Structural Analysis Methods I: Beam Elements - Mechanics of Structures: Variational and Computational Methods - page 193

TABLE 4.1
Transfer Matrix, Stiffness Matrix, Reduced Stiffness Matrices, and Flexibility Matrices for
Beams (Sign Convention 2)

Transfer Matrix Stiffness Martix

3
6 EI

3
2 EI

w b
θ b V b
M b /

w a
θ a V a
M a /

V a
M a /
V b
M b /

12 − 6 − 12 − 6 w a

θ a
w b
θ b

1 − 1

− 6 4 6 2

− 3
2 EI

− 3
EI

EI

01

= =

3

− 26 26

−

0 − 1 0

6 2 6 4

0 − 1 − 1

U i k i

Flexibility Matrices Reduced Stiffness Matrices

In Terms of End Moments and Tangents

−

θ ab
θ ba

2

12 M M b

M M b 21

12 θ ab

.

6 EI

2 EI

− 1 1 / 1 / 10

= g = =

− . θ ba

− 1 / 1 / 01

f i k i R

In Terms of Variables at One End

w
θ

2 V b

M b /

V b
M b /

63
32

w
θ

.

3
6 EI

2 EI

− 3 − 1 10

= g = =

− 36 .

3

0 − 1 01

k i R

f i

with

GJ
EI =

J

J ∗ =

φ = φ b − φ a

2 ( 1 + ν) I

11 0 . − 100

11 0 .

g =

− 010

.

00 − 1 001

A summary of some of the matrices discussed in this chapter is provided in Table 4.1.

Flexibility Matrices with Applied Loads
The flexibility matrices discussed here can be generalized to include the effects of applied
loading:
Case 1 (Fig. 4.4):

w V b w

0

3 2

/ 3 − / 2

1
EI

= + (4.38a)

2

− / 2

θ

M b θ

0

Case 2 (Fig. 4.5):

θ ab / M a θ

0
ab 3 − / 6

1
EI

= + (4.38b)

− / 6 / 3

θ ba

θ

0
ba

M b

where the superscript 0 indicates the terms due to prescribed loadings.