point fi(see figure). Span AB has a length L = 30 in. and carries a u ni form load. The overhang BC has a length b = 15 in. For what stiffness k of the spring will the uniform load produce no deflection at the free end C?The cantilever beam ACB shown in the figure supports a uniform load of intensity q throughout its length. The beam has moments of inertia I2and IYin parts AC and CB, respectively. Using the method of superposition, determine the deflection SBat the free end due to the uniform load. Determine the ratio r of the deflection 6Bto the deflection 3Xat the free end of a prismatic cantilever with moment of inertia /] carrying the same load. Plot a graph of the deflection ratio r versus the ratio 12 //t of the moments of inertia. (Let 7, tlxvary from I to 5.)
- A square wood platform is 8 ft × 8 ft in area and rests on masonry walls (see figure). The deck of the platform is constructed of 2-in. nominal thickness tongue-and-groove planks (actual thickness 1.5 in.; sec Appendix CL) supported on two S-ft long beams. The beams have 4 in. × (i in. nominal dimensions (actual dimensions 3.5 in. × 5.5 in.). The planks arc designed to support a uniformly distributed load n ( lb/ft" i acting over the entire top surface of the platform. I be allowable bending stress for the planks is 2400 psi and the allowable shear stress is 100 psi. W ben analyzing the planks, disregard their weights and assume that their reactions are uniformly distributed over the top surfaces of the supporting beams. (a) Determine the allowable platform load Mr. (lb/ft2) based upon the bending stress in the planks. (b) Determine the allowable platform load if-. (lb/ft-) based upon the shear stress in the planks. (c) Which of the preceding values becomes the allowable load alolow on the platform? Hints: Use care in constructing the loading diagram for the planks, noting especially that the reactions are distributed loads instead of concentrated loads. Also, note that the maximum shear forces occur at the inside faces of the supporting beams.A beam rests on supports at A and B and is loaded by a distributed load with intensity q as shown. A small gap exists between the unloaded beam and the support at C. Assume that span length L = 40 in. and flexural rigidity of the beam EI = 04 x 109lb-in2. Plot a graph of the bending moment at B as a function of the load intensity q. Hint: See Example 9-9 for guidance on computing the deflection at C.A framework A BCD is acted on by counterclockwise moment M at A (see figure). Assume that Elis constant. Find expressions for reactions at supports B and C Find expressions for angles of rotation at A, 5, C, and Z). Find expressions for horizontal deflections SÂand SD, If length LA3= L12, find length LCDin terms of L for the absolute value of the ratio |sysj=i.