In-plane bending of beams - The Behaviour and Design of Steel Structures to EC3 - page 193

y i 2

y 12

y 12 y i c

y i

0

z ic

y 1

y 1

z i 2 C

C

z 12

z 12

2

2 2

z 1

z 1

z i

A 2 = b 2 × t 2

A 2 = b 2 × t 2

(a) Centroid (b) Second moments of area

Figure 5.35 Calculation of cross-section properties.

Whenthesearesubstitutedintoequation5.56,thecentroidpositionisfoundfrom

y ic = A n y in / A

(5.57)

z ic = A n z in / A

The second moments of area about the centroidal axes are defined by

z 2 d A = ( A n z n + I n sin 2 θ n ) I y =

A

y 2 d A =

( A n y n + I n cos 2 θ n )

I z = (5.58)

A

I yz = yz d A = ( A n y n z n + I n sin θ n cos θ n )

A

inwhich θ n isalwaystheanglefromthe y axistotherectangularelement(positive
from the y axis to the z axis), as shown in Figure 5.35b, and

I n = b n t n / 12. (5.59)

These thin-walled approximations ignore small terms b n t n / 12, while the use of
the angle θ n adjusts for the inclination of the element to the y , z axes.