JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II AEROSPACE VEHICLE STRUCTURES-I Previous Question Paper (November/December - 2020 - R15)
Answer any Five Questions. All Questions Carry Equal Marks.
Main
A cantilever of length L and depth 2h is in a state of plane stress. The cantilever is of unit thickness, is rigidly supported at the end x = L, and is loaded as shown in Figure 1. Show that the stress function
Answer
For a stress function Φ to be valid in plane stress, it must satisfy the biharmonic equation ∇4Φ = 0, where ∇4Φ = ∂4Φ/∂x4 + 2∂4Φ/∂x2∂y2 + ∂4Φ/∂y4 = 0. Given Φ = Ax2 + Bx2y + Cy3 + D(5x2y3 - y5), we compute the partial derivatives:
Substituting these into the biharmonic equation: 0 + 2(60Dy) - 120Dy = 0. Since the equation is satisfied, the stress function is valid. The stress components are derived as:
Applying boundary conditions for a cantilever of length L and depth 2h (y from -h to h):
- At the free surface y = ±h, σy = 0 and τxy = 0. Setting τxy = 0 at y = ±h gives -2Bx - 10Dx(±h)2 = 0, so B = -5Dh2.
- Setting σy = 0 at y = ±h gives 2A + 2B(±h) + 10D(±h)3 = 0. Substituting B, we get 2A - 10Dh3 + 10Dh3 = 0, so A = 0.
- The shear force at the free end x = 0 is P. The shear stress resultant ∫-hh τxy dy = -P. Integrating -2Bx - 10Dxy2 from -h to h yields -2B(2h) - 10Dx(2h3/3) = -P. Since B = -5Dh2, this simplifies to 20Dxh3 - 20Dxh3/3 = -P, which is inconsistent unless the load is applied at the centroid. For a point load P at the end, the constants are determined by the moment and shear equilibrium: D = -P/(4h3), B = 5Ph2/(4h3) = 5P/(4h), and C is determined by the bending stress distribution.
With the help of mathematical expression explain the differences between plane stress and plane strain condition.
Answer
A plane stress condition occurs in thin plates where the stress components acting normal to the surface are assumed to be zero. Specifically, for a plate in the xy-plane, the stress components are defined as σz = 0, τxz = 0, and τyz = 0. The state of stress is defined by the matrix:
A plane strain condition occurs in long structural members where the strain components in the longitudinal direction are constrained or negligible. For a member along the z-axis, the strain components are defined as εz = 0, γxz = 0, and γyz = 0. While the normal stress σz is not necessarily zero, it is related to the other normal stresses by the Poisson effect: σz = ν(σx + σy).
Derive all stresses on an oblique section of a body subjected to direct stress in two mutually perpendicular directions.
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A beam is loaded as shown in the figure 2, Determine the deflection of beam by Moment Area method.
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A horizontal cantilever beam of uniform section of length l carries two point loads, W at the free end and 2W at a distance of a from the free end. Find the maximum deflection of the beam using Area-moment method.
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The cantilever beam ‘AB’ is of uniform cross section and carries a load ‘P’ at its free end ‘A’. Determine the equation of the elastic curve and deflection at point A as shown in figure 3.
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A beam of length 6 m is simply supported at its ends and carries two point loads of 48 kN and 40 kN at a distance of 1 m and 3 m respectively from left support. Find the deflection under each load and also find the maximum deflection. Take
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Derive an expression of crippling load when one end of the column is fixed and the other end is pinned or hinged.
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A beam of length 6m is simply supported at its end and carries two point loads of 48 kN and 40 kN at a distance of 1 m and 3m respectively from left support. Find deflection under each load and the point at which maximum deflection occurs. Take
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