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\2019\Column\column design file.rcdx
Analysis File	:	D:\Bentley\Common data\Bentley Communities\ACI\ACI_English_validation Sheet\2019\Beam\STAAD file\Sample model_02.std
Analysis Last Modified	:	2/25/2022 8:12:19 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	bc1 and dc1	=	Cross-sectional dimension of member core measured to the outside edges of the transverse reinforcement composing area Ash in 'in'
15	c	=	Distance from extreme compression fiber to neutral axis in 'in'
16	Cc	=	Clear cover to longitudinal reinforcement in 'in'
17	Cm	=	Factor relating actual moment diagram to an equivalent uniform moment diagram
18	D	=	Depth / diameter of column in 'in'
19	d	=	Distance from extreme compression fiber to centroid of longitudinal tension reinforcement in 'in'
20	d'	=	Distance from extreme compression fiber to centroid of longitudinal compression reinforcement,'in'
21	Ec	=	Modulus of elasticity of concrete in 'ksi'
22	EI	=	Flexural stiffness of compression member in 'lbs-in²'
23	f'c	=	Specified compressive strength of concrete cylinder in 'ksi'
24	fy	=	Specified yield strength of reinforcement in 'ksi'
25	fyt	=	Specified yield strength fy of transverse reinforcement in 'ksi'
26	hw	=	Height of entire wall from base to top of wall segment considered in 'in'
27	Icr	=	Moment of Inertia of concrete crack section
28	k	=	Effective length factor for compression member
29	lc	=	Length of compression member in a frame, measured center-to-center of the joints in the frame in 'in'
30	lg	=	Moment of inertia of gross concrete section about centroidal axis neglecting reinforcement in 'in⁴'
31	lw	=	Length of entire wall in 'in'
32	lux	=	Un-supported length for compression member along D in 'in'
33	luy	=	Un-supported length for compression member along B in 'in'
34	MCap	=	Moment capacity of section for a given NA angle at design Pu in 'kip-ft '
35	Mcr	=	Cracking Moment
36	MRes	=	Resultant design moment at a given load angle to local major axis in 'kip-ft '
37	Mc	=	Factored moment amplified for the effects of member curvature used for design of compression member in 'kip-ft'
38	Mm	=	Factored moment modified to account for effect of axial compression in 'kip-ft'
39	Mmx	=	Factored moment along D of column modified to account for effect of axial compression in 'kip-ft'
40	Mmy	=	Factored moment along D of column modified to account for effect of axial compression in 'kip-ft'
41	Mur	=	Sqrt (Mmy^2 + Mmx^2) for circular column in 'kip-ft'
42	Mux	=	Factored moment acting on a section along D in 'kip-ft' from Analysis (Momemt About Major Axis)
43	Muy	=	Factored moment acting on a section along B in 'kip-ft' from Analysis (Momemt About Minor Axis)
44	M1	=	Smaller factored end moment on a compression member in 'kip-ft'
45	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'
46	M1s	=	Factored end moment on compression member at the end at which M1 acts, due to loads that cause appreciable sidesway in 'kip-ft'
47	M1sldr	=	Smaller factored end moment on a compression member due to slenderness effect in 'kip-ft'
48	M2	=	Larger factored end moment on compression member in 'kip-ft'
49	M2min	=	Minimum value of moment M2 as per minimum eccentricity of column
50	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'
51	M2s	=	Factored end moment on compression member at the end at which M2 acts, due to loads that cause appreciable sidesway in 'kip-ft'
52	M2sldr	=	Largest factored end moment on a compression member due to slenderness effect in 'kip-ft'
53	Mnb	=	Flexure Capacity for Beam
54	Mnc	=	Flexure Capacity for Column
55	Mnty	=	Nominal Flexure strength of column at top along depth in 'kip-ft'
56	Mnby	=	Nominal Flexure strength of column at bottom along depth in 'kip-ft'
57	Mntx	=	Nominal Flexure strength of column at top along width in 'kip-ft'
58	Mnbx	=	Nominal Flexure strength of column at bottom along width in 'kip-ft'
59	Nu	=	Factored axial force normal to cross section occurring simultaneously with Vu in 'kip'
60	Pc	=	Critical buckling load in 'kip'
61	pt	=	Ratio of area of distributed transverse reinforcement to gross concrete area perpendicular to that reinforcement
62	Pω	=	Ratio of As to B x d
63	Q	=	Stability index for storey
64	r	=	Radius of gyration of cross section of a compression member in 'in'
65	Vc	=	Nominal shear strength provided by concrete in 'kip'
66	Vj	=	Shear Force acting at the joint in 'kip'
67	Vn	=	Nominal shear strength in 'kip'
68	Vn'	=	Nominal shear strength at joint in 'kip'
69	Vs	=	nominal shear strength provided by shear reinforcement in 'kip'
70	Vs permissible	=	Maximum nominal shear strength provided by shear reinforcement in 'kip'
71	Vur	=	Factored resultant shear force acting on the column in 'kip'
72	Vus	=	Factored horizontal shear in a storey in 'kip'
73	Vux	=	Factored shear at section along B in 'kip' (From Analysis)
74	Vux1	=	Shear induced due to column flexural capacity along width,'kip'
75	Vux2	=	Shear due to enhanced earthquake factor along width, 'kip'
76	Vuy	=	Factored shear at section along D in 'kip' (From Analysis)
77	Vuy1	=	Shear induced due to column flexural capacity along depth, 'kip'
78	Vuy2	=	Shear due to enhanced earthquake factor along depth, 'kip'
79	y	=	Neutral axis depth.
80	β	=	It is a Neutral Axis angle corresponding to load angle to find out MCap
81	So	=	Center to center spacing of transverse reinforcement within the length lo in 'in'
82	lo	=	Length, measured from joint face along axis of member, over which special transverse reinforcement must be provided in 'in'
83	Aeff	=	Effective area of column / wall considered for shear check in 'in²'
84	Ag	=	Gross area of concrete section in 'in²'
85	Vc max	=	Maximum nominal shear strength provided by concrete in 'kip'
86	Mpr	=	Probable flexural strength of member with or without axial load determined using properties of the member at joint faces assuming a tensile stress in the longitudinal bars of atleast 1.25fy and strength reduction factor Φ of 0.9 in 'kip-ft'
87	hwcs	=	Height of entire structural wall above the critical section for flexural and axial load in 'in'
88	Ωv	=	Overstrength factor equal to the ratio of Mpr / Mu at the wall critical section
89	ωv	=	Factor accounting for dynamic shear amplification
90	Ve	=	Enhanced Shear force including factors Ωv & ωv in 'kip'

Code References:

ACI 318 - 19

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)
19	Maximum Permissible Deformed Reinforcement Grade	:	20.2.2.4 (a)
20	Strength requirements for beam-columnjoints	:	15.4
21	Transfer of column axial force through the floor system	:	15.5
22	Gravity Column	:	18.14

Sway Calculation (Stability Index)

For Global-X Direction

Level	Load Combination	Storey Height (ft)	Gravity Load P (kip)	Relative Displacements (in)	Storey Shear (kip)	Stability Index (Q)	Sway Condition
Level	Load Combination	A	B	C	D	B x C / (A x D)	Sway Condition
0 ft to 13.77949 ft	4	13.78	713.62	0.04	101.2	0.002	Non Sway

For Global-Y Direction

Level	Load Combination	Storey Height (ft)	Gravity Load P (kip)	Relative Displacements (in)	Storey Shear (kip)	Stability Index (Q)	Sway Condition
Level	Load Combination	A	B	C	D	B x C / (A x D)	Sway Condition
0 ft to 13.77949 ft	6	13.78	713.62	0.01	22.49	0.002	Non Sway

General Data
Column No.	:	C1
Level	:	0 ft To 13.77949 ft
Frame Type	=	Non-Ductile
Response Modification Coefficient	=	3
Design Code	=	ACI 318 - 19
Grade Of Concrete (f'c)	=	C3.5	ksi
Grade Of Steel (fy) (User Defined)	=	Fy60	ksi
Grade Of Steel - Flexural Design	=	Fy60	ksi
Grade Of Steel - Shear Design	=	Fy60	ksi
Consider Ductile	=	No
Column B	=	15	in
Column D	=	24	in
Clear Cover, Cc	=	2	in
Clear Floor Height @ lux	=	350	in
Clear Floor Height @ luy	=	175	in
No Of Floors	=	1
No Of Columns In Group	=	1

Flexural Design (Analysis Forces)
Analysis Reference No.	=	501
Load Combination	=	[2] : 1.2 (LOAD 1: LOAD CASE 1) +1.6 (LOAD 2: LOAD CASE 2)
Critical Location	=	Top Joint
Put	=	141.81	kip
Muxt	=	382.49	kip-ft
Muyt	=	47.43	kip-ft
Vuxt	=	4.45	kip
Vuyt	=	-38.54	kip
Pub	=	141.81	kip
Muxb	=	-148.41	kip-ft
Muyb	=	-13.94	kip-ft
Vuxb	=	4.45	kip
Vuyb	=	-38.54	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	116.64	No Beam	No Beam	-	-	1
Top	116.64	314.96 x 17.72 x 31.5	No Beam	174.01	-	0.714

User Defined Effective Length Factor

0.87

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	45.56	No Beam	No Beam	-	-	1
Top	45.56	350.39 x 15.75 x 47.24	No Beam	468.91	-	0.103

User Defined Effective Length Factor

0.75

Check For Stability Index
Along D
Q	=	0.002
	0.002< 0.05, Column shall be designed as non-sway frame (Braced)

Along B
Q	=	0.002
	0.002< 0.05, Column shall be designed as non-sway frame (Braced)

Slenderness Check
Column Is Braced Along D
Slenderness Check along D
K	=	0.87
r	=	6.93	in
Kluy /r	=	21.98
M1	=	-148.41	kip-ft
M2	=	382.48	kip-ft
Min (40, 34 - 12 x (M1/M2))	=	38.66
	21.98 < 38.66, Column not slender along D
Column Is Braced Along B
Slenderness Check along B
K	=	0.75
r	=	4.33	in
Klux /r	=	60.62
M1	=	-13.94	kip-ft
M2	=	47.43	kip-ft
Min (40, 34 - 12 x (M1/M2))	=	37.53
	60.62 > 37.53, Column slender along B

Moment Magnification:
For Non-sway frame:

Along B
βdns	=	0.6
Ec	=	3372.05	ksi
Ig ( x 10 ⁶)	=	0.01	in⁴
EI ( x 10 ⁶)	=	6570.63	lbs-in²
M2min	=	12.41	kip-ft
Cm	=	0.4
Pc	=	941.13	kip
δns	=	1
Mc1	=	-13.94	kip-ft
1.4 MuyB	=	-19.52	kip-ft
Mc1	=	Min (-13.94, -19.52)	kip-ft
	=	-13.94	kip-ft
Mc2	=	47.43	kip-ft
1.4 MuyT	=	66.4	kip-ft
Mc2	=	Min (47.43, 66.4)	kip-ft
	=	47.43	kip-ft

Calculation of Design Moment

Direction	Manalysis	Msldr or Mc	Mdesign-final
	A	B	C
Major Axis Mux (top)	382.49	-	382.49
Major Axis Mux (bottom)	-148.41	-	-148.41
Minor Axis Muy (top)	47.43	47.43	47.43
Minor Axis Muy (bottom)	-13.94	-13.94	-13.94

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	=	141.81	kip
Mux	=	382.49	kip-ft
Muy	=	47.43	kip-ft

Φ Pn, Max Check

Grade of Steel (fy)	=	Min ( 80 , 60)	ksi
	=	60	ksi
Load Combination	=	[2] : 1.2 (LOAD 1: LOAD CASE 1) +1.6 (LOAD 2: LOAD CASE 2)
Critical Location	=	Bottom Joint
Pu	=	141.81	kip
Mux	=	-148.41	kip-ft
Muy	=	-13.94	kip-ft
Pt Calculated	=	2.11
φ Pn, Max	=	781.73	kip
		Pu < φ Pn, Max	Hence, OK

Minimum Ast Calculation

Grade of Steel (fy)	=	Min ( 80 , 60)	ksi
	=	60	ksi
Load Combination	=	[2] : 1.2 (LOAD 1: LOAD CASE 1) +1.6 (LOAD 2: LOAD CASE 2)
Pu-max from all Combinations	=	141.81	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	=	2.11
Reinforcement Provided	=	4-#8 + 10-#6

Load Angle	=	Tan^-1(Muy/Mux)
	=	7.07	deg
MRes	=	385.42	kip-ft
( φ ) MCap	=	386.42	kip-ft
Capacity Ratio	=	MRes/ MCap
	=	0.997 < 1

Shear Design (Analysis Forces)

Design for shear along D
luy	=	175	in
D	=	24	in
Check	luy > 5 x D
Critical Load Combination	=	[2] : 1.2 (LOAD 1: LOAD CASE 1) +1.6 (LOAD 2: LOAD CASE 2)
Nu	=	141.81	kip
Muy	=	382.48	kip-ft
Vuy	=	-38.54	kip
λ	=	1
φ	=	0.75
Deff	=	21.5	in
Ag	=	360	in²
Aeff	=	322.5	in²
Concrete Stress	=	59.16	ksi
ρw (50% of As provided)	=	0.012
Axial Stress	=	Min (Nu/6Ag , 0.05 f'c)
	=	0.1	ksi
φVcx Max	=	95.39	kip
φVcy	=	44.5	kip
Vuy	<	φVcy
Link For Shear Design Along D are not required

Design for shear along B
lux	=	350	in
B	=	15	in
Check	lux > 5 x B
Critical Load Combination	=	[11] : 0.9 (LOAD 1: LOAD CASE 1) -(LOAD 4: LOAD CASE 4 EQ-Y)
Nu	=	56.23	kip
Mux	=	48.96	kip-ft
Vux	=	5.78	kip
λ	=	1
φ	=	0.75
deff	=	12.5	in
Ag	=	360	in²
Aeff	=	300	in²
Concrete Stress	=	59.16	ksi
ρw (50% of As provided)	=	0.013
Axial Stress	=	Min (Nu/6Ag , 0.05 f'c)
	=	0.03	ksi
φVcy Max	=	88.74	kip
φVcx	=	32.48	kip
Vux	<	φVcx
Link For Shear Design Along B are not required

Check for Biaxial Shear

	Shear Along D	Shear Along B
Load Combination	[2] : 1.2 (LOAD 1: LOAD CASE 1) +1.6 (LOAD 2: LOAD CASE 2)
Vu (kip)	38.54	4.45
Φ Vc (kip)	44.5	41.39
Φ Vs required (kip)	0	0
Shear Reinforcement Provided
Diameter (in)	0.37	0.37
No of legs	4	5
Spacing (in)	12	12
Φ Vs (kip)	35.43	25.75
Φ Vn = (Φ Vc + Φ Vs) (kip)	79.93	67.14
Vu / Φ Vn	0.48	0.06
Requirement for Biaxial Check	Check is not Applicable

Design Of Links
Links in the zone where special confining links are not required
Normal Links
fyt	=	60	ksi
Along D
Av/s min-1	=	0.75 x Sqrt(f'c) x B x 12 / fyt
	=	0.13	in²/ft
Av/s min-2		50 x B x 12 / fyt
	=	0.15	in²/ft
Av/s min	=	Max ( Av/s min-1, Av/s min-2)
	=	0.15	in²/ft

Along B
Av/s min-1	=	0.75 x Sqrt(f'c) x D x 12 / fyt
	=	0.21	in²/ft
Av/s min-2		50 x B x 12 / fyt
	=	0.24	in²/ft
Av/s min	=	Max ( Av/s min-1, Av/s min-2)
	=	0.24	in²/ft
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	=	15	in

Provided spacing	=	12	in


Numbers of legs provided along D	=	4
Av/s provided along D	=	0.44	in²/ft
Numbers of legs provided along B	=	5
Av/s provided along B	=	0.55	in²/ft
		Hence, OK

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	---	---	3	---
Spacing	12	---	---	12	---