Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 302

Crooked beam-columns
Straight beam-columns

Accurate M max / M
Approximate M max / M

N=N y
N=N cr,y

N=N y

1.0 1.0

m M

N

0.8 0.8 N=N cr,y

m

1.0
0.5
0

N = N b,y , Rd

0.6

0.6

β m

1.0
0.5
0
-0.5
-1.0

N

0.4 0.4

M -0.5

-1.0

0.2 0.2 M=M y

M = M y

N cr,y / N y = 1/1.5

N cr,y / N y = 1/1.5

0

0

0 0.2 0.4 0.6 0.8 1.0

M/M

0 0.2 0.4 0.6 0.8 1.0

M/M

y y

(a) Straight beam-columns (b) Crooked beam-columns

Figure 7.7 First yield of beam-columns with unequal end moments.

The elastic in-plane behaviour of members in which the axial forces cause
tension instead of compression can also be analysed. The maximum moment in
such a member never exceeds M , and so a conservative estimate of the elastic
limit can be obtained from

N y + M

N

M y ≤ 1.0.

(7.14)

Thisformsthebasisforthemethodsofdesigningtensionmembersforin-plane
bending, as discussed in Section 2.4.

7.2.2 Fully plastic beam-columns

AnupperboundestimateoftheresistanceofanI-sectionbeam-columnbentabout
itsmajoraxiscanbeobtainedfromthecombinationofbendingmoment M pl , r and
axialforce N y , r whichcausesthecross-sectiontobecomefullyplastic.Aparticular
exampleisshowninFigure7.8,forwhichthedistance z n fromthecentroidtothe
unstrained fibre is less than ( h − 2 t f )/ 2. This combination of moment and force
lies between the two extreme combinations for members with bending moment
only ( N = 0), which become fully plastic at

h − 2 t f
2

2

M pl = f y b f t f ( h − t f ) + f y t w , (7.15)