Compression members - The Behaviour and Design of Steel Structures to EC3 - page 84

3.10.2 Stiffness of a braced member

When a structural member is braced so that its ends act as if simply supported
as shown in Figure 3.18a, then its response to equal and opposite disturbing end
moments M can be obtained by considering the differential equilibrium equation

EI d 2 v

d x 2 =− Nv − M .

(3.67)

Thesolutionofthiswhichsatisfiestheboundaryconditions ( v ) 0 = ( v ) L = 0when
N is compressive is

π

1 − cos π N / N cr , L
sin π N / N cr , L

v = M
N

N
N cr , L

x
L

N
N cr , L

x
L

sin π + cos − 1 ,

where

N cr , L = π 2 EI / L 2 .

(3.36)

The end rotation θ = ( d v / d x ) 0 is

π
2 N cr , L tan π

N N

N cr , L ,

θ = 2 M
NL

2

whence

(π/ 2 ) N / N cr , L
tan (π/ 2 ) N / N cr , L .

α = M

θ = 2 EI

(3.35)

L

When the axial load N is tensile, the solution of equation 3.67 is

1 − cosh π N / N cr , L
sinh π N / N cr , L

v = M
N

N
N cr , L

x
L

sinh π

N
N cr , L

x
L

+ cosh π − 1 ,

whence

(π/ 2 ) N / N cr , L
tanh (π/ 2 ) N / N cr , L .

α = 2 EI
L

(3.37)