The safety of designed urban structures is highly depending on the respond of structures to different types and magnitude of environmental loads. The safety assessment of an existing building needs to have a full consideration of all the environmental factors. Therefore, the engineers must realize the importance of the environment and take care of all the assumptions and uncertainties. One of the hot topics that researchers nowadays are interested is the influence of climate change to the engineering designs. There is a remarkable consensus in the scientific community telling us a fact that our climate is changing and engineering design failures are increasing. In this paper, we are going to have an investigation of the effect of the climate change to the safety of high rise building. The climate effects are briefly discussed at the first beginning of this paper. Then a detailed study is performed on the modeling of climate effects for the wind load. This is later utilized in a structural reliability analysis for a high rise building. The influence of climate effects to the overall safety of a building is investigated and discussed based on the analyzed results. It was found the influence of climate effect can be very significant in the design of high rise buildings.
Climate provides the human beings the context of environment which is suitable for us to live in this planet. It contains several key factors in our ecosystem including the air, temperature, water, and wind. With solar radiation and greenhouse effect, the earth can keep its surface temperature at 14°C. But in recent centuries, as various industrial activities increased, the greenhouse effect becomes severe. Over the last 150 years, the CO2 concentration has increased by 30% and the CH4 concentration has increased by 150% globally [
As climate change can lead to quite a lot of changes, many engineering designs have to be reassessed. Researchers have already concerned about the implications of the science in terms of reservoir yields, provisions for flood defenses, wind loads, soil moisture, demands on energy and water supply systems, and other factors affecting the design and construction of engineering works [
In this paper, we are going to explore the influence of climate change to the safety of high rise building. We are going to use bootstrap resampling techniques, kernel density estimate, and normal distribution analysis to measure the effect of climate change associated with wind speed. And then we will make a further comment on the design value derived from the extreme wind by using a simple linear regression model with consideration of climate change. Finally, we will do a reliability analysis for a simple engineering problem based on code BS6399 and try to see the influence of climate change on the high rise building’s reliability. Realizing that, the paper is organized as follows. First, a general picture of climate change evidence is reviewed in Section
Due to the high speed of urban development in US, the average temperature over the whole nation has increased by about 0.6°C; some areas are even 2.4°C, in the 20th century. Particularly for the western US, the temperature rose about 2 to 5°F in 20th century; see Figure
The mean temperature change in 20th and 21st centuries in western US (retrieved from
The west region is characterized by its diverse topography, ecosystem and a rapid growing population and economy. Since 1950, the region’s population has quadrupled, and most people are now living in the urban areas. With the development and population growth, temperature increase is much more obvious than all the other places in the US [
Model predictions for the climate change for western US in 21st century (retrieved from
Thus, catastrophe events like flooding are likely to occur more often since the melting of snowpack and heavy precipitation can happen more easily when temperature increases. This turned some areas to a hazardous region. Particularly in the Colorado State, thunderstorms are now more frequent in the eastern plains during spring and summer [
In the wind load analysis, we usually utilize the safety factor to overestimate the wind load for a safe design. A nominal design wind load is an extreme load with specified probability of being exceeded during a given time interval [
Many probabilistic methods have been developed to estimate the extreme wind speed. Despite the consideration of wind direction effect, the ASCE initially provides an assumption that the extreme wind speeds followed a Frechet distribution with a tail length parameter [
Wind loading problem is often met in engineering design works especially for the high rise building. It is closely related to the environmental loadings on a building. In some particular areas, such as coastline buildings, the design may even need to consider hazardous wind load such as hurricanes.
In practice, wind loading pressure is always assumed to obey the concept of “kinetic pressure”
For more convenient use, we may use the effective loading
Wind is very common in our daily life. Thus, it is reasonable to assume wind speed follows a normal distribution. Of course, if we consider the seasonal effect or directional effect, the model may need to be modified.
In this study, we use the data of maximum monthly wind speed in the western US for the year 2009 in the analysis [
Graph of normal fitting for the extreme wind speed data.
The estimated mean value of wind speed is 23.74 m/s and the standard deviation is 4.235. The normal fitting curve is arbitrarily good, but the peak of the probability density function (PDF) is not well approximated. So in order to test the validity of this normal fitting, we conducted a Chi-square goodness-of-fit test; see Table
Chi-square test.
Precipitation (cm) | Daily mean temperature (°F) | ||||||
---|---|---|---|---|---|---|---|
Denver | Grand Junction | Pueblo | Alamosa | Colorado Springs | Denver | Grand Junction | Pueblo |
1.30 | 1.52 | 0.84 | 14.7 | 28.1 | 29.2 | 26.1 | 29.3 |
1.24 | 1.27 | 0.66 | 22.5 | 31.7 | 33.2 | 34.1 | 34.6 |
3.25 | 2.54 | 2.46 | 32.7 | 37.8 | 39.6 | 43.4 | 41.8 |
4.90 | 2.18 | 3.18 | 40.8 | 45.3 | 47.6 | 50.9 | 49.9 |
5.89 | 2.49 | 3.78 | 50.4 | 54.6 | 57.2 | 60.5 | 59.7 |
Kernel density estimation is a simple nonparametric way to estimate the probability distribution of a random variable [
Based on this idea, we have got the final results from Matlab by using a kernel density estimation method [
Kernel density estimate results.
Cases | Number of mesh | Min (m/s) | Max (m/s) | 95% extreme value | Bandwidth |
---|---|---|---|---|---|
1 | 100 | 15 | 35 | 34.74 | 3.94 |
2 | 100 | 10 | 40 | 36.10 | 3.53 |
3 | 100 | 0 | 50 | 38.35 | 3.72 |
PDF graph of kernel density estimation for three cases.
As shown in the table, the case having interval between 0 and 50 m/s can provide a more smoothing probability density function. This is because a wider interval can include out-of-domain points (even outliers). However, for narrow intervals, they cannot handle out-of-domain data. It can be seen that the increasing of the mesh number can also help to predict the extreme value more accurately.
The kernel density estimate is helpful for us to know the distribution of extreme values. It provides a good understanding of the distribution type. However, we want to check more on the confident intervals for the extreme value estimate.
Since the collected wind speed data is too limited in our analysis, it is better to use a resampling technique to estimate the random variable’s properties. Bootstrap method is selected as the tool for this study. Bootstrap resampling method has the following three important properties: (
Generally, the variance estimator can be easily calculated as
Bootstrap results.
Number of bootstrap resamples | Mean | Standard deviation | Estimated interval |
---|---|---|---|
100 | 23.735 | 0.857 | 22.325~25.145 |
1000 | 23.797 | 0.820 | 22.448~25.146 |
10000 | 23.818 | 0.890 | 22.354~25.282 |
Histogram of bootstrap resamples for different sizes.
As the results show, the 95% confident interval for the maximum wind speed is around 22.325~25.282. It can further estimate the standard deviation in the bootstrap resampling; see Table
Bootstrap estimates.
Number of bootstrap resamples | Mean | Standard deviation | Estimated interval |
---|---|---|---|
100 | 4.137 | 0.407 | 3.468~4.806 |
1000 | 4.144 | 0.502 | 3.318~4.970 |
10000 | 4.118 | 0.475 | 3.337~4.898 |
As the study is going to see how the climate change can affect the final reliability analysis, we are now checking the changing of extreme wind speed with respect to precipitation and the temperature on a daily basis. Here the climate data from the observation station Rand Junction is analyzed for a reference [
The investigation is analyzed through a simple linear regression model. From the output, we have found that the wind speed has little effect by the precipitation, as indicated by
Regression fitting results.
Wind speed versus precipitation | Wind speed versus temperature |
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Estimated equation: | Estimated equation: |
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Standard variance = 0.6445 | Standard variance = 0.3105 |
SSE = 4.1535 | SSE = 0.9643 |
Now we are looking at the future prediction of the wind speed for our engineering design. Due to model-based projections for a mid-range emissions scenario, the global average temperature is likely to rise by about 2 to 6°F (about 1.2 to 3.5°C). For this mid-range emissions scenario, the models used for this paper that the average warming over the US would be in the range of about 5 to 9°F (about 2.8 to 5°C) [
Climate future predictions.
Extreme wind speed (m/s) | Mean wind speed (m/s) | Temperature (°C) | Increased temperature (°C) | Expected future wind speed (m/s) |
---|---|---|---|---|
38.35 | 5.5 | 41.9 | 44.7~46.9 | 39.37~40.17 |
The approximated future extreme wind speed is an interval. The confidence level is at 95%. This modeling of wind speed has many assumptions [
In the following, we use a simple example to demonstrate the proposed approaches. Determine the deflection of a simple one storey block in a high rise building with plan dimension 20 m × 20 m. The height of one storey is 5 m. The openings of the building are closed. It is erected in a city at an altitude 100 m and is 50 km from the coast. A general view of this building is illustrated in Figure
Schematic illustration of the problem.
Based on BS6399, the wind speed can be calculated by
By applying this load to the building, the deflection can be calculated as
Some of the coefficients are given in a nonvariant form. These include the altitude factor
Factor uncertainties.
Mean value | COV | |
---|---|---|
Direction factor | 0.8 | 0.2 |
Seasonal factor | 0.65 | 0.3 |
Factor | 3.6 | 0.15 |
Elasticity modulus (kN/m2) | 200 | 0.2 |
For solving this problem, we are going to use gradient projection method, numerical integration, and Monte Carlo simulation. Meanwhile, the reliability results will be compared by using the future predicted wind speed.
First, the Monte Carlo simulations are performed to estimate the failure probability of this problem. By using a sample of 100000 simulations, the results of the performance function are shown in Figure
Summary of the Monte Carlo simulation.
The distribution of the performance function values is quite like a normal distribution. The major difference is the skewness of performance function. The failure probability is the total probabilities of performance function having negative values. From the simulated data, it shows a 3.43% failure probability for the present condition and 4.84%–5.91% for the future condition. Meanwhile, we can do an estimation of the performance equation based on simple calculations by using the Taylor’s equation
Thus, we could simply obtain the reliability index
Another accurate but tedious way to solve this reliability problem is the direct numerical integration [
In the numerical integrations, we have used the trapezoidal method, which takes the area of each trapezoidal segment in the division of the joint probability equation. Because the numerical calculation is a very complicated process, we have used fewer steps to obtain the result. We have set an interval for each variable to do the integration. Then, the integration is conducted to calculate the probabilities of the performance function value when it is less than zero. In order to make the calculation more accurate, we tried to set the interval in the centre of the domain; see Table
Integration domain.
Factors | Interval | Number of divisions | Breadth of trapezoids |
---|---|---|---|
Directional factor | 0.2~0.4 | 100 | 0.002 |
Seasonal factor | 0.3~1.0 | 100 | 0.007 |
Factor | 2~6 | 100 | 0.040 |
Elasticity modulus | 100~300 | 100 | 2 |
The estimate of failure probability is 0.0211 for the present condition and 0.0325–0.0409 for the future condition, which are very close to the result from Monte Carlo simulation. The difference between these two values may come from the numerical errors. Anyway, both methods are suitable to do the reliability analysis in this problem.
Correlation problems can be very common in engineering designs [
We assumed some positive correlation coefficients between the factors
Graph of failure probability with correlation effect.
From the comparison between Monte Carlo simulation and numerical integration, we can see that the failure probability increases as the correlation increases. Both methods show the same pattern.
Besides the effect of correlation, the selection of different distribution types will also affect the reliability analysis. In the numerical analysis, it will change some formulas in the joint probability function. The random number generation is changed in the Monte Carlo simulation for every random variable. Nevertheless, the changing is not difficult to manipulate. But the result may deviate quite a lot. This warns us that if there are some wrong assumptions for random variables’ distributions, it may lead to an unexpected failure in our design. Here the lognormal distribution is used to check how the final result will change when distribution type is changed. The detailed information is provided in Table
Information of lognormal models.
Mean value | COV | Lognormal | Lognormal | |
---|---|---|---|---|
Direction factor | 0.8 | 0.2 | −0.24275 | 0.198042 |
Seasonal factor | 0.65 | 0.3 | −0.47387 | 0.29356 |
Factor | 3.6 | 0.15 | 1.269809 | 0.149166 |
Elasticity modulus | 200 | 0.2 | 5.278707 | 0.198042 |
The result from a Matlab programming shows a lower failure probability value 0.016 for the present condition and 0.0251–0.032 for the future condition when using lognormal distributions. Even by using a Monte Carlo simulation, the result is still lower than the original case having normal distribution random variables. A rough understanding of this result is that the shape of a lognormal distribution may “concentrate” more at the low values. But lognormal distribution may be more realistic as the value of the factors cannot go to negative value. If we change the distribution to the other type, the result will change again. The distribution of each variable is an assumption, and the calculation of reliability index is highly dependent on this.
From the above analysis we can see that the climate change may result in different reliability values for our engineering design. The transformation of the climate change to a wind speed variation is the first stage, and then it can be put into the reliability analysis as an uncertain climate factor as wind load. Here we have no information about the distribution of the wind speed change. An interval is utilized for the analysis. But this already shows that the climate change should be a big concern in engineering problems especially for high rise building with long expected lives. A general comparison for the climate change effect is shown in Table
Effect of climate change to changes in reliability.
Present condition | Future prediction | Increased percentage | |
---|---|---|---|
Extreme temperature (°C) | 41.9 | 44.7~46.9 | 6.7~11.9 |
Extreme wind speed (m/s) | 38.35 | 39.37~40.17 | 2.7~4.7 |
Failure probability | |||
Monte Carlo simulation | 0.0343 | 0.0484~0.0591 | 41.1~72.3 |
First-order estimation | 0.00504 | 0.01004~0.01455 | 99.2~188.9 |
Numerical integration | 0.0211 | 0.0325~0.0409 | 54.0~93.8 |
Other effects | |||
Correlation ( | 0.034 | 0.0486~0.0588 | 42.9~72.9 |
Lognormal distribution | 0.016 | 0.0251~0.032 | 56.9~100 |
Obviously, the climate effect is a significant factor in the structural safety. The failure probability has increased by nearly 50–75% by just increasing the temperature of 4°C. Although this temperature change takes a long time, it indicates that we should not ignore this problem. The influence of climate change may not only come from the wind load. For the real high rise building, we need to consider more effects, like the cumulative damage, material deterioration, steel corrosion, and many other factors that are related to climate. Thus, for a professional design and good maintenance of buildings, new approaches or corrections must be made into our design code and considerations.
The investigation shows that the climate change induced uncertainties can be well handled by the current proposed statistical approaches. The wind speed increasing rate can be modeled as random variables and then processed in the structural safety analysis. This viable approach is demonstrated to be more reliable compared to the deterministic approach. The room for indeterminacy in probabilistic models reduces the risks of too optimistic conclusions and this can also help to prevent rough assumptions [
In this paper, a simple wind load problem is used to investigate the influence of climate change to reliability analysis of high rise building. Several sampling methods are utilized to estimate the extreme wind speed. A simple linear regression model is applied to predict the future extreme wind speed by considering the climate effect. Finally, a reliability analysis for a simple wind load problem by using the predicted value is performed. A further deep view to see how the result can change with the changes of correlation and distribution properties is also discussed. Normal fitting is proved to be a bad approach when lacks of enough information. Kernel density estimation is a good use in this situation, but the bandwidth and function type need to be clarified. Bootstrap resampling method can predict a reliable confidence interval for the extreme values from the data sample. The wind speed generally has a linear relation to the daily mean temperature. This can help us to do a rough approximation of the future wind speed by considering the climate change. The reliability result shows that the failure probability may be amplified even there is a small increase in the mean atmospheric temperature.
The authors declare that they have no competing interests.
This study was financially supported by the National Natural Science Foundation of China (Grant no. 51278368), the Natural Science Fund of Hubei Province (Grants nos. 2013CFC103 and 2012FKC14201), the Scientific Research Fund of Hubei Provincial Education Department (Grant no. D20134401), Youth Talent Foundation of Hubei Polytechnic University (Grant no. 13xjz07R), Natural Science Fund of Hubei Polytechnic University (Grant no. 13xjz03A), and the Innovation Foundation in Youth Team of Hubei Polytechnic University (Grant no. Y0008).