Structural Analysis Methods I: Beam Elements - Mechanics of Structures: Variational and Computational Methods - page 192

The flexibility coefficient, f ij , is the displacement at DOF i due to a unit force applied
at DOF j . The coefficients f ii and f ji ,i j are referred to as direct and cross-flexibility

coefficients, respectively. In the definition of a flexibility coefficient, force and displace-
ment are used in a generalized sense. That is, for example, a unit load applied at j may
cause a rotation at i , and a unit moment at j may cause a deflection at i . For the case at
hand, where v R =

=

fp R , with p R = M b ] T and v R =

w R θ R ] T = w θ

] T , or

[ V b [ [

w
θ

V b
M b

3

/ 3 −

2

/ 2

1
EI

= (4.33)

−

2

/ 2

where the flexibility coefficients are given in Fig. 4.4. It should be clear from the f ij shown
in Fig. 4.4 that for this case both i and j of f ij refer to the same location of the beam, i.e.,
the right end. The stiffness coefficients can be found using k R =

f − 1 or

− 1

− 1

EI 12

1
EI

3

/ 3 −

2

/ 2 /

3

6 /

2

k R = = (4.34)

2 2

− / 2 6 / 4 /

If the axial effects are included, the reduced stiffness matrix for a cantilevered beam
element would be

=

/

N b
V b
M b

EA 0 0 u

w
θ

0 EI /

3

6 EI /

2

(4.35)

0 6 EI /

2

4 EI /

where (Fig. 4.5a) u = u b − u a , the extension of the bar element.

Flexibility and Stiffness Matrices in Terms of End Moments
Case 2 of this section corresponds to the element of Fig. 4.5. For this configuration, the
flexibility matrix and corresponding stiffness matrix are found to be

EA

− u

θ ab
θ ba

0 0 N a

M a
M b

=

3 EI

−
6 EI

v R = fp R = 0 (4.36a)

−
6 EI 3 EI

0

=

EA

N a
M a
M b

00 − u

4 EI 2 EI

p R = k R v R = 0 θ ab

θ ba

(4.36b)

2 EI 4 EI

0

with u a .

If the bar is subject to bending and torsion, the reduced stiffness matrix would appear
in nondimensional form as

u = u b −

=

M a
M b
M tb

42 0
24 0
00 J ∗

θ ab
θ ba
φ

EI

(4.37)

p R = k R v R