fem nov-dec 2006 to 2013

B.TECH. Fifth Semester Examination 2013Mechanical Engineering

Subject: Finite Element Method

Max. Marks:70Answer the following questions, Assume data suitably, if missing.

ME0502-13

C D gc.)

Min. Marks: 25

Time:3 hrs

residuals to obtain a twoExplain in brief the weighted residual methods. Use Galerkin's method of weightedparameter approximation to the solution of the differential equation,

0<x<Zn .y(0)=200*=so(t*"os x)-o.o5Y

0r

Explain the weak form of the weighted residual statement. Develop the weak form and solve the following.

';r - -dulAEYI*ax=O : a(0):0 ' AE+l -0dx' Ml,=, .

For the three - bar truss shown in Figure 1, determine the nodal displacements and the stress in each member.

Find the support reactions also. Take modulus of elasticity as 210 GPa.

Or

Derive the quadratic and cubic shape functions in terms of natural co-ordinates mapped between -1 to +1 and

0 to 1 of a one dimensional element. Write a note on cornpatibility and completeness requirements.Write short notes on Gauss-Legendre's quadrature. Evaluate the following integral using 2 X 2 G-L -Q.

-- ?r oi1 =

J.| ffi.* x(0,6), y{0,4)

Or

A wall of an industrial cven consists of three different materials, as depicted in Figure 2. The first layer is

composed of 5 cnr of insulating cement with a clay binder that has a thermal conductivity of 0.08 W/m' K. The

second layer is made from 15 cm of 6-ply asbestos board with a thermal conductivity of 0.074 Wm K. The exterior

consists of 10-cm cornmon brick with a thermal conductivity of O.72 Wm2 K. The inside wall temperature of the

oven is 200'C, and the outside air is 30'C with a convection coefficient of 40 Wm2 K. Determine the temperature

distribution along the composite wall.zP/

^ \+-,t Derive the Hermite shape functions for 1-D, 2 noded beam elements from first principle

@\\\Y' Derive the shape functions ofa. 6-noded triangular elementb. 8-nocied quad element

/i V Derive the stiffness matrix for plane stress condition using linear triangular elements.

\2 0rDifferentiate between consistent mass matrix and lumped mass matrix and derive an expression for lumped

mass for one dimensional bar element.

ME0502-13

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I o llgJnB Tech. 7th Sem" (CSWU) Examination, Spring 2013

Subiect: Finite Element Method

Branch: Mechanical Engineering

Max. Marks;80 Min, Pass Marks: 28\

Answer the following questions. Assume data suitably, if missing. Time: Three i{gurs

1) a. How is FEM different from FDM? With a suitable example explain the formulation of finite element equations ny air\ppro9;hAssumdsuitable data for the example' use 1-D analysis'

,:1rrr" the defrections cng the ol* ,n

I,, 94 steel plate is subjected to an axial load, as shown in Fig.1. A,pproximate the deflections and average stresses aLvplate

is t / t6 in th ick and has a modulus of elasticity E = 29 X 7O6 lb / inz "

Or

For the spring system shown in Fig.2, write down the elemental stiffness matrices, global stiffness matrix and determine the,#&1tr '

-OrUse Galerkin's method of weighted residuals to obtain a one-term approximation to the solution of the differential equation,

f 1tt, - ,,,,r. - -u fi .r r -a' I tir);: r1 li- il.ti,r:l

^ *#.Oerive the cubic shape functions in terms of natural co'ordinates mapped between -1 to +1" and 0 to 1 of a one dimensional

A { ^ement.

Write a note on compatibility and completeness requirements.

< b. in the accompanying figure (Fig.4), the deflection of nodes 2 and 3 are 0.02 mm and 0.025 mm, respectively. What are thedeflections at point A and point B, provided that linear elements were used in the analysis?

../^ A.rfZO-ft-tall post is used to support advertisement signs at various locations along its height, as shown in Fig.S. The post is made of

(\) " structural steel with a modulus of elasticitv of E = 29 X 106 lb/in2. Not considerinq wind loading on the signs and using linear elemen\-'l we determined that the downward deflections of the post at the points of load application are

fu,) I o II ,,, I I r.:,2 x to-r I

1,:i=lattoxlo-.,lint,.J It',+rox ro-rJ

Determine the deflection of point A, tocated at the midpoint of the middle member, using (a) global shape functions, and (b) natura

sh7p6 functions.,/r/

^dyS.Derive the shape functions of a six-noded triangular element.

(,Gfi6p:'r"r the two-dimensional plate loaded as shown in Fig" 5, determine the displacements of nodes 1 and 2 and the element stresses--4..9 using plane stress conditions.

Or

The beam shown in Fig. 7 is a wide flange W310 x 52 with a cross-sectional area of 6650 sq. rnm and depth of 317 mm. The secondmoment of area is 11-8.6 x 106 mma" The modulus of elasticity of the beam is 200 GPa. Determihe the vertical displacement at nodeand the rotations at nodes 2 and 3.

5) Write short notes on any twoa. Principle of Minimum potential energy. ,b, Compatibility and completeness requirements of interpolation functions.

,"ffiWffi,- h.r

0

Or

ft,- di#acement of each node and the reaction forces./'Z) Zl .gl L simOf;, plane truss is made of two identical bars (with E, A, and L), and loaded as shown in Fig.3. Find displacement of node 2 an

U 1tr.e2{e1chbar.bVffiite short notes on Gauss-Legendre's quadrature. Evaluate the following integral using G-L-Q.

c"

d"

Transformation methods for determining eigenvalues and eigenvecto,rs,

furnped mass matrices O r

Deriue the element rnass matrices for bar, truss, CST and axisyrnmetric triangular elements using consistent mass matrix

approach.

Fig.1 Fig.2

Fig" 3 Fig.4

driO75-nl

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B-Tech Seventh Semester Examination, Nov-Dec.2012Subject: Finite Element MethodBranch: Mechanical Engineering

' Max. Marks: 80Assume data suitably, if missing. All Questions carry equal marks unless stated otherwise.

ttll,tlsr4ll]*41.

and

b. Using Rayleigh-Ritz method, determine the displacement of the midpoint of the rod shown in Fig. 1.

or

3) a. Derive the shape functions,,of a nirte-noded quadrilaterql element in natural co-ordinates.

or

5) Write short notes on any twoa. Principle of Minimum potential energy.

b. Transformation methods for determining eigenvalues and eigenvectors.

c. Compatibility and completeness requirements of interpolation functions.

d. Formulation of axisymmetric problems

MEL27I':

Min. Pass Marks: 28

Time: Three Hours

1) a. Use Galerkin's method of weighted residuals to obtain a one-term approximation to the solution of the differential

elt r*,i- - lllr' = q fl.l r: "- n

elt-t)

c. For the spring system shown iffib*rite down the elemental stiffness matrices, global stiffness matrix and determine the

displacement of each node and t\#a/ction forces. @

a. Discuss the properties of shape functions. Derive the shape functions of cubic one-dimensional element in global, local and

natural co-ordinates @

b. For the plane truss shown '@

transform ,n" "?Jl"n,

stiffness matrix of each element into global reference frame and

assemble the global stiffness matrix. Determine the deflection of node 2 in global co-ordinate system and stresses in eacb^element. For both efements, Area =0.5 sq. inch, E = 30 x 106 psi. (?

or

c. The front window of a car is defogged by supplying warm air at 90"F to its inner surface. The glass has a thermal conductivity

of k = 0.8 Wm"C with a thickness of approximately 114 in. With the supply fan running at moderate speed, the heat transfer

coefficient associated with the air is 50 W/mt. K. The outside air is at a temperature of 20"F with an associated heat trarisfer

coefficient of 110 Wm'. K. Determine (a) the temperatures of the inner and outer surfaces of the glass and (b) the heat loss

through the glass if the area of the glass is approximately 10 ft2. @

@@ fft" temperature Ais}lU-u{on in a fin is modeled using triangular element. The nodal temperatures and their corresponding

positions are showni{,p€.y'.)What is the temperature at X=2}5 and Y - L.1 cm. Determine the components of tempejalglegiadients for this elerfrdfi-t and determine the locations of 70 "C and 75 'C isotherms.,/z ej/

and

c. We have used two-dimensional rectangular gements to model the stress distribution in a thin plate. The nodal

element belonging to the plate are shown i(f igr_ jwhat is the value of stress at the center of this element?

+) @Oiscuss Gauss-Legendre quadrature in brief. Evaluate integral I = I(4x'z -2x +S)dx using GLQ two-point sampling "rH2

and

HJr"x1r,r":f finite element formulation of plane stress problems using triangular elements, deriving the stiffness matri@

c. The beam shown ,@,, a wide-flange w 18 x ll, *i*r a cross-sectional area of 10.3 in2 and a depth of !7.7in. The

second moment of area is 510 in*. The beam is subjected to a point load of 2500 lb. The modulus of elasticity of the beam -E = 2

X 105 lb/in2. Use a two-element model and calculate the deflection of the midpoint of the beam. @

stresses for

@

@

oP;r

Derive the element mass matrices for bar, truss, C$\and axisymmetric triangular elements using consistent mass matrix' :-t - z'-'approach. ,,' (9

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Semester:B.E"V8lSury."Braneh:ffieehanicalEngE'Subject: Finite Element lVletlrod eode: 337753 {37}

Total Theory Feriods: 50 Tota! Tutorial Periods: 12

Tobl Marks in End Sernester Exam: 80

MinimumnumberofClassteststobeconducted:2

UNIT I

Historicalbackground.needfol'sdud;'irlgFinlteE'lementMethod'comparisonr'vitholhermethods-thebasic

concepts of FEM. basic equations in Elastic.ity, Variational methods of approximation (Ra;'leigh Ritz

method, method of weighted residuals), potential energ)' tbnnulations" the Finite Elerrent Method' Saint

Venant,s principle. r,on Mises strei;s, matrix displacemenr lbrmulatiorl. element shapes, nodes' nodal

runknou'n !. coordinale si'siems.

I'NIT IID.,r,,,-11,nia! .lrqne {lr'ction. convefgence requirements. clerir.'atitln trsing lloivnorlials- shape iLnction f'or

serenclipitl,famil), ciernests. Herrnite polynomial as sltape furlctiotrs, trtlnsltucliott ol:,I'rape function bv

r.ir,gt tlcliirg lecluticltlc, Strarn ciispiacement matrir'

LItitr'F Ii{i:..ier'tr.i sitif int:ss equatiorr. assembling stifihess equ2{ii,r1.q try clirect irp,irrot'rcir' calerkin's rneihoci' virtLral

rici-il rTletl,cd. r,aliational method- Discritization oi'a slructt'tre. liitlir.'rletneil! lrnalr'sis f(lr ilianc sti-css artcl

'; I: rj ii : ;:';,1 ; 1r1.,: r l 1:1 r 1 :

i i.t i l i'

iiniit e ie:iiir-riii :nai1'sis ol'Lrais at-F'J t; Ll"sc's, isoparlltne lric lirrtllrillrtiort' ctlolclinirlc transl'clrtnaticin' basic

:ii i'1 rress tt ratl'i:l- tl irrtlerica I i nlegration'

i,ir !T Y

1,,,ral-r,sis of beams anC rigirJ l'rarnes, Dynarnic analvsis using Finite t,lerlents for vibratlon pi-obierns'

riltenr:aiue problelrs. e'tc., introduction to noniineat analysis'

.!-EXT.BGOKS:

(1) Finiie Element Analysis by P- Seshu

u?.|Firtte Dlemer'ts in Engineering b;'T'R' Chandrr'rpatrla & A'D Btlegirr-'du

Vatfinit. Element Analysis b; S.S Bhe-'il'atti

R.EFERENCE BOCNS .:

( l) Finite Elemeni Method by J N. Reddy

(2) Appliedfinite Diement Analysis b-v L'J'segerlind

(3) The Finite Elenrent Method b;' O'C' Zienkiewicz

. . '.]]:j

SHASHI PATEL (NIT RAIPUR) MECHANICAL ENGG. www.nitrrmech12.webs.com

B.TECH. Seventh Semester Examination 201_2

Mechanical EngineeringSubject: Finite Element Method

s'a J 3

( t: r= e.1

M E11715

Max. Marks: B0 Min. Marks: 2gAnswer the following questions, Assume data suitably, if missing. Time: 3 hrs

.Q:1. With a suitable example explain the formulation of finite eiement equations by direct approach. Assume suitabiedata for the example. Use 1-D analysis.

OrDetermine the circumference of a circle of radius 'r' using the basic principles of finite eiement method

[14]

Q.2. For the plane truss shown in Fig. 1, transform the element stiffness matrix of each element into global referenceframe and assemble the global stiffness matrix. Determine the deflection of node 2 in global co-ordinate systemand stresses in each element. For both elements, Area =0.5 sq. inch, E = 30 x 106 psi.

itI

'rt1'n ,rlrr

{"Oro}d"L=:_-f- '1 .-LJ{4,D/6)i'l: ,,'t'4i'

--^d';€A r qI(f'\sir-t"l:i

r ts-oc, {g Fig. 1

or I14lExplain in brief the weighted residual methods. Use Galerkin's method of weighted residualsto obtain a one-termapproximation to the solution of the differential equation,

Jfu4. - l0xi e5 &_* E* I4r- fto)=y,tt)'rr

Q.3. Derive the quadratic shape functions in terms of natural co-ordinates mapped between -1 to +1 and 0 to 1 of a

one dimensional element. Write a note on compatibility and completeness requirements.or [I4)

Derive the strain displacement relation matrix from the first principles of a triangle element. Estimate the shapefunctions of a triangular element at the point P(22,44) of a CST with the coordinates 1(0,0), 2{46,8) and 3(18,62).All dimensions are in mm.

pz{. Derive the shape functions for 2 - Q isg-lqgIlgfic four noded quadrilateral elements in terms of Global, Local

and natural coordinates.Or

Derive the Jacobean matrix f or 2D axis symmetric problems.

Q,6. Derivethe stiffness matrix for plane str-e5 condition using linear triangular elements.Ar

[r4]

tL4lDerive the ls;1ajt1$tpe functions for 1-D, 2 noded beam elements from first principle

#y{O,tturentiate between consisientmass mairix anci iurnped mass mairix and derive an expression for lumpecl mass

for one dimensional bar element.

Explain the importan.. of consideration of *"ignlrtin one Gaussian qu.dr.,rru formula? Evaluate *nu tto'

following integral using two point sampling formulation of G-L-Q.tttl

B= j j 6.** rSr* r;rdrds

-? *l


o"l jji hME I 1715

0ci--"ltn'B.TECH. Seventh Semester Examination 2011

Mechanical Engineering A


Max. Marks: 80Answer the foliowing questions, Assume data suitably, if missing.

Min. Marks: 28Time: 3 hrs.

'/',\fiiscuss briefly the various steps involved in FEA. Figure 1a depicts a tapered elastic bar subjected to an applied tensile load p

:- at one end and attached to a fixed support at the other end, The cross-sectional area varies linearly from Ao at the fixedsupport at x = 0 to Aa /2 at x = L. Calculate the displacement of the end of the bar (ilby using three bar elements of equallength and similarly evaluating the area at the rnidpoint of each, and {ii} using integration to obtain the exact solution.

'

Compare the results;Or l"L{l

A shaft is made of three parts as shown in Figure 1b. Parts AB and CD are made of the same material with modulus of rigidity. of 9.8 X 103 ksi, and each has a diameter of 1.5 in. Segment BC is made of material with a rnodulus of rigidity of 11.2X 103 ksi

and has a diameter of 1 in. The shaft is fixed at both ends. A torque of 2400 lb-in is applied at C. Using three elements,deterrnine the angle of twist at B and C and the torsional reactions at the boundaries,

ra^

a.zr%r the truss shown in Fig. 2, 0r = 45o and 02 = 0o, transform the element stiffness matrix of each element into globalv reference frame and assemble the global stiffness matrix.or [141

Explain in brief the weighted residual methods, Use Galerkin's method of weighted residuals to obtain a one-termapproximation to the solution of the differential equation t

11 +v=e, oszr Ida- .V{o)*Orytl)*l

Q.3. Derive the quadratic shape functions in terms of natural co-ordinates mapped between -L to +1 and 0 to 1 sf a onedimensional element. Write a note on compatibility and completeness requirements.

0r lt4lThe circular rod depict"ed in Figure 3 has an outside diameter of 60 mm, length of 1m, and is perfectly insulated on itscircumference. The left half of the cylinder is aluminum, for which k* = 200 W/m-'C and the right half is copper having k* =389Wm-"C. The extreme right end of the cylinder is maintained at a temperature of 80"C, while the left end is subjected to a

heat input rate 4000 Wlm 2. Using four equal-length elements, determine the steady-state temperature distribution in thecylinder.

,/St +.9{riue the shape functions in natural co-ordinates for any one of the following

i. Linear quadrilateral elements.ii. Linear triangular elements.

,Or [14]Using two dimensional triangular elements, the temperature distribution in a fin is modeled as shown in Figure 4. What is thevalue of the temperature at X = 2.15 cm and Y = L.1cm? Determine the components of temperature gradients for thiselement. Also determine the location of 70 "C and 75 "C.

,r'-Q.Sy'erivethe stiffness matrix for plane stress condition using linear triangular elements.\-,/ Af

Consider an overhang frame as shown in Figure 5. The frame is rnade of steel, with E = 30 X

deformation of thy'frame under the given distributed load.

[14]106 lb/in2, Determine the

Q.6. Usingtwo equal-length finite elements, determine the natural frequencies of the circular shaft as shown in Figure 6,

[10]Write short notes on Gauss-Legendre's quadra,ulu. furf rrlt ,t," totto*ing integral using two point sampling formulation of G-

L-Q.

-$

'' Ii,'", 'iwdt


Oct-Nov 2011

ME11715

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,= 89-'Yo6o44ME10715

B.TECH Seventh Semester Examination Nov - Dec 2010Mechanical Engineering


Max. Marks:8O Min. Marks: 28

Answer the followins questions. Assume data suitablv. if missinq. Time: 3 hrs

Q.1al Discuss in brief the difference between FEM, FVM and FDM. t6)b] A system of three linearly elastic springs supporting three equalweights, W, suspended in a vertical plane is

shown in fig. 1. Treating the springs as finite elements, find the vertical displacement of each weight. (10)

Or

Calculate the elongation of the end of a circular linearly tapering bar subjeeted to an axial load P as well as self

weight as shown in fig. 2. Assume three bar elements of equal length for modeling the bar. (10)

Q.2 al The members of the truss shown in fig.3 have a cross-section of 2.3 in2 and are made of aluminum alloy (E= 10'O X

106 lb/in2). Deterrnine the deflection of point A, stress in each member and the reaction forces using FE approach.(10)

ArA wall of an industrial oven consists of three different materials. The first layer is composed of 5 crn of insulating

cement with a clay binder that has a thermal conductivity of 0.08Wm.K. The second layer is made from L5 cm of6-ply asbestos board with a thermal conductivity of O.A74 W/m.K. The exterior consists of 10 cm comrnon brick

with a thermal conductivity of 0.72 Wm.K. The inside wall temperature of the oven is 200oC, and the outside air

is 30oC with a convection coefficient of 4O W/mz.K. Determine the temperature distribution along the composite

wall.bl Give a comparative mathematicaltreatment on weighted residual methods.

Q.3 al Derive the shape functions of quadratic quadrilateral elements'

b] Two dimensional triangular elernents are used tc determine the stress distribution in a machine part and their

corresponding positions and nodal stresses are shown in fig. 4. What is the value of stress at x=2.15 cm and Y = 1'1,

cm. Determine the location of constant-stress lines for stress values of 8.0 GPa and 7.7 GPa.Or

(8)

A two-dimensional triangular plane stress element made of steel, with modulus of elasticity E=200 GPa andpoisson's ratio p=6,32 is shown in fig" 5. The element is 3mm thick and the co-ordinates of the nodes are given in

cm. Determine the stiffness and load matrices under the given conditions. (8)

e.4 Derive the conductance matrices and load matrices for two dimensional conduction problems using bilinear

rectanguiar element. (16)

Or

al Explain (any two) briefly (8)

i. lso-parametric elements. ii. Different co-ordinate systems.

iii. Geometric isotropy. iv. Compatibility and completeness requirements of interpolation functions.

bl Evaluate the integral t =it*, +3x + Z)dx using Gauss-Legendre two point sampling formula' {S)

2

e.5 Explain the finite element formulation of beam elements with two degrees of freedom at each node using hermite

functions. (16)

Or

Find the natural frequencies of the longitudinal vibrations of the constrained stepped shaft as shown in fig. 6 and

(10)(6)

(B)

compare the results obtained uslng lumped and consistent mass matrix approach. (16)

712


M810715

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212


CSVTU

Time: Three Hours

NOTE: Aftempt any five guesfrbns. Assume suitable data if required.

TsnB+o I

Code: ME e 97 i5B. Tech. (Seventh Semester) Examination (April-May) 2010

Subiect: Finite Element Methods

Branch : Mechanical Engineering

Max. Marks: 80

Min. Pass Marks: 28

;:\{*Fa (a) Find the stiffness matrix for the simple

L/anO modulus of elasticity E.

,Q2 Discuss in brief

(a) Element shapes with neat sketches, and',

,.<61 Coord inate systems

.A.\ A3 Determine the shape functions for the Constant Strain Triangle (CST). Use/'\ /-I j runcilons.

'IOOO kt{lrrr

20$0 kN./er'

gl} 'kl,l

fig"

f 4 \ qF Using Rayleigh-Ritz method determine the expressions for the deflection and bendingI .5 | rL_-/ moments in a simply supported beam subjected to uniformly distributed load over entire

beam element of span L, moment of lnertia I

(b) Determine the values of (i) Lr3 Lz dA and (ii) Lt'Lr'Ls2 dA

(10)

(06)

(08)

(08)

polynomial

(16)

Q4 Determine the shape function foi'a 4 noded rectangular element. Use natural ccordinate

'system.

(16)

Q5 A set of springs connected together as shown below is subjected to some axial load of

1O kN, 20 kN at node points 1 ans 4, Determine the displacements of node 1 ,2 and 4.

(16)

span. Find the deflection and moment at midspan and exact soluiicns.

(16)


09715

ts"r\\\\\'\

Q7 Derive the expression for consistent load vector due to self weight in a CST element.

(16)

Qg (a) Write in brief about coordinate transformation. (Og)

(b) Write about lsoparametric, superparametric and subparametric elements. (08)

Q9 A beam of length 10 m, fixed at one end and supported by roller at the other end carries

a 20 kN concentrated load at the center of the span. By taking the modulus of elasticity of

material as 200 GPa and moment of inertia as 24 x 10-06 ma, determine

(i) Deflection under load

(ii) Shear force and bending moment at mid span

(iii) Reactions at support

(16)


CSVTUB. Tech. (Seventh Semester) Examlnation (Nov-Dec)

Subject: Finite Element Methods

Branch : Mechanical EngineeringTime: Three Hours

NorE: Attempt any five quesfrbns. Assume suitabte data if required.

$i;#Code: ME09715

2009

Max. Marks: 80

Min. Pass Marks: 28

Ql Find stiffness matrix for a two nodded element having one node of a beam element andthe other node as combination of bar and beam element.

Q2 (a) Classify element shapes with neat freehand sketches. What is the differen.- o"*"t::'Lagrange and Serendipity family of elements? (08)

(b) Write general form of two and three dimensional Polynomial shape function. Expressthem upto cubic element terms by a triangle and tetrahedron. (08)

Q3 Find shape function for a two nodded element having one node of a bar element andother node of a beam element.

(16)

Q4 Determine the displacement and stress in a bar of uniform cross section due to selfweight only when suspended vertically using three and four terms for the approximatingpolynomial. verify the expressions for total extension with exact value.

(16)

Q5 A thin steel plate of uniform thickness 1 inch and total length 24 inch is suspendedvertically. The upper part of plate is having uniform width of 5.25 inch and lower part is havinguniform width of 3.75 inch. Upper and lower parts are of equal lengths of 12 inch each. young,s

modulus E = 30 x 106 lb/inz and weight density p = 0.2g36 lb/in3. ln addition to its self weight, theplate is subjected to point loads P = 100 lb at its mid point and e = 50 lb at lower end. Analyzethe plate after modeling it with two elements. Find the global stiffness matrix, stresses in eachelement and support reactions.

(16)

Q6 A triangular truss ABC having hinge support at A and roller support at B, supporls avertical downward load of 150 kN at C. The members AC and BC are equally inclineci to basehorizontal member AB. The member AB is of length 800 mm with area of cross section 1S00mm?. The vertical height of the truss is 400 mm. Both members AC and AB have cross sectionatarea of 2000 mm2. Taking E = 200 GPa, determine the nodal displa""*5rf,, and the stresses ineach member.

*

(16)


Q7 A beam of length 10 m, fixed at one end and supported by a roller at the other end carriesa 20 kN concentrated load at midspan. Taking E = 200 GPa and I = 24 x 10-06 ma; determine

(i) Deflection under load' (ii) Shear force and bending moment at mid span

(iii) Reactions at supports.

(16)

Q8 Discuss in brief:

(i) Coordinate Transformation

(ii) Basic Theorems of lsoparametric concept

(iii) LinearConstitutiveEquations

(iv) Types of NaturalCoordinates

(16)


CSVTU Code:B. Tech. {seventh Senresten) Exarnination {April-May) 200g


Branch : Mechanical Engineering

Tirne: Three Hours Max. Marks:80

Min. Pass Marks: 28

NOTE: Attempt any five guesfions. Assume suitable data if required.

Ql Explain the terms 'Plane Stress' and ' Plane Strain' problems. Give constitutive laws for

these cases.

QA-z By direct stiffness matrix approach, determine stiffness matrix for truss element

(16)

(r6)

8g'/ {a) Derive the expressions for natural eoordinates for a two nodded element in terms of {when range is -1 lo 1@7

{08}(b) integrate the following over the entire length I of the element

t^(i) [ L1'dx (ii) J 1,3[-" dx

Qe LJsing generalized coorejinate approaeh, find shape functions

elernent.

85\-"" Using Lagrange Polynomial, find shape funetierns for

(i) Three noded bar element

(ii) Five noded bar element

QG Determine strain displacement matrix for CST element.

L,Jt'1 '.i I ",J | " j

(srof

for two noddecl bar I truss

(161

(16)

(16)

{:

PTC


Q7 Assemble equations of equilibrium for the spring system shown below by direct appro'ach: '

j F ; "-* f, -*f.

"l.i'*- 4.,/../ ". ,^ -\"/.,'...,?... --L---..-.\ "..\.,,',u*.'..:--.-. " '../,."'.^u;,.r14-r.-:,

3*'1 k.;2g"3*+u. lru, -*u,

(16)

AV' Derive the expression for consistent load vector due to self weight in a CST element.(16)

Qg Discuss in brief:

(i) Coordinate Transformation..;ais(ii) Basic Theorems of lsoparametric concept , :

(16)


.-,-Arrnll l*. <( t1 1'l\. r"B.Tech. ($eventh Semester) Examination (Nov-Dec) 200g


B ran c h : Mechanical Engineering

; , ,;' .)t ') F ().t!,/ tt'tttt/

nl Jtv./

Code: ME087/5CSVTU

Tirne: Three Hours

NorE: Attempt any five questions. Assume suitabte data if required,

Q3 U*ing Rayleigh-Ritz nreihcd dcterr.nlne th* expresei*nsslmpty supp*rted bee"m subject*d t* unifarmly <tristributed loadmoment at midspan and compare with exact solutions.

tI

iI

l

I

?4 in.

_l

Max. Marks:80

Min. Pass Marks: ZB

Q{ Derive the equations of equilibrium in case of a three dimensional stress system.

(16)

Q2 Determine the shape functions for 4 noded rectangular elernent. Use natural coordinate system.

{16}

for defisciion ..eii"id_bending n"iaments in eover entire span: Find the deflection and

{16}

Q4 The thin steel plate shown in figure has a uniform thickness t = 1 in., young,s modulusE 130 x 106 psi, and lveight density p = 0"2836 lb/in.3. ln addition to its self weight, the olate is subjectedto a point lciad F = 100|b atits.midpoint.

{i) Modelthe plate with two finite etenrents.

(ii) Write down expressions for the element

stiffness matrices and element body force vectors.(iii) Assemble the structural stiffness matrix K endglobal load vector F.

{iv) Evaluate the global displacement vector 6.

{v} Determine the stresses in each element.(vi) Find the reaction force at the support.

FTQ

(16)


PTO

Q5 The elements of four bar truss shown in figurehas Young's modulus E = 2g"5 x 106 psi and crossseetional area A* = 1 in.2. Complete the foilowing:

(i) Write down expressions for the elementstiffness matrix for each element.(ii) Assemble the structural stiffness matrix K forthe entire truss.

(iii) Evaluate the global displacement vector 6.(iv) Determine the stresses in each element.(v) Find the reaction force at the support.

Y

I8s

iIl. Ar@

*+ 0r

Qn

t| ,0,

A

u4

1L->O,

8$ {*} Uslng serendlpity cencept

(b) By degrading

technique develop shape

function for the seven nodedrectangular element from the

eight noded element as Ishown:

t 87 Using virtualdisplacement principle, deterrnine the forces developed g1

the three bar truss shgwn in figure below:

E=29.5x10'psia = t.tt:nl

,20 000 lb,.'.++l'

(10)

(r6)

25 0C0 tb

(16)

fgd shape fr:nctions for quadratic serendipity familSi element. t06i

ai'.J'\l\t\-l\lt\'1-r1"soXf-_*:_+_\Di._ a" e"\{

t,a4 /t7 / zokN1/t/x/a/1/31/1/1/)/1/1/

'weight in a CST element.Q8

{1s}


AICTE Code087025S

B.Tech(SeventhSemester)BxaminationNov-Dec.2008Subject: Finite Element Methods (Elective-Il)

Branch: Mechanical EngineeringM*x Marks:100

Time: Three Hours Minimum pass Martrs:rs

NOTE: Attempt anyfive questions. A!! questions ,::fltf! equal marks Assume suitable data ifrequired

Q.1 By direct stiffness matrix approach determine stiffness matrix for truss 20

element.

Derive expressions for natural coordinates in a constant strain Triangle 2a

(CST) element.

Determine the shape funbtions for 4 noded rectangles element' IJse naturai

coordinates sYstem.

using serendipity concept find shape functions fro quadratic serendipity

family element.

Derii,e stiilness matrix for a cST ele,inent by direct approach'

I)eterrnics ihe r;ir-en:;icr: li a ta'nerr:d bar r,'f iength 5c0n:m 'jue i" seif

weight and a concentrated toa,i oi 400 N applieci at its lower end' Given

the width at top fixed end as 150mrn, width at lower free end as 75mm

and thiokness as 20mm.

Diseuss

(a) Coordinate transformation

(b) lsoperimetric, super parametrie & sub parametric elements

(c) Basic Theorem of isoperimetric concept'

Derive the expression for consistent load, which varies linearly from Pr at

node 1 to Pz at node 2 on a beam element of length le'

Q.2

Q.3

Q.4

Q.s

Q.6

Q.7

Q.8

20

20

^r /-l

20

20

20


4T2Find natural coordinates for triangular element in terms of coordinates of nodes.

uf{Qdi'" Assemble the elennent properties fon a bar with unif*rmly varying area subject to sei? v,,cight.*dian'reter

at support as br and at free end below ss b2. Treat it as single element.

.r'' 0R

L€l Vy:rte in brief about coordinate transformation.

i lpywrite about tsoparametric, superparametric and subparametric elements.{

.".".F { ( }zalt ,**'

FO" *

Determine the shape function for a 4 noded rectangular element, Use naturat coordirhii(vstem.

oR 4t\Using serendipity concept find shape functions for quadratic serendipity fami'ly element.

{20}

Find stiffness matrix for CST element by direet approaeh.

OR.

Show that in elasticitii ptoblems Galerkin's method turns out tr: be the principle of virtr"lal work.

Q5 A beam of length 10 m, fixed at one end and suppofted by roller at the other end carries a 20 kN

concentrated load at the center of the span, By taking the modulus of elasticity of material as 200 GFa and

moment of inertia as 24 x 10-06 ma, determine


(ii) Shear fgrce and bending moment at,mid span

(iii) Reactions at support l

Code:8+V025 52008

Max. Marks: 100

Min. Pass Marks: 35

tt(

AICTE

tW" Determine stiffness matrix for a truss element.v-OR

B,Tech. (Seventh Semester) Examination (April-May)


B rnnch : Mechanical Engineering

. Time: Three Hours

'NOTE: Attempt all questions. Assume suitable data if required.

-i- -" t. ^t dRe

{2e}

{r0}

{10}

(20)

077025-S


' u_J "1

AICTE

Time: Three Hours

Note: Attempt any fiverequired.

B.Tech. (Seventh Semester) Examinatlon Dec-Jan 2007-08Subject: Finite Element Method (Elective fl)

Branch: Mechanical Engineering

Q1 AtteptRt any two of the following:

(fi'Derive the equations of equilibrium in case of a three dimensional stress system. 10

(b) State and explain generalized Hooke's law. 10

(c) Briefly explain various attempts made to reduce memory requirement in storing

stiffness matrix.

Explain the situations where yCIu need 4 noded and 5 noded triangular. elements.

degradation technique der,ive the shape functions for 5 noded triangular elemgnt.

berive the shape functions for the nine-node rectangular element shown :

2S

'A set of springs connected together as shown below is subjected to some axial load of

10 kN, 20 kN at node points 1 ans 4. Determine the displacements of node 1,2 and 4.

3S

ByJA?

Q32A

Q41,'

:$ffi-P3*-*1&!{*u. 3tr&0 kr-l/m f* "' I

:1*_--r,hrl,.*--- t at 3 | r SOO kNrrn ILr&r) lcNJrn {JJr\"lv{ff---1'.1\'."r

SfA-lt{

1* n 4 ,\---{1f-v "- i

20

{,,r t nnl' $,1u"'-

Code :077025 M

Max Marks: 100Min Pass llllarks: 35

questions. All questions carry equal marks. Assume suitable data if

Fig. 1

rr0


PTO

Q5 Determine the deflection and stresses in the tlrree different rods as shown:

FsU.61'X1t15li',fiyrm"

:tf){} ri rr}1

2tfl) rnrn

H*0 rnrrEeirF-3x1r35r.il/lr !f,tr z

the eartesian coorciinate,ef the point F({ = 0.5, n = 0.6} as shown

{9, 1) ,--. .. .-*t X

Fis. 6 a

2A

in\J{rJ ia) Determine

figure below:

v

{b) Find t!"re end moments and reactions

positive joint displacement at x = 0'

t-r/it {*

"3for the beam shown tn

{0

figure below with

{ 1-- *----t

*.I

*ltft.2lqr

ris. 6 b

:10

Derive the local finite element stiffness matrix for a beam with combined transver$e

--r-rfiI

._3_ )r4'-L;

a

'Fig.

Q7

loading and axial force.20


Jiir*T'E

Tirsle: Three H*urs

Note: Attempt any fir.'e questions' Aii questionsrequired"

Qt Atternpt any two of the following:

.. .l

J jt ' '':;' "

lJ.'T-eeh. {Severrtir Ses:',e*ter} E*am!natEon $ec-.}e*n i{Siv'i -rif}

Sulrject: Frmif* Efe'r**nt f{fetfued {Efe*ttve fliBran*lr : Ft'reehcnicaE Hngine*ring

*,:cie :ij? 'i i-i:i 5 rv1

(a) Derive the equations of equiiibrium in case of a three dimensional stress systern.

(b) State and explain generalized Hooke's law.

(c) Briefly explain vai-ious aitenrpts nrade to reduce memory requirement irr storing

stiffness mairix.

Explain the situations v,'t1ere you need 4 noded and 5 nodeci triangular elenrenis.

degradation technique derive ihe shape functions for 5 noded ti-iangular elernent.

Derive the ghape functions for the nine-node redangular element shown :

"1

F'ig. 3

J A set af spr-ing* connected tr:geth*r as shown belavu is subjectecj ta same axiar! i*eci

10 kN, 20 irf{ at node points i arrs 4. Deternrirre the displacen"rents af node 1,2 ai'r-t r'

L4ax fsrlarF.,s: 1er0

lt'ilrL Fass F,ir I i::: 3 5

carry equal nrarks. Assume suitable ejate,: !f

JS

tu

I {r.

By

rQzftl

Ui

lJ{ll trN''nt, ' - -'ii,'"i;ui-_

- ilX{jr,j l":i,11,, Iii--4,\rr,i,,r,'.".- $ :

i) *5 f i l)fl'0 hiN irr: ! i:*$ k I'litrl f'l* ',r,,!'rlt'r-----,1-. lr',iir';l.*---a1.,1 I irr

1

l-.+' tN'k$irIE. 4'

iJtl

Q3

r.79 - :'-'-------ltiiI I i6;: i'.J *r f.-i-. i

l ilo'l ij_ _- I ii l"i €--:' -l- 1,, I--", --1" a'r-;*--Btx I i i I.-, ' o* *" *l alz

I lr' I

vrtof f\| !(-/


PTO

iz-":rt(t cs l\_---'

."

Deiermine the defiection and stresses in the tltree different rods as shown:

t.-, \ \

\, L/

| 4 zomntE;=O.r:rxru5 l*Al-

.\i;;i;;;l o'l l"'-" /v

E:*,i,"*"s{{.U$'l. T Bnft ..h rJ "L- 3 '1

ft

&I

\v". \){u

- j+ + il a o'1'p e 1a23* ',.rtlf-Fi;:

r{/rnr,rzr i l*"0o'..,*, eu cr-) {-1.b

Cartesian coordinate of the point P(€ = 0.5, q = 0'6) as shown

20

ir. '?

ta) Determine the

figure below:

vtt

II

- .t lY++^ ; '--f; -'-.- I !

Fii+. 6 a-

t0

{b} Finci the end moments and reactions for the beam shown in figure beronr wir#

positive joint displacement at x = 0'

10

Derive the localflnite element stiffness matrix ior a beam with eornbined transversc

'' | 20

loading and axial force. *+J * _ {

Fig.

YQ7


7,f",4+^

Code: 067025.1

Max. Marks: 100

Min. Pass h{arta: 35

moment of inertia I and

B.E. ( seventh seTi$f?Affition' Nov'-Dec' 2006


Branch: Mechanical En gineering

Time: Three Hours

Ql(a)FindthestiffnessmatrixforthesimplebeameieirrentofspanL.modulus of elasticitY E'

t-(b) Derennine the values of tii {

i'r3 Lz dA and fiilflr'? L22 L32 dA

Q2 (a). Determine the shape functions for the constant Strain Triangle' use polynomial functions' 10

(b)UsingLagrangepollrromialfindshapefunctionsforthreenodded'oarelement'Alsodrawits1Guanarinnilong the length of element'

Q3 (a) using virtuai displacerneirt principle, determine the lorces deveioped in the thiee bar truss as

shoun below:

OR

Using Rayleigh-Ritz method determine the expressions fo.r.the oeT::llending moments in a

simply supporred b;;U#Jt".""ii"r-ry distributed load over entire span' Find the deflection

;;;"#;t al midspan and exact solutions' 2a

e4 A thin plate of uniform thickness 20 mlg and le'gth 500 mm is hanged verticalry. The width for

haif the length of plate, near the ,,rpport] i, 'ii;.J*-';"hiL,

1ot other harf width is 100 mm' In addition

to the seif weight, the plate is subjected io u point loacl ef 4_00 N at mid*Jepth. The Young's modulus

E = zx 10s N/mm2 and unit weight r :;J; ioi. ),'ll'' Anaivse-th:.1]1t:-:fttr nrodeling it rvith

two eiements and find the siresses ii-r gach elemeni' Deietmine support reactions aiso' ' ;

10

10

1-lIL

08

C

(b) Discuss in brief about trpe of discontiriui

:i \l.{\1 ''-7z3o*'-^--.--.--:'4+-- e.n --!:1

I11.."t .'41/=) /'ta/j,/.:1

:l:l/:l/+'

ties in a strt

'^rAY. t. v'PTO


{a)

(b)

| :t j:.

!,,:n: I

ll,l,i1.'

r 'l::':li l r'ii ..,:jl.I

!. 1.

OR

Explain with equations, the coordinate transformation for isoparametric elements.10

Discuss in brief the basic theorems of isoparametric concept.10

e5 A bearr, of length l0 m, fixed at one end and suppoflcd b1'a rolle^r at the.other end, carries a 20 klrr

clncentrated load at the center oqthe span. By taking the modulus of elasticity of material as 200 Gpa

una *o*"nt of inertia as24 x 10-6 ma. determine:


(ii) Shear force and bending moment at mid span

(iii) Reactions at suPPons

,OR

Fincl elercenr stiffrress ald mass matrices for a rod subjected to axial vibrations.


fem nov-dec 2006 to 2013

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