# Area properties of concrete cross-section

A general integral over an area of arbitrary shape is given as

Hmn = ∫ ym zn dA

with m and n as positive integers. Setting m and n equal to 0, 1 or 2, the integral will give values of the area integrals A, Sy, Sz, Iy, Iz and Iyz which are defined as area properties.

During calculation all section types (open as well as closed) are converted to polygon shape and after that the section properties are easily calculated using numeric integration (see article). # Area properties of reinforcement in cross-section

Section properties for reinforcement composed of N bars with cross-section area Asj and positions (yj, zj) are calculated as a summation:

Nmn

Hmn = Σ yj zj Asj

j = 1

With m and n equal to 0, 1 or 2, the summation will give values of the area integrals A, Sy, Sz, Iy, Iz and Iyz which are defined as area properties. # Area properties of composite cross-section

Section properties for the composite section are calculated using the section properties of net concrete section (i.e. with area corresponding to reinforcement subtracted) and section properties of reinforcement considering the difference in modulus of elasticity. This section is also referred to as the transformed section.

The difference in modulus of elasticity is expressed by the factor:

α = Es / Ec

Summation of areas and static moments around coordinate axes gives property values of the transformed section:

A = Acn + α As

Sy = Scny + α Ssy

Sz = Scnz + α Ssz

Coordinates for the centroid of the transformed area are then easily found by

ycg = Sy /A zcg = Sz /A

Moments of inertia are then calculated using Steiner’s theorems:

Iycg = Icny + α Isy – zcg2 A

Izcg = Icnz + α Isz – ycg2 A

Iyzcg = Icnyz + α Isyz – ycg zcg A