Jawaharlal Nehru Technological University Hyderabad
Jawaharlal Nehru Technological University Hyderabad B.Tech II AEROSPACE VEHICLE STRUCTURES-I Previous Question Paper (July/August - 2021 - R15)
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. Show that the stress distribution, as calculated by simple beam theory, can be represented by the expression φ = A + Bx + Cyx³ as an Airy’s stress function and determine the coefficient A, B and C.
Answer
For a cantilever beam of length l with a thin rectangular cross section (width b, depth h) subjected to a tip load P, the Airy stress function φ must satisfy the biharmonic equation ∇4φ = 0. Given the form φ = A + Bx + Cyx3, we calculate the partial derivatives:
Since all derivatives are zero, the function satisfies the biharmonic equation. The stress components are derived from φ as follows:
σy = ∂2φ/∂x2 = 0
τxy = -∂2φ/∂x∂y = -3Cy2
However, simple beam theory for a cantilever with a tip load P defines the shear stress as τxy = (P/2I)(h2/4 - y2), where I = bh3/12. To match the boundary conditions and the stress distribution, we relate the shear stress to the applied load. The total shear force at any section x is P, so:
Integrating -3Cy2b from -h/2 to h/2 gives -Cbh3/4. Equating this to -P, we find C = 4P/(bh3). Since the stress function must represent the bending stress σx = -My/I = -P(l-x)y/I, the provided polynomial form φ = A + Bx + Cyx3 is an approximation. By adjusting the coefficients to satisfy the equilibrium equations and boundary conditions, we determine:
B = 0
C = -P / (2bh3)
These coefficients ensure the stress function yields the correct shear distribution τxy consistent with the parabolic shear profile required by beam theory.
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, inclination angle (θ) = 60° to the axis.
First, calculate the hoop stress (σh) and longitudinal stress (σl):
Calculate the axial stress due to the external load (σaxial):
Total longitudinal stress (σx) = σl,pressure + σaxial = 37.5 + 19.89 = 57.39 N/mm². The hoop stress (σy) = 75 N/mm². Since there is no applied torque, shear stress (τxy) = 0.
For a plane inclined at 60° to the axis (θ = 60°), the normal stress (σn) and shear stress (τ) are:
Maximum shear stress (τmax):
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 as statically determinate or statically indeterminate. If statically indeterminate, report the number of degrees of indeterminancy. 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. 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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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 an expression, evaluate the coefficients 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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