Second-order effects in horizontally loaded reinforced concrete columns

Pędziwiatr, Janusz; Musiał, Michał

doi:10.2478/sgem-2023-0022

Figures & Tables

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

Relative stiffness-relative moment diagram.

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

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
K_r	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
m_0ED,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
m_0ED,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
f_cm [MPa]	38	43	48	53	58	63	68	78	88	98
ɛ_c1 [‰]	2.2	2.25	2.3	2.4	2.45	2.5	2.6	2.7	2.8	2.8
E_cm [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
η_max	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
N_B/N_Ed	7.76	4.60	3.10	2.26	2.45	1.38	0.88	0.61	1.53	0.86	0.55	0.38
m_Ed/m_0Ed	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%_

ɛ_c	ɛ_d	ɛ_s1	ɛ_s2	n_c	n_s1	n_s2	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
m_Ed/m_0Ed	1.15	1.29	1.49	1.82	1.71	3.70	-	-	2.93	-	-	-
m_0ED,max	0.202	0.180	0.156	0.127	0.195	0.09	-	-	0.086

Second-order effects in horizontally loaded reinforced concrete columns

Figures & Tables

Figure 1:

Figure 2:

Figure 3:

Figure 4:

Figure 5:

Exemplary results of iterative determination of 2nd order moment for 6_0 m high column_

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

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

Basic parameters of concrete grades_

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

m*0ED,max values for three methods used_

Exemplary results of κd=f(m) relation determination for n=0_8, concrete C30/37, ρ1=ρ2=1%_

Exemplary results of iterative determination of 2nd order moment for 12_0 m high column_

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_

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