AS 5850 Finite Element AnalysisAS 5850 Finite Element Analysis

InstructorProf. K. V. N. Gopal

Department of Aerospace EngineeringIIT Madras

Two-Dimensional Linear Elasticity

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Equations of Plane Elasticity - 1

( , ), ( , )x x y yu u x y u u x y

, , 2y yx xxx yy xy

u uu ux y y x

strain- displacement relations (infinitesimal strain)

*{ } [ ]{ }B uin matrix form as

where * 0[ ]

0

Tx yB

y x

{ } { 2 }T

xx yy xy

displacement field

stress – strain relations (linear elastic)

0{ } [ ]({ } { })D

{ } { }Txx yy xy

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Equations of Plane Elasticity - 2

matrix of elastic constants for orthotropic materials

11 12

12 22

66

0[ ] 0

0 0

d dD d d

d

Plane Stress Plane Strain

111

12 21

222

12 21

12 12 22 66 12

(1 )

(1 ),

Ed

Ed

d d d G

1 1211

12 12 21

2 2122

21 12 21

12 12 22 66 12

(1 )(1 )(1 )

(1 )(1 )(1 )

,

Ed

Ed

d d d G

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Equations of Plane Elasticity - 3Stress equilibrium equations over a small sub-domain (finite element)

, 0 , 1,2 e

ij j i ef t dxdy i j

[ *] { } { } 0

e

TeB f t dxdy

in matrix form

Substituting constitutive relations and strain-displacement relations

0[ * ] [ ] [ * ]{ } [ * ] [ ]{ } { } 0e

T TeB D B u B D f t d x d y

To obtain weak form or weighted-integral use the weight function

1

2

0{ }

0w

ww

satisfying ˆ[ ]{ } { }N t t or 0 2ˆ[ ][ ]([ *]{ ( )} { }) on N D B u s t

1ˆ( ) ( ) { ( ) ( )} on Tx yu s u s u s u s

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Equations of Plane Elasticity - 4

The above integral can be simplified by integrating by parts once andusing the divergence theorem to give the weak form of the equation

0

([ *]{ }) [ ][ *]{ } { } { } { } { }

([ *]{ }) [ ]{ } 0e e e

e

T T Te e e

Te

B w D B u t dxdy t w t ds t w f dxdy

t B w D dxdy

-

Finite Element ModelThe weak/variational form clearly shows that {u} is the primary variable and the secondary variable are the vectors {f} and {t} and they can be approximatedby the Lagrange family of interpolation functions and at least linear.

The interpolation for the variables in the matrix form are

{ } { }{ }, { } { }{ }x x y yu u

1 2{ } { }n ...

0{ } [ * ] [ ] [ * ]{ } [ * ] [ ]{ } { } 0e

T T Tew B D B u B D f t d x d y

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Equations of Plane Elasticity - 5{ } is the vector of element nodal displacementsTherefore if

1 1 2 2{ } { } ... n n Tx y x y x y

1 2

1

00{ }

00 0n

n

N

.....

{ } { }{ }u N

Further let [ ] [ *]{ }B B

The equation can be written as [ ]{ } { } 0K F

[ ] [ ] [ ][ ] e

TK B D B dxdy

0{ } { } { } [ ] [ ]{ } { } { }e e e

T T TF N f dxdy B D dxdy N t ds

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Potential Energy of an Elastic System

0 01 { } [ ]{ } { } [ ]{ } { } { }2

{ } { } { } { } { } { }e e

T T Tp

V

T T T

D D dV

u F dV u T ds d P

(1) (2) (3)

(4) (5) (6)

Consider a linear elastic body with volume V and surface area S subjectedto conservative loads.

(1) - strain energy term due to mechanical strain(2) - initial/thermal strains(3) - residual stresses(4) - work due to body forces(5) - work due to surface tractions(6) - work due to point/concentrated loads with consistent

displacements at the point of application.

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Potential Energy of an Elastic System

0p

u

The solution {u} is that which minimizes the total potential energy

or 0p

0

{ } 0 [ *]{ }2

xxx

yxy

xu

yy B uuy

y x

Substituting for {} in terms of the displacement vector in p

0 01{ } [ *] [ ][ *]{ } { } [ *] [ ]{ } { } [ *]{ }2

{ } { } { } { } { } { }e e

T T T T Tp

V

T T T

u B D B u u B D u B dV

u F dV u T ds d P

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Potential Energy of an Elastic System

0p

u

Applying

0 0[ *] [ ][ *]{ } [ *] [ ]{ } [ *]{ } { } { } 0e e

T T

V

B D B u B D B dV F dV T ds

or re-arranging the terms

0 0[ *] [ ][ *]){ } ([ *] [ ]{ } [ *]{ } { }) { }e

T T

V V

B D B u dV B D B F dV T ds

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Equations of Plane Elasticity - 6

The assembly of the element level equations to give the global equationsfollows the same procedure as that for single variable except that theprocedure is applied to both degrees of freedom at each node.

During assembly, the continuity of the primary variables at each nodeIs used to obtain the global degrees of freedom.

The application of flux equilibrium along the inter-element boundary (not coinciding with the domain boundary on which traction is defined, will eliminate the tractions at the interior nodes.

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Equations of Plane Elasticity - 7

Linear triangular element - 2 dofpn and total 6 dofpe

Linear Quadrilateral- 2 dofpn and total 8 dofpe

In a linear triangular element, the strains are constant throughout the elementHence it is known as a constant strain triangular (CST) element

In a linear quadrilateral element, the first derivatives are not constant.

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Ref: M. Asghar Bhatti, ‘Fundamental Finite Element Analysis and Applications’

normal pressure q = 20 lb/in2

L = 4 in. longh1 = 2 in. h2 =1 in. (free end)t = 0.25 in. E = 104 lb/in2 =0.2

Data given

Determine displacement distribution over eachelement. Using these compute element strains,stresses, principal stresses and effective stress.

coarse 4 elementmesh of plane stresstriangular elements

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a BracketProcedure1. For the data given and the chosen mesh use the appropriate Lagrangian

interpolation function and substituting in the element level equations compute the element level matrices and vectors. (appropriate Gaussian quadrature)

2. Using the appropriate mapping between global node numbers and element nodenumbers, obtain the global equations.

3. Apply BCs and solve the global equations (several methods depending on thenature of the matrix, comparative values of the matrix elements etc).

4. Use the nodal displacements to obtain the displacement distribution in each element.

Stress Computation

5. From the nodal displacement distribution in each element, the contribution to nodalstrains and stresses at each node is obtained. The average stress at the node is computed from

6. The contribution of all the surrounding elements is averaged or the element stressesare also extrapolated from values computed at Gaussian points of each element.

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Boundary Conditions and Nodal Solution

Impose essential boundary conditions on the global level equations to get the reducedsystem of equations in terms of the remaining unknowns. Solve for the nodal values.Compute reactions and verify overall equilibrium

Nodes 1 and 2 are fixed. Thus the essential boundary conditions are as follows:

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a BracketDisplacement distribution and element strains and stress values

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Stress Analysis of a Bracket

Is the chosen mesh suitable for the problem?

Examining the stress results, we find even though sides 1 and 4 share a common side, there is significant difference in the values of the effective stress. Similarly for elements 3 and 4. This means the chosen mesh is too coarse for the problem and needs to berefined.

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Solution of Linear Equations