### 6.1 ACI 318-11 Moment Magnifier Method

The procedure described in this section was used to compute the slender column strength from the ACI moment magnifier approach. Note that the strength and stiffness reduction factors (*ϕ* and *ϕ*_{
k
} for ACI) were taken equal to 1.0 in this study.

The first step in computing the ACI ultimate strength of a slender column that is part of a braced frame is to determine the cross section strength, which is represented by an axial load-bending moment (*P*–*M*) strength interaction diagram, similar to the one shown in Fig. 7. The cross section strength interaction diagram was defined by 102 points that were computed using the compatibility of strains and the equilibrium of forces acting on the cross section. For computing the ACI cross section strength, it was assumed that (a) the strains are linearly proportional to the distances from the neutral axis; (b) the maximum concrete strain *ε*_{
cu
} = 0.003 exists at the extreme compression fiber as given in ACI 318-11; (c) the compressive stress in concrete is represented by a rectangular stress block as defined in ACI 318-11; (d) the specified concrete strength is used in computing the maximum concrete stress in the stress block; and (e) the concrete is assumed to have no strength in tension.

To develop the points on the cross section strength interaction diagram, the strain at the extreme compression fiber was held constant at *ε*_{
cu
} = 0.003, while the strain at the extreme fiber on the opposite face was incremented from a strain that equaled the maximum computed tensile strain at pure bending up to a strain that was equal to the uniform compressive strain required across the entire cross section for pure compression. The summation of forces acting on concrete and reinforcing steel at each increment of strain generated one point on the cross section axial force-bending moment interaction diagram. The entire interaction diagram for a column cross section (*ℓ/h* = 0) similar to one shown in Fig. 7 was defined by 102 points, as stated earlier.

The ACI moment magnifier procedure for slender columns uses the moment magnifier δ_{
ns
} and the larger of the two column end moments *M*_{2} obtained from a conventional elastic frame analysis to compute the magnified moment *M*_{
c
} (*M*_{
max
}), which includes second-order effects occurring along the height of the column:

{M}_{max}={M}_{c}={\mathit{\delta}}_{ns}{M}_{2}={C}_{m}\mathit{\delta}{\phantom{\rule{1.0pt}{0ex}}}_{1}\phantom{\rule{1.0pt}{0ex}}{M}_{2}\ge {M}_{2}

(1)

In Eq. (1), *δ*_{
ns
} is the moment magnifier for columns that are part of braced (nonsway) frames; *M*_{2} is the larger of the two factored end moments (*M*_{1} and *M*_{2}) computed from a conventional elastic frame analysis and is always taken as positive; *C*_{
m
} is the equivalent uniform moment diagram factor; and *δ*_{1} is the moment magnifier for the same columns when subjected to axial load and equal and opposite (equivalent) end moments causing symmetrical single curvature bending. Chen and Lui (1987) explain that the *C*_{
m
} and δ_{1} for pin-ended columns subjected to end moments can be derived from the basic differential equation governing the elastic in-plane behavior of a column. For design purposes, ACI has adopted a simplified and widely accepted approximation of δ_{
ns
}:

{\mathit{\delta}}_{ns}={C}_{m}{\mathit{\delta}}_{1}=\frac{0.6+0.4{M}_{1}/\phantom{{M}_{1}{M}_{2}}\phantom{\rule{0.0pt}{0ex}}{M}_{2}}{1-\frac{{P}_{u}}{{\mathit{\varphi}}_{k}{P}_{c}}}\ge 1.0

(2)

In Eq. (2), *P*_{
u
} is the applied axial load under consideration; *ϕ*_{
k
} is the stiffness reduction factor specified as 0.75 in ACI 318-11 but taken as 1.0 for this study; and *P*_{
c
} is the critical buckling load computed from

{P}_{c}=\frac{{\mathit{\pi}}^{2}EI}{{\left(K\ell \right)}^{2}}

(3)

In Eq. (3), *ℓ* is the column length; *K* is the effective length factor; and *EI* is the effective flexural stiffness. For computing the effective flexural stiffness (*EI*) of tied slender reinforced concrete columns for short-term loads (*β*_{
d
} = 0), the ACI Code permits the use of Eq. (4):

EI=0.2{E}_{c}{I}_{g}+{E}_{s}{I}_{rs}

(4)

where *I*_{
rs
} = moment of inertia of the longitudinal reinforcing bars taken about the centroidal axis of the column cross section. The commentary of ACI 318-11 (2011) permits the use of the Jackson–Moreland Alignment Chart, which is based on Eq. (5), for determining the effective length factor *K* for columns in braced frames:

\frac{{G}_{A}{G}_{B}}{4}{\left(\frac{\mathit{\pi}}{K}\right)}^{2}+\left(\frac{{G}_{A}+{G}_{B}}{2}\right)\left(1-\frac{\mathit{\pi}/K}{tan\left(\mathit{\pi}/K\right)}\right)+\frac{2tan\left(\mathit{\pi}/2K\right)}{\left(\mathit{\pi}/K\right)}-1=0

(5)

In Eq. (5), *G*_{
A
} and *G*_{
B
} are the relative stiffnesses of the column at upper and lower joints, respectively, and were computed as the ratios of the sum of stiffnesses of columns (∑(*EI/ℓ*)_{
col
}) meeting at the joint A or B to the sum of stiffnesses of beams (∑(*EI/ℓ*)_{
bm
}) meeting at the same joint. A graphical representation of Eq. (5) (Jackson–Moreland Alignment Chart) is given in Fig. 8, which shows the range of *K* examined in this study.

For *G*_{
A
} and *G*_{
B
}, *EI* values were computed from 0.7*E*_{
c
}*I*_{g(col)} and 0.35*E*_{
c
}*I*_{g(bm)} for columns and beams, respectively, as permitted by the ACI Code. For frames used in this study (Fig. 1), the upper end of the column (Joint A) has no beams framing into it. The upper end of the column is either pin-ended or fix-ended and, therefore, *G*_{
A
} is theoretically infinity or zero, respectively. To avoid numerical problems in solving Eq. (5), *G*_{
A
} was set equal to 1,000 when the upper end of the column was pin-ended and taken as 0.001 when the upper end of the column was fix-ended.

Equation (1) can be used to obtain the bending moment resistance of a column in a frame for a given level of axial load (*P*_{
u
}) directly from the cross section strength interaction diagram. To do this, the cross section bending moment resistance (*M*_{
cs
}) is substituted for the magnified column moment (*M*_{
c
}) in Eq. (1). Then, the larger of the two end moments (*M*_{2}), which can be applied to the column at the given axial load *P*_{
u,
} is computed by solving Eq. (1) for *M*_{2}:

{M}_{2}=\frac{{M}_{cs}}{{\mathit{\delta}}_{ns}}

(6)

To generate the column axial load-bending moment interaction diagram (Fig. 7), the cross section bending moment resistance *M*_{
cs
} for each level of axial load (*P*_{
u
}) was divided by δ_{
ns
}. Note that the maximum axial load that can be applied to a slender column is less than the pure axial load resistance of the cross section (*P*_{
o
}) and is also less than the column critical load resistance (*P*_{
c
}) computed from Eq. (3). Hence, the points on the column strength interaction curve were generated for *P*_{
u
} values that were lower than both *P*_{
o
} and *P*_{
c
}. Note that, for reinforced concrete columns examined in this study, *P*_{
o
} was computed from 0.8[(0.85{f}_{c}^{\prime})(*A*_{
g
} − *A*_{
rs
}) + *f*_{
y
}*A*_{
rs
}], as permitted by ACI 318-11 (2011), where {f}_{c}^{\prime}, *f*_{
y
} = specified compressive and yield strengths of concrete and reinforcing steel, and *A*_{
g
}, *A*_{
rs
} = areas of the gross concrete cross section and of the longitudinal reinforcing steel.

For an *M*_{1}*/M*_{2} ratio, *M*_{2} values were computed from the procedure described above for all levels of axial load (*P*_{
u
}) that were lower than or equal to both *P*_{
o
} and *P*_{
c
}. This generated the column axial load-bending moment interaction diagram for the *M*_{1}*/M*_{2} ratio under consideration. Repeating the step for all desired *M*_{1}*/M*_{2} ratios generated a series of column strength interaction curves. Four of such curves for *M*_{1}*/M*_{2} = 1.0, 0.5, 0.0 and −0.75 are shown in Fig. 7. The ACI axial load strength (*P*_{u(des)}) of a column was then computed from linear interpolation of points on these interaction diagrams, using the first order *M*_{1}*/M*_{2} and *e/h* ratios determined earlier for that column from the theoretical procedure described in the preceding section.

### 6.2 Modified ACI Moment Magnifier Method with Alternative (Nonlinear) *EI* Equation

The procedure outlined above is applicable only when *EI* is computed from the ACI *EI* equation (Eq. (4)) or from a similar *EI* equation used for calculating *P*_{
c
} from Eq. (3). This is because the ACI *EI* from Eq. (4) remains constant regardless of the magnitude of end moments and, therefore, *P*_{
c
} also remains constant. As a result, the moment magnifier (δ_{
ns
}) remains constant for a given column. However, *P*_{
c
} is strongly influenced by the effective flexural stiffness (*EI*), which varies due to the nonlinearity of the concrete stress–strain curve and cracking along the height of the column among other factors. Based on extensive analyses of 11,550 simulated and 128 physically-tested reinforced concrete columns, Tikka and Mirza (2005) proposed an *EI* design equation for short term loads, reproduced here as Eq. (7), that is dependent upon the end eccentricity ratio (*e/h*), making *EI* both variable and nonlinear:

EI=\left(0.47-3.5\frac{e}{h}\left(\frac{1}{1+\mathit{\beta}\frac{e}{h}}\right)+0.003\frac{\ell}{h}\right){E}_{c}{I}_{g}+0.8{E}_{s}{I}_{rs}

(7)

where *β* = 7.0 for columns with *ρ*_{
col
} ≤ 2 %; and *β* = 8.0 for columns with *ρ*_{
col
} > 2 %. For developing Eq. (7), Tikka and Mirza (2005) examined the practical ranges of a number of variables that could possibly affect the effective flexural stiffness of reinforced concrete columns. They found that the column *e/h*, *ℓ/h* and *ρ*_{
col
} had major, significant, and minor effects, respectively, on the column *EI* and, hence, included these variables in Eq. (7). The variable and nonlinear nature of Eq. (7) affects *P*_{
c
} which, in turn, affects δ_{
ns
}. Therefore, an iterative approach was used to determine the slender column strength interaction diagram when Eq. (7) was used in lieu of Eq. (4) for *EI* in the moment magnifier procedure.

The smaller of the cross section pure axial load strength (*P*_{
o
}) and the column critical load strength (*P*_{
c
}) was used to establish the upper limit for the axial load levels to be used in determining the slender column strength interaction diagram. For each level of axial load (*P*_{
u
}), the end eccentricity (*e*) was iterated until *e* × *P*_{
u
} × δ_{
ns
} = *M*_{
cs
} was satisfied within a tolerance of 0.01 %. The moment magnifier (δ_{
ns
}) was computed from Eq. (2) for a given *M*_{1}/*M*_{2} ratio for each iteration of end eccentricity using the *EI* computed from Eq. (7) and the effective length factor (*K*) computed from Eq. (5). This generated one point on the column strength interaction curve for the *M*_{1}/*M*_{2} ratio under consideration. Repeating this step for all axial load levels generated the entire strength interaction curve for the *M*_{1}/*M*_{2} ratio under consideration. Such column strength interaction diagrams were generated for a series of *M*_{1}/*M*_{2} ratios and were used for computing the modified ACI axial load strength (*P*_{u(des)}) of a column from linear interpolation, using the first-order *M*_{1}/*M*_{2} and *e/h* ratios calculated for that column from the theoretical procedure described in an earlier section.

### 6.3 Modified ACI Moment Magnifier Method with Alternative (Simplified) Equation for *K* Factor

A simple equation for the effective length factor was proposed by Duan et al. (1993) for columns in nonsway frames:

K=1-\frac{1}{5+9{G}_{A}}-\frac{1}{5+9{G}_{B}}-\frac{1}{10+{G}_{A}{G}_{B}}

(8)

In addition, the Commentary to ACI 318-05 (2005) permitted the use of expressions proposed by Cranston (1972), where *K* was taken as the smaller of the following for columns in nonsway frames:

K=0.7+0.05\left({G}_{A}+{G}_{B}\right)\le 1.0

(9a)

K=0.85+0.05{G}_{\text{min}}\le 1.0

(9b)

in which *G*_{min} was the smaller of *G*_{
A
} and *G*_{
B
}. A comparison of *K* computed from Eq. (5) (Jackson–Moreland Alignment Chart), Eq. (8) (Duan et al. 1993) and Eq. (9) (Cranston 1972) is shown in Fig. 9. The following observations can be made from Fig. 9: (a) Cranston’s expressions produce effective length factors that are very conservative compared to the values obtained from the Jackson–Moreland Alignment Chart when the upper joint is fix-ended (*G*_{
A
} = 0); (b) Duan’s equation produces effective length factors that are almost the same as those obtained from the Jackson–Moreland Alignment Chart when the upper joint is fix-ended (*G*_{
A
} = 0); and (c) when the upper joint is pin-ended (*G*_{
A
} = ∞), both Duan’s and Cranston’s equations produce conservative results compared to the effective length factor computed from the Jackson–Moreland Alignment Chart.

To investigate the effect of the *K* factor computed from Duan et al. (1993) on the strength of slender reinforced concrete columns, Eq. (8) was used in place of Eq. (5) and the rest of one of the two moment magnifier procedures described previously was followed, depending on whether the ACI equation (Eq. (4)) or the alternative equation (Eq. (7)) was used for calculating *EI*. No further analysis was performed with Cranston’s equation (1972), because it produced very conservative values of *K* for fix-ended columns in Fig. 9a and similar values of *K* as those produced by the Duan et al. equation for pin-ended columns in Fig. 9b.