Question Paper Code : A1327
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(AUTONOMOUS) Affiliated to JNTUH, Hyderabad B. Tech VI Semester Tutorial Question Bank 2013 - 14 (Regulations: VCE-R11) FINITE ELEMENT METHODS (Mechanical Engineering) PART-A (2 Mark Questions) Unit-I 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Define the term finite element. What is meant by discretization of domain? What is meant by stiffness matrix? Define Bar element. Give the advantage and limitation of Ritz vectors. What are the properties of stiffness matrices? What are compatible elements? What is the significance of node numbering? What is the difference between plane stress and plane strain problems? Explain in detail steps involved in analysis of a typical problem using FEM. With Neat sketch derive the Transformation matrix between Element Local Displacements to Global Displacements. Write down the Constitutive Relationship for Plane Stress Case. Define Singular matrix. What is linear static analysis? Define Stiffness Matrix. Define plane strain analysis. Differentiate between FEM and FEA. List any four popular software FEA packages. Write the stress-strain relations for the plane strain case in matrix form. Write material matrix (D-Matrix) for 3-D problem. Define body force and surface force with an example for each. Give an expression for the total potential energy functional for 3-Dimensional elastic body subjected to body forces, surface forces and point loads. Give the relationship between natural co-ordinates and Cartesian co-ordinate system in case of 1-D quadratic element. What are the properties of shape functions? Define node, primary node, secondary node and internal node. The shape function assumes _______ value at one node and __________ at the other node. Unit-II What is Truss? What is local and global coordinate system? Write down the expression of 2D element stiffness matrix for a truss element. Write down the expression of 3D element stiffness matrix for a truss element. Mention the degrees of freedom for a Beam element? Write down the expression for the potential energy of the beam. Explain the Hermit Shape Functions. Write down the expression of element stiffness matrix for a beam element. Write the stiffness matrix for a plane frame element. What is the basic difference between bar and beam elements? What is the basic difference between bar and beam elements? Give stiffness matrix of a simple beam element.
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What is meant by stability analysis? What is the difference between explicit and implicit solution of assembled matrix? Write down the expression of element load vector for a beam element. Write down the expression of element shear force for a beam element. Write down the expression of element bending moment for a beam element. Write down the expression of stiffness matrix for one dimensional bar element. Define total potential energy. Write down the expression of shape function N and displacement U for 1-D bar element. Express quadratic shape functions of a line element. Sketch a typical beam element showing degrees of freedom. How will you obtain the potential energy of a system? Differentiate between local co-ordinates and global co-ordinates in case of plane truss. Differentiate between plane truss and space truss element. Differentiate between C0 and C1 continuity. Write all the Hermite shape functions for 2 noded beam element. Give the expression for a stiffness matrix for a beam element. Write the transformation matrix for 1-D plane truss element. Sketch a space truss element in local and global coordinate system. Unit-III Write a strain displacement matrix for CST element. Wite down the expression for shape functions for a constant strain triangle element. What is the necessity for Natural coordinate system? What do you mean by C1 continuity? What do you mean by axisymmetric problem? How is a quadratic triangular element different from linear triangular element? List any four commonly used axisymmetric elements. When is an element called as isoparametric element? Write down the stiffness matrix for two dimensional constant strain triangle elements. What is the difference between the strong and weak forms of system equations? Explain the important properties of CST element. Write down the stiffness matrix for two dimensional CST elements. Write down the displacement function equation for CST element. What are serendipity elements? Define the sub-parametric, iso-parametric and super parametric elements. Write down the stiffness matrix equation for four noded iso-parametric quadrilateral elements. What are shape functions and what are their properties? What are the conditions for a problem to be axisymmetric? Give the strain displacement matrix equation for an axisymmetric triangular element. Write short notes on Axisymmetric problems. What are the convergence requirements for the finite element solutions? Define CST, LST, QST elements. Write the shape functions for 2-D CST element. Define iso parametric, sub parametric and super parametric elements. Differentiate between h-convergence and p-convergence. What is meant by axisymmetric solid? Write down the expression for shape functions for axisymmetric triangular element. State the conditions to be satisfied in order to use axisymmetric elements. Sketch ring shaped axisymmetric solid formed by a triangular and quadrilateral element. The sum of all weights of Gauss points in Gaussian quadrature technique must be ____. Explain why? Unit-IV Sketch a linear and Quadratic hexahedral element. State the purpose of interpolation models. Sketch a cubic three –dimensional element. Define the term “Static condensation”. Explain the analogies between structural, heat transfer and fluid mechanics. Why is a convergence criterion very important in finite element method?
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What is the characteristic matrix of 1D potential flow element? What are the two objectives of thermal analysis? What is mode of heat transfer? Write down the finite element equation for one dimensional heat conduction with free end convection. State the conditions to be satisfied in order to use axisymmetric elements. Write down the expression for stiffness matrix for an axisymmetric triangular element. Write down the general Helmholtz equation. Write down the expression for governing equation of one dimensional heat conduction. Write down the expression for governing differential equation of one dimensional heat transfer in thin fins. Write down the expression for governing differential equation of two dimensional heat transfer in thin fins. Write down the expression for one point Gaussian quadrature method. 1 x Using two point Gaussian quadrature formula solve
e dx.
1 Write down the expression for one point Gaussian quadrature method. Write down the expression for two point Gaussian quadrature method. With a sketch, write the boundary conditions for one dimensional heat transfer in thin fins. Give the expressions for element matrix and heat rate vector for a thin fin. Write a note on application of FEM in solving scalar field problems. List the method of describing the motion of fluid. Write the finite element equation used to analyse two dimensional heat transfer problem. Write the finite element equation used to analyse two dimensional torsionproblems. Give an expression for B matrix to solve torsion problem. List the different types of elements used to analyse a uniform shaft subjected to torsion. Differentiate between 1-D analysis of a fin and 2-D analysis of a thin plate. Write the boundary conditions for torsion problems. Unit-V Define the Hamilton’s principle. Differentiate consistent and lumped mass matrices. Write down the expression for element mass matrix for one dimensional bar element. Write down the expression for element mass matrix for truss element. Write down the expression for element mass matrix for constant strain triangular element. Write down the expression for element mass matrix for axisymmetric triangular element. Write down the expression for element mass matrix for quadrilateral element. What are the advantages of lumped matrix over consistent matrix? What is meant by a mode in dynamic analysis? Differentiate skeletal and continuum structures. What are the advantages of lumped matrix over consistent matrix? Explain the convergence criterion used in FEM. Explain P – refinement & H-refinement in FEM Name any four commercial Finite element packages. Mention the advantages of Finite Element Method. Finite element method is an approximate method (true/false). Also justify your answer. Mention any four FEA popular software packages. Write down applications of FEM. State any two non-linear problems in Finite Element Analysis. Write the common governing differential equation for a field problem. List the type of dynamic analysis problems. What are the types of eigen value problems. Give the expression for element mass matrix for 1-D bar element. Give the expression for element mass matrix for truss element. Differentiate between lumped and consistent mass matrices. Give the equation for characteristic polynomial. Differentiate between bar and a beam element.
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Write the boundary conditions for eigen value problems. Differentiate between eigen value and eigen vectors. Mention any two method of extracting eigen value and eigen vectors. PART-B (5 Mark Questions) Unit-I A bar under axial load is shown below, determine the de°ection at load applied by discretizing the bar into two elements, compare the solution with theoretical solution and comment. E =210GPa L=30cm,Load P =10000N diameter=2cm
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Determine the nodal displacements, elements stresses for the stepped bar shown in the figure below.
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Derive *D+ matrix for the Plane Stress problem using generalized Hooke’s Law relations.
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Find the deflection at the free end of the tapered plate shown in the figure below. Calculate the element stresses.
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Derive *D+ matrix for plane strain problem using Hooke’s Law relations. Determine the nodal displacements and element stresses for the stepped bar shown in the figure below.
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Explain i) Elimination approach ii) Penalty approach for the treatment of boundary conditions in an elasticity problem and enumerate the relative advantages of each method.
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With a suitable example explain the formulation of finite element equations by direct approach. Assume suitable data for your example. With a help of a neat block diagram, explain the model based simulation process of finite element method. Find displacement of the midpoint of the rod as shown. E=1,A=1,Body force per unit volume g=1,Length = 2
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Explain different approaches of getting the finite element equations and Explain the natural and geometric boundary conditions. A long rod is subjected to loading and temperature increase of 300C. The total strain n at a point is measured to be 1.2× 10−5 If E=200GPa and α=12 × 10−6/0C. Determine stress at the point. Derive stresses and strains relations and Derive equivalent nodal force vectors. For the beam and load shown in figure 1 determine slope at 2 and 3 and at the mid point of the distributed load. E=200GPa,I=5 × 106mm4.
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Write notes on the following: Weighted Residual method. Initial and Boundary value problems. Type your Question here Define and derive the Hermite shape functions for a two nodded beam element? Determine the circumference of a circle of radius ‘r’ using the basic principles of finite element method. Derive stiffness equations for a bar element from the one dimensional second order equation by variated approach. Find the deflections and support reactions for the beam shown in figure. Take E = 200 GPa.
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An axail load P = 300 x 103 N is applied to the rod as shown in figure obtain
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(a) The assembled stiffness matrix and force vector. 21. 22.
(b) The nodal displacements and element stiffness. List and explain the steps in solving an FEM problem. Differentiate between plane stress and plane strain problems. Also state the stress-strain relations for both.
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What are the advantages of FEM over other methods? Derive strain-displacement matrix [B] and element stiffness matrix [K] for 1-D linear element. Derive the expressions for a nodal load vector in a two noded bar element due to a) Body force b) Surface force. List all the properties of global stiffness matrix. Derive and sketch quadratic shape functions for 1-D bar element. For the bar shown in the fig.1, determine the nodal displacements and the reactionforces using -4 2 direct stiffness method. Take E = 70 GPa; A = 2 X 10 m ; K = 2000 KN/m. P = 8 KN 2m
K
2m Fig. 1
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The displacement field for a 2 noded 1-D bar element is given by U = N1q1+ N2q2 where q1 and q2 are nodal displacements and N1and N2are shape functions. Evaluate (1) ξ, N1and N2 at P (x = 750 mm) given x1 = 300 mm and x2= 1200 mm. (2) ifq1 = 0.09 mm andq2 = 0.15 mm, determine the value of displacement at point P. Obtain displacement at node 2, reactions, stresses and strains in the circular solid stepped bar as shown in Fig.2. Take E1 = 70 GPa, E2 = 200 GPa for the element (1)and (2)
ф0.01m
ф0.02m
0.2 m
0.3 m Fig.2
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Unit-II Find the defection at the load and the slopes at the ends for the steel shaft shown in figure consider the shaft to be simply supported at the bearings A and B I1=1.25x105 mm4 I2=4x104mm4.
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Find the nodal displacements and element stresses in a plane truss shown in the figure below.
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Find the vertical deflection and slope at the nodes for the beam shown in the figure below.
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Find the displacements at the nodes and element stresses in a plane truss shown in the figure given below.
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Find the vertical deflection and slope at the nodes for the beam shown in the figure below.
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E = 100 Gpa I = 6 106 mm4 Find the nodal displacements and element stresses in a plane truss shown in the figure below.
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Find the slopes at all nodes for the beam shown in the figure below.
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Find the nodal displacements and element stresses in a plane truss shown in the figure below.
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Find the vertical deflection and slope at all nodes of the beam shown in the figure below. E = 200GPa
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I = 4106mm4.
For the truss structure shown in figure with indicated load, calculate the stress in each element.
Consider the truss element with the coordinates 1(10, 10) and 2(50, 40). If the displacement vector is q = [15 10 21 43 ]T mm, then determine the vector q, stress in the element and stiffness matrix if E=70GPa and A=200mm2 Compute the support reaction at the other end of continuous beam shown in figure 1. use the concept of boundary element EI=400 units.
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The (x, y) coordinates of the nodes i, j and k of a triangular element are (1, 1), (4, 2) and (3, 5) respectively. The shape functions of a point P located inside the element are given by N1 = 0.15 and N2 = 0.25. Determine the x and y coordinates of the point P. For the pin jointed truss shown in figure1. If E = 200 Gpa, determine (a)Element stiffness matrices . (b) Global stiffness matrix . (c) Stress in the element 1. (d) Strain in the element 2.
The (x, y) coordinates of the nodes i, j and k of a triangular element are (1, 1), (3, 2) and (3, 5) respectively. The shape functions of a point P located inside the element are given by N1 = 0.12 and N2 = 0.25. Determine the x and y coordinates of the point P. Estimate the displacement vector, stresses and reactions for the truss structure as shown below Figure
Note: - Area is not given and assumed as A(e) = 1mm2 ‘E’ is not given. Assumed as E=2×105 N/mm2 Derive the element stiffness matrix for the plane truss element. A beam of length 2 m is fixed at both ends. Estimate the deflection at the center of the beam where load is acting vertically downward of 10 kN. Divide the beam into two elements. Compare the solution with theoretical calculations. Take E = 2 × 1011 N/m2, A = 250 mm2. Consider the truss element with the coordinates i(10,10) & q(50,40) If the displacement vector is q=[15 10 21 43]T mm, then determine (a) The trace vector F (b) Stress in each element (c) Stiffness matrix if E= 70 GPA and A= 200 mm2. An elastic bar is having a uniform cross sectional of area `A' mm2 and length `L' mm. It is _xed at one end and other end is allowed to move along the axis of the elastic bar. A force `F' KN is acting at the free end and the Youngs Modulus is `E'
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N/mm2. Calculate the displacement at the free end. Calculate the nodal displacements, element stresses and reaction forces for the truss shown infig.3.Take E =70 GPa. Cross sectional area A = 200 mm2 for all the truss members. Y
800 mm
X 600 mm
300 P = 50 KN
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Fig. 3 Derive element stiffness matrix for a truss element in global co-ordinate system. Derive and sketch Hermite shape functions for a beam element. A uniform cross-sectional beam of length 1 m is fixed at one end and supported by a roller at the other end carries a concentrated load of 20000 N at the mid point of the beam. calculate the deflection under the load. Also find the shear force, bending moment and their directions. Given E = 200 GPa; I = 25X10-6 m4. Derive element stiffness matrix and load vector for a beam element. Determine the deflection and slope under uniformly distributed load and point load for the beam as shown in Fig.4. Also state the number of elements used, number of degrees of freedom per node and the size of the global stiffness matrix K. Given E = 200 GPa; I1 = 4X10-6 m4; I2 = 2X10-6 m4. Y
24 KN/m
P = 50 KN
X
2m
1m
1m
I1 Fig.4 Roller support I2 27. 28.
Sketch a two-dimensional truss element in a) a local co-ordinate system and b) a global co-ordinate system. For the two member truss shown in Figure 5. Determine nodal displacements and stresses in each member. 2
Take E = 70 GPa,
1X10-4 m2
1X10-4 m2
1
3 1m Fig.5.
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Sketch a three-dimensional space truss element in a) a local co-ordinate system and b) a global coordinate system. Derive element stiffness matrix for a space truss element in global co-ordinate system. Unit-III Triangular elements are used for stress analysis of a plate subjected to in plane load. The components of displacement parallel to (x, y) axes at the nodes i, j and k of an element are found to
be (- 0.001, 0.01), (-0.002, 0.01) and (- 0.002, 0.02) cm respectively. If the (x, y) coordinates of the nodes I, j and j are (20, 20), (40, 20) and (40, 40) in cm respectively, find (a) the distribution of the (x, y) displacement components inside the element and (b) the components of displacement of the point (xp , yp ) = (30, 25) cm. 2.
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The nodal coordinates and the nodal displacements of a triangular element, under a specific load condition are given below: X i = 0, Y i = 0, X j = 1 mm , Y j = 3 mm, X k = 4 mm, Y k = 1, u1 = 1 mm, u2 = 0.05 mm, u3 = 2 mm, v1 = 0.5 mm , v2 = 1.5 mm and v3 = -1 mm. If E = 2 X 105 N / mm2 and μ = 0.3. Find the stresses in the element. Derive strain displacement [B] matrix for a 3 noded Triangular element? The nodal coordinates of the triangular element are shown in figure. At the interior Point P
The X coordinate is 3.3 and the shape function at node 1 is N1 is 0.3. Determine the shape functions at nodes 2 and 3 and also the y coordinate of the point P. 5.
Calculate the element stiffness matrix [k(e)], element stress vector {(e)} and element force vector {F(e)} for a plane stress element shown in the figure given below.
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The nodal coordinates for an axisymmetrical element are given below. Evaluate [ B ] matrix for the element. The coordinates are in centimeters. Ri = 10.0 Rj = 14.0 Rk =14.0 Zi = 2.0 Zj = 2.0 Zk = 4.0 A body force of 20 N/cm3 acts on the above element in the negative R direction. Evaluate body force vector.
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A fin of size 2.5 cm X 25 cm X 250 cm extends from a wall. If the wall temperature is maintained at 9000C and the ambient temperature is 300C, determine the temperature distribution in the fin 1-D elements along x-direction. Take k = 35 W/mK and h = 120 W/m2K.
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List some disadvantages of using 3-D isoparametric elements. Derive the strain displacement matrix for a tetrahedron element.
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What do you mean by a aximetric problem? If the displacement functions u = 3x2 + 60 xy - 20 y2 = -4 x2 - 20xy + 5 y2 then find the strains Ex, Ey and
xy at a point P (-1, 1).
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Heat generated in a large plate (0.8 W/m K) at the rate of 4000 W/m3. The plate is of thickness 25 cm. The outer surface is exposed to an ambient air with a heat transfer coefficient of 20 W/m2K at 300C. If the inside surface temperature is 5000C, calculate the temperature at a distance of 10 cm from the inner wall. Assume cross sectional area is 62.5 mm2. The nodal coordinates of the triangular element are 1 (1,1) , 2 (4,2) , 3 (3,5). at the interior point P, the x coordinate is 3.3 and N1 is 0.3 . Determine N2, N3 and y coordinate at point P. Find the deflection at the load of the steel shaft as shown in figure 1: take E = 200 Gpa
Consider the truss element with the coordinates i(10,10) & q(50,40) If the displacement vector is q=[15 10 21 43]T mm, then determine (a) The trace vector F (b) Stress in each element (c) Stiffness matrix if E= 70 GPA and A= 200 mm2. Consider a beam with uniform distributed load as shown in the figure4. Estimate the deflection at the centre of the beam. E = 200 Gpa ; A = 25 mm ×0 25 mm.
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The nodal coordinates of the triangular element are shown in figure5. At the interior Point P the x coordinate is 3.3 and the shape function at node 1 is N1 is 0.3. Determine the shape functions at nodes 2 and 3 and also the y coordinate of the point P
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An axisymmetric ring element is shown in _gure 2. Derive the matrices, [B] and [D]. Take E = 2_105 N/mm2and _ = 0.33.
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Explain the convergence criteria in _nite element analysis. Write about pre-processor, processor, and post-processor in any FEM software. Consider the truss element with the coordinates i(10,10) & q(50,40) If the displacement vector is q=[15 10 21 43]T mm, then determine (a) The trace vector F (b) Stress in each element (c) Sti_ness matrix if E= 70 GPA and A= 200 mm2. Using the general approach of displacement function, derive the force-displacement relationship and element sti_ness matrix. for a truss bar element. For the two bar truss shown in figure, determine the nodal displacement, element stresses and support reactions.
Show that the element strain-displacement matrix is constant for a 3 noded CST element. Derive element stiffness matrix and load vector for a CST element. 1. The nodal coordinates of a triangular element are shown in fig.1.At the interior point P, the x-coordinate is 3.3 and N1 = 0.3. Determine N2, N3 and the y-coordinate at point P.
y3(4,6) .P
2(5,3) 1(1,2) 24.
Derive weights and gauss points for 2-point methodof Gauss quadrature and evaluate using 2 point guass quadrature formula and check the same with the
hence exact
result. 25. 26.
What are the advantages of using the numerical integration? Explain one-point formula.
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Using isoparametric representation, derive shape functions for Axisymmetric triangular element. Derive shape functions for four nodedisoparametricelement. Obtain all the shape functions for a nine-noded quadrilateral element. Differentiate between Lagranjian and serendipity family of elements. Unit-IV Explain the procedure for solving transient heat conduction problem using Finite Element Analysis. Determine the nodal temperatures in the multi layered wall shown in the figure below. Using one
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Evaluate using 3 point guass quadrature formula.
dimensional steady state heat transfer analysis.
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k1= 4 w/cm-0C k2 = w/cm-0C. A metallic fin, with thermal conductivity of 360 W/m K, 0.1 cm thick and 10 cm long extends from a plane wall whose temperature is 2350C. Determine the temperature distribution and amount of heat transfer from the fine the air at 200C with a heat transfer coefficient of 9 W/m2 K. Take width of the fin is 1 m? A Uniform steel fin of length 25 cm, with a rectangular section 5 cm X 2.5 cm. If the heat transfer takes place by convection from all the sides while the root of the fin is maintained at 3000C, determine the temperature distribution in the fin. Assume k = 25 W/m K, h = 250 W/m2 K and T = 250C. Estimate the temperature distribution in 1-D fin as shown in figure by making into two
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elements? Explain the methodology for the treatment of all three boundary conditions in a
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1-D heat transfer element? 7.
A composite slab consists of 3 materials of different conductivities i.e 20W/m K, 30 W/m K , 50 W/m K of thickness 0.3 m, 0.15 m and 0.15 m respectively. The outer surface is 200C and the inner surface is exposed to the convective heat transfer coefficient of 25W/m2 K, 8000C. Determine the temperature distribution with in the wall.
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A metallic fin with thermal conductivity K=360W/moC, 0.1 cm thick and 10 cm long extends from a plane wall whose temperature is 235oC. Determine the temperature distribution and the amount of heat transferred from the fin to the air at 20oC with h = 9W/m2 oC. Take the width of fin to be 2 m. A composite wall consists of three materials. The outer temperature is T=20oC. Convection heat transfer takes place on the inner surface of the wall with T =800oC and h = 25W/m2 oC .Determine the temperature distribution in the wall. K1=20W/moC , K2=30W/moC ,K3=50W/moC ,L1=0.3m ,L2=0.15m ,L3=0.15m . A long bar of rectangular c/s, having the thermal conductivity of 1.5 w/moC is subjected to the boundary conditions shown in figure. Determine the temperature distribution in the bar.
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Heat is generated in a large plate (K=0.8 W/m0C) at the rate 4000 W/m3. The plate is 25 cm thick. The outside surfaces of the plate are exposed to ambient air at 300C with a convective heat transfer coefficient of 20W/m2 0C. Determine the temperature distribution in the wall. Shown in fugure
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Calculate the temperature distribution and the heat dissipating capacity of a fin shown in figure 2. The thermal conductivity of the material is 200W/m k. The surface heat transfer coefficient is 0.5 W/m2k. The ambient temperature is 30oC. The thickness of the fin is 1 cm the width at the base is 2 cm and at the tip is 1 cm and varies uniformly along the length, what changes takes place in the heat dissipation capacity of the fin. Derive the element conductivity matrix and load vector for solving 1- D heat conduction problems, if one of the surfaces is exposed to a heat transfer coefficient of h and ambient temperature of T∞? Estimate the temperature distribution in 1-D fin as shown in figure6 by making into two elements?
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One side of the brick wall of width 5 m, height 4 m and thickness 0.5 m is exposed to a temperature of - 250C while the other surface is maintained at 320C. If the thermal conductivity is 0.75 W/m K and the heat transfer coefficient on the colder side is 50 W/m2 K. Determine (a) The temperature distribution in the wall and (b) Heat loss from the wall. What are the basic steps involved in _nite element analysis and explain them briey. For the truss structure shown in _gure 8 with indicated load, calculate displacenments and stress in each element.
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For the three stepped bar shown in fig. 2, the fits snugly between the rigid walls at room temperature. The temperature is then raised by 300C. Determine the displacements at nodes 2 and 3, stresses in the three sections.
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a. Explain in detail the applications of isoparametric elements in two and three dimensional stress analysis.
b. Using Gaussian quadrature evaluate the following integral 20.
A long bar of rectangular cross-section, having thermal conductivity of 1.5W/moC is subjected to the boundary conditions as shown in figure. Determine the temperature distribution in the bar.
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Write the governing differential equations for one dimensional heat transfer and discuss the various types of boundary conditions used in solving heat transfer problems. Derive element conductivity matrix for one dimensional heat flow element. Comment on the forced and natural boundary conditions for heat transfer problems. Derive element matrices and heat rate vectors for heat transfer through one dimensional thin films. Explain boundary conditions for 2-D heat conduction using a sketch. A metallic fin, with thermal conductivity k =360W/m.0C, 0.1 cm thick and 10 cm long, extends from a
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plane wall whose temperature is 2350C. determine the temperature distribution and amount of heat transferred from the fin to the air at 200C with h = 9 W/m2.0C. Take the width of fin to be 1 m. Derive the governing differential equation for 2-D steady state heat conduction. Derive B matrix for torsion problem using triangular element. Using Galerkin approach, derive the stiffness matrix for torsion problems. Discuss scalar field problems with suitable examples. Unit-V
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Explain briefly about Eigen value problem. Determine the natural frequencies and mode shapes for the rod shown in fig. using the characteristic polynomial technique. Assume E = 200 GPa and mass density7850.kg/m3. L1=L2=0.3m,A1=400mm2,A2=250mm2
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Explain Finite Element formulation for obtaining eigen values of a stepped bar in axial vibrations. Distinguish between consistent mass matrix and Lumped mass matrix. Explain Gaussian Quadrative method.
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Discuss the methodology to solve the Eigen value problem for the estimation of natural frequencies of a stepped bar?
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Consider the axial vibrations of a steel bar shown in the figure:
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Develop global stiffness and mass matrices,
b)
Determine the natural frequencies?
Derive the elemental mass matrix for two noded beam element and 2-noded frame element? Evaluate the eigen values, eigen vectors and natural frequencies of a beam of cross section 360 cm2 of length 600 mm. Assume young’s modulus as 200 Gpa, density 7850 kg/m3 and moment of Inertia of 3000 mm4. Make into two elements of 300 mm length each Derive the elemental mass matrix for 1-D bar element and 1-D plane truss element? Determine the natural frequencies and mode shapes of a stepped bar as shown in figure using the characteristic polynomial technique. Assume E = 25 Gpa and density is 7850 kg/m3.
Determine the natural frequencies and corresponding mode shapes for the fig. shown below. = 7850 kg/m3, E = 2 x 1011 N/m2.
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Derive the expressions for the mass and stiffness matrices of a plate using triangular elements. Determine the Eigen values and Eigen vectors for the stepped bar shown in figure. E=30times 106 N/m2 ,specific weight = 0.283 Kg/m3 A1= 1 m2 A2= 0.5 m2 L1=10 m. L2 =5 m.
Find the natural frequencies and the corresponding mode shapes for the longitudinal vibrations for the stepped bar. Assume A1 = 2A and A2 = A ; I1 = I2 = I ; E1 = E2 = E. Determine the natural frequencies of a simply supported beam of length 800 mm with the cross sectional area of 75 cm X 25 cm as shown in the figure2. Take E= 200 Gpa and density of 7850 kg/m3.
Discuss the methodology to solve the Eigen value problem for the estimation of natural frequencies of a stepped bar? Derive the elemental jumped and consistant mass matrices for 1-D bar element and 1-D plane truss element? Evaluate the eigen values, eigen vectors and natural frequencies of a beam of cross section 360 cm2 of length 600 mm. Assume young’s modulus as 200 Gpa, density 7850 kg/m3 and moment of Inertia of 3000 mm4. Make into two elements of 300 mm length each. Consider the axial vibrations of a steel bar shown in the _gure 1: (a) Develop global sti_ness and mass matrices, (b) Determine the natural frequencies?
Evaluate the eigen values, eigen vectors and natural frequencies of a beam of cross section 350 cm2 of length 700 mm. Assume young’s modulus as 200 Gpa, density 7850 kg/m3 and moment of Inertia of 3000 mm4. Make into two elements of 300 mm length each. Obtain the eigen values and eigen vectors for the cantilever beam of length 2m using consistant mass for translation dof with E = 200GPa, ρ = 7500kg/m3. Explain the formulation of finite element model for dynamic analysis.
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Derive element mass matrix for 1-D bar element. Derive element mass matrix for 1-D truss element. Derive element mass matrix for 2-D CST element. Derive element mass matrix for 1-D beam element. Explain the different methods to extract eigen values and eigen vectors. List all the properties of eigen vectors. Derive element stiffness matrixmatrix for 1-D bar element. Compute the eigen value and vectors for the cantilever beam using 1-D element. Take E = 200GPa, breadth, b = 0.01m, height, h = 0.02m, length, l = 1m and density ρ = 7860Kg/m3. Explain consistent mass matrix and lumped mass matrix with examples.
