Scale effects on structural dynamic properties and responses: illustrated on arch dams

2.1. Theoretical basis

Operating deflection shape (ODS), a specific type of structural vibration pattern, is defined as the deflection of a structure at a particular frequency [21]. ODS is the actual vibration displacement pattern of a structure that vibrates in the steady-state condition in response to a specific structural loading [22, 23]. When the loading is a harmonic excitation at the natural frequency of the structure, the particular ODS is commonly referred to as a mode shape from the perspective of experimental modal analysis. A scanning laser Doppler vibrometer (LDV) can be used to measure the vibration response for generating an ODS. The use of the LDV as a vibration transducer offers various advantages because of its remarkable capabilities of automation, high sensitivity, high spatial resolution, noncontact, etc., compared with conventional tools [24, 25]. These features can improve the quality and efficiency of ODS testing.

The motion equation for a linear structure with sinusoidal excitation forces acting on the structure is as follows:

where $j=\sqrt{-\text{1}}$, $t$ is time; $M$, $C$, and $K$ are the mass, damping, and stiffness matrices, respectively; $x$ is the displacement vector, and $\mathrm{R}\mathrm{e}$ is the real part of the forces. The sinusoidal forces are applied with possibly complex magnitude vectors $f^m$ at the excitation frequencies ${\omega }^m$, where the superscript $m$ denotes the $m$th forcing term. To solve Eq. (1) for the steady-state response, the displacement response is assumed to be:

Thus, the velocity response is:

The scanning function of a LDV can measure structural vibration patterns effectively. The model in the study was initially excited with a periodic chirp of various frequency bandwidths to determine the resonant frequencies. The resonant frequencies were obtained based on the peaks in the frequency response function. When the model was excited at a resonant frequency, the maximum deflection was acquired by the LDV, leading to a mode shape of the structure.

2.2. Experimental procedure and results

Xiaowan Arch Dam in China, the world's second highest arch dam at 292 m (958 ft), is used as the prototype to investigate the scale effects on structural dynamic properties. One 1/1500-scale units model and another 1/3000-scale units model are designed to model the full-scale Xiaowan Arch Dam as accurately as possible. The large model measures 600 mm×300 mm×250 mm and the small model measures 300 mm×150 mm×125 mm in the length, width, and height directions, as shown in Fig. 1. Each model consists of the dam body and the foundation. The dam body is made of organic glass and the foundation material is concrete. The material parameters of the organic glass are: Young’s modulus 6.4 GPa; density 1190 kg/m³; Poisson’s ratio 0.21. The material parameters of the concrete are: Young’s modulus 14.0 GPa; density 1930 kg/m³; Poisson’s ratio 0.17. Two contraction joints of the model are simulated by two 1 mm wide cracks. The base excitation at a natural frequency is exerted using an electromagnetic vibration system controlled by a COMET USB shaker control system. While the model is vibrating, data acquisition and analysis are performed using a LDV (PSV-400) to measure the mode shape of the model. These sequential operations establish the basic procedure of the experimental modal analysis.

Fig. 11/1500-scale (large) and 1/3000-scale (small) models of full-scale Xiaowan Arch Dam

Experimental modal analysis is conducted to compare the vibration patterns between the two scale models. An excitation at the natural frequency of the model is introduced at an acceleration level of 5 m/s². The first-order and second-order resonant frequencies of the two models are obtained using a LDV, as listed in Table 1. The corresponding mode shapes are plotted in Figs. 2 to 5. From the table and figures, it can be seen that the mode shapes of the two models are similar, but the first-order natural frequency of the small model is 114.5 % higher than that of the large model, and the second-order natural frequency of the small model is 65.1 % higher than that of the large model. The differences between the models for the first and the second natural frequencies are occasioned by the scale effect of the models.

Fig. 2First-order mode shapes of large model in experimental analysis

Fig. 3Second-order mode shapes of large model in experimental analysis

Fig. 4First-order mode shapes of small model in experimental analysis

Fig. 5Second-order mode shapes of small model in experimental analysis

2.3. Numerical simulation results and discussion

The dynamic properties of the scale arch dam models are numerically analyzed by the finite element (FE) method using ANSYS. Elastic FE models of two scale arch dams are built using the same geometrical and material parameters of the dam body and the foundation as those used in the experiment in Section 2.2. The results of the numerical simulation with the FE mesh illustrated in Fig. 6 are provided in Table 2 and Figs. 7 to 10. From the table and figures, it can be seen that that the mode shapes of the two models are similar, but the first-order natural frequency of the small model is 100.8 % higher than that of the large model, and the second-order natural frequency of the small model is 78.9 % higher than that of the large model.

Fig. 6FE Mesh of arch dam model

The difference between the numerical simulation and the experimental analysis results for the first-order natural frequency of the large model is 2.2 %; the difference for the second-order natural frequency is 15.2 %. The difference between the numerical simulation and the experimental analysis results for the first-order natural frequency of the small model is 5.6 %; the difference for the second-order natural frequency is 7.4 %. It can be seen that the results from the experiments and the numerical simulation are comparable under the considerations of the approximation of numerical models to the real structures.

Fig. 7First-order mode shape of large numerical arch dam model

Fig. 8Second-order mode shape of large numerical arch dam model

Fig. 9First-order mode shape of small numerical arch dam model

Fig. 10Second-order mode shape of small numerical arch dam model

3.1. Theoretical basis

The motion equations of a structure with $N$ degrees of freedom and viscous damping coefficients can be expressed as:

where $\left[M\right]$, $\left[D\right]$, and $\left[K\right]$ represent the $n$×$n$ mass, damping, and stiffness matrices, respectively; $\ddot{x}$, $\dot{x}$, and $x$ are the acceleration, velocity, and displacement vectors, respectively; and $f\left(t\right)$ is the time-dependent applied force vector. A scale structure usually causes a change in the mass $\mathrm{\Delta }\left[M\right]$, damping $\mathrm{\Delta }\left[D\right]$, and stiffness $\mathrm{\Delta }\left[K\right]$ of the original structure:

where the subscript $s$ labels the scale structure. The frequency response function matrix of the scale structure $\left[H\right(\omega ){]}_s$ is obtained as:

Thus, the vibration response of the scale structure can be expressed as:

3.2. Experimental procedure and results

The same models as used in the dynamic property study are considered here. The characteristics of the random vibration responses of the two scale models are experimentally evaluated, with the arrangement of measuring points diagrammatically presented in Fig. 11.

Fig. 11Diagrammatic sketch of measuring points

The excitation equipment used is again an electromagnetic vibration system controlled by a COMET USB shaker control system; data acquisition and analysis are again performed using a LDV (PSV-400). The test is conducted by exciting band-limited white noise vibration of the shaking table along the horizontal direction. The excitation is introduced as random vibration with bandwidth ranging from 20 Hz to 2000 Hz and a constant acceleration power spectral density level. To determine the random response characteristics of the two scale models, five levels of power spectral density of acceleration, namely 0.1 (Case 1), 0.2 (Case 2), 0.3 (Case 3), 0.4 (Case 4), and 0.5 (m/s²)²/Hz (Case 5), are used.

The typical amplitude/time history of velocity is plotted in Fig. 12. The maximal random vibration responses at the different measuring points of the large model at five excitation levels are shown in Table 3 and Fig. 13. The maximum responses of the small model are shown in Table 4 and Fig. 14.

Fig. 12Typical amplitude/time history of velocity

Fig. 13Maximum vibration responses of large model at five excitation levels in experimental analysis

Table 3 and Figs. 13 demonstrate the similarity of the maximum velocity responses in the vertical and horizontal directions of the large model of the arch dam for Cases 1 to 5. Clearly, the maximum velocity responses in the vertical direction increase from the bottom to the top of the dam body in all cases. The maximum vibration responses occurring at the center top of the dam have similar patterns for all cases. The maximum velocity responses in the horizontal direction decrease from the center to the border of the dam body in all cases.

Table 4 and Fig. 14 demonstrate the similarity of the maximum velocity responses in the vertical and horizontal directions of the small model of arch dam for Cases 1 to 5. The maximum velocity responses in the vertical direction increase from the bottom to the top of the dam body in all cases. The maximum velocity responses in the horizontal direction decrease from the center to the border of the dam body in all cases. The maximum vibration response occurs at the center top of the dam in all cases. The differences in the maximum random vibration responses between the large and small models at five excitation levels are shown in Table 5 and Fig. 15.

Fig. 14Maximum vibration responses of small experimental model at five excitation levels in experimental analysis

Fig. 15Differences in maximum responses between large and small models in experimental analysis

Table 5 and Fig. 15 demonstrate that the differences in the maximum velocity responses of the structure between the large and small models are not obvious for all cases: the maximum difference is 9.3 mm/s, occurring at measuring point 6.

3.3. Numerical simulation results

The same FE arch dam models as used in the dynamic properties analysis are employed for the numerical simulation, yielding the dynamic responses of the models. The typical velocity pattern of the structural dynamic response is shown in Fig. 16. The maximum random vibration responses at the different measuring points of the large model at different excitation levels in the numerical simulation are shown in Fig. 17 and in Table 6. The maximum responses of the small model are exhibited in Table 7 and Fig. 18.

Fig. 16Numerically obtained typical velocity pattern of scale arch dam models

Table 6 and Fig. 17 demonstrate the similarity of the maximum velocity responses of the large model in the vertical and horizontal directions in numerical simulation for Cases 1 to 5. Notably, the maximum velocity responses of the large model in numerical simulation have the same orderliness as those of the large model in the experiment.

Fig. 17Maximum vibration responses of large numerical model at different excitation levels

Fig. 18Maximum vibration responses of small numerical model at five excitation levels

Table 7 and Fig. 18 demonstrate the similarity of the maximum velocity responses of the small model in the vertical and horizontal directions in numerical simulation for Cases 1 to 5. Observably, the maximum velocity responses of the small model in numerical simulation have the same orderliness as those of the small model in the experiment. The differences in the maximum random vibration responses between the large and small models at five excitation levels in the numerical simulation are exhibited in Table 8 and Fig. 19.

Fig. 19Differences in maximum responses between large and small numerical models

Fig. 20Maximum velocity comparative charts for Case 1 in numerical simulation

a) Large model

b) Small model

Table 8 and Fig. 19 demonstrate that the differences in the structural maximum velocity responses between the large and small models in numerical simulation are not obvious for all cases. The maximum differences occur at measuring point 6 for all cases; the value is –13.2 mm/s. Maximum velocity comparative charts between experiment and numerical simulation are plotted in Figs. 20 to 24 for Cases 1 to 5, respectively. These charts provide comparisons between (a) the large model and (b) the small model.

Fig. 21Maximum velocity comparative charts for Case 2 in numerical simulation

a) Large model

b) Small model

Fig. 22Maximum velocity comparative charts for Case 3 in numerical simulation

a) Large model

b) Small model

Fig. 23Maximum velocity comparative charts for Case 4 in numerical simulation

a) Large model

b) Small model

Figs. 20 to 24 demonstrate that the maximum velocity responses of the structures are always greater in experiment than in numerical simulation. The differences in the structural maximal velocity responses at the measuring point near the border are greater than those at other locations. The maximum differences in the structural maximum velocity responses of the large model between experiment and numerical simulation are 10.0, 14.4, 16.2, 18.2, and 20.6 mm/s, respectively. The maximum differences in the structural maximal velocity responses of the small model between experiment and numerical simulation are 10.1, 16.5, 17.5, 21.6, and 25.4 mm/s, respectively. These differences between the experimental and numerical results are occasioned by the simplification of the numerical simulation model. From comparisons of the results of experiment and numerical simulation, the numerical analysis is proven an effective method for evaluating dynamic responses under a scale effect study.

Fig. 24Maximum velocity comparative charts for Case 5 in numerical simulation

a) Large model

b) Small model