Structural Analysis Methods I: Beam Elements - Mechanics of Structures: Variational and Computational Methods - page 186

Note that all displacements are now located on the right-hand side while the forces are on
the left-hand side. Equation (4.11) can be written for a beam element as

. −

V a

12 EI 6 EI 12 EI 6 EI

− − w

a 3 2 3 2

.

k aa k ab
k ba k bb

v i

M a

6 EI 4 EI 6 EI 2 EI

− θ

a 2 2

= =

······ ··· ······ ······

.

w

12 EI 6 EI 12 EI 6 EI

V b − b

3 2 3 2

.

θ

M b

6 EI 2 EI 6 EI 4 EI

− b

2 2

p i k i v i = (4.12)

This is, of course, the same as the beam stiffness matrix derived in Chapter 3, Example 3.11.
It follows from the form of this relationship that a stiffness element k ij , e.g., k 11 =

3 ,
can be considered to be the force developed at coordinate i due to a unit displacement at
coordinate j , with all other displacements equal to zero. These “coordinates” are usually
referred to as degrees of freedom (DOF). More precisely, the DOF are the independent dis-
placement components necessary to fully describe the spatial position of a structure. The
number of DOF depends on the modeling of the structure for analysis. In static analyses, we
analyze each element before the overall structure is treated, and thus, reduce the behavior
of the element to selected DOF at each end of the element. Some of these end DOF can be
ignored if it is known that the response of the structure does not depend heavily on these
DOF. This situation occurs with rigid frames, for example, where displacements due to
uniform axial strain are usually significantly smaller than the displacements resulting from
bending.
It is possible to rewrite the stiffness matrix of Eq. (4.12) giving a form that is often more
convenient. A redefinition of the vectors p i

12 EI /

and v i leads to the modified form

V a
M a

12 − 6 − 12 − 6 w

a
θ

=

/
V b
M b /

EI − 64 62

a
w b
θ b

(4.13)

− 26 26

−

3

62 64

p i k i v i =

The complete description of an element should include a vector representing applied
loads on the element. They can be calculated as the reactions of a beam element with fixed
ends. Another possibility, which is presented in the following paragraph, is to obtain the
loading vector by a transformation of the transfer matrix. The element stiffness matrix
including its element loading vector can be expressed as

= +

V 0
a
M a
V 0
b
M 0
b

V a
M a
V b
M b

k 11 k 12 k 13 k 14 w a

θ a
w b
θ b

k 21 k 22 k 23 k 24

(4.14)

k 31 k 32 k 33 k 34

k 41 k 42 k 43 k 44

p i k i v i

= − p

As mentioned in the previous paragraph, it is necessary to include with the stiffness
matrix a vector to account for applied loading. Normally, this vector would account for
only the loading applied between the ends, since end loadings are inserted using the