Closed-form solutions of dynamic vibration equations of seismically excited structures

Figures & data

Figure 1. Idealized SDOF mathematical model.

Figure 2. Idealised MDOF mathematical model.

Figure 3. Records of earthquake acceleration used for this study

Table 1. Ground motion records used to excite building models (Abd-Elhamed et al., 2018).

Earthquake	Date	Station	PGA(g)	$M$	$D_{ss}$ (km)	Soil class
El Centro	18.05.1940	CA – Array Sta 9; Imperial Valley Irrigation District	0.341	6.9	12.2	D, C
Kobe	17.01.1995	0KJMA	0.8210	6.9	0.6	B, B
Loma Prieta	17.10.1989	Los Gatos, CA, NCJ066	0.19	6.9	43.4	B, NA
Kocaeli	17.08.1999	SKR SAKARYA	0.41	7.4	3	D, B

Figure 4. Equilibrium of forces for SDOF model.

Figure 5. Idealized elastic-perfectly plastic behaviour.

Figure 6. (a) first, (b) second, (c) $n-1$ and (d) $n$ floors.

Table 2. Peak structural response quantities SDOF building model.

Earthquake	Newmark’s method			Exact solutions			% Error
	$u_{\mathit{Newmark}}$	${\dot{u}}_{\mathit{Newmark}}$	${\ddot{u}}_{\mathit{Newmark}}$	$u_{\mathit{Exact}}$	${\dot{u}}_{\mathit{Exact}}$	${\ddot{u}}_{\mathit{Exact}}$	$u$	$\dot{u}$	$\ddot{u}$
El Centro	0.0158	0.3317	8.9537	0.0158	0.3328	9.0624	0.00	0.33	1.20
Kobe	0.0415	0.6798	12.6686	0.0419	0.6977	12.945	0.95	2.57	2.14
LomaPrieta	0.0239	0.4578	9.1851	0.0238	0.4593	9.248	0.42	0.33	0.68
Kocaeli	0.0191	0.4057	11.301	0.0192	0.4088	11.308	0.52	0.76	0.06

$u$ = peak displacement [m], $\dot{u}$= peak velocity [m/s], $\ddot{u}$= peak acceleration [m/s²]

Figure 7. Induced displacement, velocity and acceleration time-histories of the SDOF system with respect to the (a) El Centro, (b) Kobe, (c) Loma Prieta and (d) Kocaeli records.

Table 3. Inertial and elastic properties of the 3-DOF structure.

$i$	$m_i$ [kg]	$k_i$ [$\frac{\mathrm{N}}{\mathrm{m}}$]	$c_i$ [$\frac{\mathrm{N}.\mathrm{s}}{\mathrm{m}}$]
1	$30\times {10}^3$	$3.46\times {10}^6$	26000
2	$25\times {10}^3$	$0.9\times 3.46\times {10}^6$	23400
3	$20\times {10}^3$	$0.8\times 3.46\times {10}^6$	20800

Table 4. Peak structural response quantities three-DOF building model.

Earthquake	Newmark iteration method			Exact solutions			% Error
	$u_{\mathit{Newmark}}$	${\dot{u}}_{\mathit{Newmark}}$	${\ddot{u}}_{\mathit{Newmark}}$	$u_{\mathit{Exact}}$	${\dot{u}}_{\mathit{Exact}}$	${\ddot{u}}_{\mathit{Exact}}$	$u$	$\dot{u}$	$\ddot{u}$
1st storey
El Centro	0.0658	0.477	4.2134	0.0664	0.4592	4.2924	0.90	3.88	1.84
Kobe	0.1681	1.362	11.9236	0.1679	1.3903	11.9025	0.12	2.04	0.18
Loma Prieta	0.0454	0.3598	4.9231	0.0462	0.3638	4.8946	1.73	1.10	0.58
Kocaeli	0.0691	0.3797	5.1292	0.0694	0.3888	5.1601	0.43	2.34	0.60
2nd storey
El Centro	0.1188	0.6824	5.0637	0.1199	0.7041	5.0608	0.92	3.08	0.06
Kobe	0.3225	2.1483	13.7969	0.3301	2.1152	13.9508	2.30	1.56	1.10
Loma Prieta	0.0702	0.4179	3.8065	0.0710	0.4277	3.8557	1.13	2.29	1.28
Kocaeli	0.1233	0.6655	4.9226	0.1260	0.6865	4.9592	2.14	3.06	0.74
3rd storey
El Centro	0.1552	0.8069	7.2661	0.1570	0.8102	7.4451	1.15	0.41	2.40
Kobe	0.4600	3.1066	22.7379	0.4651	3.1376	22.9682	1.10	0.99	1.00
Loma Prieta	0.1033	0.5752	5.8431	0.1025	0.5886	5.8705	0.78	2.28	0.47
Kocaeli	0.1502	0.8747	5.6486	0.1535	0.8936	5.5755	2.15	2.12	1.31

$u$ = peak displacement [m], $\dot{u}$= peak velocity [m/s], $\ddot{u}$= peak acceleration [m/s²]

Figure 8. Induced displacement, velocity and acceleration time-histories for first, second, and third storey of MDOF system with respect to the El Centro earthquake.

Figure 9. Induced displacement, velocity and acceleration time-histories for first, second, and third storey of MDOF system with respect to the Kobe earthquake.

Figure 10. Induced displacement, velocity and acceleration time-histories for first, second, and third storey of MDOF system with respect to the Loma Prieta earthquake.

Figure 11. Induced displacement, velocity and acceleration time-histories for first, second, and third storey of MDOF system with respect to the Kocaeli earthquake.

Abd-Elhamed, A., Shaban, Y., & Mahmoud, S. (2018). Predicting dynamic response of structures under earthquake loads using logical analysis of data. Buildings, 8(4), 61. doi:10.3390/buildings8040061

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