Direct Variational and Weighted Residual Methods: Classical Trial Function Methods - Mechanics of Structures: Variational and Computational Methods - page 481

can be computed using a finite element analysis. For each
element, approximate the warping function

The warping function ω( y, z )

ω by

N ω = ω T N T

ω( y, z ) = N j ω j = (7.87)

j

where N j are the shape functions that form the vector N and ω

j are the nodal values of

ω

which form the vector ω .

The derivations of ω are

∂ω
∂ y = ∂

N
∂

∂ω
∂ z = ∂

N
∂ y ω z ω

(7.88)

and

N T
∂

N T
∂

∂
∂

ω T ∂

∂
∂

ω T ∂ ω T N T

δω = N δ ω = δ

y δω = δ z δω = δ

(7.89)

y z

Substitute these into Eq. (7.85)

ω T ∂

ω z ∂ dA

N T
∂

N T
∂

N T
∂

N T
∂

∂ N

∂ y + ∂

∂ N

∂

y ∂ ω T k i ω p i

A δ

+ − = δ ( − ) = 0 (7.90)

y z z y z

where k i and p i

are the element stiffness and ”quasi-loading” vector, respectively, defined

as

∂ dA

N T
∂

N T
∂

N T y ∂∂ y N

N T z ∂∂ z N dA

∂ N

∂ y + ∂

∂ N

∂

k i = = +

y z z

A A

(7.91)

z ∂ dA

N T
∂

N T
∂

z N T y ∂ −

y N T z ∂ dA y ∂ p i =− − =−

y z

A A

These expressions can be used to define the stiffness matrix and “quasi-loading” vector for
particular elements. The same relationships for k i

and p i

are derived in Pilkey (2002) using

Galerkin's method.

References

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