JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III FINITE ELEMENT TECHNIQUES Previous Question Paper (July/August - 2023 - R16)
Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, Answer all questions. iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
PART - A
(25 Marks)
Write down the different coordinate systems used in finite elements.
Answer
In finite element analysis, the primary coordinate systems used are:
- Global Coordinate System: A single, fixed reference frame used to define the geometry and boundary conditions of the entire structure.
- Local (Element) Coordinate System: A coordinate system defined specifically for an individual element to simplify the formulation of element stiffness matrices.
- Natural (Intrinsic) Coordinate System: A dimensionless coordinate system (often denoted by ξ, η, ζ) where the coordinates range from -1 to +1, used to simplify numerical integration and the definition of shape functions for isoparametric elements.
State the methods used to solve the differential equations in finite element method.
Answer
In the finite element method, differential equations governing physical phenomena are solved primarily using the following weighted residual methods:
- Point Collocation Method: The residual is set to zero at specific discrete points within the domain.
- Subdomain Method: The integral of the residual is set to zero over several sub-regions of the entire domain.
- Least Squares Method: The integral of the square of the residual is minimized over the domain to determine the unknown parameters.
- Galerkin Method: The most widely used approach in FEM, where the weighting functions are chosen to be the same as the shape functions used to approximate the solution, setting the integral of the weighted residual to zero.
Write down the polynomial function used to define the displacement of beam element
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Write down the stiffness matrix used for a truss element.
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What is the degree of polynomial used for displacement in CST element?
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Express the shape functions of a CST element using area coordinates.
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Write down the two point formula used in Gaussian quadrature.
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Differentiate between sub parametric and iso parametric elements with suitable examples.
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Write down the element matrix associated to convective heat transfer of an 1D element.
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Write down the consistent mass and lumped mass matrices for a bar element. Which one of them is more accurate in analysis?
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PART - B
(50 Marks)
For a tapered plate of uniform thickness t=10 mm as shown in figure 1. Find the displacement at the nodes by forming into two element model. The bar has mass density ρ = 7300kg/m3 Young's modulus of E =2×105 N/mm2. In addition to self-weight, the plate is subjected to a point load P=10 kN at its centre. Also determine the reaction force at the support.
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Derive the quadratic shape functions for a bar element with two end nodes and one midpoint node.
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Discuss the properties of the above shape functions. Also plot all the shape functions along the length of the elements.
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Determine the deflection at the nodes for the truss shown in Figure 2. Also find the stress and forces in each member and state its nature. Also find the reactions at the supports. Take E = 70GPa; Cross Section area for all 500mm2.
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Determine the rotations at the node B and C for the beam loaded as shown in Figure 3. Take E= 210 GPa and I = 6×106 mm4.
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A CST element with nodes i, j, and k located at (200, 200), (400, 200) and (400, 400), respectively in the standard x-y plane is one of the many triangular elements used to find stresses in a plate subjected to in-plane (x-y plane) forces. The displacements in the x-y directions of the nodes i, j, and k are found to be (-0.01, 0.1), (-0.02, 0.1) and (-0.02, 0.2), respectively. The co-ordinates and displacements are given in mm. Determine the strain components εxx, εyy and γxy.
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Evaluate the integral I = ∫ dx / (1 + √tan x) from π/6 to π/3 using Gaussian two point formula.
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Determine the element stresses for the axis symmetric element shown in Figure 4. The nodal displacements are [r1; r2; r3; z1; z2; z3]T = [0, -0.2, 0.6, 0, -0.1, 0.8]T mm. Take E=210GPa, ν= 0.25. The unit for coordinates is mm.
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For the rectangular element, assuming plane stress conditions, Evaluate matrices [J] and [B] at the point ξ=0 and η=0 if E=210GPa, ν= 0.25. the nodal displacement is [x1, x2, x3, x4, y1, y2, y3, y4]T = [0, 0.0508, 0.1524, 0, 0, 0.0762, 0.0813, 0]T mm.
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A composite wall consists of three materials as shown in Figure 5. the right side wall temperature is 20oC. Convection heat transfer takes place on the left side of the wall surface with T∞ = 800oC and h = 25W/m2C. Determine temperature distribution in the wall. Use three finite elements.
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Determine the natural frequencies and mode shape of axial vibration of a stepped bar shown in Figure 6. Take E=210GPa and density ρ = 7800kg/m3.
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