Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 668

It is cle ar that th is matrix reduces to the stiffness matrix for an Euler-Bernoulli matrix as
ε =

Then A = 4 ,B = 2 ,D =

of Eq. (11.51) permit
the formation of stiffness matrices expressed in terms of polynomials in

| N 0 | / EI → 0 . 0 . The expansions in ε

ε , rather than in

terms of transcendental functions.
The applied distributed loading p i 0 p i 0 can be

obtained by integration of Eqs. (11.23) or from a power series expansion. Loading functi on s
for several distributed loadings are listed in Tables 4.4 and 11.1. See Example 11.5 for p i 0
for some prescribed imperfections.

of the basic relationship p i k i v i

= −

Bar in Tension ( N 0 > 0)

. −

N a u a

EA / 0 0 EA / 0 0

2 .

w a

V a

(

A + B ) + D ] EI

/

3

− (

A + B )

/ − (

A + B ) + D ] EI

/

3

− (

A + B )

/

2

[2 EI 0 [2 EI

.

M a θ a

A EI A + B )

2

B EI

/ 0 ( EI / /

=

...... . ... ......... .........

N b u b

.

EA / 0 0

V b w b

.

A + B ) + D ] EI

3

A + B )

2

Symmetric [2 ( / ( EI /

M b θ b

A EI /

p i k i v i

(11.52)

=

where

ε( ε − ε ε)

sinh cosh 2

15 ε

11
6300 ε

2 4

A = A =

+ − +···

4

2 ( cosh ε − 1 ) − ε sinh ε

ε(ε − ε)

sinh 1

30 ε

13
12 600 ε

2 4

B = B =

− + −···

2

( ε − ) − ε ε

2 cosh 1 sinh

2

ε ( 1 − cosh ε) 1

10 ε

1
1400 ε

2 4

A + B = A + B =

6 + − +···

2 ( cosh ε − 1 ) − ε sinh ε

N
EI

D = ε

2 2

=

− ε

3 sinh

ε

6
5 ε

1
700 ε

2 4

(

A + B ) + D =

(

A + B ) + D =

+ − +···

2 2 12

2 ( cosh ε − 1 ) − ε sinh ε

(11.53)

| Note that the axial force N occurs in both the matrix k i EI ) and

the force vector p i . Use of this matrix in a stability study of frames is discussed later in this
chapter.
The corresponding load vectors are readily calculated, or can be taken from Tables 11.1
or 4.3.
The parameters A ,B ,A +

(through ε = N 0 | /

B ,D , 2 A + B ) ± D , are plotted in Fig. 11.14 as functions (

of ε (Chwalla, 1959).