Global second order effects in reinforced concrete buildings: simplified criterion based on Galerkin’s method

Cunha, Rafael Nunes da; Vieira, Camila de Sousa; Gomes, Rainara Souza; Silva, Ludmila dos Santos da; Mendes, Ludmilson Abritta; Amorim, David Leonardo Nascimento de Figueiredo

doi:10.1590/S1983-41952024000600003

Rafael Nunes da Cunha

formal analysis

methodology

writing

Universidade Federal de Alagoas – UFAL, Programa de Pós-graduação em Engenharia Civil, Maceió, AL, Brasil

https://orcid.org/0000-0003-2503-6758

Camila de Sousa Vieira

formal analysis

methodology

writing

Universidade Federal de Sergipe – UFS, Laboratório de Modelagem Matemática em Engenharia Civil, São Cristóvão, SE, Brasil

https://orcid.org/0000-0002-5371-191X

Rainara Souza Gomes

formal analysis

writing

Universidade Federal de Sergipe – UFS, Laboratório de Modelagem Matemática em Engenharia Civil, São Cristóvão, SE, Brasil

https://orcid.org/0000-0003-3856-8999

Ludmila dos Santos da Silva

formal analysis

writing

Universidade Federal de Sergipe – UFS, Laboratório de Modelagem Matemática em Engenharia Civil, São Cristóvão, SE, Brasil

https://orcid.org/0009-0001-8011-9460

Ludmilson Abritta Mendes

data curation

formal analysis

methodology

writing

Universidade Federal de Sergipe – UFS, Laboratório de Modelagem Matemática em Engenharia Civil, São Cristóvão, SE, Brasil

https://orcid.org/0000-0002-8202-7623

David Leonardo Nascimento de Figueiredo Amorim

conceptualization

supervision

formal analysis

methodology

writing

Universidade Federal de Alagoas – UFAL, Programa de Pós-graduação em Engenharia Civil, Maceió, AL, Brasil

Universidade Federal de Sergipe – UFS, Laboratório de Modelagem Matemática em Engenharia Civil, São Cristóvão, SE, Brasil

https://orcid.org/0000-0002-9233-3114

Corresponding author: David Leonardo Nascimento de Figueiredo Amorim. E-mail: davidnf@academico.ufs.br

Conflict of interest: Nothing to declare.

Editors: Mauro Real, Guilherme Aris Parsekian.

Frame	Type	Height	Cross-Section			γ_z	$M_{2} / M_{1}$	$κ_{o}$
Frame	Type	Height	Beam	Column	Shear-Wall	γ_z	$M_{2} / M_{1}$	$κ_{o}$
1	2	48	0.20×0.60	0.20×0.50	4.00×0.20	1.1150	1.48	0.78
2	1	24	0.15×0.40	0.25×0.30	1.50×0.25	1.1871	1.67	0.73
3	1	30	0.15×0.40	0.25×0.30	1.50×0.25	1.3225	2.67	0.53
4	2	54	0.20×0.60	0.20×0.50	4.00×0.20	1.1515	1.71	0.69
5	2	51	0.20×0.60	0.20×0.50	4.00×0.20	1.1325	1.58	0.72
6	1	36	0.15×0.40	0.25×0.30	2.00×0.25	1.2934	2.49	0.55
7	1	42	0.15×0.40	0.25×0.30	2.00×0.25	1.4587	6.46	0.24
8	2	66	0.20×0.60	0.20×0.50	4.00×0.20	1.2360	1.96	0.65
9	2	75	0.20×0.60	0.20×0.50	4.00×0.20	1.3287	2.81	0.50
10	2	81	0.20×0.60	0.20×0.50	4.00×0.20	1.4063	4.20	0.36
11	2	78	0.20×0.60	0.20×0.50	4.00×0.20	1.3656	3.34	0.43
12	2	84	0.20×0.60	0.20×0.50	4.00×0.20	1.4513	5.75	0.27
13	1	39	0.15×0.40	0.25×0.30	2.00×0.25	1.3626	3.40	0.43
14	2	69	0.20×0.60	0.20×0.50	4.00×0.20	1.2642	2.16	0.59
15	2	78	0.20×0.60	0.20×0.50	4.00×0.25	1.3344	2.96	0.48
16	2	81	0.20×0.60	0.20×0.50	4.00×0.25	1.3716	3.58	0.41
17	2	84	0.20×0.60	0.20×0.50	4.00×0.25	1.4127	4.62	0.33
18	2	81	0.20×0.60	0.20×0.50	4.00×0.29	1.3493	3.26	0.44
19	2	84	0.20×0.60	0.20×0.50	4.00×0.30	1.3823	3.95	0.37
20	2	87	0.20×0.60	0.20×0.50	4.00×0.30	1.4242	5.26	0.29
21	2	78	0.20×0.60	0.20×0.50	4.00×0.23	1.3459	3.10	0.46

Frame	Type	Height	Cross-Section		γ_z	$M_{2} / M_{1}$	$κ_{o}$
Frame	Type	Height	Beam	Column	γ_z	$M_{2} / M_{1}$	$κ_{o}$
1	2	48	0.20×0.60	0.20×0.50	1.1690	1.537	0.77
2	1	24	0.15×0.40	0.25×0.30	1.3767	2.128	0.68
3	1	30	0.15×0.40	0.25×0.30	1.5432	3.185	0.52
4	1	36	0.15×0.40	0.25×0.45	1.4735	2.757	0.57
5	2	51	0.20×0.60	0.20×0.50	1.1850	1.615	0.74
6	1	39	0.15×0.40	0.25×0.45	1.5496	3.617	0.46
7	1	42	0.15×0.40	0.25×0.45	1.6440	4.458	0.40
8	2	57	0.20×0.60	0.20×0.50	1.2201	1.815	0.69
9	1	27	0.15×0.40	0.25×0.30	1.4526	2.547	0.61
10	1	33	0.15×0.40	0.25×0.30	1.6575	4.132	0.44
11	2	45	0.20×0.60	0.20×0.50	1.1562	1.445	0.81
12	2	60	0.20×0.60	0.20×0.50	1.2481	1.640	0.77
13	2	63	0.20×0.60	0.20×0.45	1.3169	1.922	0.7
14	2	69	0.20×0.60	0.20×0.45	1.3738	2.252	0.63
15	2	69	0.20×0.60	0.20×0.40	1.4704	2.935	0.53
16	1	33	0.15×0.40	0.20×0.35	1.5836	3.376	0.5
17	1	36	0.15×0.40	0.20×0.35	1.6849	4.661	0.4
18	1	36	0.15×0.40	0.25×0.35	1.6226	3.838	0.46
19	1	39	0.15×0.40	0.20×0.60	1.4640	2.712	0.57
20	1	39	0.15×0.40	0.20×1.00	1.2887	1.949	0.68
21	2	60	0.20×0.60	0.20×0.55	1.2196	1.545	0.8

	All frames			1.10 < γ_z ≤ 1.30
	γ_z	ζ_g	MASTAN2	γ_z	ζ_g	MASTAN2
Mean (m)	0.913	1.016	0.984	0.643	0.689	0.679
Standard deviation (m)	0.353	0.408	0.384	0.176	0.192	0.186
PBIAS (%)	-7.726	3.191	-	-5.704	1.463	-
MAE (m)	0.071	0.032	-	0.037	0.010	-
MAPE (%)	7.493	2.799	-	5.680	1.387	-

	All frames			1.10 < γ_z ≤ 1.30
	Normality	Homoscedasticity	Mean test	Normality	Homoscedasticity	Mean test
MASTAN2	0.7032	0.8134	-	0.8476	0.9792	-
γ_z	0.6920		0.5389	0.8261		0.7118
ζ_g	0.6426		0.7922	0.8119		0.9222

		All frames			1.10 < γ_z ≤ 1.30
		γ_z	ζ_g	MASTAN2	γ_z	ζ_g	MASTAN2
Positive bending moments	Mean (kNm)	346.957	391.210	420.686	371.514	402.443	422.000
	Standard deviation (kNm)	171.925	187.554	191.702	134.868	147.031	157.716
	PBIAS (%)	-21.250	-7.535	-	-13.589	-4.860	-
	MAE (kNm)	73.729	29.600	-	50.486	19.929	-
	MAPE (%)	18.161	7.454	-	11.464	4.294	-
Negative bending moments	Mean (kNm)	-558.924	-604.748	-641.229	-589.743	-622.229	-653.500
	Standard deviation (kNm)	176.326	190.243	198.959	142.792	153.055	162.840
	PBIAS (%)	-14.726	-6.032	-	-10.811	-5.026	-
	MAE (kNm)	82.305	36.481	-	63.757	31.271	-
	MAPE (%)	-12.765	-5.652	-	-9.445	-4.521	-
Shear forces	Mean (kN)	336.957	356.690	370.257	374.314	389.600	399.643
	Standard deviation (kN)	99.926	106.168	109.858	79.871	85.037	89.500
	PBIAS (%)	-9.883	-3.803	-	-6.767	-2.578	-
	MAE (kN)	33.300	13.567	-	25.329	10.043	-
	MAPE (%)	8.911	3.642	-	6.099	2.366	-

		All frames			1.10 < γ_z ≤ 1.30
		Normality	Homoscedasticity	Mean test	Normality	Homoscedasticity	Mean test
Positive bending moments	MASTAN2	0.4929	0.8805	-	0.8157	0.9341	-
	γ_z	0.2294		0.1970	0.8077		0.5319
	ζ_g	0.2686		0.6173	0.8600		0.8144
Negative bending moments	MASTAN2	0.7066	0.8657	-	0.3461	0.9531	-
	γ_z	0.5465		0.1637	0.7491		0.4511
	ζ_g	0.5631		0.5471	0.8146		0.7177
Shear forces	MASTAN2	0.3809	0.9146	-	0.4611	0.9645	-
	γ_z	0.2537		0.3103	0.3867		0.5867
	ζ_g	0.1434		0.6862	0.4641		0.8332

	All frames			1.10 < γ_z ≤ 1.30
	γ_z	ζ_g	MASTAN2	γ_z	ζ_g	MASTAN2
Mean (kNm)	648.105	715.810	651.852	714.771	766.500	718.900
Standard deviation (kNm)	276.276	302.021	277.736	377.730	412.326	386.295
PBIAS (%)	-0.578	8.935	-	-0.578	6.210	-
MAE (kNm)	9.576	63.957	-	17.014	47.600	-
MAPE (%)	1.782	9.830	-	2.979	6.785	-

	All frames			1.10 < γ_z ≤ 1.30
	Normality	Homoscedasticity	Mean test	Normality	Homoscedasticity	Mean test
MASTAN2	0.0084	0.9511	-	0.0059	0.9942	-
γ_z	0.0106		0.8999	0.0056		0.7983
ζ_g	0.0160		0.3924	0.0055		0.6093

		All frames			1.10 < γ_z ≤ 1.30
		γ_z	ζ_g	MASTAN2	γ_z	ζ_g	MASTAN2
Positive bending moments	Mean (kNm)	624.443	671.376	675.586	722.986	777.829	784.086
	Standard deviation (kNm)	325.409	357.521	360.394	264.453	296.758	300.973
	PBIAS (%)	-8.190	-0.627	-	-4.261	0.230	-
	MAE (kNm)	51.143	6.419	-	16.943	5.671	-
	MAPE (%)	6.735	1.122	-	4.107	1.440	-
Negative bending moments	Mean (kNm)	-510.548	-552.362	-552.129	-561.014	-605.871	-604.171
	Standard deviation (kNm)	153.271	173.281	171.044	130.558	154.080	151.865
	PBIAS (%)	-8.144	0.042	-	-5.211	-0.293	-
	MAE (kNm)	41.581	5.919	-	20.586	2.414	-
	MAPE (%)	-7.214	-0.999	-	-5.012	-0.657	-
Shear forces	Mean (kN)	336.167	355.490	355.310	369.357	390.471	389.629
	Standard deviation (kN)	86.734	96.241	95.223	61.981	73.026	71.925
	PBIAS (%)	-5.694	0.051	-	-3.375	-0.183	-
	MAE (kN)	19.143	2.771	-	9.471	1.071	-
	MAPE (%)	5.096	0.719	-	3.241	0.404	-

		All frames			1.10 < γ_z ≤ 1.30
		Normality	Homoscedasticity	Mean test	Normality	Homoscedasticity	Mean test
Positive bending moments	MASTAN2	0.0127	0.8679	-	0.2628	0.9914	-
	γ_z	0.0110		0.3924	0.3457		0.8770
	ζ_g	0.0124		0.8405	0.3257		0.9932
Negative bending moments	MASTAN2	0.1858	0.8410	-	0.3583	0.9924	-
	γ_z	0.1873		0.4117	0.3234		0.7469
	ζ_g	0.2498		0.9965	0.3315		0.9851
Shear forces	MASTAN2	0.0475	0.8889	-	0.1229	0.9946	-
	γ_z	0.0345		0.2683	0.1049		0.8080
	ζ_g	0.0655		0.9599	0.1147		0.9894

	All frames			1.10 < γ_z ≤ 1.30
	γ_z	ζ_g	MASTAN2	γ_z	ζ_g	MASTAN2
Mean (kNm)	486.852	523.933	526.638	533.157	576.557	581.957
Standard deviation (kNm)	220.585	241.850	247.061	205.253	230.338	235.291
PBIAS (%)	-8.172	-0.516	-	-4.973	-0.032	-
MAE (kNm)	40.024	6.438	-	14.914	2.814	-
MAPE (%)	6.751	1.460	-	4.674	0.880	-

	All frames			1.10 < γ_z ≤ 1.30
	Normality	Homoscedasticity	Mean test	Normality	Homoscedasticity	Mean test
MASTAN2	0.0085	0.8306	-	0.7764	0.9893	-
γ_z	0.0089		0.3143	0.8326		0.8185
ζ_g	0.0099		0.8209	0.8149		0.9988

	All frames			1.10 < γ_z ≤ 1.30
	γ_z	ζ_g	MASTAN2	γ_z	ζ_g	MASTAN2
Mean (kNm)	3.37E+04	3.75E+04	3.41E+04	3.83E+04	4.25E+04	3.89E+04
Standard deviation (kNm)	1.83E+04	2.07E+04	1.83E+04	1.54E+04	1.79E+04	1.53E+04
PBIAS (%)	-1.253	9.094	-	-2.850	4.439	-
MAE (kNm)	5.65E+02	3.41E+03	-	5.73E+02	9.60E+02	-
MAPE (%)	2.204	8.806	-	3.223	4.608	-

	All frames			1.10 < γ_z ≤ 1.30
	Normality	Homoscedasticity	Mean test	Normality	Homoscedasticity	Mean test
MASTAN2	0.2215	0.8118	-	0.3108	0.9879	-
γ_z	0.2072		0.9407	0.4249		0.9213
ζ_g	0.2032		0.5744	0.4019		0.8725

- E I \frac{d^{4} v (x)}{d x^{4}} + E S v (x) {(\frac{d^{2} v (x)}{d x^{2}})}^{2} + E S v (x) \frac{d v (x)}{d x} \frac{d^{3} v (x)}{d x^{3}} + 2 p \frac{d v (x)}{d x} + 2 E S \frac{d v (x)}{d x^{2}} {(\frac{d v (x)}{d x})}^{2} - p L (1 - \frac{x}{L^{2}}) \frac{d^{2} v (x)}{d x^{2}} + \frac{1}{2} E S \frac{d^{2} v (x)}{d x^{2}} {(\frac{d v (x)}{d x})}^{2} = - q

∴ \{\begin{matrix} K_{i j} = \int_{0}^{L} [E I \frac{\partial^{4} ϕ_{i} (x)}{\partial x^{4}} ϕ_{j} (x) + 2 p \frac{\partial ϕ_{i} (x)}{\partial x} ϕ_{j} (x) - p L (1 - \frac{x}{L}) \frac{\partial^{2} ϕ_{i} (x)}{\partial x^{2}} ϕ_{j} (x)] d x \\ F_{j} = \int_{0}^{L} q ϕ_{j} (x) d x \end{matrix}

M_{2} = \frac{- 108 E I q L^{2} [- 21 L^{9} p^{3} + 8215 E I L^{6} p^{2} - 638550 (E {I)}^{2} L^{3} p + 9147600 (E {I)}^{3}]}{259 L^{12} p^{4} - 140352 E I L^{9} p^{3} + 18993312 (E {I)}^{2} L^{6} p^{2} - 632681280 (E {I)}^{3} L^{3} p + 1975881600 (E {I)}^{4}}

N S E = 1 - \frac{\sum_{t = 1}^{n} {(κ_{p}^{t} - κ_{o}^{t})}^{2}}{\sum_{t = 1}^{n} {(κ_{o}^{t} - κ_{o}^{A v g})}^{2}}

[1] Corresponding author: David Leonardo Nascimento de Figueiredo Amorim. E-mail: davidnf@academico.ufs.br

Brasil

Brasil

Global second order effects in reinforced concrete buildings: simplified criterion based on Galerkin’s method

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Abstract