Strength Reduction Factor for Axial-Flexure

The ACI 318-02 implementation in RAM Concrete utilizes Appendices B and C. The ACI 318-05 implementation utilizes the body of the code and does not refer to Appendices B and C.

ACI 318-02 and ACI 318-05 allow the calculation of the strength reduction factor for the design of members subjected to axial-flexural loads according to Appendix C. When this approach is taken, all provisions of Appendices B and C must be used in lieu of the body of the code. This results in the following modifications to the body of the code that are pertinent to RAM Concrete:

C.3.2.2 replaces 9.3.2.2. Using this approach, the strength reduction factor for members resisting axial/flexural load is a function of the ultimate axial load acting on the cross section.
The load combinations of C.2 replace the load combinations of 9.2.
Section B10.3.3 replaces Sections 10.3.3, 10.3.4, and 10.3.5, with the exception of 10.3.5.1.

This is the approach that was taken in the RAM Concrete implementation of ACI 318-02.

The ACI 318-05 implementation in RAM Concrete utilizes Section 9.3, the main body of the code, in calculating strength reduction factors. In ACI 318-05, Section 9.3.2.2, the calculation of the strength reduction factor used for the design of members subjected to a combination of compression-flexure or tension-flexure is dependent on where the section lies relative to a tension-controlled or compression-controlled state. ACI 318-05 defines a compression-controlled section in 10.3.3, and a tension-controlled section in 10.3.4.

Figure below shows the strain condition for a rectangular cross section where the extreme compression fiber has reached the ultimate concrete strain, ε _cu.

Cross section illustrating tensile strain in extreme tension fiber, ε_t, at ultimate concrete strain.

Referring to Figure above, a section is compression-controlled if:

ϵ_{t} \leq ϵ_{c, limit}^{+}

where

ε⁺_c,limit

the compression-controlled strain limit, which may be taken as 0.002 for grade 60 steel. RAM Concrete uses ε⁺_c,limit = 0.002 for all situations.

Referring to Figure above, a section is tension-controlled if:

ε_t≥ 0.005

The following condition represents a transition region between a compression-controlled section and a tension-controlled section.

ε⁺_c,limit ≤ ε_t ≤ 0.005

Note: RAM Concrete Beam does not consider member axial force during the design and thus all tabulations of c and ε_t are performed without the consideration of axial force.

Following from these definitions, Section 9.3.2 defines the strength reduction factors for use in member design as follows:

Section 9.3.2.1 – Tension-controlled sections: Φ = 0.90

Section 9.3.2.2 – Compression-controlled sections:

Members with spiral reinforcement conforming to 10.9.3: Φ = 0.70
Other reinforced members: Φ = 0.65

For sections in the transition region, the following equation is used to calculate the strength reduction factor:

ϕ = 0.65 + (0.90 - 0.65) (\frac{ϵ_{t} - ϵ_{c, limit}^{+}}{0.005 - ϵ_{c, limit}^{+}}) \leq 0.90

Section 9.3.2.2 – Shear and torsion: Φ = 0.75

Section 9.3.2.4 – Bearing on concrete (except for post-tensioned anchorage zones and strut-and-tie models): Φ = 0.65

Strength Reduction Factor for Axial-Flexure

Cross section illustrating tensile strain in extreme tension fiber, εt, at ultimate concrete strain.

Cross section illustrating tensile strain in extreme tension fiber, ε_t, at ultimate concrete strain.