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Second-order effects in horizontally loaded reinforced concrete columns

Janusz Pędziwiatr
Janusz Pędziwiatr
 y 
Michał Musiał
Michał Musiał
   | 31 dic 2023
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Article Category: Original Study
Publicado en línea: 31 dic 2023
Páginas: 1 - 13
Recibido: 26 jun 2023
Aceptado: 21 sept 2023
DOI: https://doi.org/10.2478/sgem-2023-0022
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© 2023 Janusz Pędziwiatr et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1:

Basic Euler model.
Basic Euler model.

Figure 2:

σc=σc (ɛc) diagram for C30/37 concrete with equation approximating this relation.
σc=σc (ɛc) diagram for C30/37 concrete with equation approximating this relation.

Figure 3:

Relative stiffness-relative moment diagram.
Relative stiffness-relative moment diagram.

Figure 4:

Relative curvature κd=f(m) versus relative moment for n=0.8, concrete C30/37, ρ1=ρ2=1%.
Relative curvature κd=f(m) versus relative moment for n=0.8, concrete C30/37, ρ1=ρ2=1%.

Figure 5:

Beam division taking fixing conditions into account: a) corbel column, b) pin-supported column.
Beam division taking fixing conditions into account: a) corbel column, b) pin-supported column.

Exemplary results of iterative determination of 2nd order moment for 6.0 m high column.

Iteration Quantity [−] Cross section no. Moment increment [%]
0 1 2 3 4 5 6
0 m0 0.2500 0.2083 0.1667 0.1250 0.0833 0.0417 0.0000
κd 0.0022 0.0016 0.0011 0.0007 0.0004 0.0002 0.0000
w 0.0000 0.0022 0.0076 0.0151 0.0240 0.0337 0.0438
Δm 0.0219 0.0208 0.0181 0.0144 0.0099 0.0051 0.0000
1 m 0.2719 0.2291 0.1848 0.1394 0.0933 0.0467 0.0000 8.760
κd 0.0025 0.0019 0.0013 0.0008 0.0005 0.0002 0.0000
w 0.0000 0.0025 0.0088 0.0177 0.0282 0.0396 0.0515
Δm 0.0258 0.0245 0.0213 0.0169 0.0117 0.0060 0.0000
2 m 0.2758 0.2328 0.1880 0.1419 0.0950 0.0476 0.0000 1.417
κd 0.0026 0.0019 0.0013 0.0008 0.0005 0.0002 0.0000
w 0.0000 0.0026 0.0091 0.0182 0.0290 0.0407 0.0529
Δm 0.0265 0.0252 0.0219 0.0174 0.0120 0.0061 0.0000
4 m 0.2766 0.2336 0.1887 0.1425 0.0954 0.0478 0.0000 0.048
κd 0.0026 0.0019 0.0013 0.0008 0.0005 0.0002 0.0000
w 0.0000 0.0026 0.0091 0.0183 0.0291 0.0409 0.0532
Δm 0.0266 0.0253 0.0221 0.0175 0.0121 0.0061 0.0000
5 m 0.2766 0.2336 0.1887 0.1425 0.0954 0.0478 0.0000 0.009
κd 0.0026 0.0019 0.0013 0.0008 0.0005 0.0002 0.0000
w 0.0000 0.0026 0.0091 0.0183 0.0291 0.0409 0.0532
Δm 0.0266 0.0253 0.0221 0.0175 0.0121 0.0061 0.0000
6 m 0.2766 0.2336 0.1887 0.1425 0.0954 0.0478 0.0000 0.002
κd 0.0026 0.0019 0.0013 0.0008 0.0005 0.0002 0.0000
w 0.0000 0.0026 0.0091 0.0183 0.0291 0.0409 0.0532
Δm 0.0266 0.0253 0.0221 0.0175 0.0121 0.0061 0.0000
7 m 0.2766 0.2337 0.1887 0.1425 0.0954 0.0478 0.0000 0

Selected calculation results for different column heights l and relative forces n, obtained using nominal curvature method.

Quantity n=0.1 n=0.5 n=0.8
l [m] 6.0 8.0 10.0 12.0 6.0 8.0 10.0 12.0 6.0 8.0 10.0 12.0
l0 [m] 12 16 20 24 12 16 20 24 12 16 20 24
Kr 1.0 1.0 0.669
d/r0 =κ0 d 4.83·10−3
d/r=κd 4.83·10−3 3.23·10−3
Δm 0.028 0.049 0.077 0.111 0.139 0.247 0.386 0.556 0.149 0.265 0.413 0.595
m0ED,max 0.204 0.183 0.155 0.121 0.194 0.086 - - 0.104 - - -

Selected calculation results for different column heights l and relative forces n, obtained using general method.

Quantity n=0.1 n=0.5 n=0.8
l [m] 6.0 8.0 10.0 12.0 6.0 8.0 10.0 12.0 6.0 8.0 10.0 12.0
m0ED,max 0.22 0.215 0.205 0.195 0.32 0.28 0.23 0.17 0.33 0.27 0.19 0.13
m*0ED,max 0.22 0.21 0.205 0.195 0.29 0.26 0.22 0.17 0.24 0.18 0.16 0.12

Basic parameters of concrete grades.

Grade C30 C35 C40 C45 C50 C55 C60 C70 C80 C90
fcm [MPa] 38 43 48 53 58 63 68 78 88 98
ɛc1 [‰] 2.2 2.25 2.3 2.4 2.45 2.5 2.6 2.7 2.8 2.8
Ecm [GPa] 32 34 35 36 37 38 39 41 42 44
A 0.884 0.830 0.766 0.713 0.670 0.633 0.602 0.552 0.501 0.471
B −0.207 −0.198 −0.189 −0.174 −0.167 −0.160 −0.148 −0.137 −0.128 −0.128
C −0.025 −0.059 −0.104 −0.120 −0.146 −0.167 −0.167 −0.189 −0.213 −0.243
k 1.945 1.868 1.761 1.712 1.641 1.583 1.566 1.490 1.403 1.320
ηmax 1.591 1.556 1.522 1.458 1.429 1.280 1.154 1.037 1.000 1.000

Selected calculation results for different column heights l and relative forces n, obtained using nominal stiffness method.

Quantity n=0.1 n=0.5 n=0.8
l [m] 6.0 8.0 10.0 12.0 6.0 8.0 10.0 12.0 6.0 8.0 10.0 12.0
l0 [m] 12 16 20 24 12 16 20 24 12 16 20 24
λ 75 100 125 150 75 100 125 150 75 100 125 150
k2 0.044 0.059 0.073 0.088 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
NB/NEd 7.76 4.60 3.10 2.26 2.45 1.38 0.88 0.61 1.53 0.86 0.55 0.38
mEd/m0Ed 1.15 1.29 1.49 1.82 1.71 3.70 - - 2.93 - - -

m*0ED,max values for three methods used.

n=0.1 n=0.5 n=0.8
l [m] 6.0 8.0 10.0 12.0 6.0 8.0 10.0 12.0 6.0 8.0 10.0 12.0
(1) 0.204 0.183 0.155 0.121 0.194 0.086 - - 0.104 - - -
(2) 0.202 0.180 0.156 0.127 0.195 0.090 - - 0.086 - - -
(3) 0.220 0.210 0.205 0.195 0.290 0.260 0.220 0.170 0.240 0.180 0.160 0.120

Exemplary results of κd=f(m) relation determination for n=0.8, concrete C30/37, ρ1=ρ2=1%.

ɛc ɛd ɛs1 ɛs2 nc ns1 ns2 m m/κd κd Notes
[‰] [‰] [‰] [‰] [−] [−] [−] [−] [−] [−]
0.456 0.456 0.456 0.456 0.715 0.043 0.043 0.000 181.5 0.00000 whole cross section compressed (part 1 in figs 3 and 4)
0.490 0.422 0.428 0.484 0.715 0.040 0.045 0.011 181.5 0.00006
0.600 0.315 0.341 0.574 0.714 0.032 0.054 0.047 181.4 0.00026
0.700 0.222 0.265 0.657 0.714 0.025 0.061 0.079 181.1 0.00043
0.800 0.131 0.192 0.739 0.713 0.018 0.069 0.110 180.7 0.00061
0.900 0.043 0.121 0.822 0.712 0.011 0.077 0.140 180.2 0.00078

Exemplary results of iterative determination of 2nd order moment for 12.0 m high column.

Iteration Quantity [−] Cross section no. Moment increment [%]
0 1 2 3 . 9 10 11 12
0 m 0.18 0.165 0.15 0.135 . 0.045 0.03 0.015 0
κd 0.001 0.001 0.001 0.001 . 0.000 0.000 0.000 0.000
w 0.000 0.001 0.005 0.010 . 0.065 0.077 0.089 0.101
Δm 0.051 0.050 0.048 0.046 . 0.018 0.012 0.006 0.000
1 m 0.231 0.215 0.198 0.181 . 0.063 0.042 0.021 0.000 28.10
κd 0.002 0.002 0.001 0.001 . 0.000 0.000 0.000 0.000
w 0.000 0.002 0.007 0.015 . 0.103 0.122 0.140 0.159
Δm 0.080 0.079 0.076 0.072 . 0.028 0.019 0.009 0.000
2 m 0.260 0.244 0.226 0.207 . 0.073 0.049 0.024 0.000 12.59
κd 0.002 0.002 0.002 0.002 . 0.000 0.000 0.000 0.000
w 0.000 0.002 0.009 0.019 . 0.129 0.152 0.175 0.199
Δm 0.099 0.098 0.095 0.090 . 0.035 0.023 0.012 0.000
3 m 0.279 0.263 0.245 0.225 . 0.080 0.053 0.027 0.000 7.60
κd 0.003 0.002 0.002 0.002 . 0.000 0.000 0.000 0.000
w 0.000 0.003 0.010 0.022 . 0.148 0.175 0.201 0.228
Δm 0.114 0.113 0.109 0.103 . 0.040 0.027 0.013 0.000
. . . . . . . . . . .
7 m 0.323 0.306 0.287 0.265 . 0.095 0.063 0.032 0.000 2.56
κd 0.003 0.003 0.003 0.002 . 0.000 0.000 0.000 0.000
w 0.000 0.003 0.013 0.028 . 0.196 0.230 0.265 0.300
Δm 0.150 0.148 0.144 0.136 . 0.052 0.035 0.018 0.000
. . . . . . . . . .
11 m 0.347 0.330 0.310 0.287 . 0.103 0.069 0.035 0.000 1.52
κd 0.004 0.004 0.003 0.003 . 0.001 0.000 0.000 0.000
w 0.000 0.004 0.015 0.032 . 0.224 0.264 0.304 0.344
Δm 0.172 0.170 0.165 0.156 . 0.060 0.040 0.020 0.000
. . . . . . . . .. . .
15 m 0.365 0.348 0.327 0.303 . 0.110 0.073 0.037 0.000 1.17
κd 0.004 0.004 0.004 0.003 . 0.001 0.000 0.000 0.000
w 0.000 0.004 0.017 0.036 . 0.248 0.291 0.336 0.380
Δm 0.190 0.188 0.182 0.172 . 0.066 0.044 0.022 0.000
16 m 0.370 0.353 0.332 0.307 . 0.111 0.074 0.037 0.000 1.32
κd 0.005 0.004 0.004 0.003 . 0.001 0.000 0.000 0.000
w 0.000 0.005 0.018 0.038 . 0.258 0.303 0.349 0.395
Δm 0.197 0.195 0.188 0.178 . 0.068 0.046 0.023 0.000
17 m 0.377 0.360 0.338 0.313 . 0.113 0.076 0.038 0.000 1.96

Results of calculations of maximum 1st order moment m0ED,max for different column heights l and relative forces n done acc. to standard stiffness method.

Quantity n=0.1 n=0.5 n=0.8
l [m] 6.0 8.0 10.0 12.0 6.0 8.0 10.0 12.0 6.0 8.0 10.0 12.0
mEd/m0Ed 1.15 1.29 1.49 1.82 1.71 3.70 - - 2.93 - - -
m0ED,max 0.202 0.180 0.156 0.127 0.195 0.09 - - 0.086
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