JNTU
JNTU B.Tech III FINITE ELEMENT METHODS Previous Question Paper (July - 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)
List out the different types of boundary conditions. Give examples.
Answer
The two primary types of boundary conditions in finite element analysis are:
- Essential (Dirichlet) Boundary Conditions: These involve the direct specification of the primary variable (e.g., displacement, temperature, or potential) at the boundary. An example is a fixed support where displacement is set to zero (u = 0).
- Natural (Neumann) Boundary Conditions: These involve the specification of the derivative of the primary variable, typically representing a flux or force. An example is an applied point load or a specified heat flux on a surface.
State the properties of stiffness matrix.
Answer
The stiffness matrix [K] in finite element analysis possesses the following fundamental properties:
- Symmetry: The stiffness matrix is always symmetric, meaning Kij = Kji. This property arises from the application of Betti's reciprocal theorem or the principle of minimum potential energy.
- Positive Definiteness: For a stable physical system, the stiffness matrix is positive definite, ensuring that the strain energy stored in the element is always positive for any non-zero displacement vector.
- Singularity: The stiffness matrix of an individual element is singular (its determinant is zero) because it represents a body that can undergo rigid body motion without internal strain. It only becomes non-singular after the application of appropriate boundary conditions to constrain these rigid body modes.
Write the element stiffness matrix for truss element.
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Compare beam and truss element.
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Define CST.
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Write the element stiffness matrix for four noded isoparametric element.
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What is convection matrix?
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What do you understand by a torsion problem?
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Define lumped mass matrix.
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Discuss the mesh generation techniques.
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Part B
(50 Marks)
Explain about the Numbering scheme used in Finite Element Method.
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Derive the shape functions of 1-D bar element.
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A load P=60x103N is applied on a bar. Determine: (a) Displacement field and (b) Stresses. Take E= 20GPa, Gap=1.2mm, A=250mm2.
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Estimate the nodal displacements in each member of the truss. Take P= 50kN and E=200GPa.
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A beam of 4 m length is subjected to point loads at the distances of 2 m and 4 m from the fixed end of 10 kN and 20 kN respectively. Calculate the deflection at the center of the beam, if modulus of elasticity of beam material E = 2 x 105 MPa and A = 400 mm2.
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The coordinates of the nodes 1, 2 and 3 of a triangular element are (1, 1), (8, 4) and (2, 7) in mm. The displacements at the nodes are q1 = 1 mm, q2 = 3 mm, q3= -2 mm, q4 = -4 mm, q5 = 2 mm and q6 = 5 mm. Determine (i) Strain-Displacement matrix [B] (ii) Strains εx, εy and shear strain γxy.
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Check what order of the Gauss quadrature could exactly integrate the following: Φ = (2+ 3x+5x3+ 8 x6)
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A metallic fin with thermal conductivity k=360 W /m °C, 0.001 m thick and 0.1m long, extends from a plane wall whose temperature is 235°C. Determine the temperature distribution. Take air at 20°C with h= 9 W/m2°C. Take the width of fin to be 1 m.
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Heat is generated in a large plate (thermal conductivity k = 0.8 W/m°C) at the rate of 4000 W/m3. The plate is 0.25 m thick. The outside surfaces of the plate are exposed to ambient air at 30 °C with a convective heat transfer coefficient of 20 W/m2°C. Determine the temperature distribution in the wall.
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Evaluate the eigen values, eigen vectors and natural frequencies of a beam of cross section 360 cm2 and of length 600mm. Assume Young’s modulus as 200GPa, density 7850 kg/m3 and moment of inertia of 3000mm4. Make into two elements of 300 mm length each.
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Discuss the various FEM software's.
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What are the convergence requirements?
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