Reg. No. :

Question Paper Code : 31043

' B.E.lB.Tech. DEGREE EXAMINATION, MAY/JUNE 2013.

Sixth Semester

Mechanical Engineering

O8O]20032 - FINITE ELEMENT ANALYSIS

(Common to Automobile Engineering)

Time : Three horus

(Regulation 2008)

Answer AI-L questions.

PART A- (10 x 2 = 20 rnarks)

Maximum: 100 marks

two dimensional heat

1.

2.

5.

b-

7.

State the advantages of Gaussian elimination technique.

What is Ritz method?

3. State the significance of shape function.

10. What are force vectors? Give an example.

\4rhat is post processing? Give an example.

\&?rat is meant by primary and secondary node?

Distinguish between CST and LST elements.

Write the finite element equation used to analysetransfer problem.

State the applications of axisymmetric elements.

When are isoparameteric elements used?

I

I

I

I

8.

9.

PART B - (5 x l6 = 80 marks)

11. (a) (i) Discuss the imporLance of FEA in assisting design process.

(ii) Solve the ordinary differential equation

( a.' u\l* { l+rOx'z=0 for 0<*<1\d*" )

(6)

Subject to the boundry conditions y(O) = y(t) : O using the Galerkin

method with the trial functions Nr(")=O; lf.(")="(r-"'). (10)

Or

Discuss the factors to be considered iri descretisation of a domain.(b) (,

(ii)

(10)

(6)

the nodal(16)

. Solve the following equations using the gauss elimination method.2xr+3xr+xr=9xr + 2xr + Sxr.= 6

.3"r+*r+2xB=O-12. (a) Fig.l shows the pin-jointed configuration. Determine

displacements and stresses in each element.

t000 N

(b)

Fig.1

Or

For the beam shownin Fig.2, determine

(, The slopes at node 2 and 3 and

(ii) Vertical deflection at the mid-point of the distributed load. All theIemenets have E = 200 GPa and I = 5 x 106 mm4. (16)

)

Fig. 2

2

24 kN/m

31043

13. (a) Compute the finite element equation for the LST element shown in Fig.3.(16)

Thickness = 2 mmE = 200 GPau=0.3

Plane stress

lNrmmr-

srz3{5'

Fig. 3

,o(b,) Deterru-ine the element matrices and vectors for the LST element shown

in Fig.4.The nodal coordinates are i (1, 1), j (5, 2) and k (3, 5). Convectiontakes place along the.edge jk.

K = 7.5 Wmm"Ch = 0.15 WmmFC4" = 30"C

Fig. 4

14. (a) Triangular elements are used for the stress analysis of plate subjected toinplane loads. The (x. y) coordinates of nodes i, j and k of an element aregiven by (2, 3), (4, 1), and (4, 5) mm respectively. The nodaldisplacements are given as :

ur = 2.0 mm, u2 = 0.5 mm, us = 3.0 mm

vr = 1.0 mm, v2 = 0.0 mm, v: = 0.5 mm

Determine element stresses. Let E = 160 GPa, Poisson's ratio = 0.25 andthickness of the element t = 10 mm. (16)

k{r

1,,

Or

31043

(b) (i) What are the non-zero strain and stress components ofaxisymmetric element? Explain. (4)

(i1) Derive the stiffness matrix of an axisyrnmetric element usingpotential approach. (12)

15. (a) (l) Consider the isoparametric quadrilateral element with nodes 1 - 4at (5, 5), (L1,7), (L2, 15), and (4, 10) respeciively. Compute theJacobian matrix and its determinant at the element centroid. (10)

(i, Use Gaussian quadrature with two points to evaluate the integral

ib.,*r(*.,))a*

The Gaussian ppints are ! 0.5774 and weights at the two pointsare equal to unity. (6)

Or

(b) Thg nodal displacements of a rectangular element having noilal \-/coorilinates (O, 0), (4, O), (4,2) a1d (0,2) are : u1 = 0 mm, vr = 0 mm,u2"= 0.1 mm, v2 =.0.05 --, ,r = 0.05 mm, ve = - 0.05, ua = 0 andv4 = 0 mm respectively. Determine the stress matrix at r = 0 and s = 0' using the isoparametric formulation. Take E = 210 GPa and Poisson'sratio = 0.25.

v

4 31043

Reg. No. :

a.

6.

7.

8.

Question Paper Code: D 2308

B. E./B.Tech. DEGREE EXAMINATION, APRII-,/MAY 20 10.

Seventh Semester

Mechanical Engineering

ME 1401 - INTRODUCTION OF FINITE ELEMENT ANALYSIS

(Common to Automobile Engineering and Mechatronics Engineering)

Time : Three hours Maximum : 100 marks

, Answer AJ,L questions.

PART A (10x2=20marks)

1.

2.

4.

What is the lirnitation of using a frnite difference method?

List the various methods of solving boundary value problems.

Write down the interpolation function of a freld variable for three-nodetriangular element.

Highlight at least two rules to guide the placement of the nodes whenobtaining approximate solution to a differential equation.

List the properties of the global stiffness matrix.

List the characteristics of shape functions.

€t'What do you mean by the terms : c0, c' and co continuityT:'

Write down the nodal displacement equations for a two dimensional triangularelasl,icity element.

9. List the required conditions for a problem assumed to be axisJrmmetric.

10. Name a few boundary conditions involved in any heat transfer analysis.

PARTB-(5x16=80marks)

11. (a) Discuss the foliowing methods to solve the given differential equation :

gd'Y.^ -1141"1=oax,

with the boundary conditions y(0)=0 and y(H)=0

(i) Variational method

(ii) Collocationmethod'Or

(b) For the spring system shown in nryI" ]' calculate the global stiffness

matrix, displaceients of nodes 2 and 3; the reaction forces at nocle 1 and

4. Also calculat"-ii" f*t"t in the spring 2' Assume' h = fr3 = 100 N/m'

A, = 200 N/m, ut=ut =0 andP = 500 N'

gure 1 SPring SYstem AssemblY

12. (a) Determine the joint displacements' the joint leaetions' element forces

and element stresses of the giveo truss elements'

Figure 2 Tluss with aPPlied load

i.'

Element A Ecm2 N/m2 m

I 32.2 6.9e10 2'54

2 58.7 2D.7el0 2'54

3 25.8 2O.7el0 3'59

Connection2to32ta I1to3

90

0

135

Or

D 2308

(b) Derive the interpolation function for the one dimensionar rinear elementwith a length 'L' and two nodes, one at each end, designated as ,i, and .j,.Assume the origin ofthe coordinate system is to the lelt of nOde.i,.

Figure B the one-dimensional linear element

13. (a) Determine three points on the 50.C contour lioe forelement shown in the Figure 4. The nodal values@;=54'C, Qr =56"C and (D_ =46C.

the rectangulararc oi = 42"C,

j -=>

s

';.''* -l

I

tI

m

(8, 3)

Figure 4 Nodal coordinates of the rectangular element

Or

(b) The simply supported beon shown in Figure b is subjected to a uniformtransverse load, as shown. Using two equal-Iength eiements and work-equivalent nodal loads obtain a finite erement soirution for the deflectionat mid-span and co"'pare it to the solution given by elementary beamtheory.

Figure 5 uniformly loaded bea"'

D 2308

74. (a) For the plane strain element shown in the Figure 6, thenodal displacements are given as : z1 =0.005mm, uz=0.0O2 rrry::,

zs =0.0 mm, zr =0.0mm, u5 =0.004 mm, u6 =0.0mm. Determine theelement stresses. Take E = 200 Gpa and ,/ = 0.3. Use unit thickness forplane strain.

(s,5) (2s, 5)

' Figure 6 Tiiangular ElementOr

(b) Determine the'element stiffness matrix and the thermal load vector for. the plane stress element shown in Figure ?. the element experiences

20oC increase in temperature. Take E = 15e6 N/cm2, y =0.25 , t = 0.5 cm

and a = 6e-6fC.

;'xir

ti

(r 5, t5)

(r,3)h

,-O.5cmE = t5<ro6) N/ca?r, = O25, * 51to{1rc

(o. o) (2, o)

Figure 7 Tliangular elastic elements

15. (a) Use Gaussian quadrature to obtain an exact value of the integral.11

r= f [tr' -!)(s-r)zdrds.JJ'

()rDefine the following terms rrith suitable examples :

(i) Plane stress, Plane strain(ii) Node, Element and Shape functions(iii) Iso-parametric element(iv) Axisymmetric analysis.

-1-1

(b)

D 2308

Reg. No. :

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER./DECEMBER 2009.

Seventh Semester

Mechanical Engineering

ME 1401 _ INTRODUCTION OF FINITE ELEMENT ANALYSIS

(Regulation 2004)

' (Common to Automobile Engineering and Mechatronics Engineering)

Time : Three hor:rs fvfs:.imr rrn I 100 marks

1.

. Answer ALt questions.

PARTA-(10x2=20marks)

Distinguish betweea 1D bar element and lD beam element.

.What is Galerkin method of approdmation?

. What are CST & LST elements?

i What is a.shape function?

State the properties of stiffness matrix.

Yit., dgyl the governing differential equation for a two dimensional steady-state heat transfer problem.

10. Distingrrish between essential boundary conditionsconditions.

natural boundary

2.

3.

4.

5.

6.

7.Whatismeantbyaxi.symetricfieldproblem?Giveanexample

8. Distingriish between plane stress and plane strain problems.

9. what are the differences between 2 Dimensional scalar variabre and vectorvariable elements? ------- *-- ':--"'

and

11. (a) (i)

PARTB-(5x16=80marks)

What is constitutive relationship? Express the constitutive relationsfor a linear elastic isotropic material including initial stress andstrain. (6)

(ii) Consider the differential equation (dzy ldxz )+ 400x2 =O for0<r<L subject to boundary conditions y(0)=0;y(1)=0.The functional corresponding to this problem, to be extremized isgiven by

1

t = [ l-o.S@y t ar)2 + 400x2 yl0

Find the solution of the problem using Rayleigh-Ritz method byconsidering a two-term solution as y(r) = crx(l - x) + crx'(l - *,).

(10)

Or(b) (i) A physical phenomenon is governed by the differential equation /

(dzu /itx21 - ]]}x2 = 5 for 0 <r <1.

The boundary conditions are given by u.,(0)=ro(1)=0. By taking a. two-term trial solution as w(x) = C it@) + C zfz@) with

fr(x)=x(x-l) and fr(t)= rz(x -1), frnd the solution of .theproblem.using the Galerkin method. (10)

Solve. the following system of equations using Gauss elimination- method. (6)

xr+3x2+zxi =L3

-2)ci t x2- x, =-3' -5*r+x2+lxa =6

12. (a) The stepped bar shown in Fig. 1 is subjected to an increase intemperature, AZ = 80"C. Determine the displacements, element stresses

(ii)

A=E=

Bronze2400 mm283 GPa

Fig. 1.

Aluminimum1200 mm270 GPa

O.r

aB = 18.9x104 /'CaA) = 23x:r}4 l'Cds =LL.I xLO4 l"CP, = q0kN

P2 = 75kNA? = 80'C

Steel600 mm2200 GPa

2

F?-8oo

mm

P 1421

(b) Consider a two-bar truss supported by a spring shown in Fig. 2. Bothbars have E = 210 GPa and A =b.0x10-am2. Bar one has a length ofb mand bhr two has a length of I0 m. The spring stiffness is A = 2,kN/m.Determine the horizontal and vertical displacements at the joint 1 andstresses in each bar.

& = 2000 kN/m

25 kN

13. (a) (,

Fic.?.

The (r, y) co-ordinates of nodes, i, j, and ft of a triangular elementare given by (0, 0), (3, 0) and (1.S, 4) ,,,- .""ps.1ively. Evaluate theshape functions N1,N2 and Ns at an interior.p oini p (2,2.b) ,.mfor the element.

(ii) For the same triangular element,relation matrix B-

(4)

obtain the strai'r-displacement

Compute element matrices and vectors for the. element shown in Fig. 3,when the edge kj experiences convection heat loss.

fr =-10 Wcm2 "KT_ -- 40ec

Fie.3.

(J2)

Or

i(b)

*/*; iw)<mz

14. (a) The triangular element shown in Fig. 4 is subjected to a constantpressure 1b N/mm2 along the edge ij. Assum e E = 200 GPa, Poisson's

iatio p = 0.3 and thickness of the element = 2 mm' The coelficient of

thermal expansion of the material is d =2xl}4 l"C and A?=50oC'Determine the constitutive matrix (stress-strain relationship matrix D)

and the nodal force vector for the element.

Thickness= 2 mmE = 200GPa

P=0.3d = 2x].o4 /"CAT = 50"C

(nn)

. Fig.4.Or

(b) Thi (*, y) co-ordinates of nodes, i, j, and & of an axisfurmetric triangular" element are given by G, a), (6, 5) and (5, 8) cm respectively' The element

displacement (in cm) vector is given as g = [0.002, 0'001, 0'001, 0'004,

-0.003, 0.007]r. Determine the element strains'

ian (global) coordinates of the corner nodes of a

quadriliteral element are given by (0, -1), (-2,3), (2,4) and (5' 3)'

Find the coordinate transformation between the global and local(natural) coordinates. Using this' determine the Cartesian

. coordinates of the point defrned by (r, s) = (0.5, 0.5) in the gilobal

coordinate system. (8)

(ii) Evaluate the integral

It = [Q+x+x2\dx and compare with exact results.

-1

.Or

(8) , -1

(b) (i) The Cartesian (global) coordinates of the corner nodes of anisoparametric quadrilateral element,.are given by (1' 0)' (2; 0),(2.8, 1.5) and (15, 1). Find its Jacobidii-{'.,fiatrix' (12)

(ii) Distinguishbetween subparametiis and superparametricelements.

P t42r

Reg. No. :

c 3387

B.E.lB.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2008.

Seventh Semester

Mechanical Engineering

ME 1401 - INTRODUCTION OF FINITE ELEMENT ANALYSIS

(Common to Automobile Engineering and Mechatronics Engineering)

(Regulation 2004)

Time : Three hours Mnximum : 100 marks

2.

Answer ALL questions.

PARTA - (t0x2=20marks)

1. Wiite the potential energy- for bea- of span ,L, simply supported at ends,subjected to a concentrated load .p, at mid span. Assume EI constant.

What do you mean by higher order elements?

what is constitutive Law and give constitutive raw for axi-s5rmmetricproblems?

Explain the important properties of CST element.

Give one example each for plaae stress and plane strain problems.

Write the stiffiress matrix for the simple beam element given below.

4.

-5.

tt

.Write the Lagrangean shape functions for a 1D, 2 noded eiement

What are the advantages of natural coord.inates over global co_ordinates?

L Define Isoparametric elements?

10. Write the natural co-ordinates for the point'P'of the triangular element. Thepoint 'P' is the C.G. of the triangle.

11. (a)

PARTB-(5x16=80marks)

Determine the expression for deflection and bending moment in a simplysupported beam subjected to uniformly distributed load over entire span.Find the deflection and moment at midspan and compare with exactsolution using Rayleigh-Ritz method.

Use y =a, sin(il r/1)+o, sin (an r /r).

12. (a)

Or

Derive the equation of equilibrium in case of a three dimensional stresssystem.

Derive the shape function for a 2 noded beam element and a 3 noded barelement.

(b)

(b) Write theconduction

Or

mathematical formulation for a steady state heat transferproblem and derive the stiffness and force matrices for the

satne,

13. (a) Find the expression for nodal vector in a CST element shown inFig. 13 (a) subject to pressures {, on side 1.

Fig. 13 (a)

Or

(b) Determine the shape functions for a constantelement in terms of natural coordinate system.

strain triangular (CST)

74. (a) tr'or the CST element given- below Fig.displacement matrix. Take t = 20 mm, E = 2 x

14. (a) assemble strain-105 N/mrn2.

(b)

.(i) Use Gaussbd

! [ "ta"at .

00

numerically

and also

integrate

(10)

a linear(6)

(100,100) (400,100)

(n =2) to

derive the shape

15. (a)

(6)

(5)

(5)

c 3387

Reg. No. :

v 4240

B.E.lB.Tech. DEGREE EXAMINATION,

(Common to Automobile Engineering and

(Regulations 2004)

Time : Tl:ree hours

Seventh Semester

Mechanical Engineering +-t

ME 1401 - INTRODUCTION OF FINITE ELEMEN"X;AIYSIS

1.

2.

3.

i4.

5.

Answer ALL questions.

. PARTA_(10x2=20marks)

List any four advantages of finite element method.

W.rit^e^1he potential energy for beam of span T,, simplysuDJected to a concentrated load,F at mid span. Assume E

'What are called higher order elements?

Write briefly about CST element.

Ih"t- i.^ the governing differential equation fortransfer?

6. Differentiate: Local axis and Global axis.

7. What is an equivalent nodal force? ,. . , ,,1#

t"#. *l!'}8. Give one example each for plane stress and pf"rr" ii".in

9. What do you mean by Constitutive Law and giveaxj-symmetric problems?

tO. State the basic laws on whieh isoparametric concept is

PARTB-(5x16=80marks)

11. (a) Explain the Gaussian elimination method for solving of simultaneousIinear algebraic equations with an example.

Or

(b) A cantilever beam of length L is loaded with a point ioad at the free end.Find the maximum deflection and maximum bending moment usingRayleigh-Ritz method using the function Y = A$'cos(m/2L)1 . Given :

EI is constant.

12. (a) (, Derive the shape functions for a 2D beam element.

(ii) Derive the shape functions for 2D truss element.

Or

(b) Each of the five bars of the pir jointed truss shown in Figure 12 (b) has across sectional area 20 sq.cm. and E = 200 GPa.

Figure 12 (b)

(i) Form' the equation F = KU where K is the assembled stiffnessmatrix of the structure. (10)

(ii) Find the forces in all the five members. (6)

13. (a) Find the temperature at a point P (1, 1.5) inside the triangular elementshown with the nodal temperatures given as Tr = 4O'C, T.t = 34"C, oadT, = AtrC. AIso determine the location of the 4?C contour line for thetriangular element shown in Figure 13 (a). (16)

(0,01 tQ.0't

Figure 13 (a)

Or

(8)

(8)

r;

x(,.3)

v 4240

(b) Calculate the -element

stjffness matrix and thermal forceptane stress element shown in Figure 18 (b). The elementrise of 10"C.

v

vector for theexperiences a

(16)

I

tu,u, (2,0)

Figure 18 (b)

74. (a) Derive thc constarrt-strain triangular. equations-

(l.l) t =5mmE= l5 x tdlVmn2p-- 0.25o.= 6 x !0'6

element's stiffness matrix and(16)

Or

Linear-Strain triangular element,s stiffness matrix and

' (16)

f (x) = 19 *,rr*) - (Bx2 t to) + (+x3 / 100) _ (_5.r4 /1000) +between 8 and 12. Use Gaussian euadrature Rule. (16)

(b)

15. (a) .

Derive theequations.

Integrate(6jr5 /10000)

Or

G) Derive element stiffness matrix for a Linear Isopa.arnet.ic euadrilateralelement. (16)

v 4240

ents-

/ theased

( rb,

tion(8)

on(8)

Reg. No. :

R 3478

B.E.lB.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2007.

Seventh Semester

Mechanical Engineering

ME 1401 - INTRODUCTION OF FINITE ELEMENT ANALYSIS

.(Cornmon to Automobile Engineering and Mechatronics Engineering)

(Regulations 2004)

Time : Three hours Maximum : 100 marks

r]

" Answer ALL questions.

PARTA-(10x2=20marks)

State the principle of minimum potential energy.

Define shape functions.

What do you mean by Constitutive Law?

Differentiate CST and LST elements.

What are the advantages of natural coordinates?

How thermal loads are input in finite element analysis?

Why polynomial type of interpolation functionstrigonometric functions?

over

8.

9.

10.

Write short notes on AxisJrmmetric problems.

What do you mean by isoparametric formulation?

What are the types of non linearity?

PARTB-(5x16=80marks)

11. (a) Compute the value of central deflection (Figure 1) by assuming

, _ (a sinz x) ..L

The beam is uniform throughout and carries a central point load P.

12. (a)

tr'igu.re 1

Or

Write short note on Galerkin's method.

Write briefly about Gaussian Elirnination.

Derive the'shape functions for a 2D been element,

Derive the shape functions for 2D truss element.

(b) (8)

(8)

(8)

(8)

(i)

(ii)

(i)

iio

(b)

13. (a)

Or

Why higher order elements are needecl? Determine the shape functions ofan eight noded rectangular element.

For the constant Strain Triangular element shown in Figure 2, assemblestrain-displacement matrix. Take , = 20 rn"' and Z = 2x105N/mm2.

Figure 2

Or

G) The temperature at the four corners of a four-noded rectangle areTpTz,Ts and ?n . Determine the consistent load vector for- a 2-Danalysis, aimed to determine the thermal stresses.

2 R 3478

74' (a) *:;-lfi.f5:;:n,f:" the erement stirrness matrix ror an

Or

(, Explain the terms "plane stress, ald ..plane strain, problems. Giveconstitutive laws for these cases. (S)

(i, Derive the equations of equilibrium in case of a three dimensionalsYstem' (g)

l:r:!Tn the strain-displacement matrix for the linear quadrilateralerement as shown in figure B .t c""." foirri ; = ;;?;;; and s = _5Z7BE .

G)

15. (a)

(b)

4,5

Figure 3

Or

Derive element stiffneds maelement. , Ltrix for a Linear Isoparametric euadrilateral

R 3478