Anna University, Chennai

PART A (10x2=20)

FINITE ELEI'vI:NTANALYSIS UNIT-1

1 What is meant by Finite Element Analysis?

2 Why polynomial type of inte rpolation function s is mostly used in FEM?

3. What are 'h' and 'p' versions of Finite Element method?

4. Name the weighted residuals te chni ques.

5. What ismeantby postprocessing?

Ii Distinguish between essential boundary conditions and natural boundary conditions.

7. Name any4 FEA softwares?

& What are the general constituents of finite element software?

9. What is Rayleigh-Ritz method?

lD. Duri ngdi scretization, menti on the places where it is necessary to place a node? PARTB

1 Solve the simultaneous systems of equation using Gauss-Elimination method.

3x + Y -z = 3; 2x - &( + z = 3; x - 2y + 9z = 8 (16)

2 Th e foil owing differential equation is available fora physi cal phenomenon:

d'y/dx' -10<2 =5; o sx s1The boundary conditions are:y(O) =Oand y(1) =O.By usingGalerkin's method of weighted residuals, fi nd an approximate solution of the above differential equation .(16)

3. A si mply supported beam subjected to uniformly di stributed load 'qo' ove r entire span and also poi nt load of magnitude 'P' at th e centre of the span. Cal culate the bendng moment and deflection at mi d span by usi n gRayleiitl-Ritz method and compare with the exact sol ution. (16)

4. A bar of uniform cross section is cI amped at one en d and left fre e atthe other end and it i s subjected to a uniformly distributed load of qO- over its entire run. Cal culate the displacement and stress in th e bar usingthree term polynomial. Compare with exact solutions. Also, determine the stresses and

di spiacements if the bar i s clamped at both ends. (16)

5. Determi ne a 2param eter sol uti on of the following usingthe Galerkin's method and compare it with th e exact sol uti on. (16)

d'y/dx' =-COSltX,O$x$l,u(O) =0, u(1) =0

liS 01 ve the following equation u sing a 2 param eter tri al sol utron by:

a) Point coli ocation method. b) Galerkin's method.

dy/dx + y =0. osxsi, y(O) = 1 (16)

7.The following differential equation is available fora physi cal phen omen on.

d'y/dx' +50=0. O$xS10 Trial function isy=alx (io x], y (0) =0. y(lO) =0. Findthevalue ofa1usingall the weighted residual te chniques and compare the solutions. (16)

a Solve the given equations by Gauss-Elimination method.

2x+4y+2z =15;2x+y+2z =-5;4x+y-2z =0(6)

9.Li st th e advantages, di sadvantages and applications of FEM. (8)

lD.Explain thevarious weighted residual techniques.(16)

PART A (10.X2=20.)

1 Define shape function.

2 How doyou calculatethe sizeofthe global stiffness matrix?

3. What are the characteristics of shape function?

4. Write down the expression of stiffness matrix for one di men sional bar element.

5. State th e prope rties of stiffn e ss matrix. fi Whatistruss?

7. Defin e total potential energy.

& State the prindpleofminimum potential energy.

9. Write down the expression of shape functi on Nand di splacement u forl-D bar el ernent.

ill Write down the expression of stiffness matrix foratruss element. PARTS

1Derive an expression for shape functi on and assemble the stiffness matrix forbendi ng in beam

elements. (16)

2Derive an expression of shape fun cti ons and the stiffness matrix for one di mensional bar element based on global co-ordinate approach. (16)

3.A two noded truss element is shown in the fi g. The nodal displacements are u1=5 mm an d u2 = 8 mm.

Cal culate the displacements at x = l/4, l/3 an d l/2 (16)

4. A steel plate ofuniformthid<ness :Dmm issubjectedtoapointioadof4aJ N at mid depth asshown in fi g. The pi ate is al so subjected to self-weight. If E = 2*10.' *N/mm2 *and density =0.8"10.-' N/mm'.

Cal culate the displacement at each nodal point and stresses i n each element. (16)

5. An axial I oad of4*lL6 N is applie d at D"C to the rod as shown in fi I> The temperature i sthen raj sed

to *ere. *Cal cui ate nodal displacements, stresses in each material an d reacti ons at each nodal poi nt. Take

E.I=o..7*lO' *N/mm2; *E" ee I=2'lO' N/mm' ;0.1=23*10' *1°C; *0" ee 1=12"10' /"C.(l6)

fi Why higher orderelements are needed? Determine the shape fun ctions of an eight noded re ctangular element. (16)

7.Derive the stiffness matrix fortwo dimensional truss el ements.(l6)

&Determi ne the slope and deflection of a cantil ever beam subjected to a uniformly di stributed load q

ove rthe entire span and a poi nt load P actingon its free e nd.(16)

9.Determi ne the slope and deflection of a simply supported beam subje cted to a uniformly distributed load q overthe enti re span anda point load P actingat its mi d span. (16)

lD.Consi derthe bar as shown in Rg_ Cal culate the following (i) Nodal eli splacements (ii) Element

stre sses (iii) Support reactions. Take E=aJOGPaand P=4OO<N.(16)

3:nnm' P 6OOmm'

:!On m :!On m 400m m

11 Consideratapersteel plate of uniform thickness, t=:Dmm as shown in Fig. The Young's modulus of the plate, E=Jl)GPa and weight densityp=082>clD-' *N/mm'. *In additi on to its self weight, the plate is subjected to a point load P =IDlN at its mi d poi nt. Cal cuiate the following by modelling the pi ate with

two fj nite elements:

(i) Global force ve ctor {F). (ii) Global stiffness matrix [K). (i ii) Displacements i n each element. (iv) *Stre *sses in each element (v) Reaction force atthe support. (16)

tsomm

300mm

600mm

P

75mm

1 Write down the expression for shape functions fora constant strai n triangular element

2 Write a strai n displ acement matrix for CSTelement

3. What is LST element?

4_ What is OST element.

5_What is meant by plane stress analysis_

6. What is CST element?

7_Write down the stiffness matrix equation for 2- D CSTelement.

& Define plane stress analysis.

9. Write adisplacementfunction equation forCSTelement.

10. Write down the stress- strain relationship matrix forpl ane stress con dition.

PARTS

1 A wall of 0.6 m thi ckness havingthermal conductivity of 12 W/mK. The wall is to be i nsul atedwith a material ofthi ckness 0.06 m havi ngan average thermal condu ctivity of0.3 W/mK_ The i nnersurface temperature is llXO °c and outsi de of the in sulation is exposed to atmosphe ricairat DOC with heat transfer coeffi cient of D W/m'K. Calculate the nodal temperatures (16)

2 A furnace wall is made up of three layers, inside Iayerwith thermal conductivity &5 W/mK, the middl e

Iaye rwith condu ctivity 0.25 W/mK, the oute rIayerwith condu ctivity 0.08 W/mK. The respective

thi cknesses of the inner, middle and outer layerare 25 em, 5 em and3 em re sp.The i nsi de temperature of the wall is a::o C an d outside of the wall is exposed to atmosphericairat D "C with heattran sfer

coeffi cient of 45 W/m'K. Determine the nodal temperatures. (16)

3. An aluminium alloyfi n of7mm thi ck an d 50 mm protrudes from a wall, whi ch is maintained at:Lal "C, The ambient airtemperature i s 22°C. Th e heattransfercoeffi cient an dthermal conductivity of the fin material are 140 W/m2K and 55 W/mK respectively. Determine the temperature di stri bution offi n.{l6)

4. A steel rod of diameter d =2cm, length l=5 cm and thermal conductivity k=50 W/mC is exposed at

one endto a constant temperature of 3J) "C. The otherend i s in ambi ent air of temperature J) "C with

a convecti on coefficient ofh = 100 W/m'oC. Dete rmine the temperature atthe midpoint of the rod (16)

5. Cal co Iate the temperatu re di stri bution ina 1- D ftn. Th e fi n is re ctangu lar i n shape an dis :Lal m m Ion g,

40 mm wide and10 mm thick. One endofthefin isfixedandotherendisfree.Assumethatconvection

heat loss occurs from the end of the fi n. Use 2 el ements. The temperature at fixeden di s:Lal °c. Take k

= 0.3 W/mmoC; h = 10-3 W/mm'C; T(amb) = J) °c. (16)

6. A metallicfin, with therma conductivity k =36) W/mc, 0.1 cm thi ck and 10 em long, extends from a plane wall whose temperature i s 23SC. Determine the temperature distribution and amount of heat transferred from the fin tothe ai rat :DC with h = 9 W/m'C. Take wi dth of the fi n to be 1 m, (16)

7.Eval uate the stiffness matrix forthe CSTelement shown in fig Assume pi ane stress condition. Take, t =

J) mm, E = 2"10' N/mm2 and m = 0.25. Th e coordi nates are given in mm.

&Assembl e the strarn-dtspacement matrix forthe CSTel ement shown in fig. Take, t = 25 mm and E =

210 Gpa.

9.Derive an expression for shape function forconstrain strain triangular element.

IDDetermi ne the shape functi ons Nl, N 2 an d N3 at the interior poi ntP forthe triangul ar element shown infig.

1. What are the conditions for a problem to be axisymmetric.

2. Give the strain-displacement matrix equation foran axisymmetric triangular element

3. Write down the stress-strain relationship matrix for an axisymmetric triangular element.

4. Write short notes on Axisymmetric problems.

5. What are the ways in which a 3-D problem can be reduced to a 2-D approach.

6. What is axisymmetric element?

7. Give the stiffness matrix equation foran axisymmetric triangular element.

8. What are the 4 basic elasticity equations?

9. Write down the displacement equation for an axisymmetric triangular element.

10. Write down the shape functions foran axisymmetric triangular element.

11. What is meant by Finite Element Analysis?

12. Why polynomial type of interpolation functions is mostly used in FEM?

13. What are 'h' and 'p' versions of Finite Element method?

14. Write down the expression of stiffness matrix for one dimensional barelement.

15. What are the constituents of total potential energy?

16. Distinguish between essential boundary conditions and natural boundary conditions.

17. Define shape function. Whatare the characteristics of shape function?

18. What are the general constituents offinite element software?

19. State the properties of stiffness matrix.

20. State the principle of minimum potential energy. PART8

1. The nodal co-ordinates for an axisymmetric triangular are given below: r, = 15 mm, z, = 15 rnrn : r, =

25 mm, z, =15 mm; r, = 35 mm ,z, = 50 mm. Determine (81 matrix for that element. (16)

2. Evaluate the temperature force vectorfor the axisymmetric triangular element shown in the fig. The element experiencies a 15'C increases in temperature.

Take 11 = 10*10"/'C, E = 2*10' N/mm', m = 0.25. (16)

3. Determine the stiffness matrix forthe element shown in fig. The co-ordinates are in mm. Take E=

2*10' N/mm2 and m = 0.25. (16)

4. Derive the expression forthe element stiffness matrix for an axisymmetric shell element. (16)

5. For the axisymmetric elements shown in fig, determine the element stresses. Let E = 210Gpa and m

=0.25. The co-ordinates are in mm. (16)

u, **=0.05 ****mm; ****Wi ****=0.03 **mm.u, **=0.02 **mm.w, **= ****0.02 **mm.u, **=Omm; ****W1 ****= ****0 mm**

6. Derive the strain - displacement matrix (8) for axisymmetric triangular element (16)

7. Derive the stress-strain relationship matrix [01 for the axisymmetric triangular element. (16)

8. Calculate the element stiffness matrix and the thermal force vector for the axisymmetric triangular element shown in fig. The element experiences a 15'C increase in temperature. The coordinates are in mm. Take 11 = 10*1O"/'C; E =2*10' N/mm';m =0.25 (16)

9. For the axisymmetric elements shown in fig, determine the element stresses. Let E = 210Gpa and m

= 0.25. The co-ordinates are in mm. (16) The nodal displacements are:

ul = 0.05 mm; wi = 0.03 mm u2 = 0.02 mm; w2 = 0.02 mm u3 = 0 mm ; w3 = 0 mm

10. Derive the element stiffness matrix fora linear isoparametric quadrilateral element{16)

11. Establish the strain - displacement function matrix forthe linear quadrilateral element as shown in fig at gauss point r =0.S7735 and s = -0.5mS. (16)

12. The integral *fl-l *(r' + 2f + **1) **drcan be evaluated exactly by two point gaussian quadrature. Examine the effect on the result if three point integration is applied.(16)

13. For a 4 noded rectangular element shown in fig. determine the following:

Jacobian matrix,Strain displacement matrix,Element stresses. Take E= 2"10' Nlmm' ; m = 0.25;

u ='0,0,0.002,0.003,0. 005,0.003,0,0 I; € =0; IJ = 0. Assume plane stress condition(16)

5. Derive the shape functions for4 noded rectangular parent element by using natural coordinate system and coordinate transformation. (16)

6. Evaluate the integral I = *fl-l *(2 + x + x') dx using gauss quadrature method and compare with exact solution. (16)

7. Evaluate the integral, I = *fl-l *cos *nx/2dx *by applying 3 point gaussian quadrature and compare

with exact solution. (16)

8. Evaluate the integral I = *fl-1 *(3" + x' *+1/x+2) *dx using one point and two point gauss quadrature. Compare with exact solution. (16)

9. For the isoparametric quadrilateral element shown in fig determine the local coordiantes of the point P which has cartesian coordinates (7,4). (16)

Anna University, Chennai