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UNIT II: ONE-DIMENSIONAL PROBLEMS

CHAPTER 3: ELEMENT MODELING AND ANALYSIS Lesson 11: THE GALERKIN APPROACH Following the concepts introduced in Chapter 2, we introduce a virtual displacement field φ = φ(x)

(11.1)

and associated virtual strain

ε (φ ) =

dφ dx

(11.2)

where φ is an arbitrary or virtual displacement consistent with the boundary conditions. Galerkin’s variational form, given in Eq. 6.11, for the one-dimensional problem considered here, is

∫ σ ε (φ )Adx − ∫ φ T

L

L

T

fAdx − ∫ φ T Tdx −∑ φ i Pi = 0 L

(11.3a)

i

This equation should hold for every φ consistent with the boundary conditions. The first term represents the internal virtual work, while the load terms represent the external virtual work. On the discretized region, Eq. 11.3a becomes.

∑ ∫e e

T

Eε (φ )Adx −∑ e

e

∫φ e

T

fAdx − ∑ e

∫φ e

T

Tdx −∑ φ i Pi = 0

(11.3b)

i

Note that ε is the strain due to the actual loads in the problem, while ε(φ) is a virtual strain. similar to the interpolation steps in Eq. 8.7b, 9.5, and 9.7, we express

φ = Nψ ,

[

]

ε (φ ) = Bψ

(11.4)

2 T

where ψ = ψ 1 ,ψ represents the arbitrary nodal displacements of element e. Also the global virtual displacements at the nodes are represented by

ψ = [ψ 1 ,ψ 2 ,.....ψ N ]

T

(11.5)

Element Stiffness Consider the first term, representing internal virtual work, in Eq. 11.3b. Substituting Eq. 11.4 into Eq. 11.3b, and noting that ε = Bq, we get 46

∫ε e

T

Eε (φ )Adx = ∫ q T B T EBψ Adx

(11.6)

e

In the finite element model (Lesson 8), the cross-sectional area of element e, denoted by Ae, is constant. Also B is a constant matrix. Further, dx = (λe / 2)dξ . Thus,



T T ∫ ε Eε (φ )Adx = q  Ee Ae

 T e =q kψ = ψTkeq

e

λe T 1  B B ∫ dξ ψ 2 −1 

(11.7a) (11.7b)

where ke is the (symmetric) element stiffness matrix given by

k e = E e Ae λe B T B

(11.8)

Substituting B from Eq. 9.6, we have

ke =

E e Ae  1 λe − 1

− 1 1

(11.9)

Force Terms Consider the second term in Eq. 11.3a, representing the virtual work done by the body force l in an element. Using φ = Nψ and dx = e dξ and noting that the body force in the element is 2 assumed constant, we have 1 λe T T T (11.10a) ∫e φ fAdx = −∫1ψ N fAe 2 dξ = ψT f e

(11.10b)

where

1   ∫ N 1 dξ  Aλ f   (11.11a) f e = e e −11  2    ∫ N 2 dξ  −1  is called the element body force vector. Substituting for N1 = (1-ξ)/2 and N2 = (1+ξ)/2, we 1

obtain

∫ N1dξ = 1. Alternatively,

−1 1

∫N

2

1

∫ N dξ 1

is the area under the N1 curve = ½ × 2 × 1 = 1 and

−1

dξ = 1. Thus,

−1

47

Ae λe f 1  2 1 The element traction term then reduces to

fe =

(11.12b)

∫φ

(11.13)

T

Tdx = ψ T T e

e

where the element traction-force vector is given by Tλ 1 Te = e   2 1

(11.14)

At this stage, the element matrices ke, fe, and Te have been obtained. After accounting for the element connectivity (in Fig. 8.3, for example, ψ = [ψ1, ψ2]T for element 1, ψ = [ψ2, ψ3]T for element 2, etc.), the variational form

∑ψ e

T

k e q − ∑ψ T f e − ∑ψ T T e − ∑ψ i Pi = 0

can be written as

e

e

ψT (KQ-F) = 0

(11.15)

i

(11.16)

which should hold for every ψ consistent with the boundary conditions. Methods for handling boundary conditions are discussed in the next lesson. The global stiffness matrix K is assembled from element matrices ke using element connectivity information. Likewise, F is assembled from element matrices fe and Te. This assembly is discussed in detail in the next lesson.

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