Bending and Axial Loads - Unified Theory of Concrete Structures - page 78

The second method is to solve Equation (3.22) by the formula of quadratic equation, giving

( n

k = ρ ) 2 + 2 n ρ − n ρ (3.23)

Equation (3.23) is, of course, the same as (3.16). This method requires an engineer to
remember Equation (3.23) or to have this formula at hand.
Once the neutral axis, represented by k , is known, the moment can be calculated directly
from the equilibrium equation (3.3) and the curvature from the compatibility equation (3.6).

3.1.3 Bending Rigidities of Cracked Sections
3.1.3.1 Singly Reinforced Rectangular Sections

For a homogeneous elastic member, the bending rigidity EI is the product of the modulus of
elasticity of the material E and the moment of inertia of the cross section I . For a cracked
concrete member, we will define the bending rigidity E c I cr as the product of the modulus of
elasticity of concrete E c and a cracked moment of inertia I cr . In this section we will derive I cr
for singly reinforced rectangular sections.
By definition, the bending rigidity of a member is the bending moment per unit curvature

M /φ . Hence, we can write

M
φ

E c I cr = (3.24)

From the strain diagram in Figure 3.1(b) the curvature φ can be expressed by

ε c
kd

φ = (3.25)

From the stress diagram in Figure 3.1(c) the moment M can be obtained by taking moment
about the tensile rebar:

1
2

f c kjbd 2

=

M (3.26)

Substituting φ and M from Equations (3.25) and (3.26) into Equation (3.24) and noticing

that f c = E c ε c and j

= − /

(1 k 3) results in:

bd 3 1

2 k 2 1

k
3

I cr = − (3.27)

Equation (3.27) shows that the cracked moment of inertia I cr can be calculated if the k value
is determined from Equation (3.23), or directly from Equation (3.22).

3.1.3.2 Doubly Reinforced Rectangular Sections

The transformed area concept can be used to great advantage in finding the cracked moments
of inertia for doubly reinforced sections and flanged sections. These types of sections occur
regularly in continuous beams, as shown in Figure 3.2. In a continuous beam the cross-sections
are usually doubly reinforced at the column faces, while those at the midspan are often designed
as flanged sections.