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Steel Designers' Manual - 6th Edition (2003)
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\{ e}= [ B]{ A}= [ B][ C] { d }
1 e
Finite element method 321
(9.29)
where [B] is the transformation matrix.
Using Equations (9.26) and (9.29), the following expression for U in terms of the
nodal point displacement vector {d e } is obtained:
1
-1, -1
U = { d } [ C] Ú Ú[
B] [ D][ B] d( vol)
C d
2
(9.30)
Differentiation of U with respect to the nodal displacements yields the stiffness
matrix [K e ]:
e , TÈT ˘
K t C ÍÚ
B D B dA
C
Î
˚
[ ]= [ ] [ ] [ ][ ]
(9.31)
where t is the thickness of the element (assumed constant).
The calculation of the stiffness matrices is generally carried out in two stages. The
first stage is to calculate the terms inside the square brackets of Equation (9.31) i.e.
the integration part. The second stage is to multiply the resulting integrations by the
inverse of the transformation matrix [C] and its transpose.
Equation (9.31) can now be written as
e
, T
[ K ]= t[ C] [ Q][ C]
-1 -1
-1 -1
Ú
T
where [ Q]= [ B] [ D][ B] dA
A
A
[ ][ ] { }
e T T T e
˙[ ]
(9.32)
The simplest elements for plane stress analysis have nodal points at the corners
only and have two degrees of kinematic freedom at each nodal point, i.e. u and v.
This type of element proves simple to derive and has been widely used. The
simplest elements of this type are rectangular and triangular in shape.
A triangular element with nodal points at the corners is shown in Fig. 9.28(a).The
displacement function of this element has two degrees of freedom at each nodal
point and the displacements are assumed to vary linearly between nodal points.This
results in constant values of the three strain components over the entire element;
the displacement functions are
u = a1 + a2x + a3y (9.33)
v = a4 + a5x + a6y
The rectangular element with sides a and b, shown in Fig. 9.28(b), is used with the
following displacement functions:
u = a1 + a2x + a3y + a4xy
v = a5 + a6x + a7y + a8xy (9.34)