Stress and strain of uncracked sections - Concrete Structures: Stresses and Deformations - page 25

where A i , B i and I i are respectively the area of the i th part and its rst and

second moment about an axis through O. A reinforcement layer may be
treated as one part.
Equations (2.7) and (2.8) may be rewritten in the matrix form

fi

N
M

A
B

B
I

ε O

= E ref

ψ

(2.12)

This equation may be used to are known; or

when N and M are known the equation may be solved for the axial strain and
curvature:

fi nd N and M when ε O and ψ

− 1

ε O 1

E ref

A
B

B
I

N
M ψ =

(2.13)

The inverse of the 2 × 2 matrix in this equation is

A
B

B
I

−1

1 I

−

− B

A

=

B 2 )

(2.14)

( AI − B

Substitution in Equation (2.13) gives the axial strain at O and the curvature

ε O 1

E ref ( AI

I
−

− B

A

N
M ψ =

B 2 )

(2.15)

− B

When the reference point O is chosen at the centroid of the transformed
section, B

= 0 and Equation (2.15) takes the more familiar form

ε O 1

E ref

N / A
M / I ψ =

(2.16)

2.3.1 Basic equations
The equations derived above give the stresses and the strains in a cross-
section subjected to a normal force and a bending moment (Fig. 2.1).
Extensive use of these equations will be made throughout this topic in
analysis of reinforced composite or non-composite cross-sections. Because of
this, the basic equations are summarized below and the symbols de fi ned for

easy reference:

ε = ε O + ψ y σ = E ( ε O + ψ y ) (2.17)

N = E ( A ε O + B ψ ) M = E ( B ε O + I ψ ) (2.18)