The beam in the figure below is subjected to a load P= 3.9 kN at its end. Young's modulus is 210 GPa and the moment of inertia for the beam's cross-section is 5x10° mm². If a = 1.5 m and b= 1.3 m. determine: A |-| a B b
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- 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 composite beam made of aluminum and steel is shown in the figure. The moduli of elasticity are TA= 75 GPa and Es= 200 GPa. Under the action of a bending moment that produces a maximum stress of 50 M Pa in the aluminum, what is the maximum stress xs in the steel? If the height of the beam remains at 120 mm and allowable stresses in steel and aluminum are defined as 94 M Pa and 40 M Pa, respectively, what heights h and h. arc required for aluminum and steel, respectively, so that both steel and aluminum reach their allowable stress values under the maximum moment?The cross section of a bimetallic strip is shown in the figure. Assuming that the moduli of elasticity for metals A and B are EA=168 GPa and EB= 90 GPa, respectively, determine the smaller of the two section moduli for the beam. (Recall that section modulus is equal to bending moment divided by maximum bending stress.) In which material does the maximum stress occur?
- A U-shaped cross section of constant thickness is shown in the figure. Derive the following formula for the distance e from the center of the semicircle to the shear center. Also, plot a graph showing how the distance e (expressed as the non dimensional ratio e/r varies as a function of the ratio b/r. (Let b/r range from 0 to 2.)A cantilever beam is subjected to a concentrated moment at B, The length of the beam L = 3 m and the height h = 600 mm. The longitudinal strain at the top of the beam is 0,0005 and the distance from the neutral surface to the bottom surface of the Iva m is 300 nun. Find the radius of curvature, the curvature, and the deflection of the beam at B.The cross section of a sign post of constant thickness is shown in the figure. Derive the formula for the distance e from the cent crime of the wall of the post to the shear center S: where I2. = moment of inertia about the z axis. Also, compare this formula with that given in Problem 6.9-11 for the special case of ß = 0 here and a = h/2 in both formulas.
- The cross section of a beam made of thin strips of aluminum separated by a lightweight plastic is shown in the figure. The beam has width b = 3.0 in., the aluminum strips have thickness t = 0.1 in., and the plastic segments have heights d = 1.2 in. and 3d = 3.6 in. The total height of the beam is h = 6.4 in. The moduli of elasticity for the aluminum and plastic are EM= 11 X 106 psi and Ep= 440 X 10* psi, respectively. Determine the maximum stresses trAiand pin the aluminum and plastic, respectively, due to a bending moment of 6,0 kip-in.Determine the shape factor f for a cross section in the shape of a double trapezoid having the dimensions shown in the figure. Also, check your result for the special cases of a rhombus (b1= 0) and a rectangle (b1= b2).Find expressions for shear force V and moment M at v = L/2 of beam AB in structure (a). Express V and M in terms of peak load intensity q0and beam length variable L. Repeat for structure (b) but find Fand M at m id-span of member BC.
- Two identical, simply supported beams AB and CD are placed so that they cross each other at their midpoints (sec figure). Before the uniform load is applied, the beams just touch each other at the crossing point. Determine the maximum bending moments (mab)max* and (MCD)max beams AB and CD, respectively, due to the uniform load if the intensity of the load is q = 6.4 kN/m and the length of each beam is L = 4 m.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.)The cross section of a steel beam is shown in the figure. This beam is subjected to a bending moment M having its vector at an angle 8 to the - axis. Determine the orientation of the neutral axis and calculate the maximum tensile stress tiand maximum compressive stress tcin the beam. Assume that e = 22.5° and M = 4.5 kN · m. Use cross-sectional properties Ix=93.14 × 106 mm4, Iy= 152.7 X 10e mm4, and 9 = 27.3º.