Slenderness Effects for Concrete Columns in Sway Frame - Moment Magnification Method

Slender Concrete Column Design in Sway Frame Buildings

Evaluate slenderness effect for columns in a sway frame multistory reinforced concrete building by designing the first story exterior column. The clear height of the first story is 13 ft-4 in., and is 10 ft-4in. for all of the other stories. Lateral load effects on the building are governed by wind forces. Compare the calculated results with the values presented in the Reference and with exact values from spColumn engineering software program from StructurePoint.

Figure 1 – Reinforced Concrete Column Cross-Section

Contents

1. Factored Axial Loads and Bending Moments. 2

1.1. Service loads. 2

1.2. Load Combinations – Factored Loads. 2

2. Slenderness Effects and Sway or Nonsway Frame Designation. 3

3. Determine Slenderness Effects. 4

4. Moment Magnification at Ends of Compression Member 5

5. Moment Magnification along Length of Compression Member 11

6. Column Design. 16

6.1. c, a, and strains in the reinforcement 16

6.2. Forces in the concrete and steel 17

6.3. ϕP_n and ϕM_n. 17

7. Column Interaction Diagram - spColumn Software. 19

8. Summary and Comparison of Design Results. 28

9. Conclusions & Observations. 30

Code

Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary (ACI 318R-14)

Reference

Notes on ACI 318-11 Building Code Requirements for Structural Concrete, Twelfth Edition, 2013 Portland Cement Association, Example 11-2

Design Data

f_c’ = 6,000 psi for columns in the bottom two stories

= 4,000 psi elsewhere

f_y   = 60,000 psi

Slab thickness = 7 in.

Exterior Columns = 22 in. x 22 in.

Interior Columns = 24 in. x 24 in.

Beams = 24 in. x 20 in. x 24 ft

Superimposed dead load = 30 psf

Roof live load = 30 psf

Floor live load = 50 psf

Wind loads computed according to ASCE 7-10

Total building loads in the first story from structural analysis:

    D   = 17,895 kip

    L   = 1,991 kip

    L_r = 270 kip

    W = 0 kip, wind loads in the story cause compression in some columns and tension in others and thus would cancel out.

1. Factored Axial Loads and Bending Moments

1.1. Service loads

Table 1 - Exterior column service loads
Load Case	Axial Load, kip	Bending Moment, ft-kip
		Top	Bottom
Dead, D	622.4	34.8	17.6
Live, L	73.9	15.4	7.7
Roof Live, Lr	8.6	0.0	0.0
Wind, W (N-S)	-48.3	17.1	138.0
Wind, W (S-N)	48.3	-17.1	-138.0

1.2. Load Combinations – Factored Loads ASCE 7-10 (2.3.2)

Table 2 - Exterior column factored loads
ASCE 7-10 Reference	No.	Load Combination	Axial Load, kip	Bending Moment, ft-kip		MTop,ns ft-kip	MBottom,ns ft-kip	MTop,s ft-kip	MBottom,s ft-kip
				Top	Bottom
2.3.2-1	1	1.4D	871.4	48.7	24.6	48.7	24.6	---	---
2.3.2-2	2	1.2D + 1.6L + 0.5Lr	869.4	66.4	33.4	66.4	33.4	---	---
2.3.2-3	3	1.2D + 0.5L + 1.6 Lr	797.6	49.5	25.0	49.5	25.0	---	---
	4	1.2D + 1.6Lr + 0.8W	722.0	55.4	131.5	41.8	21.1	13.7	110.4
	5	1.2D + 1.6Lr - 0.8W	799.3	28.1	-89.3	41.8	21.1	-13.7	-110.4
2.3.2-4	6	1.2D + 0.5L + 0.5Lr + 1.6W	710.9	76.8	245.8	49.5	25.0	27.4	220.8
	7	1.2D + 0.5L + 0.5Lr - 1.6W	865.4	22.1	-195.8	49.5	25.0	-27.4	-220.8
2.3.2-6	8	0.9D + 1.6W	482.9	58.7	236.6	31.3	15.8	27.4	220.8
	9	0.9D - 1.6W	637.4	4.0	-205.0	31.3	15.8	-27.4	-220.8

2. Slenderness Effects and Sway or Nonsway Frame Designation

Columns and stories in structures are considered as non-sway frames if the increase in column end moments due to second-order effects does not exceed 5% of the first-order end moments, or the stability index for the story (Q) does not exceed 0.05.           ACI 318-14 (6.6.4.3)

∑P_u is the total vertical load in the first story corresponding to the lateral loading case for which ∑P_u is greatest (without the wind loads, which would cause compression in some columns and tension in others and thus would cancel out). ACI 318-14 (6.6.4.4.1 and R6.6.4.3)

V_us is the factored horizontal story shear in the first story corresponding to the wind loads, and Δ_o is the first-order relative deflection between the top and bottom of the first story due to V_u.                                             ACI 318-14 (6.6.4.4.1 and R6.6.4.3)

From Table 2, load combinations (2.3.2-4 No. 5 and 6) provide the greatest value of ∑P_u.

ASCE 7-10 (2.3.2-4)

ASCE 7-10 (2.3.2-6)

ACI 318-14 (Eq. 6.6.4.4.1)

Thus, the frame at the first story level is considered sway.

3. Determine Slenderness Effects

ACI 318-14 (Table 6.6.3.1.1(a))

ACI 318-14 (19.2.2.1.b)

For the column below level 2:

For the column above level 2:

For beams framing into the columns:

Where:

ACI 318-14 (19.2.2.1.b)

ACI 318-14 (Table 6.6.3.1.1(a))

ACI 318-14 (Figure R6.2.5)

 (Column essentially fixed at base)                                                                        ACI 318-14 (Figure R6.2.5)

Using Figure R6.2.5 from ACI 318-14 à k = 1.9 as shown in the figure below for the exterior columns with one beam framing into them in the directions of analysis.

Figure 2 – Effective Length Factor (k) Calculations for Exterior Columns with One Beam Framing into them in the Direction of Analysis (Sway Frame)

ACI 318-14 (6.2.5a)

Where:

ACI 318-14 (6.2.5.1)

4. Moment Magnification at Ends of Compression Member

A detailed calculation for load combination 4 (gravity plus wind) is shown below to illustrate the procedure. Table 3 summarizes the magnified moment computations for the exterior columns.

ACI 318-14 (6.6.4.6.1b)

Where:

ACI 318-14 (6.6.4.6.2)

ACI 318-14 (6.6.4.6.2(b)) will be used for comparison purposes with results obtained from spColumn model. However, (a) and (c) can also be used to calculate the moment magnifier.

∑P_u is the summation of all the factored vertical loads in the first story, and ∑P_c is the summation of the critical buckling load for all sway-resisting columns in the first story.

ACI 318-14 (6.6.4.4.2)

Where:

ACI 318-14 (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 318-14 (Table 6.6.3.1.1(a))

ACI 318-14 (19.2.2.1.a)

β_ds is the ratio of maximum factored sustained shear within a story to the maximum factored shear in that story associated with the same load combination. The maximum factored sustained shear in this example is equal to zero leading to β_ds = 0.               ACI 318-14 (6.6.3.1.1)

For exterior columns with one beam framing into them in the direction of analysis (12 columns):

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 318-14 (6.6.4.4.4(b))

k = 1.9 (calculated previously).

For exterior columns with two beams framing into them in the direction of analysis (4 columns):

ACI 318-14 (Figure R6.2.5)

 (Column essentially fixed at base)                                                                        ACI 318-14 (Figure R6.2.5)

Using Figure R6.2.5 from ACI 318-14 à k = 1.71 as shown in the figure below for the exterior columns with two beams framing into them in the directions of analysis.

Figure 3 – Effective Length Factor (k) Calculations for Exterior Columns with Two Beams Framing into them in the Direction of Analysis

For interior columns (8 columns):

ACI 318-14 (Table 6.6.3.1.1(a))

ACI 318-14 (19.2.2.1.a)

For the column below level 2:

For the column above level 2:

For beams framing into the columns:

Where:

ACI 318-14 (19.2.2.1.a)

ACI 318-14 (Table 6.6.3.1.1(a))

ACI 318-14 (Figure R6.2.5)

 (Column essentially fixed at base)                                                                        ACI 318-14 (Figure R6.2.5)

Using Figure R6.2.5 from ACI 318-14 à k = 1.81 as shown in the figure below for the interior columns.

Figure 4 – Effective Length Factor (k) Calculations for Interior Columns

With 8-#8 reinforcement equally distributed on all sides and 24 in. x 24 in. column section à I_se = 439.1 in.⁴.

ACI 318-14 (6.6.4.4.4(b))

For load combination 4:

ASCE 7-10 (2.3.2-3)

ACI 318-14 (6.6.4.6.2(b))

ACI 318-14 (6.6.4.6.1)

ACI 318-14 (6.6.4.6.1)

P_u = 722.0 kip

A summary of the moment magnification factors and magnified moments for the exterior column for all load combinations using both equation options ACI 318-14 (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. Note: The designation of M₁ and M₂ is made based on the second-order (magnified) moments and not based on the first-order (unmagnified) moments.

Table 3 - Factored Axial loads and Magnified Moments for Exterior Column
No.	Load Combination	Axial Load,  kip	Using ACI 6.6.4.4.4(a)			Using ACI 6.6.4.4.4(b)
			δs	M1, ft-kip	M2, ft-kip	δs	M1, ft-kip	M2, ft-kip
1	1.4D	871.4	---	24.6	48.7	---	24.6	48.7
2	1.2D + 1.6L + 0.5Lr	869.4	---	33.4	66.4	---	33.4	66.4
3	1.2D + 0.5L + 1.6 Lr	797.6	---	25.0	49.5	---	25.0	49.5
4	1.2D + 1.6Lr + 0.8W	722.0	1.37	60.6	172.3	1.53	62.7	189.7
5	1.2D + 1.6Lr - 0.8W	799.3	1.37	23.0	-130.1	1.53	20.9	-147.5
6	1.2D + 0.5L + 0.5Lr + 1.6W	710.9	1.39	87.5	330.9	1.55	92.0	367.9
7	1.2D + 0.5L + 0.5Lr - 1.6W	865.4	1.39	11.5	-280.9	1.55	7.0	-317.9
8	0.9D + 1.6W	482.9	1.25	65.5	291.2	1.34	68.0	311.6
9	0.9D - 1.6W	637.4	1.25	-2.9	-259.6	1.34	-5.4	-280.0

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 318-14 (6.6.4.5) (Nonsway frame procedure), where C_m is calculated using M₁ and M₂ from ACI 318-14 (6.6.4.6.1) as follows:                                                                                                                          ACI 318-14 (6.6.4.6.4)

ACI 318-14 (6.6.4.5.1)

Where:

M₂ = the second-order factored moment.

ACI 318-14 (6.6.4.5.2)

ACI 318-14 (6.6.4.4.2)

Where:

ACI 318-14 (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 318-14 (Table 6.6.3.1.1(a))

ACI 318-14 (19.2.2.1.a)

β_dns is the ratio of maximum factored sustained axial load to maximum factored axial load associated with the same load combination.                                                                                                                                          ACI 318-14 (6.6.4.4.4)

For load combination 4:

ACI 318-14 (Figure R6.2.5)

 (Column essentially fixed at base)                                                                        ACI 318-14 (Figure R6.2.5)

Using Figure R6.2.5(a) from ACI 318-14 à k = 0.86 as shown in the figure below for the exterior column.

Figure 5 – Effective Length Factor (k) Calculations for Exterior Column (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 318-14 (6.6.4.4.4(b))

For load combination 4:

ASCE 7-10 (2.3.2-3)

ACI 318-14 (6.6.4.5.3a)

ACI 318-14 (6.6.4.6.4)

ACI 318-14 (6.6.4.6.4)

Since the column is bent in double curvature, M₁/M₂ is positive.                                                ACI 318-14 (6.6.4.5.3)

ACI 318-14 (6.6.4.5.2)

ACI 318-14 (6.6.4.5.4)

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

ACI 318-14 (6.6.4.5.4)

ACI 318-14 (6.6.4.5.1)

ACI 318-14 (6.6.4.5.4)

ACI 318-14 (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.

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 318-14 (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)
			δ	Mc1, ft-kip	Mc2, ft-kip	δ	Mc1, ft-kip	Mc2, ft-kip
1	1.4D	871.4	1.00	91.5	91.5	1.00	91.5	91.5
2	1.2D + 1.6L + 0.5Lr	869.4	1.00	91.3	91.3	1.00	91.3	91.3
3	1.2D + 0.5L + 1.6 Lr	797.6	1.00	83.7	83.7	1.00	83.7	83.7
4	1.2D + 1.6Lr + 0.8W	722.0	1.00	75.8	172.3	1.00	75.8	189.7
5	1.2D + 1.6Lr - 0.8W	799.3	1.00	83.9	-130.1	1.00	83.9	-147.5
6	1.2D + 0.5L + 0.5Lr + 1.6W	710.9	1.00	87.5	330.9	1.00	92.0	367.9
7	1.2D + 0.5L + 0.5Lr - 1.6W	865.4	1.00	90.9	-280.9	1.00	90.9	-317.9
8	0.9D + 1.6W	482.9	1.00	65.5	291.2	1.00	68.0	311.6
9	0.9D - 1.6W	637.4	1.00	66.9	-259.6	1.00	66.9	-280.0

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 318-14 (6.2.6)

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)
		Mc1/M1(1st)	Mc2/M2(1st)	Mc1/M1(1st)	Mc2/M2(1st)
1	1.4D	1.00^*	1.00^*	1.00^*	1.00^*
2	1.2D + 1.6L + 0.5Lr	1.00^*	1.00^*	1.00^*	1.00^*
3	1.2D + 0.5L + 1.6 Lr	1.00^*	1.00^*	1.00^*	1.00^*
4	1.2D + 1.6Lr + 0.8W	1.00^*	1.31	1.00^*	1.40 < 1.44
5	1.2D + 1.6Lr - 0.8W	1.00^*	1.40 < 1.46	1.00^*	1.40 < 1.65
6	1.2D + 0.5L + 0.5Lr + 1.6W	1.14	1.35	1.20	1.40 < 1.50
7	1.2D + 0.5L + 0.5Lr - 1.6W	1.00^*	1.40 < 1.43	1.00^*	1.40 < 1.62
8	0.9D + 1.6W	1.12	1.23	1.16	1.32
9	0.9D - 1.6W	1.00^*	1.27	1.00^*	1.37
^* Cutoff value of Mmin is applied to M1(1st) and M2(1st) in order to avoid unduly large ratios in cases where M1(1st) and M2(1st) moments are smaller than Mmin.

6. Column Design

Based on the factored axial loads and magnified moments considering slenderness effects, the capacity of the assumed column section (22 in. x 22 in. with 8-#8 bars distributed all sides equal) will be checked and confirmed to finalize the design. A column interaction diagram will be generated using strain compatibility analysis, the detailed procedure to develop column interaction diagram can be found in “Interaction Diagram – Tied Reinforced Concrete Column” example.

The axial compression capacity ϕP_n for all load combinations will be set equals to P_u, then the moment capacity ϕM_n associated to ϕP_n will be compared with the magnified applied moment M_u. The design check for load combination #4 is shown below for illustration. The rest of the checks for the other load combinations are shown in the following Table.

Figure 7 – Strains, Forces, and Moment Arms (Load Combination 4)

The following procedure is used to determine the nominal moment capacity by setting the design axial load capacity, ϕP_n, equal to the applied axial load, P_u and iterating on the location of the neutral axis.

6.1. c, a, and strains in the reinforcement

Where c is the distance from the fiber of maximum compressive strain to the neutral axis.

ACI 318-14 (22.2.2.4.2)

ACI 318-14 (22.2.2.4.1)

Where:

ACI 318-14 (Table 22.2.2.4.3)

ACI 318-14 (22.2.2.1)

ACI 318-14 (Table 21.2.2)

6.2. Forces in the concrete and steel

ACI 318-14 (22.2.2.4.1)

The area of the reinforcement in this layer has been included in the area (ab) used to compute C_c. As a result, it is necessary to subtract 0.85f_c’ from f_s’ before computing C_s:

6.3. ϕP_n and ϕM_n

The assumption that c = 12.75 in. is correct

Table 6 – Exterior Column Axial and Moment Capacities
No.	Pu, kip	Mu = M2(2nd), ft-kip	c, in.	εt = εs	φ	φPn, kip	φMn, kip.ft
1	871.4	91.5	14.85	0.00096	0.65	871.4	459.4
2	869.4	91.3	14.85	0.00097	0.65	869.4	459.7
3	797.6	83.7	13.75	0.00128	0.65	797.6	468.2
4	722.0	189.7	12.75	0.00162	0.65	722.0	474.1
5	799.3	-147.5	13.78	0.00127	0.65	799.3	468.0
6	710.9	367.9	12.61	0.00167	0.65	710.9	474.8
7	865.4	-317.9	14.76	0.00099	0.65	865.4	460.2
8	482.9	311.6	7.36	0.005	0.9	482.9	557.2
9	637.4	-280.0	11.68	0.00204	0.65	637.4	478.8

Therefore, since ϕM_n > M_u for all ϕP_n = P_u, use 22 x 22 in. column with 8-#8 bars.

7. Column Interaction Diagram - spColumn Software

spColumn program performs the analysis of the reinforced concrete section conforming to the provisions of the Strength Design Method and Unified Design Provisions with all conditions of strength satisfying the applicable conditions of equilibrium and strain compatibility and includes slenderness effects using moment magnification method for sway and nonsway frames. For this column section, we ran in investigation mode with control points using the 318-14. In lieu of using program shortcuts, spSection (Figure 8) was used to place the reinforcement and define the cover to illustrate handling of irregular shapes and unusual bar arrangement.

Figure 8 – spColumn Model Editor (spSection)

Figure 9 –spColumn Model Input Wizard Windows

Figure 10 – Column Section Interaction Diagram about the X-Axis – Design Check for Load Combination 4 (spColumn)

8. Summary and Comparison of Design Results

Table 7 - Factored Axial loads and Magnified Moments at Column Ends Comparison
No.	Pu, kip			δs			M1(2nd), ft-kip			M2(2nd), ft-kip
	Hand	Reference	spColumn	Hand	Reference	spColumn	Hand	Reference	spColumn	Hand	Reference^*	spColumn
1	871.4	871.4	871.4	N/A	N/A	N/A	24.6	24.6	24.6	48.7	48.7	48.7
2	869.4	869.4	869.4	N/A	N/A	N/A	33.4	33.4	33.4	66.4	66.4	66.4
3	797.6	797.6	797.6	N/A	N/A	N/A	25.0	25.0	25.0	49.5	49.5	49.5
4	722.0	722.0	722.0	1.53	1.38	1.53	62.7	60.6	62.7	189.7	173.5	189.7
5	799.3	799.3	799.3	1.53	1.38	1.53	20.9	23.0	20.9	-147.5	-131.3	-147.4
6	710.9	710.9	7110.9	1.55	1.39	1.55	92.0	87.5	92.0	367.9	331.9	367.8
7	865.4	865.4	865.4	1.55	1.39	1.55	7.0	11.5	7.0	-317.9	-281.9	-317.9
8	482.9	482.9	482.9	1.34	1.25	1.34	68.0	65.5	68.0	311.6	292.0	311.7
9	637.4	637.4	637.4	1.34	1.25	1.34	-5.4	-2.9	-5.3	-280.0	-260.3	280.0

Table 8 - Magnified Moments along Column Length to First-Order Moment Ratios Comparison
No.	δ			Mc1, ft-kip			Mc2, ft-kip			Mc1/M1(1st)			Mc2/M2(1st)
	Hand	Reference^*	spColumn	Hand	Reference^*	spColumn	Hand	Reference^*	spColumn	Hand	Reference^*	spColumn	Hand	Reference^*	spColumn
1	1.00	---	1.00	91.5	---	91.5	91.5	---	91.5	1.00	---	1.00	1.00	---	1.00
2	1.00	---	1.00	91.3	---	91.3	91.3	---	91.3	1.00	---	1.00	1.00	---	1.00
3	1.00	---	1.00	83.7	---	83.7	83.7	---	83.7	1.00	---	1.00	1.00	---	1.00
4	1.00	---	1.00	75.8	---	75.8	189.7	---	189.7	1.00	---	1.00	1.44	---	1.44
5	1.00	---	1.00	83.9	---	83.9	-147.5	---	-147.4	1.00	---	1.00	1.65	---	1.65
6	1.00	---	1.00	92.0	---	92.0	367.9	---	367.8	1.20	---	1.20	1.50	---	1.50
7	1.00	---	1.00	90.9	---	90.9	-317.9	---	-317.9	1.00	---	1.00	1.62	---	1.62
8	1.00	---	1.00	68.0	---	68.0	311.6	---	311.7	1.16	---	1.16	1.32	---	1.32
9	1.00	---	1.00	-66.9	---	-66.9	-280.0	---	280.0	1.00	---	1.00	1.37	---	1.37
^* Moment magnification along the length of the column is not covered by the reference

Table 9 - Design Parameters Comparison
No.	c, in.			εt = εs			φ			φPn, kip			φMn, kip.ft
	Hand	Reference^*	spColumn	Hand	Reference^*	spColumn	Hand	Reference^*	spColumn	Hand	Reference^*	spColumn	Hand	Reference^*	spColumn
1	14.85	14.85	14.85	0.00096	0.00096	0.00096	0.65	0.65	0.65	871.4	871.4	871.4	459.4	459.4	459.4
2	14.85	14.82	14.85	0.00097	0.00097	0.00097	0.65	0.65	0.65	869.4	869.4	869.4	459.7	459.7	459.7
3	13.75	13.75	13.75	0.00128	0.00128	0.00128	0.65	0.65	0.65	797.6	797.6	797.6	468.2	468.2	468.2
4	12.75	12.75	12.75	0.00162	0.00162	0.00162	0.65	0.65	0.65	722.0	722.0	722.0	474.1	474.1	474.1
5	13.78	13.78	13.78	0.00127	0.00127	0.00127	0.65	0.65	0.65	799.3	799.3	799.3	468.0	468.0	468.0
6	12.61	12.61	12.61	0.00167	0.00167	0.00167	0.65	0.65	0.65	710.9	710.9	7110.9	474.8	474.8	474.8
7	14.76	14.76	14.76	0.00099	0.00099	0.00099	0.65	0.65	0.65	865.4	865.4	865.4	460.2	460.2	460.2
8	7.36	7.36	7.36	0.00500	0.00500	0.00500	0.90	0.90	0.90	482.9	482.9	482.9	557.2	557.2	557.2
9	11.68	11.68	11.68	0.00204	0.00204	0.00204	0.65	0.65	0.65	637.4	637.4	637.4	478.8	478.8	478.8
^* Notes on ACI 318-11 Building Code Requirements for Structural Concrete, Twelfth Edition, 2013 Portland Cement Association, Example 11-2

In all of the hand calculations and the reference used illustrated above, the results are in precise agreement with the automated exact results obtained from the spColumn program.

9. Conclusions & Observations

The analysis of the reinforced concrete section performed by spColumn conforms to the provisions of the Strength Design Method and Unified Design Provisions with all conditions of strength satisfying the applicable conditions of equilibrium and strain compatibility and includes slenderness effects using moment magnification method for sway and nonsway frames.

ACI 318 provides multiple options for calculating values of k, (EI)_eff , δ_s, and δ leading to variability in the determination of the adequacy of a column section. Engineers must exercise judgment in selecting suitable options to match their design condition as is the case in the reference where the author conservatively made assumptions to simplify and speed the calculation effort.  The spColumn program utilizes the exact methods whenever possible and allows user to override the calculated values with direct input based on their engineering judgment wherever it is permissible.

In load combinations 4 to 7, M_u including second-order effects exceeds 1.4 M_u due to first-order effects (see Table 5). This indicates that in this building, the weight of the structure is high in proportion to its lateral stiffness leading to excessive PΔ effect (secondary moments are more than 25 percent of the primary moments). The PΔ effects will eventually introduce singularities into the solution to the equations of equilibrium, indicating physical structural instability. It was concluded in the literature that the probability of stability failure increases rapidly when the stability index Q exceeds 0.2, which is equivalent to a secondary-to-primary moment ratio of 1.25. The maximum value of the stability coefficient θ (according to ASCE/SEI 7) which is close to stability coefficient Q (according to ACI 318) is 0.25. The value 0.25 is equivalent to a secondary-to-primary moment ratio of 1.33. Hence, the upper limit of 1.4 on the secondary-to-primary moment ratio was selected by the ACI 318.

The moment magnification factor values δ_s calculated in this document and spColumn are different from the values calculated by the reference. ACI 318 provides three equation options to calculate the effective stiffness modulus (EI)_eff as was discussed previously in this document. Equation 6.6.4.4.4(b) is more accurate than equation 6.6.4.4(a) but is more difficult to use because I_se is not known until reinforcement is chosen. The reference used equation 6.6.4.4(a) due to its simplicity while spColumn uses equation 6.6.4.4.4(b) since an iterative procedure is used to select the optimum reinforcement configuration.

As can be seen in Table 5 of this example, exploring the impact of other code permissible equation options provides the engineer added flexibility in decision making regarding design. For load combinations 4 - 7 resolving the stability concern may be viable through a frame analysis providing values for V_us and Δ_o to calculate magnification factor δ_s and may allow the proposed design to be acceptable. Creating a complete model with detailed lateral loads and load combinations to account for second order effects may not be warranted for all cases of slender column design nor is it disadvantageous to have a higher margin of safety when it comes to column slenderness and frame stability considerations.