APPLICATION OF PROBABILTY AND STATISTICS IN CIVIL ENGINEERING

By -

MOHIT DWIVEDI (12BCL0172)

PRAVEEN VISVALINGAM (12BCL0262)

PORSELVAN (12BEM0109)

INTRODUCTIONThe course covers: Quantitative analysis of uncertainty and risk for engineering applicationsFundamentals of probability, random processes, statistics, and decision analysis, along with random variables and vectors, uncertainty propagation, conditional distributions Second-moment analysis. System reliability Bayesian analysis and risk-based decision Estimation of distribution parameters Hypothesis testing Simple and multiple linear regressions Poisson and Markov processes

RELIABILITY OF A BUILDING UNDER EXTREME WIND LOADS - CHOOSING THE DESIGN ACCORDING TO WIND SPEED

Consider the problem of designing a tall building for a certain level of reliability against wind loads. The building has a planned life of N years.

Let Vi be the maximum wind speed in year i at the site of the building variables V1, ..., VN may be considered independent and identically distributed with some common cumulative distribution FV(v).

Empirical data indicate that, at most locations, the distribution FV(v) has the form –

FV(v)=e^(−(v/u))^−k (1)

where u and k are positive parameters.

In Boston, plausible values of u and k are u = 49.4 mph and k = 6.5. Therefore, for Boston –

FV(v)=e^(−(v/49.4))^−6.5 (2) where V is in mph.

Consider now the problem of choosing the design wind speed v* for a building in Boston, such that the probability of non-exceeding v* during the design life of the building equals a target reliability value. The probability that any given v* is not exceeded in N years (the reliability of the building) is Rel(v*,N) and may be calculated as a function of v* and N as: Rel(v*,N)=[P(V1≤v*)∩(V2≤v*)∩...∩(VN≤v*)]=(FV(v*))^N

=e^(−N(v*/49.9))^−6.5 (3)

Reliability of a building in Boston against wind as a function of design wind speed v*, for exposure periods of 10 and 50 years

Conclusions For any given N, the reliability approaches 1 quite

slowly, meaning that one must design for very high wind speeds in order to attain high safety levels.

The reliability of the building against wind loads is conservative, because exceeding v* does not necessarily imply serious damage to the building. Typically, the actual strength of a building is much greater that the nominal strength, due to built-in conservatism in design codes, nominal material properties, and good construction practice. Therefore, the actual reliability may be much higher that the probability in Eq. 3.

APPLICATIONS OF POISSON

DISTRIBUTION 

Definition

A discrete stochastic variable X  is said to have a Poisson distribution with parameter λ > 0, if for k = 0, 1, 2, ... the probability mass function of X  is given by:

where e is the base of the natural logarithm (e =

2.71828...) k! is the factorial of k.

Poisson distribution The distribution was derived by the French

mathematician Siméon Poisson in 1837, and the first application was the description of the number of deaths by horse kicking in the Prussian army.

Properties

The expected value of a Poisson-distributed random variable is equal to λ and so is its variance.

The coefficient of variation is   , while the index of dispersion is 1.

 Poisson distribution 

Poisson distribution has several applications in civil engineering.

In earth quake engineering, in a specific time interval the probability of occurrence of an earth quake at a particular fault follows Poisson distribution.

Occurrence of cyclones in a particular time period follows Poisson distribution.

Example 1 From the past experience it is known that

on an average every two years 3 cyclones hit the coastal area of Andhra Pradesh and Orissa states.  If it is assumed that the cyclone hitting the coastal areas follows Poisson distribution then what is the probability of two cyclones crossing the coastal area of Andhra Pradesh and Orissa in the next two years?

THANK YOU !