The natural frequency of vibration for fixed-fixed boundary conditions for longitudinal mode 2 is: (a) 133286.5 rad/s (b) 199929.7 rad/s (c) 49982.4 rad/s (d) 99964.9 rad/s (e) 66643.2 rad/s
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- .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 beam of length 0.4 m, with circular cross-section of uniform radius 40 mm is made of an alloy material with material properties: density = 5,000 kg/m^3 Young's modulus = 90 GPa Poisson's ratio = 0.253. A 3-meter-long beam is used to support a heavy object. The object has a uniform distributed load of 6 kN/m on the entire beam. The Young’s modulus and moment of inertia of the beam are 200 GPa and 5×105 mm4, respectively. The beam is supported at three positions as shown below. (a) Label the element and node numbers (either on the figure or with a new simple sketch). (b) Determine the slopes at the three support positions of the beam.
- A 200kg machine is attached to the end of a cantilever beam of length L=2.5m, elastic modulus E=200x109 N/m2, and the cross sectional moment of inertia I= 1.8 x 10-6 m4. Assuming the mass of the beam is small compared to the mass of the machine, what is the stiffness of the beam?A beam has a bending moment of 3.5 kN-m applied to a section with a hollow circular cross-section of external diameter 3.9 cm and internal diameter 2 cm . The modulus of elasticity for the material is 210 x 109 N/m2. Calculate the radius of curvature and maximum bending stress. Also, calculate the stress at the point at 0.8 cm from the neutral axis (i) The moment of inertia = (in mm^4) ii) The radius of curvature is (in mm) (iii) The maximum bending stress is (in N/mm^2) iv) The bending stress at the point 0.8 cm from the neutral axis is (in N/mm^2)A beam has a bending moment of 2 kN-m applied to a section with a hollow circular cross-section of external diameter 3.2 cm and internal diameter 2 cm. The modulus of elasticity for the material is 210 x 10° N/m2. Calculate the radius of curvature and maximum bending stress. Also, calculate the stress at the point at 0.5 cm from the neutral axis Solution: (i) The moment of inertia = ii) The radius of curvature is (iii) The maximum bending stress is iv) The bending stress at the point 0.5 cm from the neutral axis is
- A beam has a bending moment of 3 kN-m applied to a section with a hollow circular cross-section of external diameter 3.4 cm and internal diameter 2.4 cm . The modulus of elasticity for the material is 210 x 109 N/m2. Calculate the radius of curvature and maximum bending stress. Also, calculate the stress at the point at 0.6 cm from the neutral axis Solution: (i) The moment of inertia = ii) The radius of curvature is (iii) The maximum bending stress is in (N/mm^2) Answer and unit for part 3 iv) The bending stress at the point 0.6 cm from the neutral axis is in(N/mm^2) Answer and unit for part 4A beam has a bending moment of 4 kN-m applied to a section with a hollow circular cross-section of external diameter 3.3 cm and internal diameter 2.3 cm . The modulus of elasticity for the material is 210 x 109 N/m2. Calculate the radius of curvature and maximum bending stress. Also, calculate the stress at the point at 0.8 cm from the neutral axis (i) The moment of inertia = ii) The radius of curvature is = (iii) The maximum bending stress is iv) The bending stress at the point 0.8 cm from the neutral axis isA beam has a bending moment of 2.5 kN-m applied to a section with a hollow circular cross-section of external diameter 3 cm and internal diameter 2.3 cm. The modulus of elasticity for the material is 210 x 109 N/m. Calculate the radius of curvature and maximum bending stress. Also, calculate the stress at the point at 0.7 cm from the neutral axis Solution: (i) The moment of inertia = 26024.12mm^4 ii) The radius of curvature is 2186.02mm (iii) The maximum bending stress is 1.44GPA iv) The bending stress at the point 0.7 cm from the neutral axis is
- A beam has a bending moment of 3.5 kN-m applied to a section with a hollow circular cross-section of external diameter 3.7 cm and internal diameter 2.2 cm . The modulus of elasticity for the material is 210 x 109 N/m2. Calculate the radius of curvature and maximum bending stress. Also, calculate the stress at the point at 0.6 cm from the neutral axis (i) The moment of inertia = ii) The radius of curvature is (iii) The maximum bending stress is iv) The bending stress at the point 0.6 cm from the neutral axis is Answer and unit for part 42. A square glass reinforced plastic plate 3 mm thick is simply supported along all the edges. The plate is subjected to uniform distributed load q. The elastic properties of the plate are: Ex = 40 kN/mm², E, = 8 kN/mm², Gxy = 4 kN/mm², µHxy = Hyx = 0.25. Obtain an expression for deflection function of the plate using Levy's solution. Also determine the maximum deflection of the plate. Take origin at the top left corner of the plate.A I-meter-long, simply supported copper beam (E= 117 GPa) carries uniformly distributed load q. The maximum deflection is measured as 1.5 mm. a. Calculate the magnitude of the distributed load q if the beam has a rectangular cross section (width b= 20 mm, height h= 40 mm). b. If instead the beam has circular cross section and q= 500 N/m, calculate the radius r of the cross section. Neglect the weight of the beam.