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The MasterSeries Concrete Column design calculates the Axial and Moment Capacity for the provided reinforcement from first principles. We carry out a rigorous analysis to determine the Axial load capacity for the applied moment (Nominal or real).
Note: In the following discussions and examples Fy is 460 and gm 0.95. This is all still valid when Fy is 500 and gm 0.87.
Nuz is the design ultimate capacity of a section when subjected to axial load only. This is used in calculation of deflection moments
Ncap is the design axial load capacity of a section as calculated below
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For a column under Axial only
X = D
Fs1 = Fy x .95 and
Fs2 = Fy x .95
Thus
Ncap = 0.9 x 0.45Fcu(Ac) + 0.95 x Fy x As
Short Columns
BS 8110 gives 2 formulas for the design of Short Braced Columns in 3.8.4.3. and 3.8.4.4.
N = 0.4fcuAc + 0.8Ascfy
N = 0.35fcuAc + 0.7Ascfy
These are simplified conservative equations which ignore any nominal moments.
General Design
In the MasterSeries we ignore the short column axial capacity equations. Instead we always apply a nominal eccentricity to the column and carry out a rigorous analysis to determine the Axial load capacity for the applied moment (Nominal or real).
This is the real design check. You will find that the column will always fail before you reach Nuz due to the moment. See example below.
Example
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Basic Data
Design to BS 8110: 1997
Grades Fy, Fyv, Fcu 460 N/mm˛, 460 N/mm˛, 30 N/mm˛
Restraint Conditions and Effective Lengths
x : for 400 mm deep column Top: Deep, Bottom: Deep 1.20
y : for 400 mm wide column Top: Deep, Bottom: Deep 1.20
Lc=3500 Lox=3500-(950+950)/2 2550
Loy=3500-(950+950)/2 2550
Lex=3060.0 mm, Lex/h=7.7, exx=20.0 mm
Ley=3060.0 mm, Ley/b=7.7, eyy=20.0 mm
Uni-Axial , Short, UnBraced, Column Design
Critical Case : 1 : Dead plus Live
Axial Capacity
Applied Axial N 2527.0 kN
Nuz=.45•(B•H-acs)•fcu + .95•asc•fy .45x(400x400-1608)x30 + 0.95x1608.0x460 2841.0 kN OK
Ncap=.9•.45•(B•H-acs)•fcu + .95•asc•fy.9x.45x(400x400-1608)x30 + 0.95x1608.0x460 2627.2 kN OK
X-X Moments
M nom 50.5
Design Moment 50.5 kNm
Moment Capacity Mu X-X (For N = 2527 kN)
Design Data X/h, h, b, X, Ac, Ybar 1.052, 400mm, 400mm, 420.7mm, 151462 mm˛, 189.3 mm
Bar group1:M1 fn(bars,d,%,,la,F)3 x 16, 50.0, 0.308, 437, 150.0, 263.6 39.5 kNm
Bar group2:M2 fn(bars,d,%,,la,F)3 x 16, 350.0, 0.059, 118, -150.0, 71.0 -10.6 kNm
Bar group3:M3 fn(bars,d,%,,la,F)2 x 16, 200.0, 0.184, 367, .0, 147.7 0.0 kNm
Concrete Mc=(Ac•.45•fcu)•(H/2-Ybar) (151462 x 0.45 x 30 ) x (400 / 2 - 189.3) 21.8 kNm
Mu =Mc + (M1+M2+M3) 21.8 + (39.5+-10.6+0.0) 50.7kNm
Max Moment/Mu 50.5 / 50.7 0.997 OK
By Hand, Check validity of computer Calcs.
Assuming X/h = 1.052 and X = 420.8 mm (outside section)
Prove Axial Capacity = N = 2527 kN and
Moment Capacity = 50.7 kNm as computed by computer
Bar Centres at 30 cover – 12 link – 16/2 bar = 50 mm in from face
Bar group1: (Compression Face)
3 x 16 = 603.2 mm2
Cover 50 thus 420.8-50 = 370.8 from Neutral Axis
Strain = x0.0035 x 370.8/420.8 = 0.00308 strain
Stress , x x .00308 = 616 N/mm2
Limit 460 x .95 = 437 N/mm2 max
Force F x A = 437 x 603.2 = 263.6 kN QED
Moment about Centre of Column
La = 400/2-50 = 150 263.6 kN x 0.15 39.5 kNm QED
Bar group2: (Tension Face)
3 x 16 = 603.2 mm2
Cover 50 thus 420.8 - 400 + 50 = 70.8 from Neutral Axis
Strain = x0.0035 x 70.8/420.8 = 0.00059 strain
Stress , x x .00059 = 118 N/mm2
Force F x A = 118 x 603.2 = 71.0 kN QED
Moment about Centre of Column
La = 400/2-50 = 150 71.0 kN x -0.15 (on tension side >> La -ve) -10.6 kNm QED
Bar group3: (Mid Bars)
2 x 16 = 402.1 mm2
Cover 50 thus 420.8 - 200 = 220.8 from Neutral Axis
Strain = x0.0035 x 220.8/420.8 = 0.00184 strain
Stress , x x .00184 = 367 N/mm2
Force F x A = 367 x 402.1 = 147.7 kN QED
Moment about Centre of Column
La = 400/2-200= 0 147.7 kN x -0.00 0.0 kNm QED
Concrete
Dc = 0.9 X = 0.9 x 420.7 = 378.6 mm (if Dc > h then Dc = h)
Force 0.45 x Dc x B x Fcu = .45 x 378.6 x 400 x 30 2044.4 kN
Moment about Centre of Column
La = 400/2-378.6/2= 10.7 mm
M = 2044.4 x 0.00107 21.8 kNm QED
Axial Capacity = 263.6 +71.0 + 147.7 + 2044.4 2526.7 kN QED
= N at 0.01% error
Moment Capacity = 39.5 – 10.6 + 0 + 21.8 50.7 kNm QED
= Mu
Reference: Kong & Evans: Reinforced and Prestressed Concrete, 3rd Edition. Chapter 7
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