FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK

ME2353 – FINITE ELEMENT ANALYSIS VI SEM MECHANICAL ENGINEERING REGULATION 2008

QUESTION BANK, CLASS NOTES & FORMULA BOOK PREPARED BY ASHOK KUMAR. R M.E (CAD)

R.M.K COLLEGE OF ENGINEERING AND TECHNOLOGY RSM NAGAR, PUDUVOYAL-601206

ASSIGNMENT QUESTIONS

ME2353 – FINITE ELEMENT ANALYSIS III B.E., VI Sem., ME DEC’13 – MAY’14

UNIT – I – FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS PART – A 1.1)

What is the finite element method?

1.2)

How does the finite element method work?

1.3)

What are the main steps involved in FEA.

1.4)

Write the steps involved in developing finite element model.

1.5)

What are the basic approaches to improve a finite element model?

[AU, April / May – 2011]

[AU, Nov / Dec – 2010] 1.6)

Write any two advantages of FEM Analysis.

[AU, Nov / Dec – 2012]

1.7)

What are the methods generally associated with finite element analysis?

1.8)

List any four advantages of finite element method.


1.9)

What are the applications of FEA?


1.10)

Define finite difference method.

1.11)

What is the limitation of using a finite difference method? [AU, April / May – 2010]

1.12)

Define finite volume method.

1.13)

Differentiate finite element method from finite difference method.

1.14)

Differentiate finite element method from finite volume method.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 1

R.M.K COLLEGE OF ENGG AND TECH / AQ / R2008/ ME2353 / VI / MECH / DEC 2013 – MAY 2014

1.15)

What do you mean by discretization in finite element method?

1.16)

What is discretization?

[AU, Nov / Dec – 2010]

1.17)

List the types of nodes.

[AU, May / June – 2012]

1.18)

Define degree of freedom.

1.19)

What is meant by degrees of freedom?

1.20)

State the advantage of finite element method over other numerical analysis

[AU, Nov / Dec – 2012]

methods. 1.21)

State the fields to which FEA solving procedure is applicable.

1.22)

What is a structural and non-structural problem?

1.23)

Distinguish between 1D bar element and 1D beam element. [AU, Nov / Dec – 2009, May / June – 2011]

1.24)

Write the equilibrium equation for an elemental volume in 3D including the body force.

1.25)

How to write the equilibrium equation for a finite element? [AU, Nov / Dec – 2012]

1.26)

Classify boundary conditions.

1.27)

What are the types of boundary conditions?

1.28)

What do you mean by boundary condition and boundary value problem?

1.29)

Write the difference between initial value problem and boundary value problem.

1.30)

What are the different types of boundary conditions? Give examples.

[AU, Nov / Dec – 2011]

[AU, May / June – 2012] 1.31)

List the various methods of solving boundary value problems. [AU, April / May – 2010]

1.32)

Write down the boundary conditions of a cantilever beam AB of span L fixed at A and free at B subjected to a uniformly distributed load of P throughout the span. [AU, May / June – 2009, 2011]

1.33)

Briefly explain force method and stiffness method.



1.34)

What is aspect ratio?

1.35)

Write a short note on stress – strain relation.

1.36)

Write down the stress strain relationship for a three dimensional stress field. [AU, April / May – 2011]

1.37)

State the effect of Poisson’s ratio.

1.38)

Define total potential energy of an elastic body.

1.39)

Write the potential energy for beam of span L simply supported at ends, subjected to a concentrated load P at mid span. Assume EI constant. [AU, April / May, Nov / Dec – 2008]

1.40)

State the principle of minimum potential energy. [AU, Nov / Dec – 2007, 2013, April / May – 2009]

1.41)

How will you obtain total potential energy of a structural system? [AU, April / May – 2011, May / June – 2012]

1.42)

Write down the potential energy function for a three dimensional deformable body in terms of strain and displacements.

1.43)

[AU, May / June – 2009]

What should be considered during piecewise trial functions? [AU, April / May – 2011]

1.44)

What do you understand by the term “piecewise continuous function”? [AU, Nov / Dec – 2013]

1.45)

Name the weighted residual methods.

[AU, Nov / Dec – 2011]

1.46)

What is the use of Ritz method?

[AU, Nov / Dec – 2011]

1.47)

Mention the basic steps of Rayleigh-Ritz method.

1.48)

Highlight the equivalence and the difference between Rayleigh Ritz method and the finite element method.

1.49)


[AU, Nov / Dec – 2012]

Distinguish between Rayleigh Ritz method and finite element method. [AU, Nov / Dec – 2013]



1.50)

Distinguish between Rayleigh Ritz method and finite element method with regard to choosing displacement function.

1.51)

[AU, Nov / Dec – 2010]

Why are polynomial types of interpolation functions preferred over trigonometric functions?

[AU, April / May – 2009, May / June – 2013]

1.52)

What is meant by weak formulation?

[AU, May / June – 2013]

1.53)

Define the principle of virtual work.

1.54)

Differentiate Von Mises stress and principle stress.

1.55)

What do you mean by constitutive law?[AU, Nov / Dec – 2007, April / May – 2009]

1.56)

What are h and p versions of finite element method?

1.57)

What is the difference between static and dynamic analysis?

1.58)

Mention two situations where Galerkin’s method is preferable to Rayleigh – Ritz method.

[AU, Nov / Dec – 2013]

1.59)

What is Galerkin method of approximation?

[AU, Nov / Dec – 2009]

1.60)

What is a weighted resuidal method?

[AU, Nov / Dec – 2010]

1.61)

Distinguish between potential energy and potential energy functional.

1.62)

Name any four FEA software PART – B

1.63)

Explain the step by step procedure of FEA.

[AU, Nov / Dec – 2010]

1.64)

Explain the general procedure of finite element analysis.

[AU, Nov / Dec – 2011]

1.65)

Briefly explain the stages involved in FEA.

1.66)

Explain the step by step procedure of FEM.

1.67)

List out the general procedure for FEA problems.

1.68)

Compare FEM with other methods of analysis.

[AU, Nov / Dec – 2010]

1.69)

Define discretization. Explain mesh refinement.

[AU, Nov / Dec – 2010]

1.70)

Explain the various aspects pertaining to discretization, process in finite element modeling analysis.

[AU, Nov / Dec – 2011] [AU, May / June – 2012]

[AU, Nov / Dec – 2013]



1.71)

Explain the process of discretization of a structure in finite element method in detail, with suitable illustrations for each aspect being & discussed. [AU, Nov / Dec – 2012]

1.72)

Discuss procedure using the commercial package (P.C. Programs) available today for solving problems of FEM. Take a structural problem to explain the same. [AU, Nov / Dec – 2011]

1.73)

State the importance of locating nodes in finite element model. [AU, Nov / Dec – 2011]

1.74)

Write a brief note on the following. (a) isotropic material (b) orthotropic material (c) anisotropic material

1.75)

What are initial and final boundary value problems? Explain. [AU, Nov / Dec – 2010]

1.76)

Explain the Potential Energy Approach

[AU, Nov / Dec – 2010]

1.77)

Explain the principle of minimization of potential energy. [AU, Nov / Dec – 2011]

1.78)

Explain the four weighted residual methods.

1.79)

Explain Ritz method with an example.

1.80)

Explain Rayleigh Ritz and Galerkin formulation with example.

[AU, Nov / Dec – 2011] [AU, April / May – 2011]

[AU, May / June – 2012] 1.81)

Write short notes on Galerkin method?


1.82)

Discuss stresses and equilibrium of a three dimensional body. [AU, May / June – 2012]

1.83)

Derive the element level equation for one dimensional bar element based on the station- of a functional.

[AU, May / June – 2012]



1.84)

Derive the characteristic equations for the one dimensional bar element by using piece-wise defined interpolations and weak form of the weighted residual method? [AU, May / June – 2012]

1.85)

Develop the weak form and determine the displacement field for a cantilever beam subjected to a uniformly distributed load and a point load acting at the free end. [AU, Nov / Dec – 2013]

1.86)

Explain Gaussian elimination method of solving equations. [AU, April / May – 2011]

1.87)

Write briefly about Gaussian elimination?

1.88)

The following differential equation is available for a physical phenomenon.

(


)

Boundary conditions are, x = 1

u=2

x =2 Find the value of the parameter a, by the following methods. (i) Collocation

(ii) Sub – Domain

(iii) Least Square

(iv) Galerkin 1.89)

The following differential equation is available for a physical phenomenon.

Trial function is

(

Boundary conditions are, y (0) = 0

) y (10) = 0

Find the value of the parameter a, by the following methods. (i) Collocation

(ii) Sub – Domain

(iii) Least Square

(iv) Galerkin FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 6


1.90)

Discuss the following methods to solve the given differential equation : ( ) with the boundary condition y(0) = 0 and y(H) = 0 (i) Variant method

1.91)

(ii) Collocation method.


A cantilever beam of length L is loaded with a point load at the free end. Find the maximum deflection and maximum bending moment using Rayleigh-Ritz method

( ) Given: EI is constant.

using the function

[AU, April / May – 2008] 1.92)

A simply supported beam carries uniformly distributed load over the entire span. Calculate the bending moment and deflection. Assume EI is constant and compare the results with other solution.

1.93)

[AU, Nov / Dec – 2012]

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

1.94)

( )

(

)

[AU, Nov / Dec – 2008]

Compute the value of central deflection in the figure below by assuming

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

[AU, Nov / Dec – 2007, April / May – 2009] 1.95)

If a displacement field is described by

(

)



(

)

Determine the direct strains in x and y directions as well the shear strain at the point x = 1, y =0. 1.96)


In a solid body, the six components of the stress at a point are given by x= 40 MPa, y = 20 MPa, z = 30 MPa, yz = -30 MPa, xz = 15 MPa and xy = 10 MPa. Determine the normal stress at the point, on a plane for which the normal is (nx, ny, nz) = ( ½, ½,

1.97)

1

2

)

In a plane strain problem, we have x = 20,000 psi

y = - 10,000 psi

E = 30 x 10 6 psi,  = 0.3.

Determine the value of the stress z. 1.98)

For the spring system shown in figure, calculate the global stiffness matrix, displacements of nodes 2 and 3, the reaction forces at node 1 and 4. Also calculate the forces in the spring 2. Assume, k1 = k3 = 100 N/m, k2 = 200 N/m, u1 = u4= 0 and P=500 N.

1.99)


Use the Rayleigh – Ritz method to find the displacement of the midpoint of the rod shown in figure.




1.100) Consider the differential equation to boundary conditions

( )

subject ( )

The functional corresponding to this

( )

∫

problem, to be extremized is given by

1.101) Find the solution of the problem using Rayleigh-Ritz method by considering a twoterm solution as ( )

(

)

(

) [AU, Nov / Dec – 2009]

1.102) A bar of uniform cross section is clamped at one end and left free at the other end. It is subjected to a uniform load axial load P as shown in figure. Calculate the displacement and stress in the bar using three terms polynomial following Ritz method. Compare the results with exact solutions.

[AU, May / June – 2011]

1.103) A simply Supported beam subjected to uniformly distributed load over entire span and it is subjected to a point load at the centre of the span. Calculate the deflection FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 9


using Rayleigh-Ritz method and compare with exact solutions. [AU, May / June – 2013] 1.104) A simply supported beam (span L and flexural rigidity EI) carries two equal concentrated loads at each of the quarter span points. Using Raleigh – Ritz method determine the deflections under the two loads and the two end slopes. [AU, April / May – 2009] 1.105) Analyze a simply supported beam subjected to a uniformly distributed load throughout using Rayleigh Ritz method. Adopt one parameter trigonometric function. Evaluate the maximum deflection and bending moment and compare with exact solution.

[AU, Nov / Dec – 2010]

1.106) Solve for the displacement field for a simply supported beam, subjected to a uniformly distributed load using Rayleigh – Ritz method. [AU, Nov / Dec – 2013] 1.107) Use the Rayleigh – Ritz method to find the displacement field u(x) of the rod as shown below. Element 1 is made of aluminum and element 2 is made of steel. The properties are 70 GPa

A1 = 900 mm2 L1 = 200 mm

Est = 200 GPa

A2 = 1200 mm2 L2 = 300 mm

Eal =

Load = P = 10,000 N. Assume a piecewise linear displacement.

Field u = a1 + a2x for 0  x  200 mm, and u = a3 + a4 x for 200  x  500 mm. 1.108) A fixed beam length of 2L m carries a uniformly distributed load of a w(in N / m) which run over a length of ‘L’ m from the fixed end, as shown in Figure. Calculate the rotation at point B using FEA.

[AU, Nov / Dec – 2011]



1.109) Analyze the beam shown in figure using finite element technique. Determine the rotations at the supports. Give E = 200GPa and I = 4 * 106mm4 [AU, Nov / Dec – 2013]

1.110) A rod fixed at its ends is subjected to a varying body force as shown in Figure. Use the Rayleigh-Ritz method with an assumed displacement field

( )

to find the displacement u(x) and stress σ(x). Plot the variation of the stress in the rod.

[AU, Nov / Dec – 2012]

1.111) A uniform rod subjected to a uniform axial load is illustrated in Figure. The deformation of the bar is governed by the differential equation given below. Determine the displacement using weighted residual method. [AU, April / May – 2011] FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 11


1.112) A steel rod is attached to rigid walls at each end and is subjected to a distributed load T(x) as shown below. a)

Write the expression for potential energy.

b)

Determine the displacement u(x) using the Rayleigh – Ritz method. Assume a displacement field u(x) = a0 + a1 x + a2 x2.

1.113) Derive the stress – strain relation and strain – displacement relation for an element in space. 1.114) Derive the equation of equilibrium in case of a three dimensional stress system. [AU, Nov / Dec – 2008] 1.115) What is constitutive relationship? Express the constitutive relations for a linear elastic isotropic material including initial stress and strain. [AU, Nov / Dec – 2009] 1.116) Give a detailed note on the following: (a) Rayleigh Ritz method

(b)

Galerkin method

(c) Least square method and

(d) Collocation method



1.117) Find the approximate deflection of a simply supported beam under a uniformly distributed load ‘P‘ throughout its span. Using Galerkin and Least square residual method.

[AU, May / June – 2011]

1.118) Solve the differential equation for a physical problem expressed as with boundary conditions as y (0) = 0 and y (10) = 0 using (i)

Point collocation method

(ii)

Sub domain collocation method

(iii)

Least squares method and

(iv)

Galerkin method.

[AU, May / June – 2013]

1.119) Solve the differential equation for a physical problem expressed as with boundary conditions as y (0) = 0 and y (10) = 0 using the

(

trail function

) Find the value of the parameters a1 by the

following methods. (i)

Point collocation method

(ii)

Sub domain collocation method

(iii)

Least squares method and

(iv)

Galerkin method.

[AU, Nov / Dec – 2011]

1.120) Solve the following equation using a two – parameter trial solution by the (a) Collocation method (

)

(b) Galerkin method. Then, compare the two solutions with the exact solution

y (0) = 1 1.121) Determine the Galerkin approximation solution of the differential equation FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 13


( )

()

1.122) Solve the following differential equation using Galerkin’s method.

( )

( )


1.123) A physical phenomenon is governed by the differential equation The boundary conditions are given by ( )

( )

( ) with,

( )

. By taking two-term trial solution as

( )

(

)

(

the problem using the Galerkin method.

( )

) find the solution of [AU, Nov / Dec – 2009]

1.124) Determine the two parameter solution of the following using Galerkin method.

( )

( )

[AU, Nov / Dec – 2012]

1.125) Give a one – parameter Galerkin solution of the following equation, for the two domain’s shown below.

  2u  2u     1. 2  x 2  y  

1.126) Describe the Gaussian elimination method of solving equations. [AU, April / May – 2011] 1.127) Explain the Gaussian elimination method for the solving of simultaneous linear algebraic equations with an example.




1.128) Solve the following system of equations using Gauss elimination method. [AU, Nov / Dec – 2010] x1 – x2 + x3 = 1 -3x1 + 2x2 – 3x3 = -6 2x1 – 5x2 + 4x3 = 5 1.129) Solve the following system of equations by Gauss Elimination method. 2x1 – 2x2 – x4 = 1 2x2 + x3 + 2x4 = 2 x1 – 2x2 + 3x3 – 2x4 = 3

[AU, May / June – 2012]

x2 + 2x3 + 2x4 = 4 1.130) Solve the following equations by Gauss elimination method. 28r1 + 6r2 = 1 6r1 + 24r2 + 6r3 = 0 6r2 + 28r3 + 8r4 = -1 8r3 + 16r4 = 10

[AU, Nov / Dec – 2010, 2012]

1.131) Use the Gaussian elimination method to solve the following simultaneous equations: 4x1 + 2x2 – 2x3 – 8x4 = 4 x1 + 2x2 + x3 = 2 0.5x1 – x2 + 4x3 + 4x4 = 10 –4x1 – 2x2 – x4 = 0


1.132) Solve the following system of equations using Gauss elimination method. x1 + 3x2 + 2x3 = 13 – 2x1 + x2 – x3 = –3 - 5x1 + x2 + 3x3 = 6

[AU, Nov / Dec – 2009]



UNIT – II – ONE DIMENSIONAL FINITE ELEMENT ANALYSIS PART – A 2.1)

Write a note on node numbering scheme.

2.2)

What do you mean by node and element?

2.3)

What are the types of problems treated as one dimensional problem? [AU, May / June – 2013]

2.4)

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


2.5)

Define shape function.

[AU, Nov / Dec – 2007, April / May – 2009]

2.6)

What is a shape function?

2.7)

Differentiate shape function from displacement model.

2.8)

Draw the shape function of a two noded line element.

2.9)

Draw the shape function of a two noded line element with one degree of freedom at each node.

2.10)

[AU, Nov / Dec – 2009]


[AU, Nov / Dec – 2010]

Draw the shape function for one dimensional line element with three nodes. [AU, April / May – 2009]

2.11)

State the properties of stiffness matrix.

[AU, Nov / Dec – 2009, 2010, 2011]

2.12)

List out the stiffness matrix properties.

[AU, May / June – 2012]

2.13)

State the characteristics of shape function.

[AU, May / June – 2011]

2.14)

List the characteristics of shape functions.


2.15)

When does the stiffness matrix of a structure become singular? [AU, Nov / Dec – 2012]

2.16)

State the significance of shape function.

2.17)

Write the element stiffness matrix for a two noded linear element subjected to axial loading.



2.18)

Write the stiffness matrix for the simple beam element given below. [AU, Nov / Dec – 2008]

2.19)

What are the properties of global stiffness matrix?


2.20)

Write the properties of Global Stiffness Matrix of a one dimensional element. [AU, May / June – 2012]

2.21)

Differentiate global stiffness matrix from elemental stiffness matrix.

2.22)

What do you mean by banded matrix?

2.23)

How will you find the width of a band?

2.24)

How do you calculate the size of the global stiffness matrix?

2.25)

List the properties of the global stiffness matrix.

2.26)

What is the effect of node numbering on assembled stiffness matrix?


[AU, Nov / Dec – 2013] 2.27)

Give a brief note on the following (a) elimination approach

(b) penalty approach.

2.28)

Name the factors which affect the number element in the given domain.

2.29)

State the requirements to be fulfilled by the approximate solution for its convergence towards the actual solution.

2.30)

What do you mean by continuity weakening?

2.31)

Compare the linear polynomial approximation and quadratic polynomial approximation.

2.32)

Why polynomials are generally used as shape function?

[AU, Nov / Dec – 2011]

2.33)

Why are polynomial terms preferred for shape functions in finite element method? [AU, April / May – 2011]

2.34)

What do you mean by error in FEA solution?



2.35)

What are the types of load acting on the structure?

2.36)

Define traction force (T).

2.37)

State the assumptions are made while finding the forces in a truss. [AU, Nov / Dec – 2011]

2.38)

How are thermal loads input in finite element analysis? [AU, Nov / Dec – 2007, April / May – 2009]

2.39)

What is an interpolation function?

2.40)

Why are polynomial types of interpolation functions preferred over trigonometric functions?

[AU, May / June – 2012]


2.41)

What is an equivalent nodal force?

2.42)

What are called higher order elements?


[AU, April / May – 2008, Nov / Dec – 2010, 2011] 2.43)

What is higher order element?

[AU, Nov / Dec – 2011]

2.44)

What do you mean by higher order elements?

[AU, Nov / Dec – 2008]

2.45)

Why higher order elements are required for FE analysis?

[AU, Nov / Dec – 2012]

2.46)

What are higher order elements and why are they preferred? [AU, April / May – 2011]

2.47)

What are the characteristics of shape functions?

2.48)

Plot the variations of shape function for 1 – D beam element. [AU, Nov / Dec – 2010]

2.49)

Illustrate element connectivity information considering beam elements. [AU, Nov / Dec – 2013]

2.50)

When do we resort to 1 D quadratic spar elements?


2.51)

Obtain any one shape function for a quadratic cubic spar element. [AU, Nov / Dec – 2013]



2.52)

Mention two advantages of quadratic spar element over linear spar element. [AU, Nov / Dec – 2013]

2.53)

Give a brief note on the sources of error in FEA.

2.54)

State the significance of post processing the solution in FEA.

2.55)

What do you know about radially symmetric problem?

2.56)

Write the boundary condition for a cantilever beam subjected to point load at its free end.

2.57)

For a one dimensional fin problem, what are all the boundary conditions that can be specified at the free end?

2.58)

Determine the load vector for the beam element shown in Figure

[AU, Nov / Dec – 2012] 2.59)

Write the element stiffness matrix of a truss element.

[AU, May / June – 2012]

2.60)

Sketch a typical truss element showing local global transformation. [AU, April / May – 2011]

2.61)

Differentiate global and local coordinates.

[AU, May / June – 2013]

2.62)

State the differences between a bar element and a truss element. PART – B

2.63)

What are the different types of elements? Explain the significance of each. [AU, Nov / Dec – 2010]

2.64)

Derive and sketch the quadratic shape function for the bar element. [AU, May / June – 2011]

2.65)

Derive the shape function of a quadratic 1 – D element.

[AU, Nov / Dec – 2011]



2.66)

Derive the shape functions for one dimensional linear element using direct method. [AU, May / June – 2013]

2.67)

Determine the shape function and element matrices for quadratic bar element. [AU, May / June – 2012]

2.68)

Derive the stiffness matrix and finite element equation for one dimensional bar. [AU, Nov / Dec – 2011]

2.69)

Derive the stiffness matrix and body force vector for a quadratic spar element. [AU, Nov / Dec – 2013]

2.70)

Obtain an expression for the shape function of a linear bar element. [AU, April / May – 2011]

2.71)

Derive shape functions and stiffness matrix for a 2D rectangular element. [AU, Nov / Dec – 2012]

2.72)

Consider the rod (a robot arm) as shown below, which is rotating at constant angular velocity  = 30 rad/sec. Determine the axial stress distribution in the rod, using two quadratic elements. Consider only the centrifugal force. Ignore bending of the rod.

2.73)

A link of 2m, pin – jointed at one end, is rotating at angular velocity 5 rad / sec. the cross – sectional area of link is 2 * 10-3 m2. Determine the nodal displacements using two linear spar elements. Take E = 200GPa and ρ = 7850 kg/m3. [AU, Nov / Dec – 2013]



2.74)

A steel rod of length 1m is subjected to an axial load of 5 kN as shown in figure. Area of cross section of the rod is 250 mm2. Using 1 – D element equation solve for the deflection of the bar, E = 2*105 N/mm2. Use four elements. [AU, Nov / Dec – 2010]

2.75)

A column of length 500mm is loaded axially as shown in figure. Analyze the column and evaluate the stress and strain at salient points. The Young’s modulus can be taken as E. Take A1 = 62.5mm2 and A2 = 125mm2 [AU, April / May – 2009]



2.76)

Consider a bar as shown in figure. Young’s Modulus E = 2*10 5 N/mm2. A1 = 2 cm2, A2 = 1 cm2 and force of 100 N. Determine the nodal displacement. [AU, Nov / Dec – 2010]

2.77)

Consider the bar shown in Figure Axial force P = 30 kN is applied as shown. Determine the nodal displacement, stresses in each element and reaction forces [AU, May / June – 2012]

2.78)

Consider the bar as shown in figure. Axial force P1 = 20 kN and P2 = 15 kN is applied as shown in figure. Determine the nodal displacements, stresses in each element and reaction forces.




2.79)

Find the nodal displacement and elemental stresses for the bar shown in Figure. [AU, April / May – 2011]

2.80)

An axial load P = 300 x 103 N is applied at 200C to the rod as shown below. The temperature is then raised to 600C a)

Assemble the stiffness (K) and load (F) matrices.

b)

Determine the nodal displacements and element stresses.



2.81)

The stepped bar shown in fig is subjected to an increase in temperature, T=80o C. Determine the displacements, element stresses and support reactions. [AU, Nov / Dec – 2009]

2.82)

Axial load of 500N is applied to a stepped shaft, at the interface of two bars. The ends are fixed. Obtain the nodal displacements and stresses when the element is subjected to all in temperature of 100˚C. Take E1 = 70*103 N/mm2, E2 = 200*103 N/mm2, A1 = 900mm2, A2 = 1200mm2, α1 = 23*10-6 / ˚C, α2 = 11.7*10-6 / ˚C, L1 = 200mm, L2 = 300mm.

2.83)

[AU, Nov / Dec – 2011]

Consider a bar as shown below having a cross sectional area A e = 1.2 in2 and Young’s modulus E = 30 x 106 psi If q1 = 0.02 in and q2 = 0.025 in, determine the following:



a) The displacement at the point P

b)

The strain  and stress 

c) The element stiffness matrix and

d)

The strain energy in the element.

A finite element solution using one – dimensional, two – noded elements has been obtained for a rod as shown below. T

Displacement are as follows Q  [-0.2, 0,0.6, - 0.1] , E = 1N/mm2, area of each mm element = 1 mm2, L1-2 = 50 mm, L2-3 = 80 mm, L3-4 = 100 mm. i)

According to the finite element theory, plot the displacement u(x) versus x.

ii) According to the finite element theory, plot the strain (x) versus x. iii) Determine the B matrix for element 2-3. 1 iv) Determine the strain energy in the element 1-2 using U  qT kq. 2

2.84)

Consider the bar, loaded as shown below. Determine the nodal displacements, element stresses and support reactions. Solve this problem by adopting elimination method for handling boundary conditions. (value of E = 200 x 109 N/m2).



2.85)

In the figure shown below load P = 60kN is applied. Determine the displacement field, stress and support reactions in the body. Take E = 20 kN/mm2 [AU, May / June – 2011]

2.86)

Consider the bar as shown below. Determine the nodal displacements, element stresses and support reactions. (E = 200 x 109 N/m2)

2.87)

An axial load P = 385 KN is applied to the composite block as shown below. Determine the stress in each material.



2.88)

For a vertical rod as shown below, find the deflection at A and the stress distribution. E = 100 MPa and weight per unit volume = 0.06 N/cm3. Comment on the stress distribution.

2.89)

Consider a two-bar supported by a spring shown in figure. Both bars have E = 210 GPa and A=5.0 x10-4 m2. Bar one has a length of 5m and bar two has a length of 10 m. The spring stiffness is k= 2 kN/m. Determine the horizontal and vertical displacements at the joint 1 and stresses in each bar.

[AU, Nov / Dec – 2009]



2.90)

Each of the three bars of the pin – jointed frame shown in figure has a cross sectional area of 1000mm2 with E = 200GPa. Solve for displacements. [AU, Nov / Dec – 2013]

2.91)

Find the deflection at the free end under its own weight, using divisions of a) 1 element b) 2 elements c) 4 elements d) 8 elements and e) 16 elements Then plot the number of elements versus deflection.



2.92)

For the discretization of beam elements as shown below, number the nodes so as to minimize the bandwidth of the assembled stiffness matrix (K)

2.93)

The elements of a row or column of the stiffness matrix of a bar element sum up to zero, but not so for a beam element. Explain why this is so.

2.94)

For the beam problem shown below, determine the tip deflection and the slope at the roller support.



2.95)

For the beam and loading as shown in figure. Determine the slopes at the two ends of the distributed load and the vertical deflection at the mid-point of the distributed load. Take E = 200GPa and I = 4*106 mm4

2.96)

[AU, May / June – 2011]

Find the deflection and slope for the following beam section at which point load is applied.

2.97)

Solve the following beam as shown below, clamped at one end and spring support at other end. A linearly varying transverse load of maximum magnitude of 100 N/cm applied over the span of 4 cm to 10 cm. Take EI = 2 x 107 N/cm2,

K  10  2 . EI



2.98)

Obtain the deflection at the midpoint of the beam shown below and determine the reaction.

2.99)

The simply supported beam shown in figure is subjected to a uniform transverse load, as shown. Using two equal-length elements and work-equivalent nodal loads obtain a finite element solution for the deflection at mid-span and compare it to the solution given by elementary beam theory.

[AU, April / May - 2010]

2.100) Determine the displacements and slopes at the nodes for the beam shown in figure. Take k=200kN / m, E=70GPa and I=2x10-4m4.

[AU, Nov / Dec – 2012]



2.101) Determine the nodal displacements and slopes for the beam shown in Figure. Find the moment at the midpoint of element 1.

[AU, Nov / Dec – 2012]

2.102) Determine the displacement of node 1 and the stress in element 3, for the three-bar truss as shown below. Take A = 250 mm2, E = 200 GPa for all elements.



2.103) Determine the force in the members of the truss as shown in figure. Take E = 200 GPa

[AU, May / June – 2012]

2.104) Determine the nodal displacements and the element stresses for the two dimensional loaded plate as shown in figure. Assume plane stress condition. Body force may be neglected in comparison to the external forces. Take E = 210GPa, µ = 0.25, Thickness t = 10mm.

[AU, May / June – 2011]

2.105) The loading and other parameters for a two bar truss element is shown in figure Determine

[AU, May / June – 2013] (i)

The element stiffness matrix for each element

(ii)

Global stiffness matrix



(iii)

Nodal displacements

(iv)

Reaction forces

(v)

The stresses induced in the elements. Assume E = 200 GPa.

2.106) Calculate nodal displacement and elemental stresses for the truss shown in Figure. E= 70Gpa.cross-sectional area A = 2cm2 for all truss members. [AU, April / May – 2011]



2.107) Find the horizontal and vertical displacements of node 1 for the truss shown below. Take

A = 300 mm2, E = 2 x105 N/mm2 for each element.

2.108) Each of the five bars of the pin jointed truss shown in figure below has a cross sectional area 20 sq. cm. and E = 200 GPa.

(i) Form the equation F = KU where K is the assembled stiffness matrix of the structure. (ii) Find the forces in all the five members.


2.109) Analyze the truss shown in figure and evaluate the stress resultants in member (2). Assume area of cross section of all the members in same. E = 2 * 105 N/mm2 [AU, Nov / Dec – 2010]



2.110) Determine the joint displacements, the joint reactions, element forces and element stresses of the given truss elements.

Elements A cm2


E

L

Global

N/m2

m

Node

 Degree

connection 1

32.2

6.9e 10

2.54

2 to 3

90

2

38.7

20.7e10 2.54

2 to 1

0

3

25.8

20.7e10 3.59

1 to 3

135



2.111) Determine the force in the members of the truss shown in figure. [AU, April / May – 2011]

2.112) Find the nodal displacement developed in the planer truss shown in Figure when a vertically downward load of 1000 N is applied at node 4. The required data are given in the Table.

[AU, May / June – 2012]

Element No.

Cross – Sectional area A

‘e’

cm2

1

2

√2 50

2 * 106

2

2

√2 50

2 * 106

3

1

√2.5 100

2 * 106

4

1

√2 100

2 * 106

Length l (e) cm

Young’s Modulus E(e) N/mm2



2.113) Derive the shape function for a 2 noded beam element and a 3 noded bar element. [AU, Nov / Dec – 2008] 2.114) Why is higher order elements needed? Determine the shape functions of an eight noded rectangular element.


2.115) Derive the shape functions for a 2D beam element. [AU, Nov / Dec – 2007, April / May – 2008, 2009] 2.116) Derive the shape functions for a 2D truss element. [AU, Nov / Dec – 2007, April / May – 2008, 2009] 2.117) Derive the interpolation function for the one dimensional linear element with a length “L” and two nodes, one at each end, designated as “i” and ” j”. Assume the origin of the coordinate system is to the left of node “i”. [AU, April / May - 2010]



Figure shows the one-dimensional linear element



UNIT – III – TWO DIMENSIONAL FINITE ELEMENT ANALYSIS PART – A 3.1)

Name few 2-D elements along with a neat sketch.

3.2)

State the differences between 2D element and 1D element.

3.3)

Define Lagrange’s interpolation.

3.4)

What is geometric Isotropy?

3.5)

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

[AU, May / June – 2013]

[AU, Nov / Dec – 2008] 3.6)

Write the relation to obtain the size of the stiffness matrix for a linear quadrilateral element having Ux and Uy as dof.

3.7)

Why is the 3 noded triangular element called as a CST element? [AU, Nov / Dec – 2010]

3.8)

Write down the interpolation function of a field variable for three-node triangular element.


3.9)

What is a CST element?


3.10)

Draw the shape functions of a CST element.

[AU, Nov / Dec – 2010]

3.11)

Explain the important properties of CST elements.

[AU, Nov / Dec – 2008]

3.12)

Write a note on CST element.

3.13)

Write briefly about LST and QST elements.

3.14)

What are CST and LST elements?

[AU, Nov / Dec – 2009]

3.15)

Define LST element.

[AU, Nov / Dec – 2012]

3.16)

Write the displacement function equation for CST element.

3.17)

Write the strain – displacement matrix for CST element.

3.18)

Differentiate CST and LST elements. [AU, Nov / Dec – 2007, April / May – 2009]

3.19)

Give the Jacobian matrix for a CST element and state its significance.

[AU, May / June – 2011]

[AU, Nov / Dec – 2013] FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 40


3.20)

Evaluate the following area integrals for the three node triangular element ∫

3.21)

A triangular element is shown in Figure and the nodal coordinates are expressed in mm. Compute the strain displacement matrix.

3.22)

[AU, May / June – 2012]

[AU, Nov / Dec – 2012]

What do you mean by the terms : c0,c1 and cn continuity? [AU, April / May – 2010]

3.23)

Distinguish between C0, C1 and C2 continuity elements.

3.24)

What are the different problems governed by 2D scalar field variables?

3.25)

Use various number of triangular elements to mesh the given domain in the order of increasing solution refinement.

3.26)

Define Pascal triangle.



3.27)

Write the significance of Pascal triangle in developing triangular elements.

3.28)

Distinguish one from the other of the following a) Linear and quadratic triangular elements. b) Linear and quadratic Lagrange elements.

3.29)

What do you mean by area co-ordinate method?

3.30)

State the advantage of serendipity element.

3.31)

What do you mean by wrapping?

3.32)

Write the node numbering and element connectivity table for the given domain using suitable discretization.

3.33)

Plot the variation of shape function with respect node of a 3 noded triangular element.

3.34)

Write down the nodal displacement equations for a two dimensional triangular elasticity element.


3.35)

Define a plane stress condition.

[AU, Nov / Dec – 2011]

3.36)

State the condition for plane stress problem.

3.37)

Give one example each for plane stress and plane strain problems. [AU, Nov / Dec – 2008]

3.38)

Distinguish between plane stress and plane strain problems. [AU, Nov / Dec – 2009]

3.39)

Distinguish plane stress and plane strain conditions.

[AU, Nov / Dec – 2010]

3.40)

Define plane strain with suitable example.

[AU, Nov / Dec – 2012]

3.41)

Define plane strain analysis.

[AU, Nov / Dec – 2011]

3.42)

Define a plane stress problem with a suitable example.

[AU, May / June – 2013]

3.43)

Explain plane stress problem with an example.


3.44)

Explain plane stress conditions with example.

[AU, May / June – 2011]



3.45)

Write down the strain displacement relation.

3.46)

State whether plane stress or plane strain elements can be used to model the following structures. Justify your answer.


[AU, Nov / Dec – 2012]

(a) A wall subjected to wind load (b) A wrench subjected to a force in the plane of the wrench. 3.47)

Write the assumptions used to define the given problem as plane stress problem.

3.48)

Write the assumptions used to define the given problem as plane strain problem.

3.49)

Using general stress - strain relation, obtain plane stress equation.

3.50)

Beginning with general elastic stress-strain relation, derive the plane strain condition.

3.51)

What are the differences between 2 Dimensional scalar variable and vector variable elements?

3.52)

[AU, Nov / Dec – 2009]

What are the ways by which a three dimensional problem can be reduced to a two dimensional problem?

3.53)

How to reduce a 3D problem into a 2D problem?

[AU, Nov / Dec – 2012]

3.54)

Give the stiffness matrix equation for an axisymmetric triangular element.

3.55)

What is axisymmetric element?

3.56)

Give examples of axisymmetric problems.

[AU, May / June – 2012]

3.57)

What is an axisymmetric problem?


3.58)

Write short notes on axisymmetric problems. [AU, Nov / Dec – 2007, April / May – 2009]

3.59)

What is meant by axi-symetric field problem? Given an example. [AU, Nov / Dec – 2009]

3.60)

When are axisymmetric elements preferred?

[AU, Nov / Dec – 2013]

3.61)

State the situations where the axisymmetric formulation can be applied. [AU, April / May – 2011]



3.62)

Give four applications where axisymmetric elements can be used. [AU, April / May – 2011]

3.63)

State the applications of axisymmetric elements.

[AU, Nov / Dec – 2010]

3.64)

Write down the constitutive relationship for axisymmetric problem. [AU, April / May – 2009]

3.65)

Write down the constitutive relationship for the plane stress problem. [AU, Nov / Dec – 2010]

3.66)

What do you mean by constitutive law and give the constitutive law for axisymmetric problems?

3.67)

[AU, April / May, Nov / Dec – 2008]

Specify the body force term and the body force vector for axisymmetric triangular element.

3.68)

[AU, Nov / Dec – 2013]

Give one example each for plane stress and plane strain problems. [AU, April / May - 2008]

3.69)

Explain plane strain problem with an example.

[AU, May / June – 2012]

3.70)

Give a brief note on static condensation.

3.71)

Prove that 2  0 for plane strain condition.

3.72)

Differentiate axi – symmetric and cyclic –symmetric structures.

3.73)

Differentiate axi-symmetric load and asymmetric load with examples.

3.74)

State the condition for axi-symmetric problem.

3.75)

List the required conditions for a problem assumed to be axisymmetric. [AU, April / May – 2010]

3.76)

What are the four basic sets of elasticity equations?

3.77)

Give examples for the following cases.

[AU, May / June – 2012]

a) plane stress problem b) plane strain problem c) axi-symmetric problem 3.78)

Define the following terms with suitable examples




i) Plane stress, plane strain

ii) Node, element and shape functions

iii) Axisymmetric analysis

iv) Iso – parametric element

3.79)

Define the term initial strain.

3.80)

State the effect of Poisson’s ratio in plane strain problem.

3.81)

How will the stress field vary linearly?

3.82)

Compare the changes in the D matrix evolved out of plane strain, plane stress and axi-symmetric problem.

3.83)

What do you mean by Isoparametric formulations? [AU, Nov / Dec – 2007, April / May – 2009]

3.84)

Express the shape functions of four node quadrilateral element. [AU, May / June – 2012]

3.85)

What do you understand by a natural co – ordinate system? [AU, April / May – 2011]

3.86)

What do you mean by natural co-ordinate system?

3.87)

What are the advantages of natural co-ordinates?

[AU, May / June – 2011]

[AU, Nov / Dec – 2007, April / May – 2009] 3.88)

What are the advantages of natural coordinates over global co-ordinates? [AU, Nov / Dec – 2008]

3.89)

Give a brief note on natural co-ordinate system.

3.90)

Write the natural co-ordinates for the point “P” of the triangular element. The point ‘P’ is the C.G. of the triangle.

3.91)

[AU, Nov / Dec – 2008]

Show the transformation for mapping x-coordinate system onto a natural coordinate system for a linear spar element and for a quadratic spar element. [AU, Nov / Dec – 2012]

3.92)

Define a local co – ordinate system.

[AU, Nov / Dec – 2011]

3.93)

What is area co – ordinates?

[AU, Nov / Dec – 2011]

3.94)

What do you understand by area co – ordinates?




3.95)

State the basic laws on which Isoparametric concept is developed. [AU, April / May – 2008]

3.96)

Differentiate: local axis and global axis.


3.97)

Define super parametric element.


3.98)

Explain super parametric element.

[AU, Nov / Dec – 2010]

3.99)

Define Isoparametric elements?

[AU, Nov / Dec – 2008]

3.100) Define Isoparametric elements with suitable examples 3.101) Define Isoparametric element formulations. 3.102) What do you mean by Isoparametric formulation?

[AU, April / May – 2010] [AU, Nov / Dec – 2012] [AU, April / May – 2011]

3.103) What is the purpose of Isoparametric elements? 3.104) What is the salient feature of an Isoparametric element? Give an example. [AU, Nov / Dec – 2013] 3.105) What are the applications of Isoparametric elements?


3.106) Differentiate x – y space and - space. 3.107) Write the advantages of co-ordinate transformation from Cartesian co-ordinates to natural co-ordinates. 3.108) Define Jacobian.

[AU, Nov / Dec – 2013]

3.109) What is a Jacobian?

[AU, Nov / Dec – 2010]

3.110) What is the need of Jacobian?


3.111) Write down the Jacobian matrix.

[AU, Nov / Dec – 2010]

3.112) Write about Jacobian transformation used in co-ordinate transformation. 3.113) What is the significance of Jacobian of transformation?

[AU, May / June – 2012]

3.114) Differentiate between sub-parametric, isoparametric and super – parametric elements. 3.115) Give two examples for sub parametric elements.

[AU, Nov / Dec – 2013]



3.116) Represent the variation of shape function with respect to nodes for quadratic elements in terms of natural co-ordinates. 3.117) Compare linear model, quadratic model and cubic model in terms of natural coordinate system. 3.118) Write a brief note on continuity and compatibility. 3.119) Write down the element force vector equation for a four noded quadrilateral element. 3.120) Write down the Jacobian matrix for a four noded quadrilateral element 3.121) Write the shape function for the quadrilateral element in ,  space. 3.122) Why is four noded quadrilateral element is preferred for axi-symmetric problem than three noded triangular element? 3.123) Sketch a four node quadrilateral element along with nodal degrees of freedom. [AU, April / May – 2011] 3.124) Write down the stiffness matrix for four noded quadrilateral elements. [AU, May / June – 2011] 3.125) Distinguish between essential boundary conditions and natural boundary conditions.

[AU, Nov / Dec – 2009]

3.126) Write the advantages of higher order elements in natural co – ordinate system. 3.127) What are the types of non-linearity? [AU, Nov / Dec – 2007, April / May – 2009, May / June – 2012] 3.128) State the advantage of Gaussian integration. 3.129) State the four-point Gaussian quadrature rule. 3.130) Briefly explain Gaussian quadrature. 3.131) What are the advantages of Gaussian quadrature?

[AU, April / May – 2011] [AU, Nov / Dec – 2012]

3.132) What are the weights and sampling points of two point formula of Gauss quadrature formula?

[AU, May / June – 2012]



3.133) Why numerical integration is required for evaluation of stiffness matrix of an Isoparametric element?

[AU, Nov / Dec – 2011]

3.134) When do we resort to numerical integration in 2D elements? [AU, Nov / Dec – 2013] 3.135) Write the Gauss points and weights for two point formula of numerical integration. [AU, April / May – 2011] 3.136) Write down the Gauss integration formula for triangular domains. [AU, April / May – 2009] 3.137) Evaluate the integral ∫ (

)

using Gaussian quadrature method. [AU, Nov / Dec – 2012]

3.138) Name the commonly used integration method in natural – co-ordinate system. 3.139) Write the relation between weights and Gauss points in Gauss-Legendre quadrature. PART – B 3.140) Determine the shape functions for a constant strain triangular (CST) element in terms of natural coordinate system.

[AU, Nov / Dec – 2008]

3.141) What are shape functions? Derive the shape function for the three noded triangular elements.

[AU, Nov / Dec – 2011]

3.142) Derive the element strain displacement matrix and element stiffness matrix of a CST element.


3.143) Explain the terms plane stress and plane strain problems. Give the constitutive laws for these cases.


3.144) Derive the equations of equilibrium in the case of a three dimensional system. [AU, Nov / Dec – 2007, 2008, April / May – 2009] 3.145) Derive the expression for constitutive stress-strain relationship and also reduce it to the problem of plane stress and plane strain.

[AU, Nov / Dec - 2008]

3.146) Derive the constant-strain triangular element’s stiffness matrix and equations. FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 48


[AU, April / May - 2008] 3.147) Derive the linear – strain triangular element’s stiffness matrix and equations. [AU, April / May – 2008] 3.148) Derive the stiffness matrix and equations for a LST element. [AU, Nov / Dec – 2012] 3.149) Derive the element strain displacement matrix and element stiffness matrix of a triangular element.

[AU, May / June – 2012]

3.150) A two noded line element with one translational degree of freedom is subjected to a uniformly varying load of intensity P1 at node 1 and P2 at node 2. Evaluate the nodal load vector using numerical integration. 3.151) Calculate the element stresses x,y, xy,

[AU, Nov / Dec – 2012]

1 and  2 and the principle angle  p for

the element shown below.

3.152) The nodal co-ordinates of the triangular element is as shown below. At the interior point P, the x- co-ordinate is 3.3 and N1 = 0.3. Determine N2, N3 and the y – coordinate at point P.



3.153) The (x,y) co-ordinates of nodes i, j and k of a triangular element are given by (0,0), (3,0) and (1.5,4) mm respectively. Evaluate the shape functions N1, N2 and N3 at an interior point P (2, 2.5) mm for the element. For the same triangular element, obtain the strain-displacement relation matrix B.

[AU, Nov / Dec – 2009]

3.154) For the triangular element shown below, obtain the strain – displacement relation matrix B and determine the strains x ,y and xy.

3.155) Derive the expression for nodal vector in a CST element subjected to pressures P x1, Py1 on side 1, Px2, Py2, on side 2 and Px3, Py3 on side 3 as shown in figure. [AU, Nov / Dec – 2013]



3.156) Consider the triangular element show in Figure. The element is extracted from a thin plate of thickness 0.5 cm. The material is hot rolled low carbon steel. The Nodal co-ordinates are xi =0, yi = 0, xj =0, yj = -1, xk =0, yk = -1 cm,. Determine the elemental stiffness matrix. Assuming plane stress analysis. Take µ = 0.3 and E = 2.1*107 N/cm2

[AU, May / June – 2012]

3.157) Derive the interpolation function 14 for the quadratic triangular element as shown below.

3.158) Derive the interpolation function of a corner node in a cubic serendipity element. 3.159) Find the expression for nodal vector in a CST element shown in figure subjected to pressures Px1 on side 1.

[AU, Nov / Dec – 2008]



3.160) For the CST element given below, assemble stain displacement matrix. Take t = 20 mm and E = 2*105 N/mm2

[AU, Nov / Dec - 2008]

3.161) Calculate the value of pressure at the point A which is inside the 3 noded triangular elements as shown in figure. The nodal values are φ1 = 40 MPa, φ2 = 34 MPa and φ3 = 46 MPa, Point A is located at (2, 1.5) Assume pressure is linearly varying in the element. Also determine the location of 42 MPa contour line. [AU, May / June – 2013]



3.162) Obtain the global stiffness matrix for the plate shown in figure. Taking two triangular elements. Assume plane stress condition.

[AU, May / June – 2012]

3.163) For the constant strain triangular element shown in figure below, assemble the strain – displacement matrix. Take t = 20 mm and E = 2 x 105 N/mm2. [AU, Nov / Dec – 2007, 2013, April / May – 2009]



3.164) A CST element has nodal coordinates 1 (0,0), 2 (5,0), 3 (0,4). The element is subjected to a body force f = x3 N/m3. Determine the nodal force vector. Take the element thickness as 0.3m.

[AU, Nov / Dec – 2013]

3.165) For the plane strain element shown in the figure, the nodal displacements are given as :

u1= 0.005 mm, u2 = 0.002 mm, u3=0.0mm, u4 = 0.0 mm, u5 = 0.004 mm, u6

= 0.0 mm. Determine the element stresses. Take E = 200 Gpa and  = 0.3. Use unit thickness for plane strain.


3.166) For the two-dimensional loaded plate as shown in Figure. Determine the nodal displacements and element stress using plane strain condition considering body force. Take Young’s modulus as 200 GPa, Poisson’s ration as 0.3 and density as 7800 kg/m3.




3.167) Derive element force vector when linearly varying pressure acts on the side joining nodes jk of a triangular element shown in Figure and body force of 25N/mm2 acts downwards. Thickness = 5mm.


3.168) For the plane stress element whose coordinates are given by (100,100), (400, 100) and (200, 4000, the nodal displacements are u1 = 2.0mm, v1 =l.0mm, u2 =l.0mm, v2 =1.5mm, u3 = 2.5mm, v3 = 0.5mm. Determine the element stresses. Assume E = 200 GN/m2, µ = 0.3 and t = 10 mm. All coordinates are in mm. [AU, May / June – 2013]



3.169) A thin plate a subjected to surface fraction as shown in Figure. Calculate the global stiffness matrix. Table t = 25 mm, E = 2 *105 N/mm2 and γ = 0.30. Assume plane stress condition.

[AU, Nov / Dec – 2011]

3.170) Determine the deflection of a thin plate subjected to extensional load as shown.

3.171) Calculate nodal displacement and elemental stresses for the truss shown in Figure. E = 70 GPa cross-sectional area A = 2 cm2 for all truss members. [AU, Nov / Dec – 2012]



3.172) A thin elastic plate subjected to uniformly distributed edge load as shown below. Find the stiffness and force matrix of the element.

3.173) For the configuration as shown in figure determine the deflection at the point load applications. Use one model method. Assume plane stress condition. [AU, April / May – 2011]



3.174) Derive the expression for the element stiffness matrix for an axisymmetric shell element.


3.175) Describe the step by step procedure of solving axisymmetric problem by finite element formulation.

[AU, May / June – 2012]

3.176) Derive an expression for the stiffness matrix of an axisymmetric element. [AU, April / May – 2011] 3.177) For an axisymmetric triangular element. Obtain the [B] matrix and constitutive matrix

[AU, Nov / Dec – 2010]

3.178) Derive the stress-strain relationship matrix (D) for the axisymmetric triangular element.

[AU, Nov / Dec – 2012]

3.179) Explain the modeling of cylinders subjected to internal and external pressure using axisymmetric.

[AU, Nov / Dec – 2011]

3.180) For a thick cylinder subjected to internal and external pressure, indicate the steps of finding the radial stress.

[AU, Nov / Dec – 2010]

3.181) Derive the material property matrix for axisymmetric elasticity. [AU, Nov / Dec – 2011] 3.182) Explain Galerkin’s method of formulation for determining the stiffness matrix for an axisymmetric triangular element.

[AU, Nov / Dec – 2013]

3.183) The (x, y) coordinates of nodes i, j and k of an axisymmetric triangular element are given by (3, 4), (6, 5), and (5, 8) cm respectively. The element displacement (in FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 58


cm) vector is given as q = [0.002, 0.001, 0.001, 0.004, -0.003, 0.007]T. Determine the element strains.

[AU, Nov / Dec – 2009]

3.184) A long cylinder of inside diameter 80 mm and outside diameter 120 mm snugly fits in a hole over its full length. The cylinder is then subjected to an internal pressure of 2 MPa. Using two elements on the 10 mm length shown, find the displacement at the inner radius.

3.185) Determine the stiffness matrix for the axisymmetric element shown in fig, Take E as 2.1* 106 N/mm2 and Poisson's ratio as 0.3.

[AU, Nov / Dec – 2012]

3.186) Determine the element stresses for the axisymmetric element as shown below. Take E = 2.1 x 105 N/mm2 and = 0.25.



Use the nodal displacements as u1 = 0.05 mm

w1 = 0.03 mm

u2 = 0.02 mm

w2 = 0.02 mm

u3 = 0 mm

w3 = 0 mm

3.187) Compute the strain displacement matrix for the following axisymmetric element. Also calculate the element stress vectors. If [q] = [ 3.484

[D] = [

0

3.321

]

0

0

0]T * 10-3 cm


3.188) An open ended steel cylinder has a length of 200mm and the inner and outer diameters as 68mm and 100mm respectively. The cylinder is subjected to an internal pressure of 2MPa. Determine the deformed shape and distribution of principle stresses. Take E = 200GPa and Poisson’s ratio = 0.3 [AU, April / May – 2011] 3.189) Derive the Isoparametric representation for a triangular element. [AU, Nov / Dec – 2010] FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 60


3.190) Derive element stiffness matrix for a linear Isoparametric quadrilateral element [AU, Nov / Dec – 2007, April / May – 2008] 3.191) Derive stiffness matrix for a linear Isoparametric element. [AU, Nov / Dec – 2012] 3.192) Explain the terms Isoparametric sub parametric and super parametric elements. [AU, Nov / Dec – 2013] 3.193) Distinguish between sub parametric and super parametric elements. [AU, Nov / Dec – 2009, 2010] 3.194) Establish the shape functions of an eight node quadrilateral element and represent them graphically.


3.195) Establish any two shape functions corresponding to one corner node and one mid – node for an eight node quadrilateral element.

[AU, Nov / Dec – 2013]

3.196) Derive the shape function for an eight noded brick element. [AU, April / May – 2009] 3.197) Derive the shape functions of a nine node quadrilateral Isoparametric element. [AU, April / May – 2011, May / June – 2012] 3.198) Derive element stiffness matrix for linear Isoparametric quadrilateral element. [AU, April / May – 2009, May / June – 2011] 3.199) Describe the element strain displacement matrix of a four node quadrilateral element.

[AU, May / June – 2012]

3.200) Derive the shape function for an eight – noded quadrilateral element in ,  space. 3.201) Establish the body force and traction force (uniformly distributed) vector for a lower order quadrilateral element.

[AU, Nov / Dec – 2013]

3.202) For the Isoparametric quadrilateral element as shown below, the Cartesian coordinates of the point P are (6,4). The loads 10 KN and 12 KN are acting in X and Y directions on that point P. Evaluate the nodal equivalent forces.



3.203) Consider the quadrilateral element as shown below using the linear interpolation functions of a rectangular element, transform the element to the local co-ordinate system and sketch the transformed element.

3.204) A four noded rectangular element is shown in figure. Determine the following: (i) Jacobian Matrix (ii) Strain Displacement Matrix (iii) Element Stress. Take E = 20*105N/mm2, δ = 0.5. u=[0 0

0.003

0.004

Assume plane stress condition.

0.006

0.004

0

0]T ε = 0, η = 0. [AU, May / June – 2012]



3.205) A four nodal quadrilateral plane stress Isoparametric element is defined by nodes 1 (0,0), 2 (40,0), 3 (40, 15) and 4 (0,15). Determine the Jacobian matrix corresponding to the Gauss point (0.57735, 0.57735) for the above element. [AU, Nov / Dec – 2013] 3.206) In a four-noded rectangular element, the nodal displacements in mm are given by u1 = 0

u2 = 0.127

u3 = 0.0635

u4 = 0

v1 = 0

v2 = 0.0635

v3 = -0.0635

v4 = 0

For b = 50 mm, h = 25 mm, E = 2*105 N / mm2 and Poisson's ratio = 0.3, determine the element strains and stresses at the centroid of the element and at the corner nodes.

[AU, Nov / Dec – 2012]

3.207) Find the Jacobian matrix for the nine-node rectangular element as shown below. What is the determinant of the Jacobian matrix?



3.208) Determine the Jacobian for the (x, y) – (, ) transformation for the element shown below. Also find the area of triangle using determinant method.

3.209) Compute the element and force matrix for the four noded rectangular elements as shown below.

3.210) The Cartesian (global) coordinates of the corner nodes of a quadrilateral 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 defined by (r,s) = (0.5, 0.5) in the global coordinate system. [AU, Nov / Dec – 2009] FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 64


3.211) Consider a rectangular element as shown below. Assume plane stress condition, E  30x106 psi,

  0.3, q  [0, 0, 0.002, 0.003, 0.006, 0.0032, 0, 0]T inches

Evaluate

the Jacobian transformation (J), B matrix, and  at  = 0 and  = 0.

3.212) The Cartesian (global) coordinates of the corner nodes of an Isoparametric quadrilateral element are given by (1,0), (2,0), (2.5,1.5) and(1.5,1). Find its Jacobian matrix.

[AU, Nov / Dec – 2009]

3.213) A rectangular element has its nodes at the following points in Cartesian coordinate system (0, 0), (5, 0), (5, 5), and (0, 5). Obtain the expressions for the shape functions of the corresponding Isoparametric element. Using them obtain the elements if Jacobian matrix of transformation.

[AU, Nov / Dec – 2011]

3.214) If the coordinates of the quadrilateral are (1, 2), (10, 2), (8, 6) and (2, 10). Obtain the Jacobian and hence, find the area of the element.

[AU, Nov / Dec – 2011]

3.215) Determine the Jacobian matrix for the following quadrilateral element at x = 4.35 mm and y = 3 mm.




3.216) Consider the quadrilateral element as shown in figure. Evaluate

(ξ, η), (0, 0) and (

and

at

) using Isoparametric formulation.

3.217) Establish the strain – displacement matrix for the linear quadrilateral element as shown in figure below at Gauss point r = 0.57735 and s = -57735. [AU, Nov / Dec – 2007, April / May – 2009]



3.218) Write short notes on

[AU, Nov / Dec – 2008]

(i) Uniqueness of mapping of Isoparametric elements. (ii) Jacobian matrix. (iii) Gaussian Quadrature integration technique. 3.219) Derive the Gauss points and weights in case of one point formula and two point formula of Gauss numerical integration.


3.220) Derive the weights and Gauss points of two point formula of Gauss quadrature rule. 3.221)

[AU, May / June – 2012]

Integrate

( )

between

12. Use Gaussian quadrature rule. 3.222) Evaluate the integral = ∫ (

8 and


)

and compare with exact results. [AU, Nov / Dec – 2009]

3.223) Numerically evaluate the following integral and compare with exact one.

∫ ∫ (

)


3.224) Using natural coordinates derive the shape function for a linear quadrilateral element.

[AU, Nov / Dec – 2008]

3.225) Use Gauss quadrature rule (n=2) to numerically integrate

[AU, Nov / Dec – 2008]

∫ ∫ FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 67


3.226) Use Gaussian quadrature rule (n = 2) to numerically integrate

∫ ∫ 3.227) Evaluate ∫ (

[AU, Nov / Dec – 2012]

)

using Gauss quadrature formula. [AU, May / June – 2012]

3.228) Evaluate the integral ∫

using one point and two point

Gauss quadrature formula. 3.229) Evaluate the integral ∫


( )

using three point Gauss quadrature and

compare with exact solution. 3.230) Evaluate ∫

∫ (

[AU, Nov / Dec – 2011]

)

using Gauss numerical integration. [AU, April / May – 2011]

3.231) Use Gaussian quadrature to obtain an exact value of the integral. [AU, April / May – 2010]

∫ ∫ (

)(

)



UNIT – IV – DYNAMIC ANALYSIS USING FINITE ELEMENT METHOD PART – A 4.1) What is dynamic analysis? Give examples

[AU, Nov / Dec – 2010]

4.2) What is mean by dynamic analysis?

[AU, Nov / Dec – 2011]

4.3) List the types of dynamic analysis problems.

[AU, May / June – 2012]

4.4) Define normal modes.

[AU, May / June – 2013]

4.5) What is the principle of mode superposition technique?

[AU, Nov / Dec – 2013]

4.6) Sketch two 3D elements exhibiting linear strain behavior. [AU, April / May – 2011] 4.7) What is the influence of element distortion on the analysis results? [AU, April / May – 2011] 4.8) Determine the element mass matrix for one-dimensional, dynamic structural analysis problems. Assume the two-node, linear element.

[AU, Nov / Dec – 2011]

4.9) What are consistent and lumped mass techniques?

[AU, Nov / Dec – 2013]

4.10) Specify the consistent mass matrix for a beam element.

[AU, Nov / Dec – 2013]

4.11) Comment on the accuracy of the values of natural frequencies obtained by using lumped mass matrices and consistent mass matrices. 4.12) Explain consistent load vector.

[AU, Nov / Dec – 2012] [AU, Nov / Dec – 2010]

4.13) What do you mean by Lumped mass matrix?

[AU, May / June – 2011]

4.14) Write down the lumped mass matrix for the truss element. [AU, April / May – 2009] 4.15) What are the types of Eigen value problems?

[AU, May / June – 2012]

4.16) What is meant by mode superposition technique?

[AU, May / June – 2013]

PART – B 4.17) Derive the equation of' motion based on weak form for transverse vibration of a beam.

[AU, May / June – 2012]

4.18) Derive the governing equation for longitudinal vibration. 4.19) Derive the weak formulation for longitudinal vibration. 4.20) Derive the weak formulation for transverse vibration. 4.21) Derive the element equation for longitudinal free vibration. FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 69


4.22) Derive the element equation for transverse free vibration. 4.23) Derive the consistent mass matrix for bar element 4.24) Derive the consistent mass matrix for truss element 4.25) Derive the consistent mass matrix for CST element 4.26) Explain the direct integration method using central difference scheme for predicting the transient dynamic response of a structure.

[AU, Nov / Dec – 2013]

4.27) Derive the consistent mass matrix for a truss element in its local coordinate system. [AU, Nov / Dec – 2012] 4.28) Derive the finite equations for the time – dependent stress analysis of one dimensional bar.

[AU, May / June – 2011]

4.29) Find the natural frequencies of transverse vibrations of the cantilever beam shown in Figure using one beam element.

[AU, Nov / Dec – 2012]

4.30) Consider a uniform cross – section bar as shown in figure of length “L” made up of a material whose Young’s modulus and density are given by E and ρ. Estimate the natural frequencies of axial vibration of the bar using both lumped and consistent mass matrix.

[AU, May / June – 2005]

4.31) Determine the natural frequencies and mode shapes of transverse vibration for a beam fixed at both ends. The beam may be modelled by two elements, each of length L and cross-sectional area A. Consider lumped mass matrix approach. [AU, April / May – 2011] FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 70


4.32) Using two equal-length finite elements, determine the natural frequencies of the solid circular shaft fixed at one end shown in figure.

[AU, Nov / Dec – 2011]

4.33) Obtain the natural frequencies of vibration for a stepped steel bar of area 625mm2 for the length of 250mm and 312.5mm2 for the length of 125mm. The element is fixed at larger end

[AU, Nov / Dec – 2007]

4.34) Determine the Eigen values and frequencies for the stepped bar shown in the figure. Take E = 20 * 1010 N/m2 and self-weight = 8500 kg.m3

[AU, May / June – 2005]



4.35) Find the natural frequency of longitudinal vibration of the unconstrained stepped bar as shown in figure.

[AU, Nov / Dec – 2006]

4.36) Compute material frequencies of free transverse vibration of a stepped beam shown in figure.

[AU, May / June – 2003]

4.37) Determine the natural frequencies of transverse vibration for a beam fixed at both ends. The beam may be modelled by two elements each of length L and cross – sectional area A. The use of symmetry boundary condition is optional. [AU, May / June – 2008]



4.38) Determine the natural frequencies and mode shapes of a system whose stiffness and mass matrices are given below

[ ]

[

[AU, May / June – 2008]

]

[ ]

[

]

4.39) A vertical plate of thickness 40 mm is tapered with widths of 0.15m and 0.075m at top and bottom ends respectively. The plate is fixed at the top end. The length of the plate is 0.8m. Take Young's modulus as 200 GPa and density as 7800 kg/m3. Model the plate with two spar elements. Determine the natural frequencies of longitudinal vibration and the mode shapes.

[AU, Nov / Dec – 2013]

4.40) Find the response of the system given below using modal superposition method.

[AU, April / May – 2011] 4.41) Determine the natural frequencies for the truss shown in figure using finite element method.

[AU, May / June – 2007]



4.42) Find the natural frequencies of vibration of two element simply supported beam by taking advantage of the symmetry about the mid-point

[AU, May / June – 2007]

4.43) Formulate the mass matrix for two-dimensional rectangular element depicted in figure. The element has uniform thickness 5 mm and density ρ = 7.83 * 10-6kg/mm3. [AU, Nov / Dec – 2011]

4.44) Consider the undamped 2 degree of freedom system as shown in figure. Find the response of the system when the first mass alone is given an initial displacement of unity and realised from rest.



The mathematical representation of the system for free, Harmonic vibration is given by

[AU, May / June – 2012]

4.45) Calculate the consistent and lumped load vector for the element shown in figure. [AU, Nov / Dec – 2010]

4.46) Consider a uniform cross section bar, as shown in figure of length L made up of material whose young's modulus and density is given by E and ρ . Estimate the natural frequencies of axial vibration of the bar using both consistent and lumped mass matrices.

[AU, May / June – 2013]

4.47) Determine the Eigen values and Eigen vectors for the stepped bar as shown in figure. [AU, May / June – 2011]



4.48) Find the Eigen values and Eigen vectors of the matrix.

[

]

4.49) Find the Eigen values and Eigen vectors of the matrix.

[

]

4.50) Use iterative procedures to determine the first and third Eigen values for the structure shown in figure. Hence determine the second Eigen value and the natural frequencies of building. Finally, establish the Eigen vectors and check the rest by applying the orthogonality properties of Eigen vectors.

[AU, May / June – 2013]



UNIT – V – APPLICATIONS IN HEAT TRANSFER & FLUID MECHANICS PART – A 5.1) Write down the one dimensional heat conduction equation. [AU, April / May – 2011] 5.2) Distinguish between homogenous and non – homogenous boundary conditions. [AU, Nov / Dec – 2013] 5.3) Write down the expression of shape function and temperature function for one dimensional heat conduction.

[AU, May / June – 2011]

5.4) Write down the governing differential equation for the steady state one dimensional conduction heat transfer.

[AU, Nov / Dec – 2010, 2012]

5.5) Write down the governing differential equation for a two dimensional steady-state heat transfer problem.

[AU, Nov / Dec – 2009]

5.6) Write down the stiffness matrix equation for one dimensional heat conduction element.

[AU, Nov / Dec – 2011]

5.7) Sketch a two dimensional differential control element for heat transfer and obtain the heat diffusion equation.

[AU, Nov / Dec – 2012]

5.8) Define element capacitance matrix for unsteady state heat transfer problems. [AU, May / June – 2013] 5.9) Name a few boundary conditions involved in any heat transfer analysis. [AU, April / May – 2010] 5.10) Mention two natural boundary conditions as applied to thermal problems. [AU, April / May – 2011] 5.11) Consider a wall of a tank containing a hot liquid at a temperature T0 with an air stream of temperature Tx passed on the outside, maintaining a wall temperature of TL at the boundary. Specify the boundary conditions.


5.12) Define static condensation.

[AU, Nov / Dec – 2010]

5.13) Give the governing equation of torsion problem.

[AU, May / June – 2012]



5.14) Write the step by step procedure of solving a torsion problem by finite element method.


5.15) Outline the step by step procedure of handling torsion problem using the finite element method.

[AU, May / June – 2012]

5.16) Define streamline.

[AU, May / June – 2012]

5.17) Define the stream function for a one-dimensional incompressible flow. [AU, April / May – 2011] 5.18) List the applications of the potential flow. 5.19) List the method of describing the motion of fluid.

[AU, Nov / Dec – 2011] [AU, May / June – 2012]

5.20) State the relation between the velocity of fluid flow and the hydraulic gradient according to Darcy's law, explaining the terms involved.

[AU, Nov / Dec – 2012]

5.21) Define the stream function for a one dimensional incompressible flow. [AU, Nov / Dec – 2013] 5.22) Define the stream function for a two dimensional incompressible flow. [AU, May / June – 2013] Part – B 5.23) Write the mathematical formulation for a steady state heat transfer conduction problem and derive the stiffness and force matrices for the same. [AU, Nov / Dec – 2008] 5.24) Consider a plane wall with uniformly distributed heat source. Obtain the finite element formulation for the above case based on the stationarity of a functional. [AU, Nov / Dec – 2013] 5.25) Derive a finite element equation for one dimensional heat conduction with free end convection.

[AU, May / June – 2013]

5.26) The temperature at the four corners of a four – noded rectangle are T1, T2 T3 and T4. Determine the consistent load vector for a 2-D analysis, aimed to determine the thermal stresses.




5.27) Derive the stiffness matrix and load vectors for fluid mechanics in two dimensional finite element.

[AU, May / June – 2012]

5.28) Give the one-dimensional formulation for one-dimensional flow and derive the element stiffness matrix for the flow through a porous medium. [AU, Nov / Dec – 2012] 5.29) In the finite element analysis of a two dimensional flow using triangular elements, the velocity components u and v are assumed to vary linearly within an element (e) as

u(x ,y) = a1Ui(e) + a2Uj(e) + a3Uk(e) v(x ,y) = a1Vi(e) + a2Vj(e) + a3Vk(e) (e)

(e)

where (Ui , Vi ) denote the values of (u, v) at node i. Find the relationship between (e)

(e)

(e)

(Ui , Vi ............. Vk )which is to be satisfied for the flow to be incompressible. [AU, May / June – 2013] 5.30) Develop stiffness coefficients due to torsion for a three dimensional beam element. [AU, April / May – 2009] 5.31) Establish the finite element equations including force matrices for the analysis of two dimensional steady – state fluid flows through a porous medium using triangular element.

[AU, Nov / Dec – 2013]

5.32) Explain the potential function formulation of finite element equations for ideal flow problems.

[AU, May / June – 2013]

5.33) Find the temperature at a point P(1,1.5) inside the triangular element shown with the nodal temperatures given as T1 = 400C, TJ = 340C, and TK = 460C. Also determine the location of the 420C contour line for the triangular element shown in figure below. [AU, April / May - 2008]



5.34) Obtain the finite element equations for the following element. The thermal conductivity (k) of the material of the element is 2 W/ mK. The convective heat transfer coefficient (h) is 3 W/m2K. The ambient temperature (Tf) is 25˚ C. The thickness (t) of the material is 1mm. Assume convection along the edge ‘jk’ alone. [AU, April / May - 2011]

5.35) Compute the elemental stress vectors for the following element, assuming plane stress conditions. The nodal displacements in ‘mm’ [q] = [0 1

1

0

1

1]T. The

temperature increase in the element is 5˚C. Take E = 200 GPa and µ = 0.3. The thermal coefficient of expansion is 11 * 10-6 /˚C. The thickness of the material is 1 mm.




5.36) Calculate the element stiffness matrix and the thermal force vector for the axisymmetric triangular element as shown below. The element experiences a 15 0 C increase in temperature. Take  = 10 x 10-6 / 0C, E = 2 x 105 N/mm2 and = 0.25

5.37) Determine the temperature and heat fluxes at a location (2, 1) in a square plate as shown in figure. Draw the isothermal for 125°C. T1 = 100°C, T2 = 150°C, T3 = 200°C, T4 = 50°C

[AU, Nov / Dec – 2010]

5.38) Consider a brick wall of thickness 0.3 m, k = 0.7 W/m˚C. The inner surface is at 28˚C and the outer surface is exposed to cold air at -15˚C. The heat transfer coefficient associated with the outside surface is 40 W/m2˚C. Determine the steady state FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 81


temperature distribution within the wall and also the heat flux through the wall. Use two 1D elements and obtain the solution.

[AU, Nov / Dec – 2013]

5.39) Consider a brick wall as shown in figure of thickness L = 30cm, K = 0.7 W/m˚C. The inner surface is at 28˚C and the outer surface is exposed to cold air at -15˚C. The heat transfer coefficient associated with the outside surface is h = 40 W/m2˚C. Determine the steady state temperature distribution within the wall and also the heat flux through the wall. Use a two element model. Assume one dimensional flow. [AU, April / May – 2011]

5.40) A composite wall consists of three materials as shown in figure. The outer temperature is T0 = 20˚C. Convection heat transfer takes place on the inner surface of the wall with T∞ = 800˚C and h = 25W/m2˚C. Determine the temperature distribution in the wall.

[AU, May / June – 2011]



5.41) A composite wall is made of three different materials. The thermal conductivity of the various sections are k1 = 2 W/cm ˚C, k2 = 1 W/cm ˚ C, k3 = 0.2 = W/cm ˚C. The thickness of the wall for the section is 1cm, 5cm and 4cm respectively. Determine the temperature values of nodal points within the wall. Assume the surface area to unity. The left edge of the wall is subjected to a temperature of 30˚C and the right side of the wall is at 10˚C.

[AU, Nov / Dec – 2011]

5.42) Figure shows a sandwiched composite wall. Convection heat loss occurs on the left surface and the temperature on the right surface is constant. Considering a unit area and with the parameters given, use three linear elements (one for each layer) and (i) Determine the temperature distribution through the composite wall and (ii) Calculate the flux on the right surface of the wall. [AU, Nov / Dec – 2012]



5.43) A wall of 0.6m thickness having thermal conductivity of 1.2 W/m-K the wall is to be insulated with a material of thickness 0.06 m having an average thermal conductivity of 0.3 W/m-K. The inner surface temp is 1000˚C and outside of the insulation is exposed to atmospheric air at 30˚C with heat transfer co-efficient of 35 N/m2 K. Calculate the nodal temperature using FEA.

[AU, Nov / Dec – 2011]

5.44) A long bar of rectangular cross section having thermal conductivity of 1.5 W/m˚C is subjected to the boundary condition as shown below.

Two opposite sides are

maintained at uniform temperature of 180 0C. One side is insulated and the remaining side is subjected to a convection process with T = 85˚C and h = 50 W/m2˚C. Determine the temperature distribution in the bar.



5.45) The plane wall shown below is 0.5 m thick. The left surface of the wall is maintained at a constant temperature of 2000C and the right surface is insulated. The thermal conductivity

K = 25 W/MoC and there is a uniform heat generation inside the wall

of Q = 400 W/m3. Determine the temperature distribution through the wall thickness using linear elements.

5.46) Determine three points on the 50o C contour line for the rectangular element shown in the figure. The nodal values are i= 42o C, j=54o C, k= 56o C and m= 46o C.

5.47) Compute the steady state temperature distribution for the plate shown in the figure below. A constant temperature of T0 = 1500 C is maintained along the edge y = w and all other edges have zero temperature. The thermal conductivities are Kx = Ky = 1. Assume w = L = 1 and thickness t = 1.



5.48) A steel rod of diameter d = 2 cm, length l =5 cm and thermal conductivity K = 50 W/m˚C is exposed at one end to a constant temperature of 320˚C. The other end is in ambient air of temperature 20˚C with a convection co-efficient of h = 100 W/m2˚C. Determine the temperature at the midpoint of the rod using FEA. [AU, Nov / Dec – 2011] 5.49) Determine the temperature distribution in one dimensional rectangular cross-section as shown in Figure. The fin has rectangular cross-section and is 8cm long 4cm wide and 1cm thick. Assume that convection heat loss occurs from the end of the fin. Take h = 3W / cm˚C, h = 0.1 W / cm2˚ C,T ∞ = 20˚C.


5.50) Calculate the temperature distribution in stainless steel fin shown in figure. The region can be discretized into five elements and six nodes. [AU, April / May – 2009]



5.51) Calculate the element stiffness matrix and thermal force vector for the plane stress element shown in figure below. The element experiences a rise of 10 0C. [AU, April / May - 2008]

5.52) Calculate the temperature at the point for a three noded triangular element as shown in figure. The nodal values are T1 = 40˚C, T2 = 34˚C and T3 = 46˚C. Point A is located at (2, 1.5). Assume the temperature is linearly varying in the element. Also determine the location of 42˚C contour line.

[AU, May / June – 2011]



5.53) Determine the element stiffness matrix and the thermal load vector for the plane stress element shown in figure. The element experiences 20 oC increase in temperature. Take E = 15e6 N/cm2,  = 0.25, t = 0.5 cm and a = 6e - 6/o C.


5.54) The triangular element shown in figure is subjected to a constant pressure 10 N/mm2 along the edge ij. Assume E = 200 Gpa, Poisson’s ratio  = 0.3 and thickness of the element =

2 mm. The coefficient of thermal expansion of the material  = 2

x10-6/ oC and T = 50o C. Determine the constitutive matrix (stress-strain relationship matrix D) and the nodal force vector for the element.

[AU, Nov / Dec - 2009]

5.55) Compute the element stiffness matrix and vectors for the element shown in figure when the edge 2 – 3 and 3 – 1 experience heat loss.

[AU, May / June – 2012]



5.56) Compute the element matrices and vectors for the element shown below, when the edges jk and ik experience convection heat loss.

5.57) Compute element matrices and vectors for the elements shown in figure when the edge kj experiences convection heat loss.

[AU, Nov / Dec – 2009]



5.58) For the smooth pipe of variable cross-section as shown in Figure. Determine the potentials at the junctions, the velocities in each pipe. The potentials at the left end is 10 m and that at the right end is 2m.The permeability coefficient is 1 m/sec. [AU, April / May – 2011]

5.59) For the two dimensional sandy soil region as shown in figure. Determine the potential distribution. The potential (fluid head) on the left side is 10m and on right hand side is 0. The upper and lower edges are impermeable Kxx = 25*10-5 m/s and Kyy = 25*10-5 m/s. Assume unit thickness.


, -D C- L ~ L A ~.-O R:

Re-)

zu

,-a&.

-~

' " .

~

w < m p a p w i c d e ~ m b mab

SPWQ Z

4% *F*=o, mz

o-Lx.510 w h b

n . y c~o) = o *ui9

=c

~

~

A2ph- *FO= 0

2

$.LL~

a,-&

cLk.b

w -d

p--.

U

(1

=

>

y Cio)

(ii)

a,K C L O - X ) -----

0

@ 2

3=

a(~

Y=

GLI

inz

a

~Zfl

~caatis$& t

y C=a ~M' x . ~ i o - x,>

L

~

0 _cZ_LlD

9

M2 Q3 q[o')=O

y

Z

rrzet;ttoW,

t ~

%

3

~ p hiaL

p

P-

0 C L)

0

O-0)

=Q

C L O )- - c I D - ~ G )

~

~

zo

.

KC*) = 0

POI.bJT - GA?e L A&n ON -~ P Z...-W OD . . ~

q .= 0-1 ~ ~ o x2) z --4- = mi ~ m - z 7 L j &&

A2q .-. -

-..

=

~

b2 R

m)

-

LC-2)

= -zai

-m,+GOsQ

---.-

Q

4

QI=257 ..

. -

I

/

-

-

--

-

-

@A&K&\lV\'&t;.~~-~-

'k

s U t I L L"-*2-/

-

c ; L s (- r w -

-

*

- - d'R cw*C=>

9

@ =

2-( O X

.

3

=

-

CLI *.Can%+ - a x

&3 - n ~

&

J?,

--tb~t~ta~potaMtial

3

=

V-H

----a -- @

-

7

2

-

4

-----

LfJ"

dV--fi

&

/

-

q--a2-n2

,az

3n z -@3 ---

Lo-

243 E r n -

NUP? o

9-TON bch&jj~ & r =2i

d ~eam

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DEPARTMENT OF MECHANICAL ENGINEERING

ME2353 – FEA

ME2353 – FINITE ELEMENT ANALYSIS

FORMULA BOOK

FINITE ELEMENT ANALYSIS FORMULA BOOK BY ASHOK KUMAR .R (AP / MECH)

1


ME2353 – FEA

UNIT I – FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS WEIGHTED RESIDUAL METHODS: General Trial function:

Substitute the trial function in differential equation to get the residue function. Point Collocation Method:

( ) R(x)

Residue function

Here the point of x is between these limits. If the limit is 0 to 1 For two unknowns any two point between these limits. For example x value is

x value is

For three unknowns any two point between these limits. For example x value is

x value is

x value is

Sub Domain Method:

∫ ( ) Here the limit should be taken into two intervals. If the limit is 0 to 1 The limit varies between 0 – 0.5

∫

( )

The limit varies between 0.5 – 1

∫

( )

Least Square Method:

∫ ( ) Differentiate the R(x) with respect to a1, a2 Galerkin Method:

∫ ( )

( )


2


( )

ME2353 – FEA

Weighting function (Function associated with unknown

trial function) ( ) RAYLEIGH – RITZ METHOD: For Beam the Fourier series equation

∑

For Cantilever Bar (Axial Loading)

Where a1 a2 & a3 are Ritz parameters The Total Potential Energy of the Beam:

U

Strain Energy

H

Work Done by External Force

Strain Energy for Transverse Loading:

∫ (

)

Strain Energy for Axial Loading:

∫ (

)

Work Done by External Force:

∫

For SSB with UDL throughout its length For SSB with Point Load at its mid – point of length


3


∫

ME2353 – FEA

For SSB with UDL throughout its length and Point Load at its mid – point of length

∫

∫

For Cantilever Bar with axial loading

Bending Moment:

E

Young’s modulus

I

Moment of Inertia

Exact Solution: SSB Max Deflection with UDL throughout its length

SSB Max Bending Moment with UDL throughout its length

SSB Max Deflection with Point Load at its mid – point of length

SSB Max Bending Moment with Point Load at its mid – point of length

SSB Max Deflection with UDL throughout its length and Point Load at its mid – point of length

SSB Max Bending Moment with UDL throughout its length and Point Load at its mid – point of length


4


ME2353 – FEA

GENERAL TRIGONOMETRIC FUNCTIONS USED IN RITZ METHOD:

(

)

(

)

SPRINGS: The Total Potential Energy of the Beam:

U

Strain Energy

H

Work Done by External Force

Strain Energy:

k

Stiffness of Spring

δ

Deflection

Work done by External Force:

F

Force acting on the spring

u

Displacement


5


ME2353 – FEA

UNIT II – ONE DIMENSIONAL FINITE ELEMENT ANALYSIS Displacement Function: For 1D linear bar element

For 1D quadratic element


6

DEPARTMENT OF MECHANICAL ENGINEERING N1

Shape Function at node 1

u1

Displacement at node 1

N2


u2


N3


u3


ME2353 – FEA

Stiffness Matrix: For 1D linear bar element

[ ]

[

]


[ ]

[

]

A

Area of the element

- mm2

E

Young’s Modulus of the element

- N/mm2

l

Length of the element

- mm

General Force Equation:

{ }

[ ]{ }

{ }

Force vector (Global)

[ ]


{ }

Displacement matrix

Reaction Force:

{ } { }

[ ]{ }

{ }

Reaction force

If The Body Is Subjected To Self – Weight: For 1D linear bar element

{ }

{ }


7


ME2353 – FEA


{ } { } ρ

Unit weight density of the element

- N/mm3

A

Area of the element

- mm2

l


- mm

Stress On The Element:

FOR TAPER PLATE:

For rectangular cross section Area = Width * Thickness At any point of x (

)

Area at node 1:

Area at node 2(Applicable only for mid – point):


8


(

ME2353 – FEA

)

Area at node 3:

Average area of element 1:

Average area of element 2:

For circular cross section

TEMPERATURE EFFECTS ON STRUCTURAL PROBLEM: Stiffness Matrix:

[ ]

[

]

A

Area of the element

- mm2

E


- N/mm2

l


- mm

Thermal Load:

{ }

{

}

A

Area of the element

- mm2

E


- N/mm2

α

Coefficient of thermal expansion

- /˚C

Temperature difference Thermal Stress:

{ } Thermal Strain FINITE ELEMENT ANALYSIS FORMULA BOOK BY ASHOK KUMAR .R (AP / MECH)

9


ME2353 – FEA

For element 1

{ } Reaction Force:

{ }

[ ]{ }

{ }

Reaction force

{ }


[ ]


{ }

Displacement matrix

{ }

SPRINGS: Stiffness Matrix:

[ ] k

[

]

Stiffness of the spring

- N/mm

TRUSSES:

Stiffness Matrix:

[ ]

[

]

Ae

Area of the element

- mm2

Ee


- N/mm2

le


- mm


10


ME2353 – FEA

Length of element 1

√(

)

(

)

Stress: Stress of element 1

[

]{

}

u matrix varies according the element BEAMS: Displacement Function:

(

)

(

) (

) (

N1


u1


N2


u2


N3


u3


N4


)


11

DEPARTMENT OF MECHANICAL ENGINEERING u4

ME2353 – FEA


Stiffness Matrix:

[ ]

[

]

I

Moment of Inertia

- mm4

E

Young’s Modulus

- N/mm2

L

Length of the beam

- mm


12


ME2353 – FEA

Nodal Forces and Bending Moments: S.No

LOADING CASE

F1

m1

F2

m2

1)

2)

3)

(

)

(

(

)

)

(

)

4)


13


S.No

LOADING CASE

F1

m1

ME2353 – FEA

F2

m2

5)

6)

7)


14


ME2353 – FEA

UNIT III – TWO DIMENSIONAL FINITE ELEMENT ANALYSIS CONSTANT STRAIN TRIANGULAR ELEMENT: Shape Function:

Area:

|

|

Shape Function:

When all the shape function is given


15


ME2353 – FEA

When two or one shape function is given

(

)

(

)

(

)

(

)

Strain Displacement Matrix:

[ ] A

[

] - mm2

Area of the triangular element

Stress Strain Relationship Matrix: For Plane Stress condition:

[ ]

[

]

For Plane Stress condition:

( [ ]

(

)(

)

[

E

Young’s Modulus

γ

Poisson’s ratio

) (

) (

)

]

- N/mm2


16


ME2353 – FEA

Stiffness Matrix:

[ ]

[ ] [ ][ ]

[ ]

Strain Displacement matrix

[ ]

Stress – Strain Relationship matrix

t

Thickness of the triangular element

- mm


- mm2

To Convert Traction Force Into Nodal Force:

(

)

Nodal force

-N

Surface traction

- N / mm2

t

Thickness of the element

- mm

b

Breadth of the element

- mm


- mm2

To Convert Pressure Into Nodal Force:


17


ME2353 – FEA

If side j - k is subjected to pressure in x direction (node 2 & 3)

{ }

{

}

If side j - k is subjected to pressure in y direction (node 2 & 3)

{ }

{

}

If side i - j is subjected to pressure in x direction (node 1 & 2)

{ }

{

}

If side i - j is subjected to pressure in y direction (node 1 & 2)

{ }

{

}


18


ME2353 – FEA

If side i - k is subjected to pressure in x direction (node 1 & 3)

{ }

{

}

If side i - k is subjected to pressure in y direction (node 1 & 3)

{ }

{

}

Force vector

-N

Pressure

- N / mm2

t


- mm

L


- mm

Element Stress:

{ }

{

}

[ ][ ]{ }

[ ][ ] {

}

Normal stress in x direction

- N/mm2

Normal stress in y direction

- N/mm2

Shear Stress

- N/mm2


19

DEPARTMENT OF MECHANICAL ENGINEERING u&v

Nodal displacement

ME2353 – FEA - mm

Maximum Normal Stress:

√(

)

√(

)

Minimum Normal Stress:

Principle Angle:

Element Strain:

{ }

{

}

[ ]{ }

[ ] {

u&v

}

Normal strain in x direction

- N/mm2

Normal strain in y direction

- N/mm2

Shear Stress

- N/mm2

Nodal displacement

- mm

TEMPERATURE EFFECTS: Initial Strain: For Plane Stress condition:


20


{ }

{

ME2353 – FEA

}

For Plane Strain condition:

{ } α

(

){

}


- /˚C

Temperature difference γ

Poisson’s ratio

Element Temperature Force: { }

[ ] [ ]{

}

[ ]


[ ]


{ }

Initial strain

t


- mm


- mm2

AXIS SYMMETRIC ELEMENTS: Shape Function:


21


ME2353 – FEA

Area:

|

|


[ ] [ A

] Area of the triangular element

- mm2


22


ME2353 – FEA

Stress Strain Relationship Matrix:

[ ]

(

)(

) [

]

E

Young’s Modulus

γ

Poisson’s ratio

- N/mm2

Stiffness Matrix:

[ ]

[ ] [ ][ ]

[ ]


[ ]


r

Radius of the triangular element

- mm


- mm2

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{ }

{

}

[ ][ ]{ }

[ ][ ] {

}

Radial stress

- N/mm2

Circumferential stress

- N/mm2

Longitudinal stress

- N/mm2


23


u&w

ME2353 – FEA

Shear Stress

- N/mm2

Nodal displacement

- mm

TEMPERATURE EFFECTS: Thermal Force Vector:

[ ]

[ ] [ ]{ }

[ ]


[ ]


{ }

Strain matrix

r

Radius of the triangular element

- mm


- mm2

Strain:

{ } α

{

}


- /˚C

Temperature difference RECTANGULAR ELEMENT:

Shape Function:

(

)


24


ME2353 – FEA

ISOPARAMETRIC ELEMENT: FOUR NODED QUADRILATERAL: Shape Function:

(

)(

)

(

)(

)

(

)(

)

(

)(

)

Jacobian Matrix:

[ ] [

[ ]

]

[

]


25


ME2353 – FEA

[ (

)

(

)

(

)

(

)

]

[ (

)

(

)

(

)

(

)

]

[ (

)

(

)

(

)

(

)

]

[ (

)

(

)

(

)

(

)

]


[ ] ( ( [

) )

( ( ( (

) )

| |

[

]

) )

( ( ( (

) )

) )


( ( ( (

26

) )

) ) ( (

) ) ]


ME2353 – FEA

Stress Strain Relationship Matrix: For Plane Stress condition:

[ ]

[

]

For Plane Stress condition:

( [ ]

(

)(

)

) (

[

) (

E

Young’s Modulus

γ

Poisson’s ratio

)

]

- N/mm2

Stiffness Matrix: For Isoparametric Quadrilateral Element:

[ ]

∬[ ] [ ] [ ]

For Natural Co – Ordinates [ ]

∫ ∫ [ ] [ ][ ]

| |

[ ]


[ ]


| |

Determinant of the Jacobian matrix

ε,η

Natural Co – Ordinates

t


- mm


- mm2

SIX NODED QUADRILATERAL: Shape Function:

(

)


27


(

)

(

)

ME2353 – FEA

EIGHT NODED QUADRILATERAL: Shape Function:

(

)(

)(

)

(

)(

)(

)

(

)(

)(

)

(

)(

)(

)

( ( ( (

)( )( )( )(

) ) ) )


28


ME2353 – FEA

NINE NODED QUADRILATERAL: Shape Function:

(

)(

)

(

)(

)

(

)(

)

(

)(

)

(

)( (

)

)( (

)

)(

(

)( (

) )

)(

)

GAUSSIAN QUADRATURE:

∫ ∫

( )

( )

( )

∑

( )

( ) ( )

( )

(

)

Weight function ( )

Values of the function at pre – determined sampling points


29

DEPARTMENT OF MECHANICAL ENGINEERING Number of points n

ME2353 – FEA

Location

Weights

xi

wi

1

2

√

√ 3

4


30


ME2353 – FEA

UNIT IV – DYNAMIC ANALYSIS USING FINITE ELEMENT METHOD

LONGITUDINAL VIBRATION OF BAR: General Finite Element Equation:

{[ ]

{ }

}{ }

[ ]

[ ]

Stiffness matrix

[ ]

Mass matrix

{ }

External Force

{ }

Displacement matrix Natural Frequency

- rad / sec

Stiffness Matrix:

[ ]

[

]

A

Area of the element

- mm2

E


- N/mm2

l


- mm

Lumped Mass Matrix:

[ ]

[

]

ρ


- N/mm3

A

Area of the element

- mm2

l


- mm

Consistent Mass Matrix:

[ ]

[

]

ρ


- N/mm3

A

Area of the element

- mm2

l


- mm


31


ME2353 – FEA

TRANSVERSE VIBRATION OF BEAMS: General Finite Element Equation:

{[ ]

}{ }

[ ]

[ ]

Stiffness matrix

[ ]

Mass matrix

{ }

External Force

{ }

Displacement matrix

{ }

Natural Frequency

- rad / sec

Stiffness Matrix:

[ ]

[

]

I

Moment of Inertia

- mm4

E

Young’s Modulus

- N/mm2

L

Length of the beam

- mm

Lumped Mass Matrix:

[ ]

[

]

ρ


- N/mm3

A

Area of the element

- mm2

l


- mm

Consistent Mass Matrix:

[ ]

[

]

ρ


- N/mm3

A

Area of the element

- mm2

l


- mm


32


ME2353 – FEA

EIGEN VALUES & EIGEN VECTORS: General Finite Element Equation:

{[ ]

[ ] }{ }

[ ]

Stiffness matrix

[ ]

Mass matrix

{ }

Displacement matrix Eigen values

Determinant Method:

|[ ] [ ]

Stiffness matrix

[ ]

Mass matrix

[ ] |

Eigen values Natural Frequency of Bar Element:

Frequency of bar [ ]

- Hz

Mass matrix Eigen values Natural Frequency

- rad / sec

TRANSIENT VIBRATION ANALYSIS:


33


ME2353 – FEA

UNIT V – APPLICATIONS IN HEAT TRANSFER & FLUID MECHANICS HEAT TRANSFER: ONE DIMENSIONAL HEAT TRANSFER ON WALL: Global Stiffness Matrix When Wall Subjected To Conduction:

[ ]{ } { }


[ ]


{ }

Temperature matrix

{ }

[ ]

[

]

[

]{ }

{ }

A

Area of the wall

- m2

k

Thermal conductivity of wall

- W/mK

l

Length of the wall

-m

Global Stiffness Matrix When Wall Subjected To Conduction & Convection:

[

[

]

[

]] { }

{ }

A

Area of the wall

- m2

k

Thermal conductivity of wall

- W/mK

l

Length of the wall

-m

h

Heat transfer coefficient

- W/m2K

Atmospheric air temperature

-K


34


ME2353 – FEA

ONE DIMENSIONAL HEAT TRANSFER ON FIN: Global Stiffness Matrix When Fin Subjected To Conduction & Convection:

[

[

]

[

]] { }

{ }

(

)

A

Area of the fin

- m2

P

Perimeter of the fin

-m

k

Thermal conductivity of fin

- W/mK

l

Length of the fin

-m

h


- W/m2K

Atmospheric air temperature

-K

Heat Generation

-W

Q

TWO DIMENSIONAL HEAT TRANSFER: Stiffness Matrix When Subjected To Conduction:

[ ]

[ ] [ ][ ]

[ ]


[ ]

Material Property matrix

t


- mm


- mm2


[ ]

[

]


35


ME2353 – FEA

Area:

|

|

Material Property Matrix:

[ ] kxx & kyy

[

]

Thermal conductivity

- W/mK

Stiffness Matrix When Subjected To Conduction:

[ ]

k

[

]

Thermal conductivity

- W/mK


- mm2

Stiffness Matrix When Subjected To Conduction & Convection: If side 1-2 is subjected to convection

[ ]

[ ] [ ][ ]

[

]

If side 2-3 is subjected to convection FINITE ELEMENT ANALYSIS FORMULA BOOK BY ASHOK KUMAR .R (AP / MECH)

36


[ ]

[ ] [ ][ ]

ME2353 – FEA

[

]

[

]

If side 1-3 is subjected to convection

[ ]

[ ] [ ][ ]

Where

√(

)

(

)

√(

)

(

)

√(

)

(

)

[ ]


[ ]


t


- mm


- mm2

l

Length of the side subjected to convection

-m

h


- W/m2K

Stiffness Matrix When Subjected To Convection:


[ ]

[

]

[

]

[

]


[ ]


[ ]


37


ME2353 – FEA

Force Vector For Convection:


[ ] If side 2-3 is subjected to convection

[ ] If side 1-3 is subjected to convection

[ ] Atmospheric air temperature

-K

t


- mm

l

Length of the side subjected to convection

-m

h


- W/m2K

Force Vector If Heat Source Is Given:

[ ]

Q

Heat Source

- W/m3

V

Volume of triangular element

- m3

A

Area of triangular element

- m2

t


- mm

Force Vector If Heat Flux Is Given: If side 1-2 is subjected to heat flux

[ ] FINITE ELEMENT ANALYSIS FORMULA BOOK BY ASHOK KUMAR .R (AP / MECH)

38


ME2353 – FEA

If side 2- 3 is subjected to heat flux

[ ] If side 1-3 is subjected to heat flux

[ ] FLUID FLOW ANALYSIS: ONE DIMENSIONAL FLUID FLOW ANALYSIS: Stiffness Matrix:

[ ]

[

]

A

Area of the cross section of the pipe

- m2

kxx

Permeability coefficient

- m/s

l

Length

-m

Fluid Velocity:

[ ]{

[ ]

}

[

]

[

]{ }

For element 1

[ ]


kxx


- m/s

p

potential

- m2/s

Volumetric Flow Rate:

Q

Flow rate

- m3/s

v

Velocity

- m/s

A

Area of the cross section of the pipe

- m2


39


ME2353 – FEA

TWO DIMENSIONAL FLUID FLOW: Stiffness Matrix:

[ ]

[ ] [ ][ ]

[ ]


[ ]


t


-m


- m2


[ ]

[

]

Area:

|

|

Material Property Matrix:

[ ] kxx & kyy

[

]


- m/s


40


ME2353 – FEA

Stiffness Matrix:

[ ]

[

]

kxx & kyy

[

]


- m/s


- m2

Gradient Matrix:

[ ]{ }

{ } [ ]


p

potential matrix

- m2/s

Velocity Gradient Matrix:

[ ]{ }

{ } [ ]


{ }

Gradient matrix

- m/s

Volumetric Flow Rate: [

t

]

{

}

Fluid flow rate

- m3/s


-m


41


ME2353 – FEA

Area:

|

|


42

FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK

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