JNTU
JNTU B.Tech II THEORY OF STRUCTURES Previous Question Paper (October - 2020 - R18)
Answer any five questions. All questions carry equal marks.
Main
A brass bar having cross sectional area of 1000 mm2, is subjected to axial forces as shown in Figure 1. Find the total change in the length of the bar (E= 105 kN/mm2, A = 1000 mm2).
Answer
To find the total change in length of the brass bar, we apply the principle of superposition. The total deformation is the algebraic sum of the deformations of individual segments of the bar, calculated using the formula:
Given the cross-sectional area A = 1000 mm2 and the modulus of elasticity E = 105 kN/mm2, we first determine the internal axial force (P) in each segment by considering the equilibrium of sections. Assuming the bar is divided into segments based on the points of application of the axial forces:
- Identify the axial force in each segment by cutting the bar at each section and applying the equilibrium condition ΣF = 0.
- Calculate the deformation for each segment using ΔLi = (Pi Li) / (A E), where Pi is the internal force, Li is the length of the segment, A is the area, and E is the Young's modulus.
- Sum the individual deformations: ΔLtotal = ΔL1 + ΔL2 + ΔL3 + ...
- Ensure consistent units: since A is in mm2 and E is in kN/mm2, the product AE is in kN. Forces should be in kN and lengths in mm to obtain the total change in length in mm.
A positive result indicates an increase in total length (elongation), while a negative result indicates a decrease (contraction).
A solid steel shaft in a rolling mill transmits 20 kW of power at 2 Hz. Determine the smallest safe diameter of the shaft if the shear stress τw is not to exceed 40 MPa and the angle of twist θ is limited to 6° in a length of 3 m. Use G = 83 GPa.
Answer
To determine the smallest safe diameter, we must satisfy both the strength criterion (shear stress) and the stiffness criterion (angle of twist). Given: Power (P) = 20,000 W, Frequency (f) = 2 Hz, Allowable shear stress (τ) = 40 MPa = 40 N/mm2, Length (L) = 3 m = 3000 mm, Angle of twist (θ) = 6° = 6 × (π / 180) = 0.1047 radians, and Modulus of rigidity (G) = 83 GPa = 83,000 N/mm2.
First, calculate the torque (T) transmitted by the shaft using the power formula:
1. Strength Criterion: The maximum shear stress is given by τ = (16T) / (πd3). Rearranging for d:
2. Stiffness Criterion: The angle of twist is given by θ = (TL) / (GJ), where J = (πd4) / 32. Substituting J:
Comparing the two results, the diameter must satisfy the more restrictive condition. Therefore, the smallest safe diameter is 58.74 mm.
Draw the shear force and bending moment diagram for the beam shown in Figure 2. Also indicate the points of contraflexure if any.
Answer
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A cast iron beam has an I-section with top angle 100 mm × 40 mm, web 140 mm × 20 mm and bottom angle 180 mm × 40 mm. If tensile stress is not to exceed 35 MPa and compressive stress 95 MPa, what is the maximum uniformly distributed load the beam can carry over a simply supported span of 6.5 m.
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A hollow cylinder of 150 mm external diameter 15 mm thick and 3 m long is hinged at one end and fixed at other. Find:
The ratio of Euler and Rankin load.
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For what length are the critical loads by Euler’s and Rankin’s formulae will be equal?
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A cantilever of length ‘l’ carries a uniformly distributed load of ‘w’ per unit run for a distance 3l/4 from the fixed end. Find the slope and deflection at the free end.
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Explain statically determinate structures and statically indeterminate structures with neat sketches and examples.
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Determine the maximum bending moment and the maximum deflection of a rail road. The rail is subjected to a single wheel load of 150 kN. The elastic support for the rail has a spring constant of k=14 MN/m2. The moment of inertia of rail is 3700 × 10-8 m4, E=200 GPa. Also calculate the maximum stress in the rail assuming that the depth of the rail is 200 mm and that the distance of the centroidal axis of the cross-section of the rail from the top surface is 100 mm.
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How will you find out graphically the resultant stress on an oblique section when the body is subjected to direct stresses in two mutually perpendicular directions?
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