STATIC FEM ALGORITHM FOR A TRUSS

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STATIC FEM ALGORITHM FOR A TRUSS

Piotr Pluciński

e-mail: p.plucinski@L5.pk.edu.pl Jerzy Pamin

e-mail: jpamin@L5.pk.edu.pl

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Scope of lecture

Truss as a set of nodes and bars (truss elements)

Truss element description

Example computations for 2D truss (MathCAD)

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Computational model - discretization

Computational model

In the computational model we must guarantee:

I continuity of displacements (at nodes, where elements are connected)

I satisfaction of kinematic constraints

I satisfaction of equilibrium equations for the structure and any

substructure (e.g. a node or element)

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Computational model - discretization

Discretization process - mesh generation

Bar structure is idealized as a discrete system of elements and nodes

I Numbering nodes

I Numbering elements

I Specification of relations between elements and nodes (topology of discrete model)

e i j

1 1 2

2 1 4

3 2 4

4 4 5

5 2 5

6 2 3

7 3 5

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Truss element description (2D)

Definitions of variables describing the displacements, strain and cross-section force in the bar under tension/compression

u(x ) = {u(x )}, e(x ) = {ε 0 (x )}, s(x ) = {N(x )}

Kinematic and physical equation for a point P(x , y , z) = P(x , 0, 0) = P(x ) on bar axis

ε ₀ = du

dx → e = Lu, L = d dx

N = EA · ε 0 → s = De, D = [EA]

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Truss element description (2D)

Displacement approximation in element (local) coordinate system 0x Number of local dofs of a node ndof _w = 1 and element ndof ^e = 2.

1 2 e

L ^e

q ₁ ^e = u ^e ₁

q ₂ ê = u ₂ ê x ê , u ξ = x ê

L ^e

where ξ = x ^e

L ^e normalized coordinate q w

[1×1]

= {u w }

q ^e

[2×1]

= {q ₁ ê , q ê ₂ } = {u ₁ ê , u ê ₂ }

u(ξ)

[1×1]

= {u(ξ)} = N(ξ)

[1×2]

· q ^e

[2×1]

=

(1 − ξ) ξ ·

q ₁ ^e q ₂ ^e

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Truss element description (2D)

Strain, normal force and stiffness matrix in element (local) coordinate system 0x

e(ξ)

[1×1]

= {ε ₀ (ξ)} = LN ^e (ξ) · q ^e = B(ξ)

[1×2]

· q ^e

[2×1]

=

− _L ¹ _e _L ¹ _e ·

q ^e ₁ q ^e ₂

s(ξ)

[1×1]

= {N(ξ)} = D

[1×1] · B(ξ)

[1×2]

· q ^e

[2×1]

= [EA] ^e ·

− _L ¹ _e _L ¹ _e ·

q ₁ ^e q ₂ ^e

k ^e

[2×2]

=

L ^e

Z

0 B ^T DBdx = EA L

e

· 1 −1

−1 1

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Truss element description (2D)

Description of truss element in global coordinate set 0XY

Number of global dofs of a node NDOF _w = 2 and element NDOF ê = 4 (c = cos α ê , s = sin α ê )

Q ₁ ^e q ₁ ^e

q ₂ ^e

Q ₂ ^e

Q ₃ ^e Q ₄ ^e

x y

α ^e

Q w [2×1]

= {U w , V w }

Q ^e

[4×1]

= {Q 1 , Q 2 , Q 3 , Q 4 } =

= {U 1 , V 1 , U 2 , V 2 }

q ^e

[2×1]

=

q 1

q ₂

e

=

c s 0 0

0 0 c s

e

· 



 Q 1

Q 2

Q ₃ Q ₄







e

= T ^e

[2×4] · Q ^e

[4×1]

Q ^e

[4×1]

= T ^eT

[4×2] · q ^e

[2×1]

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Truss element description (2D)

Global stiffness matrix: (c = cos α ^e , s = sin α ^e )

K ^e

[4×4] = (T ^T kT) ^e = EA L

^e

· 





cc cs −cc −cs cs ss −cs −ss

−cc −cs cc cs

−cs −ss cs ss







e

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Flowchart of FEM algorithm for statics

Discretization

Computation of stiffness matrices and nodal load vectors for elements

Assembly

Consideration of boundary conditions

Computation of nodal displacement vector and reaction vector

Return to each ele-

ment to compute

nodal force vectors

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Example computations for 2D truss (MathCAD)

Problem definition and discretization

Q 1

Q 2

Q 3

Q ₄ Q ₅

Q 6

1

2

3

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Example computations for 2D truss (MathCAD)

Input data

Stiffness matrix k ^e

Transformation matrix T ^e

Incidence matrix

Longitudinal stiffness

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Example computations for 2D truss (MathCAD)

Computation of length, values of trigonometric functions and transformation

matrix for element 1

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Example computations for 2D truss (MathCAD)

Stiffness matrix of element 1 in local and global coordinate set

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Example computations for 2D truss (MathCAD)

Computation of length, values of trigonometric functions and transformation

matrix for element 2

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Example computations for 2D truss (MathCAD)

Stiffness matrix of element 2 in local and global coordinate set

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Example computations for 2D truss (MathCAD)

Computation of length, values of trigonometric functions and transformation

matrix for element 3

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Example computations for 2D truss (MathCAD)

Stiffness matrix of element 3 in local and global coordinate set

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Example computations for 2D truss (MathCAD)

Assembly of stiffness matrix for element 1 into global stiffness matrix

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Example computations for 2D truss (MathCAD)

Assembly of stiffness matrix for element 2 into global stiffness matrix

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Example computations for 2D truss (MathCAD)

Assembly of stiffness matrix for element 3 into global stiffness matrix

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Example computations for 2D truss (MathCAD)

Definition of load vector with point load

10 kN

Q ₆

Q 5

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Example computations for 2D truss (MathCAD)

Computation of substitute load vector due to imposed displacement ∆

∆ = 0.001 m Q 3

Q ₄

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Example computations for 2D truss (MathCAD)

Imposition of boundary conditions K → K _w , F → F _w

Q 1 = 0

Q ₂ = 0

Q 3

Q 4 = −∆

Q ₅ = 0

Q 6 





K 11 K 12 K 13 K 14 K 15 K 16

K 21 K 22 K 23 K 24 K 25 K 26

K 31 K 32 K 33 K 34 K 35 K 36

K 41 K 42 K 43 K 44 K 45 K 46

K 51 K 52 K 53 K 54 K 55 K 56

K 61 K 62 K 63 K 64 K 65 K 66











 0 0 Q 3

−∆

0 Q 6







=





 0 0 0 0 0 F s6





 +





 R 1

R 2

0 R 4

R 5

0 





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Example computations for 2D truss (MathCAD)

Computation of nodal displacements

Q ₃ = 5 · 10 ⁻⁴

Q 4 = 1 · 10 ⁻³

Q 6 = 3.111 · 10 ⁻³

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Example computations for 2D truss (MathCAD)

Determination of support reactions

R 1 = 1.667

R ₂ = 7.778 R 4 = 2.222

R 5 = 1.667

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Example computations for 2D truss (MathCAD)

Return to element to compute nodal forces - diagram of nodal forces Element 1

Element 2

Element 3

STATIC FEM ALGORITHM FOR A TRUSS