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CE 513: STATISTICAL METHODS

IN CIVIL ENGINEERING (2019)

Lecture-1: Introduction & Overview

Dr. Budhaditya Hazra

Room: N-307Department of Civil Engineering

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Schedule of Lectures• 4-5 extra classes on weekends: 5 marks bonus for full

attendance

• Grading scheme:

• Midterm 30 %

• End term: 40%

• Surprise Quizzes : 20 %

• Programming assignments: 10%

• Lectures: Wed (1-20 to 2:50) 5101;

Thurs (12-40 to 2-00 pm) 5101

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INTROCUTION

• Principle aim of design: SAFETY

• Often this objective is non-trivial

• On occasions, structures fail to perform their

intended function

• RISK is inherent

• Absolute safety can never be guaranteed for

any engineering system; a probabilistic notion

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A motivating example

F = 1 KN

EI = 10000000 Nm2

L = 2 m

mm

EI

FL

2667.0

3

3

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Uncertainties

• Can we be always certain about EI ?

• For RCC, fixing a point or a single value of E

is fraught with risks

• Are we always sure about I ? Or the dimensional

properties ? Can be risky again

• In lot of practical applications, even F cannot be known

for certain ?

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F=1+0.1*randn(100,1) ; % F is normally distributed

EI=10^7+1000*randn(100,1); % EI is normally distributed

L=2;

for i=1:100

delta(i)= F(i)*L^3/(3*EI(i));

end

Let’s consider the cantilever beam example

again, now with some uncertainties

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The displacement becomes uncertain too

0.26 0.262 0.264 0.266 0.268 0.27 0.272 0.274 0.2760

5

10

15

20

Deflection in mm

Fre

quency

of

no

of

occure

nces

Mean value = 0.2674 mm

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Practical example: Reliability based design

Stress (S) > Yield strength (R )

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Reliability example

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Probability

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Classical definition

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Sample space

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Example-1

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Issues with classical definition

• What is “equally likely”?

• What if not equally likely?

e.g.: what is the probability that sun would rise

tomorrow ?

• No room for experimentation.

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Frequency def

If a random experiment has been performed n number of times

and if m outcomes are favorable to event A, then the probability of

event A is given by

Frequency definition

Issues

What is meant by limit here?

One cannot talk about probability without conducting an experiment

What is the probability that someone meets with an accident

tomorrow?

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Notions lacking definition

Experiments

Trials

Outcomes

• An experiment is a physical phenomenon that is

repeatable. A single performance of an experiment is

called a trial. Observation made on a trial is called outcome.

• Axioms are statements commensurate with our experience.

No formal proofs exist. All truths are relative to the

accepted axioms.

Axioms of probability

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Sample space

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Axioms of probability

Rigorously speaking the

axioms of probability requires

the knowledge of measure

theory & sigma algebra

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Problems that we will study

• Random Processes: Wind, earthquake & wave loads on

engineering structures. Hydro-climatological processes like

temperature & rainfall data

• Stochastic Calculus: If 𝐹 𝑥 = 𝑥2𝑑𝐹 ≠ 2𝑥𝑑𝑥 ; where F(x)

is a stochastic function of a random process x

• Stochastic Differential Equations: 𝑑𝑥

𝑑𝑡= 𝑓 𝑥, 𝑡 + 𝐿 𝑥, 𝑡 𝑤(𝑡);

𝑤(𝑡) is a realization of white noise

• Monte Carlo Simulation

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Reference material

1. Probability, reliability and statistical methods in engineering design

by A. Halder & S. Mahadevan

1. Probability Concepts in Engineering: Emphasis on Applications to Civil and

Environmental Engineering by Alfredo H-S. Ang & Wilson H. Tang

2. Probability, Statistics, and Reliability for Engineers and Scientists by Bilal M.

Ayyub & Richard H. McCuen

4. Probability, Random Variables, and Stochastic Processes: Papoulis and Pillai

5. Probabilistic models for dynamical systems: Benaroya, Han & Nagruka