JNTU
JNTU B.Tech III FINITE ELEMENT METHODS Previous Question Paper (February/March - 2022 - R18)
Answer any five questions
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Main
Explain the step by step procedure for formulating the equations for finite element methods.
Answer
The formulation of finite element equations follows a systematic procedure to transform a continuous physical problem into a discrete algebraic system:
- Discretization: The domain is divided into a finite number of elements connected at nodes.
- Selection of Displacement Model: An interpolation function (shape function) is chosen to represent the variation of the field variable within each element.
- Derivation of Element Stiffness Matrix and Load Vector: Using the principle of minimum potential energy or the Galerkin method, the element stiffness matrix [k] and force vector {f} are derived based on the chosen shape functions and material properties.
- Assembly of Global Equations: Individual element matrices are assembled into a global stiffness matrix [K] and global load vector {F} by enforcing continuity and boundary conditions at the nodes.
- Application of Boundary Conditions: Known displacements or forces are applied to the global system to solve for unknown nodal values.
- Solution of Equations: The resulting system of linear algebraic equations [K]{u} = {F} is solved using techniques like Gaussian elimination.
- Calculation of Derived Quantities: Strains and stresses are computed from the nodal displacements using the strain-displacement and constitutive relations.
Draw the Pascal’s triangle and Pascal’s tetrahedron for the formulation of interpolations functions. Explain the salient features.
Answer
Pascal’s triangle and tetrahedron are graphical representations used to select terms for polynomial interpolation functions to ensure geometric isotropy and completeness.
Pascal's Triangle (2D): The rows represent the degree of the polynomial. For a 2D element, terms are selected as 1, x, y, x², xy, y², x³, x²y, xy², y³... This ensures that the polynomial is complete and invariant under coordinate rotation.
Pascal's Tetrahedron (3D): This is a 3D extension where terms are arranged in layers representing the polynomial degree. The terms follow the sequence 1, x, y, z, x², xy, xz, y², yz, z²... which provides the necessary basis functions for 3D elements.
Salient Features:
- Completeness: The inclusion of all lower-order terms ensures that the element can represent constant strain states and rigid body motion.
- Geometric Isotropy: Selecting terms symmetrically from the triangle or tetrahedron ensures the element behavior is independent of the local coordinate system orientation.
- Convergence: Using complete polynomials satisfies the compatibility and completeness requirements necessary for the convergence of the finite element solution as the mesh is refined.
What do you understand by the assembly of 1 D bar elements and formulate global stiffness matrix and global load vector.
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Calculate the nodal displacements and forces for the stepped bar with the stiffness values are 12 kN/m and 8 kN/m and a load of 6 kN is subjected at the end of the stepped bar and other end of the bar is fixed.
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Derive the stiffness matrix and load vector for two noded 2 DOF truss element and explain the importance.
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Calculate the deflection at the center and slopes at the ends of a fixed beam of 2 m length subjected to a UDL of 50 kN/m throughout the length and a point load of 50 kN at the centre. Take EI = 700 × 105 N-m2.
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The coordinates of the plane truss element is given as 1(0,0) and 2(20,35) mm has the displacement values {-0.03 -0.01 -0.03 0.02}T mm with the material had Young’s modulus of 200 GPa. Calculate the stiffness matrix, load vector and strain energy if the cross sectional area of the truss is 100 mm2.
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Derive the load vector for two noded beam element subjected to a uniformly distributed load.
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Calculate the strain displacement matrix for element with the coordinates 1(4,5), 2(9,2) and 3(6,8) mm. And also calculate load vector, stresses, strains and strain energy for (i) plane stress and (ii) plane strain of triangle whose nodal displacement values are u1=0.3 mm, v1 = 0.3 mm, u2 = 0.2 mm, v2 = -0.4 mm, u3 = 0.3 mm, v3 = 0.5 mm. Take E = 200 GPa, poisons ratio = 0.3 and thickness=2 mm.
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Derive the stiffness matrix for three noded axisymmetric triangular element from the first principles.
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How to make use of Gaussian quadrature method for solving two dimensional integral equations with a suitable example?
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Explain the finite element formulation for four noded quadrilateral element using isporametric condition.
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A composite slab consists of three materials having different thermal conductivities i.e 100 W/m K, 125 W/m K , 75 W/m K of thickness 0.35 m, 0.25 m and 0.5 m respectively. The outer surface is 25oC and the inner surface is exposed to the convective heat transfer coefficient of 90 W/m2 K, 850oC. Determine the temperature distribution in the wall.
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Derive the conductivity matrix for the fin element based on convective boundary condition over the entire surface.
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Derive the equilibrium equations by considering the mechanical vibration varying by following sine curve and discuss the salient points.
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What are the advantages and limitations of ANSYS commercial software over other available finite element software? Explain.
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