Figure 11-5: Assumed stress-strain relation for concrete (from O’Neill and Reese 1999)

160
Figure 11-
6: Assumed stress-strain curves for steel (from O’Neill and Reese 1999).
The derivation of the relation between bending moment, axial load, and E
p
I
p
proceeds by assuming that plane sections in a beam-column remain plane after loading. Therefore, when an axial load (P
x
) and a moment (M) are applied to a section, it results that the neutral axis is displaced from the center of gravity of asymmetrical section. The equilibrium equations for such condition can be expressed as follows
𝑃𝑃
𝑥𝑥
= 𝑏𝑏 � 𝜎𝜎 Equation 11-23)
𝑀𝑀 = 𝑏𝑏 � 𝜎𝜎 𝑦𝑦 Equation 11-24)
Where:
Σ = Stress normal in the section. y = Vertical coordinate in the section from the center of gravity of the section. Other terms are defined in Figure 11-7 fora circular section. Note that integration above considers the forces caused by σ in each of the infinitesimal horizontal bands with width (b) and thickness (dy) shown in Figure 11-7. The value of E
p
I
p
for reinforced concrete can betaken as that of the gross section. However, as the loading increases cracking of the concrete will occur, causing a significant reduction in E
p
I
p
. Further reductions occur as the bending moment further increases therefore, a modification in E
p
I
p
may be needed for accurate computations, especially if deflection controls. The numerical procedure for determining the relationship between axial load, bending moment, and E
p
I
p
of the section, considers the nonlinear stress-strain properties of the concrete and steel and the combined action of the (compressive) axial load and bending moment. The procedure, which is typically conducted in these computer programs, is summarized below.

161 The dimensions of the section, as well the amount and distribution of longitudinal reinforcement are selected. Geometrical properties (areas, reinforcement spacing, section covers, etc) must be selected.
•
The neutral axis is selected. A strain gradient Φ
ε across the section about the neutral axis is also selected. Φ
ε
is defined such that the product of Φ
ε
and distance y from the neutral axis gives the strain at this specific distance from the neutral axis. Φ
ε
, which has units of strain/length, is assumed to be constant, whether the section is in an elastic or inelastic state. This step defines the strain at every point in the section.
•
With the strain distribution in the section and with the stress-strain relationships for the steel and concrete shown earlier, the distribution of stresses across the cross-section can be computed numerically.
•
The resultant of normal stresses on the section is calculated with Equation 11-23. If the computed value is different to the applied axial load (P
x
), the position of the neutral axis is moved and the computations are repeated. This process is continued until the computed value of P
x
is equal to the applied value of P
x
•
The bending moment associated with this condition is then computed by summing moments from the normal stresses in the cross-section about a convenient point in the section (e.g., the centroidal axis or the neutral axis) using Equation From beam theory, it can be shown that E
p
I
p

= M/Φ
ε
. Therefore, a unique relationship between P
x
, M
,
and E
p
I
p
is found fora given section considering the selected amount and distribution of reinforcement, and the material properties. The process is repeated for different values of Φ
ε
•
The E
p
I
p
value for this combination of axial load P
x
is then determined.

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