1to6 FEM Quest

7/27/2019 1to6 FEM Quest

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Total No. of Questions : 6] [Total No. of Pages : 2

[3665]- 463

M.E. (Civil / Structures)

FINITE ELEMENT METHOD

(2008 Course)

Time : 4 Hours] [Max. Marks : 100

Instructions to the candidates:

1) Answer any two questions from each section.

2) Answers to the two sections should be written in separate books.

3) Neat diagrams must be drawn wherever necessary.

4) Figures to the right indicate full marks.

5) Use of electronic pocket calculator is allowed.6) Assume suitable data, if necessary.

P1511

SECTION - I

Q1) a) Explain principle of minimum potential energy and hence evaluate

element stiffness matrix of two dimensional CST element using proper

polynomial displacement function. [15]

b) How you will improve performance of CST element? How you will

make use of higher order polynomials for more accuracy? [10]

Q2) a) What is effective node numbering in finite element mesh. Derive equation

for half band width of overall stiffness matrix of a problem. [10]

b) Explain characteristics of element stiffness matrix and overall stiffness

matrix. Why you need to consider boundary conditions? [10]

c) Write requirements of displacement function in finite element

formulation. [5]

Q3) a) Derive shape functions for quadrilateral element with corner nodes usingnatural coordinate system test the correctness of all shape functions.[12]

b) If element is to be isoparametric how you will express element geometry

functions? Using 4 nodes quadrilateral and shape functions derived in

Q 3(a), develop Jacobian matrix. [13]

SECTION - II

Q4) a) Explain Paseals pyramid and its use in selecting terms in displacement

function polynomials for 3D tetrahydron element. [8]

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b) For isoparametric hyxahydron element write shape function for any node

of element write element geometry equations using the shape functions.

[12]

c) For axisymmetric element write stress-strain relations and hence obtain

(D) matrix. [5]

Q5) Write displacement functions for both ACM and BFS elements. Explain

how interelement compatibility is improved in BFS element. Verify

conformity of both the elements. [25]

Q6) How many different types of shell elements exist in finite element technique.

Draw sketch of each element with mesh pattern. How you will differentiate

shell element in comparison with plate element? [25]

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Total No. of Questions : 6]

[Total No. of Pages : 2

[4165] - 465

M.E. (Civil - Structures)


(2008 Course) (Sem. - II)

Time :4 Hours] [Max. Marks :100





4) Neat diagrams must be drawn wherever necessary.5) Use of non programmable calculator is allowed.

6) Assume suitable data, if necessary.

P2096

SECTION - I

Q1) a) For the rigid frame shown in fig. 1 determine the displacements androtations of the nodes, the elementary forces and the reactions.

All elements have E = 210 GPa, A = 1.0 102 m2 and I = 2.0 104 m4[17]

b) Explain with examples different types of co - ordinates used in finite elementmethod to define location of points in element. Hence obtain relation fornatural co - ordinates for two noded element when range is 1 to +1. [8]

Q2) a) State and Explain Convergence Requirements of displacement function.Examine whether the given displacement field for a plane stress rectangularelement satisfy the convergence criteria. [9]

u = a0

+ a1x + a

2y + a

3xy

v = a4

+ a5x + a

6y + a

7xy

SEAT No. :

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b) Determine the shape function for Linear Strain Triangular (LST) element.

Use natural coordinate system. [8]

c) Explain variational method used for formulation of element stiffness matrixwith suitable example. [8]

Q3) a) A six noded rectangular element has 4 corner nodes and one node at thecentre of the two edges parallel to x axis. The other two edges are parallel toy axis. Obtain the six shape functions using Lagrange interpolation. [10]

b) Explain the isoparametric concept and types of isoparametric elements

in finite element analysis. Discuss their advantages over other elements.[7]

c) Explain Jacobian matrix in case of four noded isoparametric quadrilateralelement. [8]

SECTION - II

Q4) a) For axisymmetric element write stress strain relations and hence obtain

element stiffness matrix. [10]

b) Explain the method of finding shape function for a hexahedral element

using natural coordinates. [10]

c) What are the applications of axisymmetric elements? [5]

Q5) a) What do you understand by C0, C1 and C2 continuity? Explain with

suitable examples. [6]

b) Write short note on Conforming and non conforming plate bending

elements. [6]c) Explain the term Midlins C0 continuity plate element and briefly explain

stiffness matrix formulation for such elements. [13]

Q6) a) Explain the concept of degenerated solid elements by suitable examples.

Write displacement fields in 4 noded degenerated shell element. [15]

b) Explain with neat sketches the various three dimensional elements used

in the analysis of shells. How will you differentiate shell element with

plate element. [10]

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SECTION - II

Q4) a) Establish relation between strain at any point with nodal displacement in

a two noded bar element using shape functions in terms of natural

co-ordinates. [6]

b) For the CST element with nodes 1(100, 100), 2(400, 100) and 3(200,400), find the element stiffness matrix by FEA. Assume t = 20 mm and

E = 2 105 N/mm2. [9]

c) Explain the Isoparametric concept in FEM. State the basic laws on which

it is developed? For the isoparametric element shown in figure 4.1,

determine Cartesian co-ordinates of the point p which has local

coordinates = 0.57735 and = 0.57735. [10]

Q5) a) What is displacement function for ACM plate bending element? Examine

nodal as well as inter element compatibility of the element. [7]

b) Derive all matrices to formulate [K] of ACM plate bending element. [8]

c) Explain BFS plate bending element and its displacement function. [10]

Q6) a) Explain with neat sketches the various 3D elements used in analysis of

shells. What are the factors to be considered in the development of shell

elements? [10]

b) What is degenerated solid element? Explain how a 3D brick element

can be reduced to shell element. [15]

tttt

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[4065]-465

M.E. (Civil) (Structures)


(2008 Course) (501409) (Sem. - II)







5) Use of non-programmable calculator is allowed.6) Assume suitable data, if necessary.

SECTION - I

Q1) a) Explain direct method and variational method used for formulation of

element stiffness matrix with suitable examples. [8]

b) For a plane truss shown in fig.1 determine the horizontal and vertical

displacements of node 1 and the stresses in each element.

All elements have E=210 GPa and A=4.0 x 2m4-10 . [17]

P1468

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Fig.1 (a.1b)


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[4065]-465 2

Q2) a) Write stress-strain matrices in case of plane stress and plane strain

formulations. [8]

b) Develop from first principles the element stiffness matrix for one

dimensional bar element with axial displacement as degree of freedom

using [17]

i) Direct approach

ii) Displacement function in polynomial form

iii) Shape function in natural coordinates.

Q3) a) What is Lagrange Interpolation Functions? State its

characteristics.Using this function, obtain the shape functions for a three

noded onedimensional element. Hence write the shape functions for anine noded two dimensional rectangular element. [12]

b) Explain Jacobian matrix in case of four noded isoparametric quadrilateral

element.Obtain strain displacement matrix. [13]

SECTION - II

Q4) a) Explain Pascals Pyramid and its use in selecting terms in displacementfunction polynomials for 3D tetrahedron element. [10]

b) Obtain element stiffness matrix of axisymmetric ring element with a

triangular cross section using cylindrical coordinates. [15]

Q5) a) Write displacement functions for both ACM and BFS elements. Verify

conformity of both the elements. [17]

b) What is Midlins theory of plate bending? [8]

Q6) a) What is the difference between plate element and shell element. [5]

b) Explain with neat sketches the various three dimensional elements used

in the analysis of shells. [8]

c) Explain membrane and bending actions in shell elements. How these two

states of stresses are considered in formulating [K] for shell element.[12]

abab


9/12

SECTION - I

Q1) a) Explain with examples different types of co-ordinates used in finite

element method to define location of points in element. Hence obtain

relation for natural co-ordinates for two noded element when range is

1 to +1. [10]

b) Prove that the natural co-ordinates are nothing but area co-ordinatesfor CST element of 2D problem. [15]

Q2) a) Define shape function. State and explain the convergence requirements

of a polynomial shape function. Obtain and plot shape function for a

three node bar element. [10]

b) Determine shape functions for a tetrahedron element used for 3D

problems in natural co-ordinates. [15]

Q3) a) What is serendipity family element? Using this concept find shape

functions of quadratic serendipity family element. [6]

b) Derive general equation for determining the stiffness of an element

using principle of minimum potential energy. [6]

c) Discuss various points to be considered while descretizing a structure

for finite element analysis. [6]

d) Derive elemental stiffness matrix for a plane truss element using

variational approach. [7]


[3865]-465

M.E. (Civil) (Structures)


(2008 Course)


Instructions to the candidates :





5) Use of electronic pocket calculator is allowed.

6) Assume suitable data, if necessary.

P1379

P.T.O.


10/12

SECTION - II

Q4) a) Explain the isoperimetric concept and types of isoperimetric elements

in F.E.A. Discuss their advantages over other elements. [5]

b) What is Jacobian matrix? For the isoperimetric quadrilateral element

shown in figure 4.1, assemble the Jacobian matrix for Gaussian point(0.57735, 0.57735). [8]

c) What is Lagrange shape function? Write shape functions for nine node

rectangular elements with central node. [12]

Q5) a) What is displacement function for ACM plate bending element?

Examine nodal as well as inter element compatibility of the element.[7]

b) Derive all matrices to formulate [K] of ACM plate bending element.[8]

c) Explain BFS plate bending element and its displacement function.[10]

Q6) a) Explain with neat sketches the various 3D elements used in analysis ofshells. What are the factors to be considered in the development of

shell elements? [10]

b) What is degenerated solid element? Explain how a 3D brick element

can be reduced to shell element. [15]

rrrr

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[3765]-463

M.E. (Civil / Structures)

FINITE ELEMENT METHOD(2008 Course) (501409) (Sem. - II)



1) Answer any two questions from each sections.

2) Answers to the two sections should be written in separate answer books.



5) Use of electronic pocket calculator is allowed.

P1712

SECTION - I

Q1) a) Explain direct and variational approach in formulating element stiffeness

matrix in FEM. [8]

b) Using proper displacement function in polynomial form for prismatic

beam element of length l and extensibility AE, obtain inverse of

coordinate matrix and hence shape functions. [17]

Q2) a) Using shape functions of CST element in area coordinates, obtainstrain-displacement matrix using following numeric data of nodal

coordinates.

Node 1 (5,5)

Node 2 (10,5)

Node 3 (5,11) [15]

b) Explain nodal and interelement compatibility draw neat sketches for

effective explaination. [10]

Q3) a) Eight noded isoparametric quadrilateral is used as element for plane

stress condition. It has four corner nodes and four midside nodes.

i) Obtain shape functions for any one corner and any one midside

node. [8]

ii) State elasticity matrix. [3]

iii) Write Jacobian matrix. [4]

b) Explain properties of element sitffness matrix with physical significance.

[10]

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[3765]-463 2

SECTION - II

Q4) Write with neat sketches following :

a) Necessity of three dimensional elements. [5]

b) Strain - displacement relations for tetrahydron element [10]c) Shape functions in case of hyxahydron element in natural coordinates.

[10]

Q5) Write polynomial displacement function for w in case of plate bending

element developed by ACM. Why this element is non conforming?

If [B] = z [Q] [A1] {}, write [Q] [25]

Q6) a) Explain membrane and bending actions in shell element. How these

two states of stresses are considered in formulating [K] for shell element.

[15]

b) Show by means of following elements finite element mesh for cylindrical

shell.

i) Four noded flat rectangular element.

ii) Eight noded curved shell element. [10]

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1to6 FEM Quest

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