1.48 Coupling degrees of freedom

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If two elements are joined at a node, the displacements must be the same

at the node. Conversely this implies that a force that acts at the node must

work against the stiffness of both elements. Hence if two degrees of freedom

are coupled, their stiffness adds as in springs working in parallel.

• This simple coupling of even the most diverse elements is the real advantage

of the FE method with regard to other numerical methods.

To understand why the stiffness adds let us first recall two rules of matrix

algebra.

a) If the columns of a unit matrix I are permuted in an arbitrary fashion

I → IP , and if a matrix K (having the same size) is multiplied from the right

by this matrix IP, the columns of K are permuted in the same way. If the

matrix K is multiplied from the left by the transposed matrix IT

P, the rows

of K are interchanged in the same way.

b) If a 2 × 2 matrix is multiplied from the right by a vector [1, 1]T the

columns of the matrix are added. If the same is done from the left, the rows

are added:

_

a b

c d

__

11

_

=

_

a + b

c + d

_ _

1, 1

_ _

a b

c d

_

=

_

a + c, b + d

_

. (1.635)

Next consider the bar in Fig. 1.163, which consists of two elements, so that

1.48 Coupling degrees of freedom 225

Fig. 1.163. Bar consisting of

two elements

⎡

⎢⎢⎣

k1 −k1 0 0

−k1 k1 0 0

0 0 k2 −k2

0 0−k2 k2

⎤

⎥⎥⎦

⎡

⎢⎢⎢⎣

u(1)

1

u(1)

2

u(2)

1

u(2)

2

⎤

⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎣

f(1)

1

f(1)

2

f(2)

1

f(2)

2

⎤

⎥⎥⎥⎦

or Kl ul = fl . (1.636)

In matrix algebra the coupling between the global, ui, and local degrees of

freedom, u(j)

i , can be written as

⎡

⎢⎢⎢⎣

u(1)

1

u(1)

2

u(2)

1

u(2)

2

⎤

⎥⎥⎥⎦

=

⎡

⎢⎢⎣

1 0 0

0 1 0

0 1 0

0 0 1

⎤

⎥⎥⎦

⎡

⎣

u1

u2

u3

⎤

⎦ or ul = Au, (1.637)

and because

uTl

Kl ul = uT AT KlAu = uT Ku (1.638)

we have

⎡

⎣

1 0 0 0

0 1 1 0

0 0 0 1

⎤

⎦

⎡

⎢⎢⎣

k1 −k1 0 0

−k1 k1 0 0

0 0 k2 −k2

0 0−k2 k2

⎤

⎥⎥⎦

⎡

⎢⎢⎣

1 0 0

0 1 0

0 1 0

0 0 1

⎤

⎥⎥⎦

=

⎡

⎣

k1 −k1 0

−k1 k1 + k2 −k2

0 −k2 k2

⎤

⎦ = K.

(1.639)

Due to the multiplication from the right, columns 2 and 3 are added; due to

the multiplication from the left, rows 2 and 3 are added. This is the algebra

which governs the assemblage of the global stiffness matrix.

Rigid elements should be modeled by formulating coupling conditions,

and not by raising the stiffness. The displacements ux, uy, uz of a node x =

(x, y, z) in a rigid element can easily be expressed in terms of displacements

ux,ref, uy,ref, uz,ref and rotations ϕx, ϕy, ϕz of a reference node

uz = uz,ref − (x − xref ) · ϕy,ref + (y − yref ) · ϕx,ref . (1.640)

Implicit formulations as in the case of a skew roller support

u•n = ux nx + uy ny + uz nz = 0 (1.641)

226 1 What are finite elements?

Fig. 1.164. Coupling of a beam and a slab

must be transformed into an explicit form to be able to distinguish between

master and slave degrees of freedom. A master degree of freedom is a genuine

degree of freedom which has its place in the structural system of equations,

while a slave is eliminated either at the element level or later, when the global

system of equations is assembled. Elimination at the element level only makes

sense if the coupling condition is a genuine property of the element, and if the

number of equations in the global stiffness matrix does not increase. Repeated

application of the explicit form can raise the rank of the matrix.

T beams

If a girder is modeled by a beam as in Fig. 1.164, the movements of the beam

must follow the movements of the slab:

⎡

⎢⎢⎢⎢⎢⎢⎣

u5

u6

u7

u8

u9

u10

⎤

⎥⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎣

0 −e 0 0

1 0 0 0

0 1 0 0

0 0 0 e

0 0 1 0

0 0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎣

u1

u2

u3

u4

⎤

⎥⎥⎦

or uB

(6) = A(6×4)uS

(4) . (1.642)

Correspondingly a modified beam element matrix is obtained

AT

(4×6)K(6×6) A(6×4) = K(4×4) , (1.643)

which can be incorporated directly into the global stiffness matrix of the slab.

A different approach to formulating coupling conditions between degrees

of freedom is provided by Lagrange multipliers. Their application is simple,

but it is often difficult to obtain stable solutions with this technique.

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