# Influence of Climate Change in Reliability Analysis of High Rise Building

Authors: Yi Zhang; Keqin Yan; Tao Cheng; Quan Zhou; Liping Qin; Shan Wang
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1]. These atmospheric changes together with human-induced natural influences have finally changed the climate. The most obvious change can be seen from the temperature. The global average temperature has warmed up by 0.6°C since the mid-1800s [2]. The Northern Hemisphere has an average temperature nearly 0.9°C higher than that few centuries before the Industrial Revolution [3]. From the recorded human development, the rising of carbonated gases concentration in the atmosphere will continue. There is no doubt the climate change will cause a significant warning to our human being’s lives.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 [4]. Close monitoring of the situation is essential if we want to prevent climate induced disasters. Climate extremes are often highly related to the changes of soil types which form the basic foundations for high rise buildings [5]. In geotechnical engineering, many geologists and geotechnical engineers are already trained to observe and predict the uncertainties of natural materials resulted from climate. The training and experiences enable them to work in the time-frames over which significant climatic change might be expected to happen [6]. Besides the material uncertainties, structural engineers need to guarantee the safety of their design in terms of “design lives.” The basic goal is trying to design structures with less vulnerable to the time-scale of significant climatic change. Concerns about the durability of structures are also encountered in the construction process [7]. The safety of structural members like the anchors in tension is strongly dependant on the climate of the geological location [8]. Environmental loading resulted structure failures can be found in [9]. Therefore, the design of structures needs people to monitor and assess the influence of climatic changes [10].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 Section2. Then, based on the collected data, the modeling of climate effects in the wind load is constructed in Sections 3 and 4. Following that, Sections 5 and 6 investigate the climate change predictions from the model established in Section 4. Section 7 then details the problem investigated in this study. The results and discussion are provided in Section 8. The conclusion is summarized in Section 9.


## 2. Review of Climate Change: Evidence from US

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 Figure1. The region becomes wetter, and some areas’ annual precipitation rose about 50%. The length of the snow season increases about 16 days in California and Nevada from 1951 to 1996 [11].Figure 1
The mean temperature change in 20th and 21st centuries in western US (retrieved fromhttp://www.scisnack.com/).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 [12, 13]. In the Hadley and Canadian General Circulation Model, the temperature is predicted to have a rise of 4°F by the 2030s and 11°F by the 2090s; see Figure 2.Figure 2
Model predictions for the climate change for western US in 21st century (retrieved fromhttp://www.scisnack.com/).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 [14]. Heavy snow from the mountains creates serious problems to the residents. As a result of that, the wind load that we are taking into account for the design works should also be expected to increase. We should look deep into the effect of climate change to our engineering design works.


## 3. Wind Load Modeling

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 [15]. Normally, 50 years are set as this time interval. For example, in Florida State, it has used a specified 50-year nominal design wind speed of 49.17 m/s for their design due to the ASCE Standards 7-93 [16]. This extreme wind speed should be selected based on probabilistic approach which could make sure the occurrence of it is rather small during its time interval. The structures can then be expected to withstand loads within the limit in the specified years without loss of integrity [17, 18].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 [19]. Further research shows Gumbel distribution could be a better model [20]. Although the approximated value is enough “safe” for the current design, it needs to be modified yearly based on new data collected. More and more efforts in the description model for the wind are needed for our engineering design.


## 4. Wind Load Equations

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”q, which can be calculated by (1)qt=12ρV2t,where the actual pressure onto the building is related to the air densityρ and the time-dependent wind velocity V(t). The wind force on a simple structure can thus be written as(2)P=qACD,whereA is the loaded area and CD is the drag coefficient. Catalogue of such coefficients can be found in various design guides and codes [21]. In engineering design works, the basic force formula can be further linearized as (3)Pt=12ρV2tACD=12ρV¯+u2ACD=12ρV¯2+2uV¯ACD.For more convenient use, we may use the effective loadingP¯=(1/2)ρV¯2ACD instead of the real force. To solve these wind loading problems, many codes and standards have been published. The British, American, and European countries are all having such relevant specifications [22]. Two very famous codes that are commonly used are CP3 and BS6399. Both of these two codes provide the equations to estimate the design wind speed [23, 24]. Obviously, we can see that the wind speed is related to quite a lot of uncertain factors.


## 5. Wind Speed Analysis

### 5.1. Wind Speed Modeling

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 [25]. The data is plotted in Figure 3.Figure 3
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 Table1. In the Chi-square test, we generally compare the observed frequencies n1,n2,…,nk from our data with the theoretical frequencies e1,e2,…,ek estimated from the distribution model. Since we have totally 24 observations and the necessary condition for Chi-square test is ei≥5, we divide the wind speed domain into four intervals for the comparison. Based on the above, we want to check the fitted model to see whether it satisfies the criteria [26](4)∑i=1kni-ei2ei<c1-α,f,in which c1-α,f is the critical value of the χf2 distribution at the cumulative probability of 1-α. The calculated value for the ∑ni-ei2/ei=1.32<1.6488 only satisfies a significant level of 20%. Therefore, we may need to utilize other approaches for a comparison.Table 1
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


### 5.2. Kernel Density Estimation

Kernel density estimation is a simple nonparametric way to estimate the probability distribution of a random variable [27]. Obviously, the choosing of a suitable kernel function and band width of the sample data can greatly affect the final histogram appearance and thus the extreme value estimate. It is reasonable to assume a normal like probability for the kernel density functions. However, the selection of an appropriate bandwidth is difficult. An unsuitable choosing of bandwidth may result in under smoothed or over smoothed problem [28]. Actually, there are hundreds of articles used a so-called minimizing Mean Integrated Squared Error method to obtain the bandwidth, where the MISE can be shown here [29]: (5)MISEf^x=E∫f^x-fx2dx.Based on this idea, we have got the final results from Matlab by using a kernel density estimation method [5]. Three different band widths are compared for the calculation. These are shown in Figure 4 and Table 2.Table 2
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 3.72Figure 4
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.


### 5.3. Bootstrap Resampling Method

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: (1) invariance under reparameterization, (2) automatic computation, and (3) higher order accuracy. It can give a good estimation without the dependencies on any extra analytical inference [30]. In this section, we will go to estimate the standard deviation for the extreme wind speed.Generally, the variance estimator can be easily calculated as(6)σ^∗=∑i=1nθ^i∗-θ-∗2n-11/2,where the θ-∗=∑i=1nθ^i∗/n. Under general assumptions, this is a consistent estimator of the true standard deviation [31]. Thus, it is utilized to construct the confident interval θ^-σ^∗zα/2,θ^+σ^∗zα/2. The results are obtained by using three different bootstrap sample numbers; see Figure 5 and Table 3.Table 3
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 22.354~25.282Figure 5
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 Table4. The estimated standard deviation of the population is around 3.337~4.898 by a 95% confident interval. Thus, by applying the Chebyshev’s inequality for a 99% probability bound, we have got the value 74.26 for PrX-μ≥α≤σ2/α2=0.01. This value is highly overestimated compared with the methods used before. But it may be the only way when there is no sufficient information.Table 4
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


## 5.1. Wind Speed Modeling

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 [25]. The data is plotted in Figure 3.Figure 3
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 Table1. In the Chi-square test, we generally compare the observed frequencies n1,n2,…,nk from our data with the theoretical frequencies e1,e2,…,ek estimated from the distribution model. Since we have totally 24 observations and the necessary condition for Chi-square test is ei≥5, we divide the wind speed domain into four intervals for the comparison. Based on the above, we want to check the fitted model to see whether it satisfies the criteria [26](4)∑i=1kni-ei2ei<c1-α,f,in which c1-α,f is the critical value of the χf2 distribution at the cumulative probability of 1-α. The calculated value for the ∑ni-ei2/ei=1.32<1.6488 only satisfies a significant level of 20%. Therefore, we may need to utilize other approaches for a comparison.Table 1
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


## 5.2. Kernel Density Estimation

Kernel density estimation is a simple nonparametric way to estimate the probability distribution of a random variable [27]. Obviously, the choosing of a suitable kernel function and band width of the sample data can greatly affect the final histogram appearance and thus the extreme value estimate. It is reasonable to assume a normal like probability for the kernel density functions. However, the selection of an appropriate bandwidth is difficult. An unsuitable choosing of bandwidth may result in under smoothed or over smoothed problem [28]. Actually, there are hundreds of articles used a so-called minimizing Mean Integrated Squared Error method to obtain the bandwidth, where the MISE can be shown here [29]: (5)MISEf^x=E∫f^x-fx2dx.Based on this idea, we have got the final results from Matlab by using a kernel density estimation method [5]. Three different band widths are compared for the calculation. These are shown in Figure 4 and Table 2.Table 2
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 3.72Figure 4
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.


## 5.3. Bootstrap Resampling Method

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: (1) invariance under reparameterization, (2) automatic computation, and (3) higher order accuracy. It can give a good estimation without the dependencies on any extra analytical inference [30]. In this section, we will go to estimate the standard deviation for the extreme wind speed.Generally, the variance estimator can be easily calculated as(6)σ^∗=∑i=1nθ^i∗-θ-∗2n-11/2,where the θ-∗=∑i=1nθ^i∗/n. Under general assumptions, this is a consistent estimator of the true standard deviation [31]. Thus, it is utilized to construct the confident interval θ^-σ^∗zα/2,θ^+σ^∗zα/2. The results are obtained by using three different bootstrap sample numbers; see Figure 5 and Table 3.Table 3
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 22.354~25.282Figure 5
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 Table4. The estimated standard deviation of the population is around 3.337~4.898 by a 95% confident interval. Thus, by applying the Chebyshev’s inequality for a 99% probability bound, we have got the value 74.26 for PrX-μ≥α≤σ2/α2=0.01. This value is highly overestimated compared with the methods used before. But it may be the only way when there is no sufficient information.Table 4
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


## 6. Climate Change Effect and Future Prediction

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 [25]. This can help to reduce the regional effect.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 byR2=0.143. Actually, the Pearson correlation coefficient for the wind speed and precipitation is only 0.378 with a P value of 0.226. Obviously, this shows that the correlation is not so severe between the wind speed and the precipitation. But the mean temperature cannot be ignored. R2 of the linear model between the wind speed and the mean temperature is 0.801. It means the mean temperature is strongly correlated with mean wind speed. Thus, in the prediction of future extreme wind speed, we will consider simple linear regression model between wind speed and mean temperature. The results are shown in Table 5.Table 5
Regression fitting results.
Wind speed versus precipitation Wind speed versus temperature Estimated equation:wind speed = 2.75 + 0.469 · precipitation Estimated equation:wind speed = 2.99 + 0.0593 · temperature R 2 = 0.143 R 2 = 0.80 1 R 2 (adj) = 0.057 R 2 (adj) = 0.708 F value = 1.66 F value = 40.24 P value = 0.226 P value = 0.000 Standard variance = 0.6445 Standard variance = 0.3105 SSE = 4.1535 SSE = 0.9643Now 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) [1]. By using this information and the extreme wind speed model established, a prediction of future extreme wind speed can be done. The results are recorded in Table 6.Table 6
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 39.37~40.17The approximated future extreme wind speed is an interval. The confidence level is at 95%. This modeling of wind speed has many assumptions [32]. The lack of enough information may result in statistical uncertainty. Nevertheless, the prediction is already a good one based on available information. The model can be developed in the future with new data collected.


## 7. Case Study: High Rise Building Storey Block

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 Figure6.Figure 6
Schematic illustration of the problem.Based on BS6399, the wind speed can be calculated by(7)Vs=Vb×Sa×Sd×Ss×Spand effective wind speed is calculated by Ve=S×Vs×Sb.By applying this load to the building, the deflection can be calculated as(8)Δ=ωl48EI=1/2ρV¯2CD×L×H48EIand this is subject to the deflection limit Δ/H≤0.0025. Thus, we can write the performance function as(9)G=0.0025-3ρVe2CDH44BEL2.Some of the coefficients are given in a nonvariant form. These include the altitude factorSa=1+0.001Δs=1+0.001×100=1.1, drag coefficient CD=1.0, the basic wind load Vb, and probability factor Sp=1.0. Moreover, assumptions of normal distributions of the variables are presented in Table 7.Table 7
Factor uncertainties.
Mean value COV Direction factorSd2 0.8 0.2 Seasonal factorSs2 0.65 0.3 FactorSb2 3.6 0.15 Elasticity modulus (kN/m2) 200 0.2For 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.


## 8. Reliability Analysis

### 8.1. Monte Carlo Simulation

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 Figure7.Figure 7
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(10)μG≈GSs¯,Sd¯,Sb¯,E¯,σG2=∑i=1N∑j=1N∂G∂XiXi¯∂G∂XjXj¯ρσiσj.And since there is no correlation between variables, we can get our value based on the noncovariance case(11)μG≈GSs¯,Sd¯,Sb¯,E¯=0.0025-3ρVe2Ss¯,Sd¯,Sb¯CDH44BE¯L2=0.001325,0.001239-0.001187,(12)σG2≈∑i=1N∑j=1N∂G∂XiXi¯∂G∂XjXj¯ρσiσj=∂G∂Sb2σSb2+∂G∂Sd2σSd2+∂G∂Ss2σSs2+∂G∂E2σE2=2.65×10-7,2.84-2.96×10-7.Thus, we could simply obtain the reliability indexβvalue byβ=μG/σG, and the failure probability can be further calculated by inversing the Gaussian distribution p=Φ-1-β=Φ-1-μG/σG. The calculated value is 5.044×10-3 for the present condition and 10.036×10-3~14.555×10-3 for the future condition. Both are smaller than the simulated result. This estimated value will be used for a reference.


### 8.2. Numerical Integration

Another accurate but tedious way to solve this reliability problem is the direct numerical integration [33]. Fortunately, since the formula is not so sophisticated, the numerical calculation is easy to be conducted.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 Table8.Table 8
Integration domain.
Factors Interval Number of divisions Breadth of trapezoids Directional factorSd2 0.2~0.4 100 0.002 Seasonal factorSs2 0.3~1.0 100 0.007 FactorSb2 2~6 100 0.040 Elasticity modulusE 100~300 100 2The 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.


### 8.3. Correlation

Correlation problems can be very common in engineering designs [34]. It arises from the dependence between loads or more frequently some assumptions in the factors. Here we can see that the wind load direction can always be related to the seasons. In this sense, it has a correlation relationship between these two factors. Thus, we are going to see how the correlations will influence the final result in this session.We assumed some positive correlation coefficients between the factorsSs2 and Sd2. Then following the same procedures, Monte Carlo simulation and numerical calculations are conducted to calculate the failure probability. The results are plotted in Figure 8.Figure 8
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.


### 8.4. Distribution Type

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 Table9.Table 9
Information of lognormal models.
Mean value COV Lognormalλ Lognormalζ Direction factorSd2 0.8 0.2 −0.24275 0.198042 Seasonal factorSs2 0.65 0.3 −0.47387 0.29356 FactorSb2 3.6 0.15 1.269809 0.149166 Elasticity modulusE 200 0.2 5.278707 0.198042The 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.


### 8.5. Effect of Climate Change to Reliability

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 Table10.Table 10
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 (ρ=0.1) 0.034 0.0486~0.0588 42.9~72.9 Lognormal distribution 0.016 0.0251~0.032 56.9~100Obviously, 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 [35]. On the other hand, the full utilization of the available information leads to more conservative conclusions compared to the deterministic approaches which may be quite bias. In view of making critical engineering designs, direct emphasis can be put on the extreme values in the estimated bounds for the structural safety assessed values. Such stochastic model can take into account various climate related uncertainties load process in a quantitative manner and provide a tool for rationalizing the prioritization design of high rise buildings.


## 8.1. Monte Carlo Simulation

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 Figure7.Figure 7
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(10)μG≈GSs¯,Sd¯,Sb¯,E¯,σG2=∑i=1N∑j=1N∂G∂XiXi¯∂G∂XjXj¯ρσiσj.And since there is no correlation between variables, we can get our value based on the noncovariance case(11)μG≈GSs¯,Sd¯,Sb¯,E¯=0.0025-3ρVe2Ss¯,Sd¯,Sb¯CDH44BE¯L2=0.001325,0.001239-0.001187,(12)σG2≈∑i=1N∑j=1N∂G∂XiXi¯∂G∂XjXj¯ρσiσj=∂G∂Sb2σSb2+∂G∂Sd2σSd2+∂G∂Ss2σSs2+∂G∂E2σE2=2.65×10-7,2.84-2.96×10-7.Thus, we could simply obtain the reliability indexβvalue byβ=μG/σG, and the failure probability can be further calculated by inversing the Gaussian distribution p=Φ-1-β=Φ-1-μG/σG. The calculated value is 5.044×10-3 for the present condition and 10.036×10-3~14.555×10-3 for the future condition. Both are smaller than the simulated result. This estimated value will be used for a reference.


## 8.2. Numerical Integration

Another accurate but tedious way to solve this reliability problem is the direct numerical integration [33]. Fortunately, since the formula is not so sophisticated, the numerical calculation is easy to be conducted.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 Table8.Table 8
Integration domain.
Factors Interval Number of divisions Breadth of trapezoids Directional factorSd2 0.2~0.4 100 0.002 Seasonal factorSs2 0.3~1.0 100 0.007 FactorSb2 2~6 100 0.040 Elasticity modulusE 100~300 100 2The 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.


## 8.3. Correlation

Correlation problems can be very common in engineering designs [34]. It arises from the dependence between loads or more frequently some assumptions in the factors. Here we can see that the wind load direction can always be related to the seasons. In this sense, it has a correlation relationship between these two factors. Thus, we are going to see how the correlations will influence the final result in this session.We assumed some positive correlation coefficients between the factorsSs2 and Sd2. Then following the same procedures, Monte Carlo simulation and numerical calculations are conducted to calculate the failure probability. The results are plotted in Figure 8.Figure 8
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.


## 8.4. Distribution Type

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 Table9.Table 9
Information of lognormal models.
Mean value COV Lognormalλ Lognormalζ Direction factorSd2 0.8 0.2 −0.24275 0.198042 Seasonal factorSs2 0.65 0.3 −0.47387 0.29356 FactorSb2 3.6 0.15 1.269809 0.149166 Elasticity modulusE 200 0.2 5.278707 0.198042The 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.


## 8.5. Effect of Climate Change to Reliability

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 Table10.Table 10
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 (ρ=0.1) 0.034 0.0486~0.0588 42.9~72.9 Lognormal distribution 0.016 0.0251~0.032 56.9~100Obviously, 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 [35]. On the other hand, the full utilization of the available information leads to more conservative conclusions compared to the deterministic approaches which may be quite bias. In view of making critical engineering designs, direct emphasis can be put on the extreme values in the estimated bounds for the structural safety assessed values. Such stochastic model can take into account various climate related uncertainties load process in a quantitative manner and provide a tool for rationalizing the prioritization design of high rise buildings.


## 9. Conclusion

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.



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*Source: 5709245-2016-12-07.xml*