Column Design: Design Calculations

COLUMN DESIGN CALCULATIONS
Project Name	:	Unassigned
Client Name	:	Unassigned
Engineer Name	:	Unassigned
Design File	:	D:\Bentley\Common data\Bentley Communities\ACI\ACI_English_validation Sheet\2014_English\Column_Shear_Ductile Special\STAAD file\Demo Model in English Unit-Column-1.rcdx
Analysis File	:	D:\Bentley\Common data\Bentley Communities\ACI\ACI_English_validation Sheet\2014_English\Column_Shear_Ductile Special\STAAD file\Demo Model in English Unit.STD
Analysis Last Modified	:	2/27/2023 3:06:19 PM

Definitions Of Terms:

	All forces in units 'kip' and 'ft'
	All reinforcement details like area, spacing in 'in'
	Neutral axis angle for resultant design moment is with respect to local major axis.

1	βdns	=	Ratio to account for reduction of stiffness of columns due to sustained axial loads
2	δns	=	Moment magnification factor for frames not braced against sidesway
3	Δo	=	First-order relative deflection between the top and bottom of the story due to Vu in 'kip'
4	∑Pu	=	Total factored vertical load in 'kip'. (Clause 6.6.4.4)
5	δu	=	Design displacement in 'in'
6	λ	=	Modification factor reflecting the reduced mechanical properties Of concrete
7	ac	=	Coefficient defining the relative contribution Of concrete strength To nominal wall shear strength
8	Ach	=	Cross-sectional area of a structural member measured to the outside edges of transverse reinforcement in 'in²'
9	Acv	=	Gross area of concrete section bounded by web thickness And length of section in the direction Of shear force considered in 'in²'
10	Aj	=	Effective cross-sectional area within a joint In a plane parallel To plane Of reinforcement generating shear In the joint in 'in²'
11	As	=	Area Of non-prestressed longitudinal tension reinforcement in 'in²'
12	Avmin	=	Minimum area Of shear reinforcement within spacing 's' in 'in²'
13	B	=	Width of column/ wall in 'in'
14	beff	=	Effective Width of column/ wall in 'in'
15	bc1 and dc1	=	Cross-sectional dimension of member core measured to the outside edges of the transverse reinforcement composing area Ash in 'in'
16	c	=	Distance from extreme compression fiber to neutral axis in 'in'
17	Cc	=	Clear cover to longitudinal reinforcement in 'in'
18	Cm	=	Factor relating actual moment diagram to an equivalent uniform moment diagram
19	D	=	Depth / diameter of column in 'in'
20	deff	=	Effective Depth / diameter of column in 'in'
21	d	=	Distance from extreme compression fiber to centroid of longitudinal tension reinforcement in 'in'
22	d'	=	Distance from extreme compression fiber to centroid of longitudinal compression reinforcement,'in'
23	Ec	=	Modulus of elasticity of concrete in 'ksi'
24	EI	=	Flexural stiffness of compression member in 'lbs-in²'
25	f'c	=	Specified compressive strength of concrete cylinder in 'ksi'
26	fy	=	Specified yield strength of reinforcement in 'ksi'
27	fyt	=	Specified yield strength fy of transverse reinforcement in 'ksi'
28	hw	=	Height of entire wall from base to top of wall segment considered in 'in'
29	Icr	=	Moment of Inertia of concrete crack section
30	k	=	Effective length factor for compression member
31	lc	=	Length of compression member in a frame, measured center-to-center of the joints in the frame in 'in'
32	lg	=	Moment of inertia of gross concrete section about centroidal axis neglecting reinforcement in 'in⁴'
33	lw	=	Length of entire wall in 'in'
34	lux	=	Un-supported length for compression member along D in 'in'
35	luy	=	Un-supported length for compression member along B in 'in'
36	MCap	=	Moment capacity of section for a given NA angle at design Pu in 'kip-ft '
37	Mcr	=	Cracking Moment
38	MRes	=	Resultant design moment at a given load angle to local major axis in 'kip-ft '
39	Mc	=	Factored moment amplified for the effects of member curvature used for design of compression member in 'kip-ft'
40	Mm	=	Factored moment modified to account for effect of axial compression in 'kip-ft'
41	Mmx	=	Factored moment along D of column modified to account for effect of axial compression in 'kip-ft'
42	Mmy	=	Factored moment along D of column modified to account for effect of axial compression in 'kip-ft'
43	Mur	=	Sqrt (Mmy^2 + Mmx^2) for circular column in 'kip-ft'
44	Mux	=	Factored moment acting on a section along D in 'kip-ft' from Analysis (Momemt About Major Axis)
45	Muy	=	Factored moment acting on a section along B in 'kip-ft' from Analysis (Momemt About Minor Axis)
46	M1	=	Smaller factored end moment on a compression member in 'kip-ft'
47	M1ns	=	Factored end moment on a compression member at the end at which M1 acts, due to loads that cause no appreciable sidesway in 'kip-ft'
48	M1s	=	Factored end moment on compression member at the end at which M1 acts, due to loads that cause appreciable sidesway in 'kip-ft'
49	M1sldr	=	Smaller factored end moment on a compression member due to slenderness effect in 'kip-ft'
50	M2	=	Larger factored end moment on compression member in 'kip-ft'
51	M2min	=	Minimum value of moment M2 as per minimum eccentricity of column
52	M2ns	=	Factored end moment on compression member at the end at which M2 acts, due to loads that cause no appreciable sidesway in 'kip-ft'
53	M2s	=	Factored end moment on compression member at the end at which M2 acts, due to loads that cause appreciable sidesway in 'kip-ft'
54	M2sldr	=	Largest factored end moment on a compression member due to slenderness effect in 'kip-ft'
55	Mnb	=	Flexure Capacity for Beam
56	Mnc	=	Flexure Capacity for Column
57	Mnty	=	Nominal Flexure strength of column at top along depth in 'kip-ft'
58	Mnby	=	Nominal Flexure strength of column at bottom along depth in 'kip-ft'
59	Mntx	=	Nominal Flexure strength of column at top along width in 'kip-ft'
60	Mnbx	=	Nominal Flexure strength of column at bottom along width in 'kip-ft'
61	Nu	=	Factored axial force normal to cross section occurring simultaneously with Vu in 'kip'
62	Pc	=	Critical buckling load in 'kip'
63	pt	=	Ratio of area of distributed transverse reinforcement to gross concrete area perpendicular to that reinforcement
64	Pω	=	Ratio of As to B x d
65	Q	=	Stability index for storey
66	r	=	Radius of gyration of cross section of a compression member in 'in'
67	Vc	=	Nominal shear strength provided by concrete in 'kip'
68	Vj	=	Shear Force acting at the joint in 'kip'
69	Vn	=	Nominal shear strength in 'kip'
70	Vn'	=	Nominal shear strength at joint in 'kip'
71	Vs	=	nominal shear strength provided by shear reinforcement in 'kip'
72	Vs permissible	=	Maximum nominal shear strength provided by shear reinforcement in 'kip'
73	Vur	=	Factored resultant shear force acting on the column in 'kip'
74	Vus	=	Factored horizontal shear in a storey in 'kip'
75	Vux	=	Factored shear at section along B in 'kip' (From Analysis)
76	Vux1	=	Shear induced due to column flexural capacity along width,'kip'
77	Vux2	=	Shear due to enhanced earthquake factor along width, 'kip'
78	Vuy	=	Factored shear at section along D in 'kip' (From Analysis)
79	Vuy1	=	Shear induced due to column flexural capacity along depth, 'kip'
80	Vuy2	=	Shear due to enhanced earthquake factor along depth, 'kip'
81	y	=	Neutral axis depth.
82	β	=	It is a Neutral Axis angle corresponding to load angle to find out MCap
83	So	=	Center to center spacing of transverse reinforcement within the length lo in 'in'
84	lo	=	Length, measured from joint face along axis of member, over which special transverse reinforcement must be provided in 'in'

Code References:

ACI 318 - 14

Sr.No	Element		Clause / table
1	Minimum area of longitudinal reinforcement for column	:	18.7.4
2	Maximum area of longitudinal reinforcement for column	:	18.7.4
3	Minimum longitudinal and transverse reinforcement for wall	:	18.10.2.1
4	Minimum diameter of transverse ties	:	25.7.2
5	Minimum spacing of transverse ties	:	25.7.2
6	Maximum spacing of longitudinal and transverse reinforcement for wall	:	18.10.2.1
7	Applicability of boundary element	:	18.10.6
8	Area and spacing of special confining reinforcement	:	18.7.5
9	Slenderness Moments	:	6.2.5
10	Shear Strength provided by concrete for column	:	22.5.5
11	Design of shear for non-ductile wall	:	11.5.4
12	Design of shear for ductile wall	:	18.10.4.1
13	Minimum Flexural Strength of Columns	:	18.7.3
14	Shear Check at Column Joint	:	18.8.4.1
15	Shear Strength of Column	:	18.3.3, 18.4 & 18.6.5
16	fs,perm	:	10.6.4
17	fc,perm	:	10.2.7.1
18	Wcr	:	Eq 4.2(a)

Sway Calculation (Stability Index)

For Global-X Direction

Level	Load Combination Analysis	Storey Height (ft)	Gravity Load P (kip)	Relative Displacements (in)	Storey Shear (kip)	Stability Index (Q)	Sway Condition
Level	Load Combination Analysis	A	B	C	D	B x C / (A x D)	Sway Condition
-8.25 ft to 0 ft	104	8.25	23927.59	0.01	221.56	0.013	Non Sway
0 ft to 10 ft	104	10	18366.13	0.03	217.09	0.018	Non Sway
10 ft to 20 ft	104	10	12126.75	0.02	162.82	0.013	Non Sway
20 ft to 30 ft	104	10	5887.36	0.01	108.54	0.006	Non Sway

For Global-Y Direction

Level	Load Combination Analysis	Storey Height (ft)	Gravity Load P (kip)	Relative Displacements (in)	Storey Shear (kip)	Stability Index (Q)	Sway Condition
Level	Load Combination Analysis	A	B	C	D	B x C / (A x D)	Sway Condition
-8.25 ft to 0 ft	105	8.25	23927.59	0.01	133.24	0.022	Non Sway
0 ft to 10 ft	105	10	18366.13	0.03	217.09	0.02	Non Sway
10 ft to 20 ft	105	10	12126.75	0.03	162.82	0.016	Non Sway
20 ft to 30 ft	105	10	5887.36	0.02	108.54	0.008	Non Sway

General Data
Column No.	:	C21
Level	:	-8.25 ft To 0 ft
Frame Type	=	Lateral
Response Modification Coefficient	=	3
Design Code	=	ACI 318 - 14
Grade Of Concrete (f'c)	=	C3.5	ksi
Grade Of Steel (Main)	=	Fy60	ksi
Grade Of Steel (Shear)	=	Fy60	ksi
Grade Of Steel - Flexural Design	=	Fy60	ksi
Grade Of Steel - Shear Design	=	Fy60	ksi
Consider Ductile	=	Yes
Type of Frame	=	Special
Column B	=	30	in
Column D	=	36	in
Clear Cover, Cc	=	2	in
Clear Floor Height @ lux	=	69	in
Clear Floor Height @ luy	=	69	in
No Of Floors	=	1
No Of Columns In Group	=	1
Strength Reduction Factor φ (Shear)	=	0.6

Flexural Design (Analysis Forces)
Analysis Reference No.	=	375
Critical Analysis Load Combination	:	104
Load Combination	=	[5] : 1.2 (LOAD 1: DEAD LOAD) +0.5 (LOAD 2: LIVE LOAD) +(LOAD 3: EQ-X)
Critical Location	=	Bottom Joint
Put	=	1174.78	kip
Muxt	=	-26.36	kip-ft
Muyt	=	-3.52	kip-ft
Vuxt	=	1.57	kip
Vuyt	=	7.12	kip
Pub	=	1185.93	kip
Muxb	=	-145.13	kip-ft
Muyb	=	9.44	kip-ft
Vuxb	=	1.57	kip
Vuyb	=	247.22	kip

Effective Length Calculation

Calculation Along Major Axis Of Column

Joint	Column Stiffness	Beam Sizes		Beam Stiffness		ψ
Joint	Column Stiffness	Beam 1 (Length x Width x Depth)	Beam 2 (Length x Width x Depth)	Beam 1	Beam 2	ψ
	lbs-ft x 10^6	in	in	lbs-ft x 10^6	lbs-ft x 10^6
Bottom	1315.07	No Beam	No Beam	-	-	1
Top	1315.07	321 x 12 x 30	321 x 12 x 30	93.88	93.88	12.782

Sway Condition (as per Stability Index)	=	Non Sway
Effective Length Factor along Major Axis	=	0.86

Calculation Along Minor Axis Of Column

Joint	Column Stiffness	Beam Sizes		Beam Stiffness		ψ
Joint	Column Stiffness	Beam 1 (Length x Width x Depth)	Beam 2 (Length x Width x Depth)	Beam 1	Beam 2	ψ
	lbs-ft x 10^6	in	in	lbs-ft x 10^6	lbs-ft x 10^6
Bottom	913.24	No Beam	No Beam	-	-	1
Top	913.24	360 x 12 x 30	360 x 12 x 30	83.71	83.71	9.955

Sway Condition (as per Stability Index)	=	Non Sway
Effective Length Factor along Minor axis	=	0.86

Check For Stability Index
Along D
Q	=	0.013
	0.013< 0.05, Column shall be designed as non-sway frame (Braced)

Along B
Q	=	0.022
	0.022< 0.05, Column shall be designed as non-sway frame (Braced)

Slenderness Check
Column Is Braced Along D
Slenderness Check along D
k	=	0.86
r	=	10.39	in
Kluy /r	=	5.71
M1	=	-26.36	kip-ft
M2	=	-145.13	kip-ft
Min (40, 34 - 12 x (M1/M2))	=	31.82
	5.71 < 31.82, Column not slender along D
Column Is Braced Along B
Slenderness Check along B
k	=	0.86
r	=	8.66	in
Klux /r	=	6.85
M1	=	-3.52	kip-ft
M2	=	9.44	kip-ft
Min (40, 34 - 12 x (M1/M2))	=	38.47
	6.85 < 38.47, Column not slender along B

Calculation of Design Moment

Direction	Manalysis	Msldr or Mc	Mdesign-final
	A	B	C
Major Axis Mux (top)	-26.36	-	-26.36
Major Axis Mux (bottom)	-145.13	-	-145.13
Minor Axis Muy (top)	-3.52	-	-3.52
Minor Axis Muy (bottom)	9.44	-	9.44

Where
A	=	Moments from analysis
B	=	Moment due to slenderness effect
C	=	Final design Moment = Maximum of (Manalysis, Maximum of (Msldr or Mc))

Final Critical Design Forces

Critical Case - Axial Load & BiAxial Bending
Pu	=	1185.93	kip
Mux	=	-145.13	kip-ft
Muy	=	9.44	kip-ft

Φ Pn, Max Check

Critical Analysis Load Combination	=	101
Load Combination	=	[2] : 1.2 (LOAD 1: DEAD LOAD) +1.6 (LOAD 2: LIVE LOAD)
Critical Location	=	Bottom Joint
Pu	=	1455.69	kip
Mux	=	-15.59	kip-ft
Muy	=	22.34	kip-ft
Pt Calculated	=	1.03
φ Pn, Max	=	2000.85	kip
		Pu < φ Pn, Max	Hence, OK

Minimum Ast Calculation

Critical Analysis Load Combination	=	101
Load Combination	=	[2] : 1.2 (LOAD 1: DEAD LOAD) +1.6 (LOAD 2: LIVE LOAD)
Pu-max from all Combinations	=	1455.69	kip
Pt required for Pu-max	=	0	%
Pt min (User Defined)	=	1	%
Minimum Pt required	=	1	%

Resultant Moment (Combined Action)
Moment Capacity Check
Pt Calculated	=	1.03
Reinforcement Provided	=	4-#8 + 18-#6

Load Angle	=	Tan^-1(Muy/Mux)
	=	3.72	deg
MRes	=	145.44	kip-ft
( φ ) MCap	=	1099.81	kip-ft
Capacity Ratio	=	MRes/ MCap
	=	0.132 < 1

Beam Capacity At Joint:

Beam Table For Shear Along B & Shear Along D

Level	=	0 ft
Analysis Reference Number	=	C21

Beams at Level: Top Joint

Beam Size	Beam angle w.r.t. column Ly	Torsion moment	Moment Capacity Beam at Top				Moment Capacity Beam at Bottom				Resultant Moment
(in)	(deg)	(kip-ft)	Mu (kip-ft)	Ast req (in²)	Ast pro (in²)	Mu cap (kip-ft)	Mu (kip-ft)	Ast req (in²)	Ast pro (in²)	Mu cap (kip-ft)	Top @ D (kip-ft)	Top @ B (kip-ft)	Bot @ D (kip-ft)	Bot @ B (kip-ft)
12 x 30	90	0.21	445.23	4.05	4.12	451.9	0	2.03	2.16	253.12	0	451.9	0	253.12
12 x 30	0	0.01	340.54	2.99	3.14	355.97	0	1.5	1.57	187.51	355.97	0	187.51	0
12 x 30	270	0.07	451.94	4.12	4.32	470.21	0	2.06	2.16	253.12	0	470.21	0	253.12
12 x 30	180	0.01	340.12	2.99	3.14	355.97	0	1.49	1.57	187.51	355.97	0	187.51	0

Effective Moment for Column

	Mu Major (Along D) (kip-ft)		Mu Minor (Along B) (kip-ft)
	Left	Right	Left	Right
Top	355.97	355.97	470.21	451.9
Bottom	187.51	187.51	253.12	253.12

Shear Calculation (Analysis Forces)	Along D	Along B
Lu (in)	69	69
Pu Top (kip)	788.73	788.73
Mnt (kip-ft)	1493.55	1239.38
Pu Bottom (kip)	797.09	797.09
Mnb (kip-ft)	1486.79	1232.81
Vu1 (kip)	518.32	429.95
Shear from Beam Capacity at Joint
Moment Mnt (kip-ft)	MAX((left,Bot + Right,Top), (left,Top + Right,Bot))
	543.48	723.33
Moment Mnb (kip-ft)	MAX((Left,Top + Right,Bot), (Right,Top + Left,Bot))
	0	0
Lu (in)	69	69
Vu2 (kip)	94.52	125.8
Critical Analysis Load Combination	100	108
Critical Load Combination	[1] : 1.4 (LOAD 1: DEAD LOAD)	[9] : 0.9 (LOAD 1: DEAD LOAD) +(LOAD 4: EQ-Y)
Nu (kip)	1240.19	788.73
Mu (kip-ft)	47.9	12.67
Vu3 (kip)	277.38	9.4
Vu' (kip)	Minimum(Vu1, Vu2)
	94.52	125.8
Design Shear, Vu (kip)	Maximum(Vu', Vu3)
	277.38	125.8
λ	1	1
φ	0.6	0.6
Deff (in)	33.5	27.5
ρw (50% of As provided)	0.006	0.006
Mm (kip-ft)	-1379.62	-747.31
φVc (kip)	226.7	192.93
Check	Vu > φVc	Vu < φVc
Link For Shear Design	Required	Not Required
Shear Links Design
Vs (kip)	(Vu - φVc) / φVc
	84.47	-
Vs Permissible (kip)	8 x sqrt(f'c) x b x deff
	475.64	-
Vs Permissible Check	Vs < Vs permissible; Hence, OK	-
Check for Minimum Shear Reinforcement		-
0.5 x φVc (kip)	113.35	-
Minimum Shear Reinforcement Check	Vu > 0.5 x φVc; Hence, Minimum Shear reinforcement required	-
Av/s minimum (in²/ft)	0.3	-
Av/s shear (in²/ft)	0.5	-
Av/s required (in²/ft)	max (Av/s minimum , Av/s shear)
	0.5	-
Link Rebar Number	3	-
Diameter of link (in)	0.37	-
Numbers of legs provided	6	-
Spacing of Link Provided (in)	12	-
Av/s provided (in²/ft )	0.66	-
Av/s provided check	Av/s required < Av/s provided; Hence, OK	-

Design Of Links
Links in the zone where special confining links are not required
Normal Links

Maximum Longitudinal Diameter	=	Dia Of Rebar
Rebar Number of bundled bar, D1	=	8
Diameter of bundled bar, D1	=	1	in
Bundled Rebar	=	No
Minimum diameter of link	>=	0.37	in
Provided Link Rebar Number	=	3
Provided Diameter of link	=	0.37	in

Criterion for spacing of normal links
Min. Longitudinal Bar dia X 16	=	12	in
48 x diameter of links	=	18	in
Min. Dimension of column	=	30	in

*Spacing Criteria as per Vs & 4Sqrt (f'c)Aeff*
Along D
Vs	=	84.47	kip
4Sqrt (f'c)Aeff	=	237.82	kip
Check	Vs <= 4Sqrt (f'c)B*d
d/2	=	16.75	in
24	=	24	in

Provided spacing	=	12	in

Provided spacing	=	12	in

Criterion for spacing of Ductile links / Links at Beam column Joint:
fyt	=	60	ksi
Along D
Av/s min-1	=	0.75 x Sqrt(f'c) x B x 12 / fyt
	=	0.27	in²/ft
Av/s min-2		50 x B x 12 / fyt
	=	0.3	in²/ft
Av/s min	=	Max ( Av/s min-1, Av/s min-2)
	=	0.3	in²/ft

Along B
Av/s min-1	=	0.75 x Sqrt(f'c) x D x 12 / fyt
	=	0.32	in²/ft
Av/s min-2		50 x B x 12 / fyt
	=	0.36	in²/ft
Av/s min	=	Max ( Av/s min-1, Av/s min-2)
	=	0.36	in²/ft
Criterion for spacing of Ductile links:
Min. Longitudinal Bar dia x 6	=	4.5	in
B / 4	=	7.5	in
So	=	6	in
Max 6 in	=	6	in
Provided Spacing	=	4	in

Special confining reinforcement as per ACI
Load Combination	=	[2] : 1.2 (LOAD 1: DEAD LOAD) +1.6 (LOAD 2: LIVE LOAD)
Pu	=	1455.69	kip
Check for Ash
0.3 x Ag x f'c	=	1134	kip
	Pu > 0.3 x Ag x f'c
f'c	=	3.5	ksi
	f'c <= 10.15
Along D
No of bars along D	=	6
S1	=	4	in
Pu	=	1455.69	kip
Ag	=	1080	in²
bc1	=	27	in
Ach	=	891	in²
AshD1	=	0.4	in²
AshD2	=	0.57	in²
AshD3	=	0.65	in²
AshD (Max (AshD1, AshD2, AshD3))	=	0.65	in²
AshD Provided	=	1.18	in²
Along B
No of bars along B	=	7
S1	=	4	in
Pu	=	1455.69	kip
Ag	=	1080	in²
dc1	=	33	in
Ach	=	891	in²
AshB1	=	0.49	in²
AshB2	=	0.69	in²
AshB3	=	0.79	in²
AshB (Max (AshB1, AshB2, AshB3))	=	0.79	in²
AshB Provided	=	1.37	in²


Provided Links	=	#4@4	c/c

Numbers of legs provided along D	=	6
Av/s provided along D	=	1.98	in²/ft
Numbers of legs provided along B	=	7
Av/s provided along B	=	2.31	in²/ft
		Hence, OK

Length of Confining Zone
Column maximum dimension	=	36	in
Clear height /6	=	11.5	in
Min. 18 in	=	18	in
Zone length	=	34.5	in

Table For Links

Note: Ductile Design of Links is Applicable Only For Boundary Elements

	Required			Provided
	Normal Design	Shear Design	Ductile Design	Normal Zone	Ductile Zone
Link Rebar Number	3	---	4	3	4
Spacing	12	---	4	12	4