Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 325

where

(µ L /π) 2 = N / N cr , y . (7.77)

The bending moment distribution can be obtained by substituting equation 7.76
intoequation7.75.Thepositionofthemaximummomentcanthenbedetermined
by solving the condition d ( − EI y d 2 w / d x 2 ) d x = 0, whence

1 − µ 2 L 2
π 2

{ (β m cosec µ L + cot µ L ) cos µ x max + sin µ x max }
cos π x max / L

N δ 0
M

= µ L
π

.

(7.78)

The value of the maximum moment M max can be obtained from equations 7.75
and 7.76 (with x = x max ) and from equation 7.78, and is given by

M max
M

= cos µ x max − (β m cosec µ L + cot µ L ) sin µ x max

( 1 − µ 2 L 2 /π 2 ) sin π x max

( N δ 0 / M )

+

. (7.79)

L

First yield occurs when the maximum stress is equal to the yield stress f y ,in
which case

N y + M max

N

= 1,

M y

or

N
N y

1 + M max
M

N y δ 0
M y

M
N δ 0

= 1. (7.80)

In the special case of M = 0, the first yield solution is the same as that given in
Section3.2.2foracrookedcolumn.Equation7.78isreplacedby x max = L / 2,and
equation 7.79 by

N δ 0
( 1 − µ 2 L 2 /π 2 ) .

M max =

The value N L of N which then satisfies equation 7.80 is given by the solution of

N L δ 0 / M y
{ 1 − ( N L / N y )/( N cr , y / N y ) } = 0.

N L
N y +

(7.81)

The value of M / M y at first yield can be determined for any specified set of
values of N cr , y / N y , β m , δ 0 N y / M y , and N / N y . An initial guess for x max / L allows
N δ 0 / M to be determined from equation 7.78, and M max / M from equation 7.79,
and these can then be used to evaluate the left-hand side of equation 7.80. This