Thin sheet metal and/or membrane are often used for roof cladding of spatial structures because of their strength and lightness (Noguchi et al., 2003). Being light and flexible, such roofing materials are vulnerable to dynamic wind actions. Since wind pressures acting on spatial structures vary spatially as well as in time, the design wind loads should be determined based on the dynamic characteristics of wind pressures. Fatigue of cladding elements, such as roofing material and its fixings, may play an important role in the wind resistant performance of cladding systems, although it is seldom considered in the design.
Roof cladding is usually designed based on the worst peak pressure coefficients irrespective of wind direction. The conventional codification provides a single peak design pressure coefficient for each roof zone considering a nominal worst-case scenario. Neither the probability distribution of the peak pressure coefficients nor the peaks other than the largest one are considered. Hence, they are not suitable for fatigue and risk-consistent designs. Building design has recently shifted to a performance-oriented one. Therefore, it is hoped to develop a new methodology that provides the peak pressure coefficients according to predetermined risk levels and the loading sequence for estimating the fatigue damage to roof cladding and its fixings. Computer simulation of wind pressure time series may be useful for this purpose.
Kumar and Stathopoulos (1998, 1999, 2001) proposed a novel simulating methodology that generates both Gaussian and non-Gaussian wind pressure fluctuations on low building roofs. Despite its simple procedure, the technique is successfully applied to fatigue analysis as well as to the evaluation of extreme pressures in a risk-consistent way. Therefore, this technology is used in this chapter and a simplification of this method is discussed. Gaussian and non-Gaussian pressure fluctuations can be simulated from the statistics of wind pressures, i.e. the mean, standard deviation, skewness, kurtosis and power spectrum. These statistical values change with location as well as with many factors related to the structure’s geometry and the turbulence characteristics of approach flow. For such a complicated phenomenon, in which a number of variables involve, artificial neural networks (simply neural networks or ANN’s) can be used effectively. Artificial neural networks can capture a complex, non-linear relationship via training with informative input-output example data pairs obtained from computations and/or experiments. Among a variety of artificial neural networks developed so far, Cascade Correlation Learning Network (Fahlman and Lebiere, 1990) is applied to the present problem. It is a popular supervised learning architecture that dynamically grows layers of hidden neurons of a fixed non-linear activation (e.g. sigmoid), so that the topology (size and depth) can also be efficiently determined.
This chapter proposes a computer-assisted wind load evaluation system for the design of roof cladding of spatial structures. Focus is on spherical domes and vaulted roofs, as typical shapes of spatial structures. The composition of the system is schematically illustrated in Fig. 1. This system provides wind loads for the design of cladding and its fixings without carrying out any additional wind tunnel experiments. An aerodynamic database, artificial neural network and time-series simulation technique are employed in the system. Finally, applications of the system to risk-consistent design as well as to fatigue design are presented.
The wind load evaluation system proposed here is based on our previous studies (Uematsu et al., 2005, 2007, 2008). It can be applied not only to spherical domes and vaulted roofs but also to any other structures. However, such a system may be more useful for designing the cladding of spatial structures because of its sensitivity to dynamic load effects of fluctuating wind pressures. The spatial variation of statistical properties and the non-normality of pressure fluctuations on spherical domes and vaulted roofs are less significant than those on flat and gable roofs. Therefore, an ANN and a time-series simulation technique can be used more efficiently for these structures. This is the reason why we focus on the cladding of spherical domes and vaulted roofs in this chapter.
2. Aerodynamic dadabase
2.1. Wind tunnel experiments
Two series of wind tunnel experiments were carried out; one is for spherical domes and the other is for vaulted roofs. The experimental conditions are somewhat different from each other. The outline of the experimental conditions is presented here.
2.1.1. Spherical dome
The experiments were carried out in a closed-circuit-type wind tunnel with a working section 18.1 m long, 2.5 m wide and 2.0 m high. Two kinds of turbulent boundary layers simulating natural winds over typical open-country and urban terrains were generated; these flows are respectively referred to as Flows ‘II’ and ‘IV’ in this chapter. The geometric scale of these flows ranges from 1/400 to 1/500, judging from the longitudinal integral scale of turbulence.
The geometry of the wind tunnel model is schematically illustrated in Fig. 2(a). The rise/span ratio (
2.1.2. Vaulted roof
The experiments were carried out in a closed-circuit-type wind tunnel with a working section 18.9 m long, 2.6 m wide and 2.1 to 2.4 m high. Two kinds of turbulent boundary layers similar to those used for spherical domes were generated; these flows are respectively referred to as Flows ‘II’’ and ‘IV’’ in this chapter.
The geometry of the wind tunnel model is schematically illustrated in Fig. 3(a). The rise/span ratio (
The experimental procedure is the same as that for spherical domes except that the wind direction is varied from 0 to 90o at a step of 5o.
2.2. Database of the statistics of wind pressures
The data from the simultaneous pressure measurements are stored on a computer in the form of pressure coefficient; the pressure coefficient
The power spectrum
In the spherical dome case, the general shape of
In the vaulted roof case, the wind pressures are affected by the wind direction. Hence, the power spectra are calculated for all pressure taps and wind directions. Fig. 7 shows sample results of comparison between experiment and formula for the power spectra at two points on a vaulted roof. Again, the agreement is generally good.
3. Artificial neural network
3.1. Spherical dome
Although the wind pressures were measured simultaneously at several hundreds points in the wind tunnel experiments, spatial resolution may be still limited from the viewpoint of cladding design. Cladding or roofing cover is sensitive to the spatial variation and fluctuating character of the time-dependent wind pressures. The turbulence of approach flow also affects the wind pressures significantly. Hence, an artificial neural network based on Cascade Correlation Learning Network (CCLN, Fahlman and Lebiere, 1990) is used to improve the resolution.
Fig. 8 illustrates the network architecture, which has a layered structure with an input layer, an output layer and a hidden layer between the input and output layers. The input vector consists of five parameters, that is, two geometric parameters of the building (
The quickprop algorithm (Fahlman, 1988) is used to train the output weights. Training begins with no hidden units. As the first step, the direct input-output connections are trained as well as possible over the entire training set. The network is trained until either a predetermined maximum number of iterations is reached, or no significant error reduction has occurred after a certain number of training cycles. If the error is not acceptable after the first step, a new hidden unit is added to the network to reduce this residual error. The new unit is added to the network, its input weights are frozen, and all the output weights are once again trained. This cycle repeats until the error becomes acceptably small.
Well-distributed representative data are required for training the network. In the above-mentioned database, pressure data at 230 locations are stored each for five
The sigmoid function represented by the following equation is used to process the net input signals and provide the output signals at hidden nodes:
Fig. 10 shows comparisons between experiment and prediction by ANN for, ,
To discuss the application of the ANN to practical situations, a comparison is made between the prediction by the ANN and the experimental data for Nagoya Dome (Fig. 11). The geometry of this building is as follows: i.e. span
3.2. Vaulted roof
Fig. 13 shows the ANN architecture for vaulted roofs. In this case, the wind direction
Fig. 15 shows comparisons between experiment and prediction by ANN for, ,
4. Time series simulation of wind pressures
4.1. Outline of the procedure
First, the application of the Kumar and Stathopoulos’s method (1999, 2001) to the present problem is discussed. The flow chart for the simulation is described in Fig. 14. The approach is based on an FFT Algorithm. The Fourier amplitude is constructed from the power spectrum
4.2. Toward simplification of the procedure
The most troublesome and time-consuming procedure is the determination of the optimum value of
4.3. Results and discussion
A comparison of the wind pressure time series between experiment and simulation is shown in Fig. 18. The spike features of pressure fluctuations are simulated well. Figure 19 and 20 summarize comparisons between experiment and simulation for the statistics of the wind pressures at two typical points on a spherical dome and a vaulted roof, respectively. Note that the averaging time for evaluating the peak pressure coefficients is 1 sec and the values in the table are all the ensemble averages of the results from six consecutive runs. A good agreement between experiment and simulation is seen for both points. Similar comparisons are made for ninety-two points shown in Fig. 21 (points on the solid lines). The results for
As mentioned above, the accuracy of the ANN prediction for
5. Application of the wind load evaluation system to wind resistant design
The wind load evaluation system proposed here can provide peak pressure coefficients according to a predetermined risk level by combining the extreme value analysis. Fig. 20 shows the probability of non-exceedence for
Furthermore, by introducing a load cycle counting method, such as the rainflow count method, the system can provide the wind load cycles for fatigue design. Fig. 25 shows a sample result on the frequency distribution of wind pressure coefficient fluctuations, represented as a function of mean and amplitude of fluctuation at the center of a dome. By combining such a result with the influence coefficients, we can easily compute the stresses or strains induced in the cladding and its fixings, which are used for evaluating the fatigue damage.
6. Concluding remarks
A computer-assisted load evaluation system for the design of roof cladding of spatial structures using an aerodynamic database, artificial neural network and time-series simulation technique has been proposed. Focus is on spherical domes and vaulted roofs as typical roof shapes used for spatial structures. The proposed methodology is capable of providing peak pressure coefficients according to pre-determined risk levels by combining the extreme value analysis; this can generate risk consistent and more economical design wind loads for the roof cladding. Furthermore, by introducing a load cycle counting method, such as the rainflow count method, the system can provide the wind load cycles to be used for fatigue design.
This chapter describes the components of the load evaluation system proposed by the author. Although there are some problems to be investigated further, the results presented here indicate that the proposed system is promising. In this chapter the subject is limited to spherical domes and vaulted roofs. However, it is possible to apply the proposed method to the cladding of any buildings, once the database of the statistics of wind pressures has been constructed based on a wind tunnel experiment and/or CFD computations.
A part of the study is financially supported by Nohmura Foundation for Membrane Structure’s Technology. The authors are much indebted to Dr. Takeshi Hongo of Kajima Technical Research Institute and Dr. Hirotoshi Kikuchi of Shimizu Corporation for providing them the wind tunnel test data. Thanks are also due to Mr. Raku Tsuruishi, Ms. Miki Hamai and Chihiro Sukegawa, who were then graduate students of Tohoku University, for assistance in constructing the neural networks.