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PROBABILISTIC MODELING IN CIVIL ENGINEERING COURSE 11376 MAY 23, 2013

DESIGN OF A SPAGHETTI BRIDGE GROUP 13 LUCA FIANCHISTI – S121877 ANNE-LAURE TAING – S121568 ESZTER KÖRTVÉLYESI – S120887 MORTEN HJORTBØL – S042015


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PROBABILISTIC MODELING IN CIVIL ENGINEERING

Preface This project is written as a part of the course, 11376 Probabilistic Modeling in Civil Engineering, which is dealing with the subjects, probability and probabilistic modeling related to construction. During the course the group has achieved insight and capabilities in a variety of tools all of which improve the ability to take vital decisions under uncertainty – In this case, to be able to calculate the risk of collapse of a bridge and to dimension a simple bridge which has a very specific probability of collapse. This report is written in the spring of 2013 by Group 13: Luca Fianchisti, Anne-Laure Taing, Körtvélyesi Eszter and Morten Hjortbøl. Each chapter of the report has been assigned to one of the team members who have had the final responsibility of its content. However, all group members have participated equally in all activities in the execution of the project. In the table below, the responsible person for each chapter or sub sections can be found: Chapter Preface, Introduction, layout and other formalities Experimental setup Descriptive statistics Model selection Model parameter estimation Model verification Formulation of design criteria Reliability based design Experimental setup for design verification Design verification (incl. Hypothesis testing) Monte Carlo Simulation Hypothesis testing Discussion Conclusion

Responsible person Morten Morten Morten Anne-Laure Anne-Laure Eszter Luca Luca Eszter Morten Luca All All


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Table of content Preface................................................................................................................................................................2 Introduction........................................................................................................................................................4 Experimental setup ............................................................................................................................................5 Registration of the ultimate bending strength...............................................................................................5 The weight of the stones ................................................................................................................................7 Thickness of the spaghetti..............................................................................................................................8 Assumptions ...................................................................................................................................................8 Descriptive statistics ...........................................................................................................................................9 Graphical representation – Histograms .........................................................................................................9 Numerical representation ............................................................................................................................11 Measures of central tendency..................................................................................................................11 Measures of dispersion ............................................................................................................................12 Measure of asymmetry and peakness .....................................................................................................13 Model selection ................................................................................................................................................14 Probability papers ........................................................................................................................................14 Method .........................................................................................................................................................14 Results ..........................................................................................................................................................15 Model parameter estimation ...........................................................................................................................19 Method of Moment......................................................................................................................................19 Maximum Likelihood Method ......................................................................................................................20 Estimation of the parameters ......................................................................................................................21 Model verification ............................................................................................................................................23 Formulation of design criteria ..........................................................................................................................26 Formulation of limit state function ..............................................................................................................26 Introduction..............................................................................................................................................26 Convolution ..................................................................................................................................................26 Structural reliability theory ..........................................................................................................................27 The Limit State Function (LSF) ......................................................................................................................28 Experimental verification of the model ...........................................................................................................30 Design verification ............................................................................................................................................31 The Monte Carlo Simulation ........................................................................................................................31 Hypothesis testing ........................................................................................................................................33 Discussion .........................................................................................................................................................35 Conclusion ........................................................................................................................................................36 Appendix...........................................................................................................................................................37 A.1 Weight of the stones ..............................................................................................................................37 A.2 Measured Ultimate breaking load .........................................................................................................38 A.3 Ultimate bending strength – Cecco (The chosen brand) .......................................................................39 A.4 Matlab code – Limit State Function, FORM ...........................................................................................40 A.5 Matlab code – Convulation ....................................................................................................................41


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Introduction In this project the aim is to dimension a simple bridge structure which will fail 10% of the times. This is done through experiments, exercises and calculations with the purpose of achieving competencies in the field of probabilistic modeling and reliability design. Since the prime focus of this project is on the probabilistic modeling and not the construction, the bridge structure should only consist of one or more pieces of dry spaghetti simply supported in two points. This will ensure that the number of uncertainty factors is kept to a reasonable level, with the possibility of actually hitting the desired target.

Figure 1 – Schematic diagram of the desired spaghetti bridge (from assignment).

The load on the bridge, which statistically should lead to collapse in 10% of the cases, consists of two randomly selected stones, taken from a collection of 100 known stones. In order to achieve such a bridge, it will therefore be necessary to link the probability of the spaghettis ultimate bending strength with the probability of the stresses caused from the weight of two stones. The conclusions that will be taken to achieve the best bridge will cover the specific "building material" spaghetti brand, optionally the number of pieces of spaghetti and last but not least, the distance between the supporting points.


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Experimental setup In order to dimension a desirable bridge structure probabilistic, it is a must to know the physical properties of the building materials' and especially their tensile strength. As the "building material" in this project consists of spaghetti, we do not know the physical properties in advance. Therefore some initial tests of the spaghetti are a necessity for the subsequent probabilistic calculations. In this section the experimental setup for gaining this data will be described. The relevant variables in order to dimension this bridge can be defined as the ultimate bending strength of any given piece of spaghetti from our chosen brand, and the expected load, that will be applied, when two stones are picked randomly from a bag with 100 known stones.

Registration of the ultimate bending strength Based on the breaking load, the distance between the supporting points and the diameter of the spaghetti, the ultimate bending strength can be derived. The relationship between these variables will be described later in this part. To ensure a sufficiently stable structure where the distance does not change between each test run, while still being flexible in its range of available distances to use, we have developed an A-shaped structure with screws placed in fixed distances from each other. The chosen distances are: 200, 180, 160, 140, 120 and 100 mm. Later when the specific distance is determined and will be tested, these divisions are not precise enough. But with this construction there will be plenty of room to add extra nails with the exact distance from each other. In the figures below drawings of the bridge can be seen.


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Figure 2 – Experimental setup (own production)

Figure 3 – Experimental setup (own production)


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The tests are carried out by placing the spaghetti on the supporting screws, after which a cup with a hook attached is applied on the spaghetti. The cup is then slowly filled with salt until the spaghetti breaks. The total weight of salt and cup therefore equal the spaghettis breaking load at this particular distance, which is then entered into an Excel sheet. In order to minimize the uncertainty when the stones are included in the test of the bridge, the stones should represent the largest share of the combined load as possible. Therefore, a light weighted polystyrene cup was used and suspended from a hook of thin steel wire. This test is made 20 times for 140, 160 and 180 mm, and for both types of spaghetti (Zara and Cecco). As mentioned the ultimate bending strength,

depends on the

breaking load, the distance between the two supports and the diameter of the spaghetti. The relationship between these variables is shown below:

Figure 4 – Testing for breaking load (own production)

Figure 5 – Calculating the ultimate bending strength (from the course material)

Where: I is the moment of inertia F is the maximum force on the spaghetti before it breaks

The weight of the stones In order to calculate the expected weight of two randomly selected stones out of one hundred known stones statistically, each stone was initially weighted individually so that the weight of two randomly “selected” stones can be calculated based on that. However, there may be a risk that the picking of the two stones might not be totally random in the final test, which could result in a wrongful verification or rejection of the design. Therefore, the weight of two randomly picked stones from the pile will also be measured a hundred times, and then compare the two hopefully equal probabilities with each other. Should the two probabilities prove too different, there will still be the possibility of changing the selecting method in the final test to a more reliable but perhaps more difficult method.


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Thickness of the spaghetti As part of the calculation of the ultimate bending strength of the spaghetti it is necessary to know its thickness. This is found by measuring the diameter of approximately 50 of each type of spaghetti and then simply use the average for each brand during the subsequent calculations.

Assumptions Besides the mentioned parameters there are a few variables which could also have an influence on the outcome but that have been considered insignificant in this context. This includes any variation in the mechanical properties and geometrical characteristics of the spaghetti, for example a change in thicknesses along the length of each piece of spaghetti, or a change in moisture content of the spaghetti, due to the fact that the package is open and exposed to ambient humidity for several months between the initial testing and the final design verification. It was also assumed that the hook will be rightly placed in the middle of the spaghetti each time by rule of thumbs, which otherwise would have an impact on the ultimate bending strength. These potentially influential parameters will be ignored in this project.

Figure 6 – Materials and equipment used in the final testing (own production)


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Descriptive statistics In order to build the desired spaghetti bridge, a number of results have been collected through the mentioned measurements and experiments. In this section, the collected data will be gathered and described in an easy and understandable manner including a graphical representation The following values were measured and tested: Parameter Stone weight

Type 1 stone 2 stones Pasta Zara

Type

Runs 100 100 Breaking load 140 mm 20 160 mm 20 180 mm 20 Pasta De Cecco 140 mm 20 160 mm 20 180 mm 20 Pasta diameter Pasta Zara 50 Pasta Cecco 50 Table 1 – The measured parameters (own production)

Graphical representation – Histograms A great way to show this kind of measurement data is through histograms. These depict the number of values in a bar graph divided into a number of intervals called bins. That way one can quickly and easily get an overview of the distribution of high and low values in the dataset. The determination of the number of bins is not trivial, since a change in the number of bins can result in a completely different visual representation and thus perhaps also of the reader's perception thereof. It is not clear which one that's the best way to determine the number of bins, therefore we have chosen to follow the calculation method, as recommended in this course:

Where: k = number of bins n = number of data samples Subsequently, it is rounded up to the first integer.  

For the weight of the stones, 100 test runs were made for each brand, providing 8 bins. For the ultimate bending strength of the spaghetti, 20 runs were made per length, per brand, providing 6 bins. o For the ultimate bending strength histograms of the three lengths separately, and with all the lengths together were made. Since the measurement of the thickness of the spaghetti will only be used to find the average, it was found unnecessary to draw a histogram of it.

The histograms below are showing the distributions of the stones' weight and spaghetti’s ultimate bending strength:


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Figure 7 – Histograms of the Ultimate Bending Strength for De Cecco and Zara divided in each length (own production)

Figure 8 – Histograms of the Ultimate Bending Strength for De Cecco and Zara all three lengths combined (own production)


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Figure 9 – Histograms of the weight of the stones based respectively 1 stone and 2 stones (own production)

Numerical representation Another way to make data understandable and comparable is to look at its individual distributions, and describe the extent to which there is symmetry, how much it spreads across the data set, or whether the values are located centrally or at either end. In the following, several of these values are found and described. Measures of central tendency To describe how the dataset in general are compared to one another, the Sample mean, Sample median and Sample mode are used. These three values are only taking the dataset as a whole into account and how they are represented on an overall scale, independent of how its own values are distributed within the dataset. Sample mean basically describes the average of the sample set, and is simply calculated by putting all the values together and divide by the amount of samples:

Where: x1, x2,‌, xn Sample median describes the sample value in the middle of the sample set, when it has been sorted. If the dataset has an odd number of samples, the sample median is the mean of the two middle ones. Finally the Sample mode describes the value which appears most times in the dataset. If one made a histogram with a bin for every distinct value, the highest bar would represent the sample mode. A sample


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set can have more than one mode. In that case, the distribution would be called bimodal or multimodal depending on, weather it contains two or more modes. The result is shown in the table below: Weight of stones 1 Stone

2 Stones

Zara 140

Zara 160

[g]

[g]

Sample mean

11,94

43,36

37,74

35,11

Sample median

9,00

40,00

38,59

35,72

Sample mode

Ultimate Bending Strength De Cecco De Cecco Zara 180 140 160

[N/mm2] [N/mm2] [N/mm2]

Diameter of spaghetti De Cecco 180

Zara

De Cecco

[N/mm2]

[N/mm2]

[N/mm2]

[mm]

[mm]

32,84

25,10

26,39

24,57

1,65

1,74

33,24

24,71

26,73

24,57

1,65

1,75

7,00 36,00 39,36 37,05 33,74 25,70 25,60 22,03 Table 2 – Calculated Sample mean, Sample Median and Sample mode (own production)

1,69

1,75

Measures of dispersion If one take a closer look at the distribution of values within a single dataset, there is a series of summations which makes it much easier to compare them with each other. These are the Sample variance, Sample standard deviation and Sample coefficient of variation: The Sample variance basically describes how much every value in the dataset in average varies from the overall average of the dataset:

The Sample standard deviation is calculated as the square root of the sample variance, and is therefore also describing how much it varies from the overall average of the dataset. The advantage of this is that the square root offsets the square in the formula, making the value directly comparable to the sample mean:

Finally the Sample coefficient of variation can be used to eliminate the overall differences in values, leaving the possibility of comparing the degree of variation between totally different dataset:

The result is shown in the table below: Weight of stones 1 Stone

2 Stones

Zara 140

Zara 160

[g]

[g]

Sample variance

85,92

403,79

3,98

4,09

Sample standard deviation

9,27

20,09

2,00

2,02

Ultimate Bending Strength De Cecco De Cecco Zara 180 140 160

[N/mm2] [N/mm2] [N/mm2]

De Cecco 180

[N/mm2]

[N/mm2]

[N/mm2]

2,65

3,90

4,39

3,66

1,63

1,98

2,10

1,91

Sample coefficient of variation 0,78 0,46 0,05 0,06 0,05 0,08 0,08 0,08 Table 3 – Calculated Sample variance, Sample standard deviation and Sample coefficient of variation (own production)


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Measure of asymmetry and peakness A third way to investigate our results is by looking in to the degree of skewness, g1 and kurtosis, g2. Which indicates, how much the distribution is “leaning” (level of asymmetry from the sample mean), and how much the distribution is peaking:

The result is shown in the table below: Weight of stones

Sample skewness Sample kurtosis

1 Stone

2 Stones

[g]

[g]

2,51

0,45

Zara 140

Zara 160

Ultimate Bending Strength De Cecco De Cecco Zara 180 140 160

[N/mm2] [N/mm2] [N/mm2]

De Cecco 180

[N/mm2]

[N/mm2]

[N/mm2]

0,17

-0,97

0,08

11,25 3,12 1,77 1,69 3,87 2,41 4,32 Table 4 – Calculated Sample skewness and Sample kurtosis (own production)

1,65

-0,46

-0,30

-0,16


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Model selection Now that data are collected on the different random variables, it has to be exploited in order to design the spaghetti bridge. For this, it is necessary to assign suitable common distribution families to the sample data.

Probability papers In order to decide on which distributions were most fitted to each random variable, probability papers were used. Example of a probability paper can be seen in Figure 10 : sample values on the x-axis, related probability on the y-axis: The probability papers are useful tools for establishing the most suitable distribution to sample data.

Figure 10 – Example of probability papers

After sorting the data in ascending values and assigning to each sample value (x1,x2,..,xi,..xn) its quantile [qi=i/(n+1)], the sample data are placed on a probability paper, with the coordinates as (xi,qi). The scale of the y-axis (and that of the x-axis as well for the lognormal distribution) of a probability paper is plotted in accordance with the common probability distribution it is designed for. If the sample data are placed on a straight line on it, the distribution related to the probability paper is adapted to our sample data.

Method After the verification that Matlab “probplot” function was appropriate for our purpose, it was decided to use that function to draw for each of the random variable probability papers related to the normal, lognormal, exponential, Gumbel max and min distributions This was done by creating for each random variable two vectors: x1=(x1,x2,..,xi,..xn) x2=(q1,q2,..,qi,..qn) Then using probplot ('distribution',[x1 x2]) to plot the probability paper. The term distribution was replaced with:  “normal” for the normal distribution  “lognormal” for the lognormal distribution  “exponential” for the exponential distribution  “extreme value” for the Gumbel min distribution.  For Gumbel max distribution, the following function was used: probplot ('extreme value',[-x1 ,x2])


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The decision to use Matlab to plot the probability paper was based on the accuracy of the method, and the fact that it permitted to try all the distributions in a reduced span of time.

Results Below are displayed the selected distributions from the probability papers that appeared to result in the most linear (value-quantile) curves:  One stone weight: lognormal distribution  Two stone weight: normal and lognormal distributions  Zara bending strength: normal and lognormal distributions  Cecco bending strength: normal and lognormal distribution

Figure 11 – One stone weight lognormal distribution probability paper (own production)

Figure 12 – Two stones weight normal distribution probability paper (own production)


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Figure 13 – Two stones weight lognormal distribution probability paper (own production)

Figure 14 – Zara breaking strength normal distribution probability paper (own production)


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Figure 15 – Zara breaking strength lognormal distribution probability paper (own production)

Figure 16 – Cecco breaking strength normal distribution probability paper (own production)

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Figure 17 – Cecco breaking strength lognormal distribution probability paper (own production)


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Model parameter estimation To further establish the probabilistic models related to the studied random variables and have a usable mathematical model of the variables, an estimation of the relevant parameters of the distributions has to be conducted. Two methods were used for this purpose.

Method of Moment The Method of Moments (MOM) bases its parameter estimation on the values of the descriptive statistics established on the sample data. It equals the "experimental" moments from the sample data to the "theoretical" moments, defined mathematically from the model. The estimates of the parameters are derived from the calculations. 

Normal distribution

Two parameters have to be estimated: and . Therefore we equal the first two moments : By definition of the two first moments, we have:

Therefore: Where 

and

are the moments of the sample data.

Lognormal distribution

Two parameters also has to be estimated

Therefore:

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Maximum Likelihood Method The Maximum Likelihood Method is based on the probability of obtaining the sample values that resulted from the experiment. The main tool in this method is the likelihood function, which is the joint probability density of the considered n samples. The likelihood function is therefore a function of the parameters of the distribution in question. The estimates of the requested parameters are the ones maximizing the likelihood function. 

Normal distribution

The corresponding likelihood function

is as follows:

The log-likelihood function is then defined because it is easier to handle :

Then the estimators are found by finding the maxima of the likelihood function, solving the following system of equations :

For the normal distribution, the estimators of the parameters obtained are :

The estimators for the parameters of the normal distribution from the two methods are the same.  Lognormal distribution The same steps are followed for the lognormal distribution, with:

The calculations yield:


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Estimation of the parameters Using the results above, the parameters from the distribution chosen in the paragraph Model Selection, are estimated from the results of the descriptive statistics.

Random variable

one stone weight(g)

Probability Distribution

lognormal

normal two stones weight(g) lognormal

normal zara ultimate breaking strength(N/mm²)

lognormal

normal cecco ultimate breaking strength(N/mm²)

lognormal

( MOM) 11,94 (estimated from

9,269414 (estimated from

43

20.09456

43,36 (estimated from

20,09456 (estimated from

35

2.752647

35 (estimated from

2.752647 (estimated from

25

2.13838

25 (estimated from

2.13838 (estimated from

(MLM)

(MOM)

(MLM)

2,285338

0.686793

0.579461

3,653615

0,441086

0,510152

3,558855

3,558838

0,078015

0,078311

3,229432

3,229358

0,084188

0,085459

2.244052

3.672259

Table 5 – Parameters estimators (own production)

The estimators from the MOM and MLM methods are consistent with each other, which demonstrate the correctness of the expressions of the estimators. Moreover, the mean value and standard deviations given from the normal and lognormal distributions are quite similar, using the following formulas:


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The results are similar to the one from the normal distribution hypothesis, which is logical from the methods used to calculate the estimators. Indeed, the two equations above were used for the MOM. The consistency of the method is thus demonstrated. If one compares the estimators of the normal distribution for the ultimate breaking strength for the two pasta brands, Zara brand seems to provide stronger spaghetti, but with a higher dispersion shows by the the higher standard deviation. Therefore, Cecco brand spaghetti appears to be more reliable to work with for the assignment.


SPRING 2013 – GROUP 13 Model verification In the previous step the probabilistic model was established, according to several random variables as stone weight and bending strength. Furthermore this probabilistic model has to be verified. Consequently the sample data and the probabilistic model have to match. This condition can be verified with graphical methods. With graphical representation the data can be assessed, communicated to the others and verified. Three types of graphical representation were used. Namely quantile-quantile plots (Q-Q Plots), probability-probability plots (P-P Plots) and cumulative distribution function (CDF). For 1 stone data set the lognormal distribution was chosen. This decision was verified with the CDF graphical method, showed in Figure 18.

Figure 18 – Lognormal distribution for one stone weight CDF (own production)

As the Figure 18 shows the 1 stone weight probability distributions follows the lognormal distribution, according to the lognormal CDF. Consequently the decision was correctly made. In case of 2 stone weights the normal distribution was chosen. This was verified with normal CDF and PP plot as well. Both graphical methods prove that the data follows the distribution line.

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Figure 19 – Two stone PP plot (own production)

Figure 20 – Two stone weight PP plot (own production)

Furthermore the distribution for the bending strength of the two different types of spaghetti was verified. For “Zara” the normal distribution was chosen, as well as for “Cecco”. Both were verified and compared to each other.

Figure 21 – Normal distribution CDF "Cecco" (own production)


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Figure 22 – Normal distribution CDF "Zara" (own production)

Figure 21 and Figure 22 shows that the chosen distributions are following normal distribution CDF function, but the “Cecco” spaghetti shows a better fit around the mean values and around the lower and upper tails as well, according to the shown CDF. As the deviations from the normal CDF is less for the “Cecco” spaghetti brand, this is going to be the chosen material in the further investigations. For the chosen spaghetti type, namely “Cecco” the verification was made with a Q-Q plot as well.

Figure 23 – "Cecco" Q-Q plot verification (own production)

As the above figure shows, there are just small deviations both in upper and lower part, but generally the data set, namely the bending strength of the spaghetti is following the distribution line. The deviations are so small that they can be neglected. Consequently the decision shows to be correct. As a conclusion the chosen distributions were verified. Furthermore it can be stated, that the two stone weights, and the bending strength of the two spaghetti types follow the normal distribution, whereas the one stone weight is following the lognormal distribution. Between the two bending strength data the “Cecco” was chosen, according to a better fitting to the CDF. Each probabilistic model was verified with more than one graphical representation, the most representative ones were presented above.


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Formulation of design criteria Formulation of limit state function Introduction In the previous sections a distribution of the random variables has been determined. According to the performed analysis the “Cecco” pasta was chosen because the cumulative density function seemed to fit better then the “Zara” pasta. The results obtained until now are: RANDOM VARIABLE 1 STONE

DISTRIBUTION FUNCTION Lognormal

2 STONES

Normal

CECCO

Normal

PARAMETERS ESTIMATION (M.O.M) (M.O.M) 25 2.138

Table 6 – Summary of distribution functions of the random variables used (own production)

In order to verify our spaghetti bridge 2 stones should be applied on it. But which of the distribution functions to use: the distribution functions of the weight of two stones or the sum of the distribution functions of the weight of one stone? To decide which one to use the convolution method has been used.

Convolution At the beginning of the experiment both samples of the weight of one and two stones respectively were collected. The collecting procedure for the two stones was carried out by picking up two stones “randomly” from the bag. With the convolution now we try to model the probability distribution function of the weight of two stones by the weight of one stone and see if the two models are close enough, that it means that our samples are random enough. As there is no correlation between the two stones they are considered to be independent. The weight of two stones is:

The cumulative distribution function of

can be written as:

and the probability density function of

can be written as:

where and are the weight of the first stone and the second stone respectively and and the respective probability density functions and is called convolution. Using the Matlab script available on “Supplement convolution” the probability density function of one stone, two stones and the convolution has been plotted:


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Figure 24 – Comparison between the probability density function of the weight of two stones (red) and the convolution (blue). The dashed line is the probability density function of the weight of one stone (own production)

It can be noted that there is a discrepancy between the two plots. In particular for low values of the weight the graph obtained with convolution has a lower probability of failure than the two stones. But when the higher values of the weight are considered, on the right side of the graph, the situation is the opposite: there is a higher probability of failure for the convolution model than the one of two stones. Moreover this difference becomes higher in the lower part of the curve, around 40g that is the part of the graph used in many engineering applications. This difference between the two density function shows that our way of choosing the two stones wasn’t random enough. Therefore the convolution function has been chosen for the bridge failure estimation.

Structural reliability theory The aim of reliability analysis is to evaluate the probability of failure of any state which leads to losses in terms of costs or lives, damages of buildings impact to the environment. In many engineering applications it is crucial to predict the probability of failure related to a certain event. In our case it is necessary to evaluate the probability of our spaghetti bridge to fail under the load of two stones. The ways to compute the probability of failure can be done by:    

Analytical solution Numerical integral First Order Reliability Method (FORM) Monte Carlo Techniques

In this case, the FORM method and the Monte Carlo simulation will be compared using the same length of the spaghetti bridge.


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The Limit State Function (LSF) Generally the probability of failure can be expressed as:

where:  

is the resistance random variable is the failure random variable

A failure event is than described by a functional relation that is the limit state function (LSF) are the realizations of the basic random variables influencing the probability of failure. The probability of failure is also expressed as:

where

is the failure domain which is defined as :

where

In the case of the spaghetti bridge the function Method provides an approximated solution. In the case of the spaghetti bridge:  

is not linear. Then the First Order Reliability

is the resistance of failure of the Cecco spaghetti bridge (N/mm2) is the stress generated by the weight of two stones (N/mm2)

Figure 25 – Determination of the stress due to the load of two stones (from course material)

The limit state function is:

In this case the random variables are not normal because the probability density function of the two stones is given by the sum of the probability density function of one stone. The probability of failure cannot be solved analytically but the Matlab script is used (Appendix


SPRING 2013 – GROUP 13 A.4 Matlab code – Limit State Function, FORM). The Matlab script “Sample_FORM.m” has been modeled according to our case. In this script file the LSF is standardized and the β parameter has been calculated. Finally the probability of failure is obtained. The aim is to find the length of the bridge so that the probability of failure is 10%. Making an iterative calculation in the Matlab file the length of the spaghetti was found to be .

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Experimental verification of the model After the analytical verification of the calculated failure, the bridge has to be verified experimentally. It is expected to fail 10 times from the 100 trials. The length of the bridge according to the FORM model is 151 mm, the experimental equipment as the bridge, stones and spaghetti package are the same as the initial data set. The bridge was extended with the new length. It is verified with 2 randomly picked stones. To achieve randomness the stones were put on the table as shown in the Figure 26, and with the help of an online random number generator the stones were picked. The experiment has two possible outcomes either the spaghetti bridge breaks, either it does not. All the 100 trials were carried out independently from each other. It was expected to have 10 failed trials, whereas the experiment showed 13 broken trials. From the collected data the probability of failure was calculated.

Figure 26 – Experimental verification (own production)

The probability of failure was calculated after 100 trials as follows.

Where: Pf - The probability of failure, nf - The number of failed experiments, N - Total number of experiment As a conclusion the number of failure it was exceeded with 3% from the expected probability of failure of 10%. This deviation it was checked by the hypothesis testing if it is acceptable or not. In the following chapter it is represented that this deviation is in the confidence interval, consequently the design can be adopted. Although the final results are not based on this test, the experiment outcomes obtained here are in accordance with the Monte Carlo simulation.


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Design verification The Monte Carlo Simulation After the statistically optimal distance between the supports have been found through the FORM, it is now possible to test the results. Besides a practical test of spaghetti in our own setup, there are a number of options to statistically test the likelihood of collapse. One possibility which will be used in this project is the technique the Monte Carlo Simulation. Monte Carlo is about looking at the the outcome of a simulation in which the length between the supports set to the obtained 151 mm, and the ultimate bending strength is tested against the weight of the two stones. As the values of these three variables is based precisely on the measured mean and standard deviations, the probability of failure should hit exactly the desired 10%, when the simulation is carried out a sufficient number of times (at least 10.000). When the Monte Carlo Simulation is performed in Excel, the function NORMINV(probability,mean,standard_dev) for the ultimate bending strength is used, as this follows the Normal distribution, while the function LOGINV(probability,mean,standard_dev) used for the stones' weight, as they follow the Lognormal distribution. Below a sample of the sheet shows how the Monte Carlo Simulation is conducted with 65,000 test run (the maximum number in Excel). The red boxes indicate that the bridge collapses.

Table 7 – Monte Carlo Simulation, Excel sheet (own production)


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As seen in the sheet in Table 7 and the graph in Figure 27 the probability converges towards 0.150. This means that there in the Monte Carlo Simulation is a collapse in 15% of the cases and not 10%, as desired. Whether this result is acceptable or must be rejected will be examined in the following section.

Figure 27 – Monte Carlo Simulation, Graph of probability in relation to the number of test runs (own production)


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Hypothesis testing In order to evaluate if the assumptions and results are correct the hypothesis testing has been made. It is essential that the uncertainty present in the assumptions and calculation methods are quantified. The approach used here is the use of confidence intervals, that is the interval 1-α within our results are correct. The first step in this procedure is to formulate a null-hypothesis expressing the true mean value of the spaghetti strength that is 10 as the request is to obtain that over 100 trials less than 10 are failures. The next step is to formulate an operating rule according to which the null hypothesis can be accepted or rejected. The Bernoulli distribution has been chosen as the representative probability density for this case. It is the probability distribution of the number of success in a number of trials n each of which yields a success with a probability p. In the spaghetti bridge case the aim is to find that in a determined number of trials n=100, the success with a probability of failure of 10%:

α is the amplitude of the confidence interval. α can be assumed equal to 0.1, so α . This shows the amplitude of the confidence interval within which the number of failure is less than . If the numbers of trials where the failure are obtained are within this interval the assumptions are correct. The procedure used for the estimation of the parameter is shown in Figure 25. As the Bernoulli distribution is a discrete probability distribution the range of the intervals cannot be estimated precisely. Using a value of a=4 the confidence interval is:

Nr. of trials

False True 1 0.000295127 0.000322 2 0.001623197 0.001945 3 0.005891602 0.007836 4 0.015874596 0.023711 5 0.033865804 0.057577 6 0.059578729 0.117156 7 0.088895246 0.206051 8 0.114823027 0.320874 9 0.130416277 0.45129 10 0.131865347 0.583156 11 0.119877588 0.703033 12 0.098788012 0.801821 13 0.074302095 0.876123 14 0.051303827 0.927427 15 0.032682438 0.960109 16 0.019291717 0.979401 17 0.010591531 0.989993 18 0.005426525 0.995419 19 0.002602193 0.998021 20 0.001170987 0.999192 21 0.000495656 0.999688 22 0.000197762 0.999886 23 7.45189E-05 0.99996 24 2.65646E-05 0.999987 25 8.97293E-06 0.999996 26 2.87594E-06 0.999999 Table 8 – Confidence interval (own production)


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binomial distribution 1-α=90% 11,99% 13,19% 13,04% 11,48%

0,14

0,06

0

1,59% 0,59% 0,16% 0,03%

0,02

0,02% 0,05% 0,12% 0,26% 0,54% 1,06% 1,93% 3,27%

3,39%

0,04

binomial distribution 5,13%

5,96%

0,08

7,43%

8,89%

0,1

9,88%

0,12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Figure 28 – Bernoulli distribution in red the confidence interval is shown (own production)


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Discussion In order to validate the results of the initial investigation, the following two tests was carried out: The experimental test and The Monte Carlo Simulation both based on the FORM model. There was a slight difference between these two, but it can be due to the choice made in the visual comparison according to the probability papers which might not be the best one. In addition the convolution showed that the stones were not randomly picked, so in the FORM model the convolution function has been chosen for the bridge failure. The deviation can be a result in the natural randomness in the stones and spaghetti material. In addition as it was stated in lecture 10, in the calculation of the probability of failure by FORM, the physical test seems to have a very high percent of approximation error, due to the approximations of the non-linear limit state function with a linear one for the investigated case. The decision to use the Monte Carlo outcomes was made because it seemed to be more reliable as the amount of trials used in the verification was 65.000 whereas in the experimental model were only carried out 100 trials. On top of that, the uncertainties which is lying within the spaghetti test are several and not always predictable. One example could be the way that we dropped the stones in the cup or whether some of the spaghetti would have broken anyway, if we waited a little bit due to long term deformation. Therefore, we believe that the Monte Carlo Simulation gives a more accurate estimation of the probability of failure of the spaghetti bridge. The process used in the FORM simulation itself even if it is faster by the computing point of view compared with the Monte Carlo Simulation, leads to some errors that they will be reflected in the final result of the spaghetti length. Therefore even if in the Monte Carlo simulation gives the probability of failure at 15% and which is out of the range of the confidence interval it can still be assumed that it is not too far to be rejected. In addition it can be said that since the probability of failure of a bridge is in the order of amplitude of 1/ , moving from 10% to 15% of the probability of failure doesn’t affect the results significantly. Concerning the hypothesis testing it can be said that the confidence interval has been calculated with the Bernoulli distribution which is a discrete distribution. Therefore the actual range is wider.


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Conclusion The aim of this report was to design a spaghetti bridge with a 10% probability of failure under the weight of two randomly picked stones from a group of 100 known stones. This design was made in three phases started with initial measurements of our building material (the spaghetti) and the load (the stones), followed by the probabilistic modeling and reliability based design and ended with a verification of the bridge. Going through these steps it was found that the weight of one stone is following the lognormal distribution, while both the weight of two stones and the bending strength is following the normal distribution. After running the tests and calculations on both brands of spaghetti, we chose the “Cecco” based on the CDF graphical representation with a length of 151 mm, found by the First Order Reliability Method, FORM. After the length was found, we ran the physical test with 2 stone picked totally randomly. The result of this based on 100 trials was a failure rate at 13%. The probability was also estimated according to the Monte Carlo simulation, which gave a probability of failure of 15% based on 65.000 runs. It can be concluded that even though the final result has a deviation from the 10%, it can’t be rejected as a valid estimation of the desired structure.


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Appendix A.1 Weight of the stones Weight of 1 stone (w1) [g] 25 27 19 22 36 23 15 39 62 39 13 29 20 33 23 30 19 21 13 13 21 16 11 10 15 11 13 10 11 14 12 15 18 9 8 11 8 11 9 9 7 6 7 6 8 9 13 11 7 11

Weight of 2 stones (w2) [g] 52 34 101 27 54 52 47 82 36 66 41 69 72 30 58 63 48 61 72 36 74 38 57 71 35 74 41 50 40 36 32 38 32 12 70 36 38 13 70 36 47 25 43 55 16 43 51 36 46 69

12 9 7 10 7 9 8 8 6 9 7 10 8 9 7 7 7 7 9 6 5 7 8 5 9 7 11 7 8 7 5 5 9 5 3 5 7 7 7 4 5 5 6 6 7 5 5 4 5 5

22 52 41 43 95 63 60 39 66 25 36 54 38 45 47 33 15 40 35 43 25 44 16 17 51 29 38 37 45 24 39 13 18 35 33 81 13 34 48 39 20 65 13 95 30 24 44 28 12 9


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A.2 Measured Ultimate breaking load Zara 140 [g] 51 52 48 44 50 50 52 51 45 45 51 47 51 50 51 52 46 46 47 49

Zara 160 [g] 37 36 38 36 40 42 39 40 41 37 41 42 37 41 42 41 38 43 42 43

Zara 180 [g] 29 32 34 34 33 34 32 37 31 34 33 34 34 32 34 32 32 32 35 34

Cecco 140 [g] 38 44 39 41 36 40 41 36 36 32 37 35 39 37 37 39 35 35 43 42

Cecco 160 [g] 34 27 36 34 39 37 36 37 33 38 38 36 34 35 33 34 31 36 39 34

Cecco 180 [g] 29 30 27 29 26 31 26 31 32 27 26 26 27 30 28 33 31 31 28 32


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A.3 Ultimate bending strength – Cecco (The chosen brand) Cecco ultimate bending strength 2 [N/mm ] 20.33 21.08 22.03 22.03 22.03 22.03 22.87 22.87 22.87 23.06 23.06 23.06 23.34 23.72 23.72 23.72 23.72 23.72 24.38 24.38 24.38 24.57 24.57 24.85 24.85 25.04 25.41 25.41 25.60 25.60 25.60 25.60 25.60 25.70 25.70 25.70 26.26 26.26 26.26 26.26 26.36 26.36 27.01 27.01 27.11 27.11 27.11 27.11 27.11 27.11 27.67 27.86 27.86 27.96 28.33 28.61 28.61 28.99 29.37 29.37

rank [-] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


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A.4 Matlab code – Limit State Function, FORM function out = LSF(x,a) % a is a design parameter out = x(1) -9.81*a*(x(2)+x(3))/(1000*pi*0.87^3); function out = GradLSF(i,x,a) % i indicates partial derivative with respect to xi delta = 1e-4; % Change, if necessary xd = x; xd(i) = x(i)+delta; out = (LSF(xd,a)-LSF(x,a))/delta;

Variable definition dist_X1 = 'norm'; mean_X1 = 25.3547; std_X1 = 2.1382; dist_X2 = 'logn'; log_median_X2 = 2.244051543; log_std_X2 = 0.686793368; dist_X3 = 'logn'; log_median_X3 = 2.244051543; log_std_X3 = 0.686793368; Parameters definition Distribution parameters 'Define by yourself' Dist_X{1} = dist_X1; % 'norm' Dist_X{2} = dist_X2; % 'logn' Dist_X{3} = dist_X3; % 'logn' ParamA_X = [mean_X1; log_median_X2; log_median_X3];% log_median_X4]; ParamB_X = [std_X1; log_std_X2; log_std_X3];% log_std_X4]; % Parameter (which you can use to represent design parameter) r=0.87; %pasta diameter (mm) I=pi*r^4/4; %inertia moment b1=(r/(4*I))* 9.81/1000; a = 151; %length % Initial checking point vector x = [mean_X1; exp(log_median_X2); exp(log_median_X3)];% exp(log_median_X4)]; % Mapping of x into U-space u(1,1) = norminv(cdf(Dist_X{1},x(1),ParamA_X(1),ParamB_X(1))); u(2,1) = norminv(cdf(Dist_X{2},x(2),ParamA_X(2),ParamB_X(2))); u(3,1) = norminv(cdf(Dist_X{3},x(3),ParamA_X(3),ParamB_X(3))); % Setting for iteration du = 999; eps = 1e-3; Niter = 0; while du > eps % Change the criterion, if necessary Niter = Niter + 1 % Matrix J J = zeros(3,3); J(1,1) = pdf(Dist_X{1},x(1),ParamA_X(1),ParamB_X(1)) / normpdf(u(1)); J(2,2) = pdf(Dist_X{2},x(2),ParamA_X(2),ParamB_X(2)) / normpdf(u(2)); J(3,3) = pdf(Dist_X{3},x(3),ParamA_X(3),ParamB_X(3)) / normpdf(u(3)); % Value of the limit state function in U-space c = LSF(x,a);


SPRING 2013 – GROUP 13 % Gradients of the limit state function with respect to x G1 = GradLSF(1,x,a); % del_G/del_x1 G2 = GradLSF(2,x,a); % del_G/del_x2 G3 = GradLSF(3,x,a); % del_G/del_x3 Gx = [G1;G2;G3]; % Directional cosines Alpha = J \ Gx; l = sqrt(sum(Alpha.^2)); Alpha = Alpha / l; % Reliability index, beta beta = - u' * Alpha; % Save current design point uold = u; % New design point in U-space u = - Alpha * (beta + c/l); % Mapping of u into original space x(1,1) = icdf(Dist_X{1},normcdf(u(1)),ParamA_X(1),ParamB_X(1)); x(2,1) = icdf(Dist_X{2},normcdf(u(2)),ParamA_X(2),ParamB_X(2)); x(3,1) = icdf(Dist_X{3},normcdf(u(3)),ParamA_X(3),ParamB_X(3));

% L1-norm of change in u du = abs(u-uold); end out{1} = x; out{2} = Alpha; out{3} = beta; out{4} = normcdf(-beta); 'finished'

A.5 Matlab code – Convulation Parameter estimation for CDF one stone ls1=λ fis1=ξ S1=weight of 1 stone ls1=2.244051543; fis1=0.686793368;

Convolution dx = 0.1; x = 0:dx:100; N = length(x); fx = pdf('logn',x,ls1,fis1); % Lognormal, median = ls1, COV=fis1 w = conv(fx,fx); g=pdf('norm',x,mean(S1),std(S1)); fy = w(1:N)*dx; figure(1); hold on plot(x,fy); plot(x,fx,':'); plot(x,g,'r'); title ('CONVOLUTION'); xlabel('weight [g]');

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Probabilistic Modeling 2013 – Group 13