5. Moment Mag. along Length

5. Moment Magnification along Length of Compression Member

In sway frames, second-order effects shall be considered along the length of columns. It shall be permitted to account for these effects using ACI 318-19 (6.6.4.5) (Nonsway frame procedure), where C_m is calculated using M₁ and M₂ from ACI 318-19 (6.6.4.6.1) as follows: ACI 318-19 (6.6.4.6.4)

ACI 318-19 (6.6.4.5.1)

Where:

M₂ = the second-order factored moment.

ACI 318-19 (6.6.4.5.2)

ACI 318-19 (6.6.4.4.2)

Where:

ACI 318-19 (6.6.4.4.4)

There are three options for calculating the effective flexural stiffness of slender concrete columns (EI)_eff. The second equation provides accurate representation of the reinforcement in the section and will be used in this example and is also used by the solver in spColumn.

Further comparison of the available options is provided in “Effective Flexural Stiffness for Critical Buckling Load of Concrete Columns” technical note.

ACI 318-19 (Table 6.6.3.1.1(a))

ACI 318-19 (19.2.2.1.b)

β_dns is the ratio of maximum factored sustained axial load to maximum factored axial load associated with the same load combination. ACI 318-19 (6.6.4.4.4)

For load combination 4:

ACI 318-19 (Figure R6.2.5.1)

(Column essentially fixed at base) ACI 318-19 (Figure R6.2.5.1)

Using Figure R6.2.5.1(a) from ACI 318-19 à k = 0.690 as shown in the figure below for the exterior columns.

08 Effective Length Factor 4

Figure 5 - Effective Length Factor (k) Calculations for Exterior Columns (Nonsway)

With 8#8 reinforcement equally distributed on all sides and 22 in. x 22 in. column section à I_se = 352.6 in.⁴.

ACI 318-19 (6.6.4.4.4(b))

For load combination 4:

ASCE 7-10 (2.3.2-3)

ACI 318-19 (6.6.4.5.3a)

ACI 318-19 (6.6.4.6.4)

Since the column is bent in double curvature, M₁/M₂ is positive.

ACI 318-19 (6.6.4.5.2)

ACI 318-19 (6.6.4.5.4)

Where P_u = 722 kip, and h = the section dimension in the direction being considered = 22 in.

ACI 318-19 (6.6.4.5.1)

ACI 318-19 (6.6.4.5.4)

ACI 318-19 (6.6.4.5.1)

M_c1 and M_c2 will be considered separately to ensure proper comparison of resulting magnified moments against negative and positive moment capacities of unsymmetrical sections as can be seen in the following figure.

09 Case of unsymmetrical section

Figure 6 - Column Interaction Diagram for Unsymmetrical Section

A summary of the moment magnification factors and magnified moments for the exterior column for all load combinations using both equation options ACI 318-19 (6.6.4.4.4(a)) and (6.6.4.4.4(b)) to calculate (EI)_eff is provided in the table below for illustration and comparison purposes.

Table 4 - Factored Axial loads and Magnified Moments along Exterior Column Length
No.	Load Combination	Axial Load, kip	Using ACI 6.6.4.4.4(a)			Using ACI 6.6.4.4.4(b)
No.	Load Combination	Axial Load, kip	δ	M_c1, ft-kip	M_c2, ft-kip	δ_s	M_c1, ft-kip	M_c2, ft-kip
1	1.4D	871.4	1.00	91.5	91.5	1.00	91.5	91.5
2	1.2D + 1.6L + 0.5L_r	869.4	1.00	91.3	91.3	1.00	91.3	91.3
3	1.2D + 0.5L + 1.6L_r	797.6	1.00	83.7	83.7	1.00	83.7	83.7
4	1.2D + 1.6L_r + 0.8W	722.0	1.00	75.8	161.6	1.00	75.8	173.2
5	1.2D + 1.6L_r - 0.8W	799.3	1.00	83.9	-119.4	1.00	83.9	-131.0
6	1.2D + 0.5L + 0.5L_r + 1.6W	710.9	1.00	84.7	308.5	1.00	87.7	333.0
7	1.2D + 0.5L + 0.5L_r - 1.6W	865.4	1.00	90.9	-258.5	1.00	90.9	-283.0
8	0.9D + 1.6W	482.9	1.00	63.8	277.9	1.00	65.6	292.4
9	0.9D - 1.6W	637.4	1.00	66.9	-246.3	1.00	66.9	-260.8

For column design ACI 318 requires the second-order moment to first-order moment ratios should not exceed 1.40. If this value is exceeded, the column design needs to be revised. ACI 318-19 (6.2.5.3)

Table 5 - Second-Order Moment to First-Order Moment Ratios
No.	Load Combination	Using ACI 6.6.4.4.4(a)		Using ACI 6.6.4.4.4(b)
No.	Load Combination	M_c1/M_1(1st)	M_c2/M_2(1st)	M_c1/M_1(1st)	M_c2/M_2(1st)
1	1.4D	1.00^*	1.00^*	1.00^*	1.00^*
2	1.2D + 1.6L + 0.5L_r	1.00^*	1.00^*	1.00^*	1.00^*
3	1.2D + 0.5L + 1.6L_r	1.00^*	1.00^*	1.00^*	1.00^*
4	1.2D + 1.6L_r + 0.8W	1.00^*	1.23	1.00^*	1.32
5	1.2D + 1.6L_r - 0.8W	1.00^*	1.34	1.00^*	1.40 < 1.47
6	1.2D + 0.5L + 0.5L_r + 1.6W	1.10	1.26	1.14	1.36
7	1.2D + 0.5L + 0.5L_r - 1.6W	1.00^*	1.32	1.00^*	1.40 < 1.45
8	0.9D + 1.6W	1.09	1.18	1.12	1.24
9	0.9D - 1.6W	1.00^*	1.20	1.00^*	1.27
^* Cutoff value of M_min is applied to M_1(1st) and M_2(1st) in order to avoid unduly large ratios in cases where M_1(1st) and M_2(1st) moments are smaller than M_min.