JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year AEROSPACE VEHICLE STRUCTURES I Previous Question Paper (July/August - 2021 - R13)
Answer any five questions. All questions carry equal marks.
Main
A beam of length l, with a thin rectangular cross section, is built-in at the end x = 0 and loaded at the tip by a vertical force P as shown in Figure 1. Show that the stress distribution, as calculated by simple beam theory, can be represented by the expression:
as an Airy’s stress function and determine the coefficient A, B and C.
Answer
For a cantilever beam of length l, depth d (from y = -d/2 to d/2), and width b, subjected to a tip load P, the Airy stress function must satisfy the biharmonic equation ∇⁴φ = 0. Substituting φ = Ay³ + Bx³ + Cyx into the biharmonic equation:
This confirms the function is valid. The stress components are derived as:
Applying boundary conditions:
- At the free end (x = l), the shear force must equal P: ∫τₓᵧ dA = -C(bd) = -P, hence C = P/(bd).
- At the surfaces (y = ±d/2), σᵧ = 0. Since σᵧ = 6Bx, this implies B = 0.
- The bending moment at any section x is M = P(l - x). From simple beam theory, σₓ = -My/I, where I = bd³/12. Thus, σₓ = -P(l - x)y / (bd³/12) = -12P(l - x)y / (bd³).
Comparing this with σₓ = 6Ay, we find A = -2P(l - x) / (bd³). Since A must be a constant for the Airy function form provided, we evaluate the stress distribution at the root (x=0) or recognize the function represents the state where A = -2Pl/(bd³), B = 0, and C = P/(bd).
A cylindrical pressure vessel has an internal diameter of 2m and is fabricated from plates 20mm thick. If the pressure inside the vessel is 1.5 N/mm² and, in addition, the vessel is subjected to an axial tensile load of 2500 kN, calculate the direct and shear stresses on a plane inclined at an angle of 60° to the axis of the vessel. Calculate also the maximum shear stress.
Answer
Given: Internal diameter d = 2000 mm, thickness t = 20 mm, internal pressure p = 1.5 N/mm², axial load P = 2500 kN = 2.5 × 10⁶ N, and inclination angle θ = 60° to the axis.
First, calculate the hoop stress (σh) and longitudinal stress (σl):
For a plane inclined at θ = 60° to the axis, the angle with the normal to the hoop stress is α = 90° - 60° = 30°. Using stress transformation equations:
The maximum shear stress is given by:
A beam ABC of length 16 m consists of spans AB and BC each 8 m long and is simply supported at A, B and C. The beam carries uniformly distributed load of 40 kN/m on the whole length. Find the reactions at the supports and the support moments.
Answer
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Classify each of the frames shown in figure 2, as statically determinate or statically indeterminate. If statically indeterminate, report the number of degrees of indeterminacy. The frames are subjected to external loadings that are assumed to be known and can act anywhere on the frames.
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Determine the maximum bending moment and the maximum deflection of a rail road. 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.
Answer
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A simply supported beam AB of span L and uniform section carries a distributed load of intensity varying from zero at A to w₀/unit length at B. If the deflected shape of the beam is given approximately by the expression, evaluate the coefficients a₁ and a₂ and find the deflection of the beam at mid-span.
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Derive an expression of critical load for a column subjected to eccentric loading.
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Write expression for the potential energy for a simply supported beam with uniformly distributed load having whole span of length 'L'.
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