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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- 3. 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 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 = 0.25
- 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 isA 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 4
- A 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 isA 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 4
- 2. 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.The cross section of a composite beam is shown below. The Young's modulus of steel is 200 GPa and for aluminum it is 70 GPa. The distance between the neutral axis (Z-axis) and the bottom surface is approximately: stepl 1s mm aluminum 30 mm 25 mm