Design of non slender columns

Example 01
Brace Non Slender Column Design

• Edge column
• 300mm square column
• Axial Load 1500kN
• Moment at top -40kNm
• Moment at Bottom 45kNm
• fck 30N/mm2
• fyk 500N/mm2
• Nominal Cover 25mm
• Floor to Floor height 4250mm
• Depth of the beam supported by the column 450mm

Mtop                     = -40kNm
Mbottom                = 45kNm
NEd                      = 1500kN

Clear height        = 4250-450
                           = 3800mm
Effective length    = lo
                            = factor * l
Factor                  = 0.85 (concise Eurocode 2, Table 5.1. This may more conservative).
lo                         = 0.85* 3800
                            = 3230mm

Slenderness λ   = lo/i

i                       = radios of gyration

                        = h/√12

λ                      = lo/( h/√12 )

                        = 3.46*lo/h

                        = 3.46*3230/300

                        = 37.3

Limiting Slenderness λlim

λlim                     = 20ABC/√n

A                      = 0.7 if effective creep factor is unknown

B                      = 1.1 if mechanical reinforcement ratio is unknown          

C                      = 1.7 - rm              

                        = 1.7-Mo1/Mo2

Mo1                  = -40kNm

Mo2                  = 45kNm  where lMo2l ≥ lMo1l

C                      = 1.7 - (-40/45)

                         = 2.9

n                       = NEd / (Ac*fcd)

fcd                    = fck / 1.5

                         = (30/1.5)*0.85
                         = 17
n                       = 1500*1000 / (300*300*17)

                         = 0.98

 λlim                     = 20*0.7*1.1*2.9/√0.98

                         = 45.1

 λlim > λ hence, column is not slender.

Calculation of design moments

MEd                 = Max{Mo2, MoEd +M2, Mo1 + 0.5M2}

Mo2                 = Max {Mtop, Mbottom} + ei*NEd

                       = 45 + (3.23/400)*1500 ≥  Max(300/30, 20)*1500

                       = 57.1kNm  >  30kNm

Mo2                = Min{Mtop, Mbottom} + ei*NEd

                       = -40 + (3.23/400)*1500 ≥  Max(300/30, 20)*1500

                       = 27.9kNm

MoEd                   = 0.6*Mo2+ 0.4*Mo1 ≥ 0.4*Mo2

                       = 0.6*57.1 + 0.4*(-27.9) ≥ 0.4*57.1

                       = 23.1 ≥ 22.84

M2                  = 0 , Column is not slender

MEd                 = Max{Mo2, MoEd +M2, Mo1 + 0.5M2}

                       = Max{57.1, 23.1 +0, -27.9 + 0.5*0} 

                       = 57.1kNm

MEd / [b*(h^2)*fck]  = (57.1*10^6) / [300*(300^2)*30]                       

                            = 0.07

NEd / (b*h*fck)        = (1500*10^6) / (300*300*30

                             = 0.56

Assume 25mm diameter bars as main reinforcement and 10mm bars as shear links

d2                  = 25+10+25/2

                     = 47.5mm

d2/h                = 47.5 / 300

                      = 0.16

Note: d2/h = 0.20 chart is reffed to find the reinforcement area, but it is more conservative. Interpolation can be used to find the exact value.

As*fyk / b*h*fck      = 0.24

                     As    = 0.24*300*300*30 / 500

                             = 1296mm2

Provides four 25mm bars (As Provided 1964mm2)

 

Check for Biaxial Bending
Further check is not required if 
0.5 ≤ ( λy/ λz) ≤  2.0 For rectangular column
and
0.2 ≥ (ey/heq)/(ez/beq) ≥ 5.0
Here λy and λz are slenderness ratios

λy is nearly equal to λz
therefore  λy/λz is nearly equal to one.
Hence, λy/λz < 2 and > 0.5 OK

ey/heq  =  (MEdz / NEd) / heq
ez/beq  =  (MEdy / NEd) / beq
    

(ey/heq)/(ez/beq) = MEdz / MEdy  Here h=b=heq=beq, column is square

MEdz  = 45kNm
MEdy  = 30kNm  Minimum moment, see calculation of Mo2 for method of calculation

Note : Moments due to imperfections need to be included only in the direction where they have the most unfavorable effect - Concise Eurocode 2

(ey/heq)/(ez/beq) = 45/30
                                = 1.5 > 0.2 and < 5
Therefore Biaxial check is required.

(MEdz / MRdz)^a + (MEdy / MRdy)^a ≤  1

MEdz            = 45kNm
MEdy           = 30kNm

MRdz and MRdy are the moment resistance in respective direction, corresponding to an axial load NEd.

For symetric reinforcement section

MRdz            = MRdy

As Provided  = 1964mm2

As*fyk / b*h*fck      = 1964*500/(300*300*30)
                             = 0.36

NEd / (b*h*fck)        = 0.56 
From the chart d2/h =0.2 

MEd / [b*(h^2)*fck]   = 0.098

MEd                       = 0.098*300*300*300*30
                             = 79.38kNm

a                               = an exponent
a                               = 1.0 for NEd/NRd = 0.1
a                               = 1.5 for NEd/NRd = 0.7
NEd                                    = 1500kN
NRd                                    = Ac*fcd + As*fyd
NRd                                    = 300*300*(0.85*30/1.5) + 1964*(500/1.15)
                                 = 2383.9kN

NEd/NRd                            = 1500/2383.9
                                 = 0.63
By interpolating
a                               = 1.44

(MEdz / MRdz)^a + (MEdy / MRdy)^a =  (45 / 79.39)^1.44 + (30 / 79.38)^1.44

                                                  = 0.69 <1
Hence, Check for biaxial bending is ok
Therefore, Provide four 25mm diameter bars.