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 15 ft6 in., and is 9 ft. 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 Slender Reinforced Concrete Column CrossSection
Contents
1. Factored Axial Loads and Bending Moments
1.2. Load Combinations Factored Loads
2. Slenderness Effects and Sway or Nonsway Frame Designation
3. Determine Slenderness Effects
4. Moment Magnification at Ends of Compression Member
4.1. Gravity Load Combination #2 (Gravity Loads Only)
4.2. Lateral Load Combination #6 (Gravity Plus Wind Loads)
5. Moment Magnification along Length of Compression Member
5.1. Gravity Load Combination #2 (Gravity Loads Only)
5.2. Load Combination 6 (Gravity Plus Wind Loads)
6.1. c, a, and strains in the reinforcement
6.2. Forces in the concrete and steel
7. Column Interaction Diagram  spColumn Software
8. Summary and Comparison of Design Results
Code
Building Code Requirements for Structural Concrete (ACI 31814) and Commentary (ACI 318R14)
Reference
Reinforced Concrete Mechanics and Design, 7^{th} Edition, 2016, James Wight, Pearson, Example 123
Design Data
f_{c} = 4,000 psi for columns
f_{y} = 60,000 psi
Slab thickness = 6 in.
Exterior Columns = 18 in. x 18 in.
Interior Columns = 18 in. x 18 in.
Interior Beams = 18 in. x 30 in. x 30 ft
Exterior Beams = 18 in. x 30 in. x 32 ft
Floor superimposed dead load = 20 psf
Floor live load = 80 psf
Roof superimposed dead load = 25 psf
Roof live load = 30 psf
Wind loads computed according to ASCE 710
Total building loads in the first story from the reference:
Table 1 Total building factored loads 

ASCE 710 Reference 
No. 
Load Combination 
∑ P_{u}, kip 
2.3.21 
1 
1.4D 
10,990 
2.3.22 
2 
1.2D + 1.6L + 0.5L_{r} 
11,400 
2.3.23 
3 
1.2D + 0.5L + 1.6 L_{r} 
10,459 
4 
1.2D + 1.6L_{r} + 0.8W 
9,882 

5 
1.2D + 1.6L_{r}  0.8W 
9,882 

2.3.24 
6 
1.2D + 0.5L + 0.5L_{r} + 1.6W 
10,100 
7 
1.2D + 0.5L + 0.5L_{r}  1.6W 
10,100 

2.3.26 
8 
0.9D + 1.6W 
7,065 
9 
0.9D  1.6W 
7,065 
Table 2  Exterior column service loads 

Load Case 
Axial Load, 
Bending Moment, ftkip 

Top 
Bottom 

Dead, D 
283.0 
34.9 
36.8 
Live, L 
42.9 
11.2 
11.8 
Roof Live, L_{r} 
10.1 
0.0 
0.0 
Wind, W (NS) 
9.0 
47.8 
46.1 
Table 3  Exterior column factored loads 

ASCE 710 
No. 
Load Combination 
Axial Load, 
Bending Moment,

M_{Top,ns} 
M_{Bottom,ns} 
M_{Top,s} 
M_{Bottom,s} 

Top 
Bottom 

2.3.21 
1 
1.4D 
396.2 
48.9 
51.5 
48.9 
51.5 
 
 
2.3.22 
2 
1.2D + 1.6L + 0.5L_{r} 
413.3 
59.8 
63 
59.8 
63 
 
 
2.3.23 
3 
1.2D + 0.5L + 1.6 L_{r} 
377.2 
47.5 
50.1 
47.5 
50.1 
 
 
4 
1.2D + 1.6L_{r} + 0.8W 
363.0 
80.1 
81 
41.9 
44.2 
38.2 
36.9 

5 
1.2D + 1.6L_{r}  0.8W 
348.6 
3.6 
7.3 
41.9 
44.2 
38.2 
36.9 

2.3.24 
6 
1.2D + 0.5L + 0.5L_{r} + 1.6W 
380.5 
124.0 
123.8 
47.5 
50.1 
76.5 
73.8 
7 
1.2D + 0.5L + 0.5L_{r}  1.6W 
351.7 
29.0 
23.7 
47.5 
50.1 
76.5 
73.8 

2.3.26 
8 
0.9D + 1.6W 
269.1 
107.9 
106.9 
31.4 
33.1 
76.5 
73.8 
9 
0.9D  1.6W 
240.3 
45.1 
40.6 
31.4 
33.1 
76.5 
73.8 
Columns and stories in structures are considered as nonsway frames if the increase in column end moments due to secondorder effects does not exceed 5% of the firstorder end moments, or the stability index for the story (Q) does not exceed 0.05. ACI 31814 (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 31814 (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 firstorder relative deflection between the top and bottom of the first story due to V_{us}. ACI 31814 (6.6.4.4.1 and R6.6.4.3)
From Table 1, load combination (2.3.22 No. 2) provides the greatest value of ∑P_{u}.
ASCE 710 (2.3.22)
ACI 31814 (Eq. 6.6.4.4.1)
Thus, the frame at the first story level is considered sway.
ACI 31814 (Table 6.6.3.1.1(a))
ACI 31814 (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 31814 (19.2.2.1.b)
ACI 31814 (Table 6.6.3.1.1(a))
ACI 31814 (Figure R6.2.5)
(Column considered fixed at the base) ACI 31814 (Figure R6.2.5)
Using Figure R6.2.5 from ACI 31814 ΰ k = 1.44 as shown in the figure below for the exterior column.
Figure 2 Effective Length Factor (k) for Exterior Column (Sway Frame)
ACI 31814 (6.2.5a)
Where:
ACI 31814 (6.2.5.1)
A detailed calculation for load combinations 2 and 6 is shown below to illustrate the slender column moment magnification procedure. Table 4 summarizes the magnified moment computations for the exterior columns.
ACI 31814 (6.6.4.6.1b)
Where:
P_{u} = 413.3kip
ACI 31814 (6.6.4.6.1b)
Where:
ACI 31814 (6.6.4.6.2)
There are three options for calculating δ_{s}. ACI 31814 (6.6.4.6.2(b)) will be used since it does not require a detailed structural analysis model results to proceed and is also used by the solver engine in spColumn.
∑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 swayresisting columns in the first story.
ACI 31814 (6.6.4.4.2)
Where:
ACI 31814 (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 31814 (Table 6.6.3.1.1(a))
ACI 31814 (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 31814 (6.6.3.1.1)
For exterior columns with one beam framing into them in the direction of analysis (8 columns):
With 8#6 reinforcement equally distributed on all sides I_{se} = 111.5 in.^{4} (Ref. uses approximate value of 150 in.^{4} in lieu of exact value calculated by spColumn).
ACI 31814 (6.6.4.4.4(b))
k = 1.44 (calculated previously).
For exterior columns with two beams framing into them in the direction of analysis (8 columns):
ACI 31814 (Figure R6.2.5)
(Column essentially fixed at base) ACI 31814 (Figure R6.2.5)
Using Figure R6.2.5 from ACI 31814 ΰ k = 1.31 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) for Exterior Columns with Two Beams Framing into them in the Direction of Analysis
For interior columns (8 columns):
ACI 31814 (Figure R6.2.5)
(Column essentially fixed at base) ACI 31814 (Figure R6.2.5)
Using Figure R6.2.5 from ACI 31814 ΰ k = 1.31 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 I_{se} = 192.6 in.^{4}
ACI 31814 (6.6.4.4.4(b))
ACI 31814 (6.6.4.6.2(b))
ACI 31814 (6.6.4.6.1)
ACI 31814 (6.6.4.6.1)
P_{u} = 380.5 kip
A summary of the moment magnification factors and magnified moments for the exterior column for all load combinations using both equation options ACI 31814 (6.6.4.6.2(a)) and (6.6.4.6.2(b)) to calculate δ_{s} is provided in the table below for illustration and comparison purposes. Note: The designation of M_{1} and M_{2} is made based on the secondorder (magnified) moments and not based on the firstorder (unmagnified) moments.
Table 4  Factored Axial loads and Magnified Moments at the Ends of Exterior Column 

No. 
Load Combination 
Axial Load, 
Using ACI 6.6.4.6.2(a) 
Using ACI 6.6.4.6.2(b) 

kip 
δ_{s} 
M_{1}, ftkip 
M_{2}, ftkip 
δ_{s} 
M_{1}, ftkip 
M_{2}, ftkip 

1 
1.4D 
396.2 
^{*} 
^{*} 
^{*} 
 
48.9 
51.5 
2 
1.2D + 1.6L + 0.5L_{r} 
413.3 
 
59.8 
63 
 
59.8 
63 
3 
1.2D + 0.5L + 1.6 L_{r} 
377.2 
^{*} 
^{*} 
^{*} 
 
47.5 
50.1 
4 
1.2D + 1.6L_{r} + 0.8W 
363 
^{*} 
^{*} 
^{*} 
1.51 
99.6 
99.9 
5 
1.2D + 1.6L_{r}  0.8W 
348.6 
^{*} 
^{*} 
^{*} 
1.51 
11.5 
45.9 
6 
1.2D + 0.5L + 0.5L_{r} + 1.6W 
380.5 
1.14 
134.2 
134.7 
1.53 
163 
164.5 
7 
1.2D + 0.5L + 0.5L_{r}  1.6W 
351.7 
^{*} 
^{*} 
^{*} 
1.53 
62.8 
69.6 
8 
0.9D + 1.6W 
269.1 
^{*} 
^{*} 
^{*} 
1.32 
130.4 
132.2 
9 
0.9D  1.6W 
240.3 
^{*} 
^{*} 
^{*} 
1.32 
64.1 
69.4 
^{*} Not covered by the reference 
In sway frames, secondorder effects shall be considered along the length of columns. It shall be permitted to account for these effects using ACI 31814 (6.6.4.5) (Nonsway frame procedure), where C_{m} is calculated using M_{1} and M_{2} from ACI 31814 (6.6.4.6.1) as follows: ACI 31814 (6.6.4.6.4)
ACI 31814 (6.6.4.5.1)
Where:
M_{2} = the secondorder factored moment.
ACI 31814 (6.6.4.5.2)
ACI 31814 (6.6.4.4.2)
Where:
ACI 31814 (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 31814 (Table 6.6.3.1.1(a))
ACI 31814 (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 31814 (6.6.4.4.4)
ACI 31814 (Figure R6.2.5)
(Column essentially fixed at base) ACI 31814 (Figure R6.2.5)
Using Figure R6.2.5(a) from ACI 31814 ΰ k = 0.81 as shown in the figure below for the exterior column.
Figure 5 Effective Length Factor (k) Calculations for Exterior Column (Nonsway)
With 8#6 reinforcement equally distributed on all sides I_{se} = 111.5 in.^{4} (Ref. uses approximate value of 150 in.^{4} in lieu of exact value calculated by spColumn).
ACI 31814 (6.6.4.4.4(b))
ASCE 710 (2.3.22)
ACI 31814 (6.6.4.5.3a)
ACI 31814 (6.6.4.6.4)
ACI 31814 (6.6.4.6.4)
Since the column is bent in double curvature, M_{1}/M_{2} is positive. ACI 31814 (6.6.4.5.3)
ACI 31814 (6.6.4.5.2)
ACI 31814 (6.6.4.5.4)
Where P_{u} = 413.3 kip, and h = the section dimension in the direction being considered = 18 in.
ACI 31814 (6.6.4.5.4)
ACI 31814 (6.6.4.5.1)
ACI 31814 (6.6.4.5.4)
ACI 31814 (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
ACI 31814 (Table 6.6.3.1.1(a))
ACI 31814 (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 31814 (6.6.4.4.4)
ACI 31814 (Figure R6.2.5)
(Column essentially fixed at base) ACI 31814 (Figure R6.2.5)
Using Figure R6.2.5(a) from ACI 31814 ΰ k = 0.81
With 8#6 reinforcement equally distributed on all sides I_{se} = 111.5 in.^{4} (Ref. uses approximate value of 150 in.^{4} in lieu of exact value calculated by spColumn).
ACI 31814 (6.6.4.4.4(b))
ASCE 710 (2.3.24)
ACI 31814 (6.6.4.5.3a)
ACI 31814 (6.6.4.6.4)
ACI 31814 (6.6.4.6.4)
Since the column is bent in double curvature, M_{1}/M_{2} is positive. ACI 31814 (6.6.4.5.3)
ACI 31814 (6.6.4.5.2)
ACI 31814 (6.6.4.5.4)
Where P_{u} = 380.5 kip, and h = the section dimension in the direction being considered = 18 in.
ACI 31814 (6.6.4.5.4)
ACI 31814 (6.6.4.5.1)
ACI 31814 (6.6.4.5.4)
ACI 31814 (6.6.4.5.1)
M_{c1} and M_{c2} are considered separately to ensure proper comparison of resulting magnified moments against negative and positive moment capacities of unsymmetrical sections.
A summary of the moment magnification factors and magnified moments for the exterior column for all load combinations using both equation options ACI 31814 (6.6.4.6.2(a)) and (6.6.4.6.2(b)) to calculate δ_{s} is provided in the table below for illustration and comparison purposes.
Table 5  Factored Axial loads and Magnified Moments along Exterior Column Length 

No. 
Load Combination 
Axial Load, kip 
Using ACI 6.6.4.6.2(a) 
Using ACI 6.6.4.6.2(b) 

δ 
M_{c1},ftkip 
M_{c2}, ftkip 
δ 
M_{c1}, ftkip 
M_{c2}, ftkip 

1 
1.4D 
396.2 
^{*} 
^{*} 
^{*} 
1 
48.9 
51.5 
2 
1.2D + 1.6L + 0.5L_{r} 
413.3 
1 
59.8 
63 
1 
59.8 
63 
3 
1.2D + 0.5L + 1.6 L_{r} 
377.2 
^{*} 
^{*} 
^{*} 
1 
47.5 
50.1 
4 
1.2D + 1.6L_{r} + 0.8W 
363 
^{*} 
^{*} 
^{*} 
1 
99.6 
99.9 
5 
1.2D + 1.6L_{r}  0.8W 
348.6 
^{*} 
^{*} 
^{*} 
1 
33.1 
33.1 
6 
1.2D + 0.5L + 0.5L_{r} + 1.6W 
380.5 
1 
134.2 
134.7 
1 
163 
164.5 
7 
1.2D + 0.5L + 0.5L_{r}  1.6W 
351.7 
^{*} 
^{*} 
^{*} 
1 
62.8 
69.6 
8 
0.9D + 1.6W 
269.1 
^{*} 
^{*} 
^{*} 
1 
130.4 
132.2 
9 
0.9D  1.6W 
240.3 
^{*} 
^{*} 
^{*} 
1 
64.1 
69.4 
^{*} Not covered by the reference 
For column design ACI 318 requires the secondorder moment to firstorder moment ratios should not exceed 1.40. If this value is exceeded, the column design needs to be revised. ACI 31814 (6.2.6)
Table 6  SecondOrder Moment to FirstOrder Moment Ratios 

No. 
Load Combination 
Using ACI 6.6.4.6.2(a) 
Using ACI 6.6.4.6.2(b) 

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^{*} 
2 
1.2D + 1.6L + 0.5L_{r} 
1.00^{*} 
1.00^{*} 
1.00^{*} 
1.00^{*} 
3 
1.2D + 0.5L + 1.6 L_{r} 
^{**} 
^{**} 
1.00^{*} 
1.00^{*} 
4 
1.2D + 1.6L_{r} + 0.8W 
^{**} 
^{**} 
1.24 
1.23 
5 
1.2D + 1.6L_{r}  0.8W 
^{**} 
^{**} 
1.00^{*} 
1.00^{*} 
6 
1.2D + 0.5L + 0.5L_{r} + 1.6W 
1.08 
1.09 
1.32 
1.33 
7 
1.2D + 0.5L + 0.5L_{r}  1.6W 
^{**} 
^{**} 
1.40 < 1.88 
1.40 < 2.08 
8 
0.9D + 1.6W 
^{**} 
^{**} 
1.22 
1.23 
9 
0.9D  1.6W 
^{**} 
^{**} 
1.40 < 1.58 
1.40 < 1.54 
^{* }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}. ^{**} Not covered by the reference 
Based on the factored axial loads and magnified moments considering slenderness effects, the capacity of the assumed column section (18 in. x 18 in. with 8#6 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 #6 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 6)
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.
Where c is the distance from the fiber of maximum compressive strain to the neutral axis.
ACI 31814 (22.2.2.4.2)
ACI 31814 (22.2.2.4.1)
Where:
ACI 31814 (Table 22.2.2.4.3)
ACI 31814 (22.2.2.1)
ACI 31814 (Table 21.2.2)
ACI 31814 (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}:
The assumed value of c = 10.65 in. is correct.
Table 7 Exterior Column Axial and Moment Capacities 

No. 
P_{u}, kip 
M_{u} = M_{2(2nd)}, ftkip 
c, in. 
ε_{t} = ε_{s} 
φ 
φP_{n}, kip 
φM_{n}, kip.ft 
1 
396.2 
51.5 
10.98 
0.00123 
0.65 
396.2 
177.0 
2 
413.3 
63.0 
11.35 
0.00110 
0.65 
413.3 
174.7 
3 
377.2 
50.1 
10.55 
0.00141 
0.65 
377.6 
179.5 
4 
363.0 
99.9 
10.25 
0.00154 
0.65 
363.0 
181.1 
5 
348.6 
33.1 
9.96 
0.00167 
0.65 
348.6 
182.6 
6 
380.5 
164.5 
10.65 
0.00137 
0.65 
380.4 
179.0 
7 
351.7 
69.6 
10.02 
0.00164 
0.65 
351.6 
182.3 
8 
269.1 
132.2 
7.26 
0.00340 
0.77 
269.2 
205.1 
9 
240.3 
69.4 
6.36 
0.00431 
0.84 
240.2 
212.2 
Since ϕM_{n} > M_{u} for all ϕP_{n} = P_{u}, use 18 x 18 in. column with 8#6 bars.
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 31814. 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 XAxis Design Check for Load Combination 6 (spColumn)
Analysis and design results from the hand calculations above are compared for the one load combination used in the reference (Example 123) and exact values obtained from spColumn model.
Table 8 Parameters for Moment Magnification along the Column Length 


Q 
k 
β_{dns} 
C_{m} 
I_{se} 
P_{c}, kip 
δ 
M_{2(min)}, ftkip 
M_{2(2nd)}, ftkip 
Hand 
0.21 
0.81^{*} 
0.82 
0.221 
111.5^{} 
2,277 
1 > 0.264 
39.3 
63.04 
Reference 
0.21 
1.00^{} 
0.82 
0.22 
150.0^{} 
1,660 
1 > 0.330 
39.3 
63 
spColumn 
 
0.81^{} 
0.82 
0.221 
111.5^{} 
2,259 
1 > 0.270 
39.3 
63.04 
^{*} From nomographs (ACI 318 charts) ^{} Conservatively estimated not using exact formulae without impact on the final results in this special case ^{} Exact formulated answer 
In this table, a detailed comparison for all considered load combinations are presented for comparison.
Table 9  Factored Axial loads and Magnified Moments at Column Ends 

No. 
P_{u}, kip 
δ_{s} 
M_{1(2nd)}, ftkip 
M_{2(2nd)}, ftkip 

Hand 
spColumn 
Hand 
spColumn 
Hand 
spColumn 
Hand 
spColumn 

1 
396.2 
396.2 
N/A 
N/A 
48.9 
48.9 
51.5 
51.5 
2 
413.3 
413.3 
N/A 
N/A 
59.8 
59.8 
63.0 
63.0 
3 
377.2 
377.2 
N/A 
N/A 
47.5 
47.5 
50.1 
50.1 
4 
363.0 
363.0 
1.51 
1.51 
99.6 
99.6 
99.9 
99.9 
5 
348.6 
348.6 
1.51 
1.51 
33.1 
33.1 
33.1 
33.1 
6 
380.5 
380.5 
1.53 
1.53 
163.0 
163.0 
164.5 
164.5 
7 
351.7 
351.7 
1.53 
1.53 
62.8 
62.8 
69.6 
69.6 
8 
269.1 
269.1 
1.32 
1.32 
130.4 
130.4 
132.2 
132.2 
9 
240.3 
240.3 
1.32 
1.32 
64.1 
64.1 
69.4 
69.4 
Table 10  Factored Axial loads and Magnified Moments along Column Length 

No. 
δ 
M_{c1}, ftkip 
M_{c2}, ftkip 
M_{c1}/M_{1(1st)} 
M_{c2}/M_{2(1st)} 

Hand 
spColumn 
Hand 
spColumn 
Hand 
spColumn 
Hand 
spColumn 
Hand 
spColumn 

1 
1.00 
1.00 
48.9 
48.9 
51.5 
51.5 
1.00 
1.00 
1.00 
1.00 
2 
1.00 
1.00 
59.8 
59.8 
63.0 
63.0 
1.00 
1.00 
1.00 
1.00 
3 
1.00 
1.00 
47.5 
47.5 
50.1 
50.1 
1.00 
1.00 
1.00 
1.00 
4 
1.00 
1.00 
99.6 
99.6 
99.9 
99.9 
1.24 
1.24 
1.23 
1.23 
5 
1.00 
1.00 
33.1 
33.1 
33.1 
33.1 
1.00 
1.00 
1.00 
1.00 
6 
1.00 
1.00 
163.0 
163.0 
164.5 
164.5 
1.32 
1.32 
1.33 
1.33 
7 
1.00 
1.00 
62.8 
62.9 
69.6 
69.6 
1.88 
1.88 
2.08 
2.08 
8 
1.00 
1.00 
130.4 
130.4 
132.2 
132.2 
1.22 
1.22 
1.23 
1.23 
9 
1.00 
1.00 
64.1 
64.1 
69.4 
69.4 
1.58 
1.58 
1.54 
1.54 
Table 11  Design Parameters Comparison 

No. 
c, in. 
ε_{t} = ε_{s} 
φ 
φP_{n}, kip 
φM_{n}, kip.ft 

Hand 
spColumn 
Hand 
spColumn 
Hand 
spColumn 
Hand 
spColumn 
Hand 
spColumn 

1 
10.98 
10.98 
0.00123 
0.00123 
0.65 
0.65 
396.2 
396.2 
177.0 
177.0 
2 
11.35 
11.35 
0.00110 
0.00110 
0.65 
0.65 
413.3 
413.3 
174.7 
174.7 
3 
10.55 
10.54 
0.00141 
0.00141 
0.65 
0.65 
377.6 
377.6 
179.5 
179.5 
4 
10.25 
10.25 
0.00154 
0.00154 
0.65 
0.65 
363.0 
363.0 
181.1 
181.1 
5 
9.96 
9.96 
0.00167 
0.00167 
0.65 
0.65 
348.6 
348.6 
182.6 
182.6 
6 
10.65 
10.65 
0.00137 
0.00137 
0.65 
0.65 
380.4 
380.4 
179.0 
179.0 
7 
10.02 
10.02 
0.00164 
0.00164 
0.65 
0.65 
351.6 
351.6 
182.3 
182.3 
8 
7.26 
7.30 
0.00340 
0.00337 
0.77 
0.76 
269.2 
269.1 
205.1 
204.0 
9 
6.36 
6.37 
0.00431 
0.00430 
0.84 
0.84 
240.2 
240.3 
212.2 
211.7 
All the results of the hand calculations illustrated above are in precise agreement with the automated exact results obtained from the spColumn program.
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 7 and 9, M_{u} including secondorder effects exceeds 1.4 M_{u} due to firstorder effects (see Table 6). 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 secondarytoprimary 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 secondarytoprimary moment ratio of 1.33. Hence, the upper limit of 1.4 on the secondarytoprimary moment ratio was selected by the ACI 318.
As can be seen in Table 6 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 7 & 9 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 by inspection of load combination #6. 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.