1. Statistical Analysis of Loads
Chapter 5
Statistical Analysis of Loads

2. Chapter 5: Statistical Analysis of Loads
Contents
5.1 Loads and Actions
5.2 Probability Models of Loads
5.3 Statistical Analysis of Loads
5.4 Representative Values of Loads
5.5 Combination of Load Effects

3. Chapter 5 Statistical Analysis of Loads
5.1 Loads and Actions

4. 5.1.1 Definitions of Loads and Actions
An action is:
An assembly of concentrated or distributed mechanical forces acting on a structure (direct actions), or
The cause of deformations imposed on the structure or constrained in it (indirect actions)
Action
Direct Action — Load
Indirect Action
Types of Actions
1. Classification according to the variation of their magnitude with time
Permanent action, which is likely to act continuously throughout a given reference period and for which variations in magnitude with time are small compared with the mean value.

5. 5.1 Loads and Actions …2
Variable action, for which the variation in magnitude with time is neither negligible in relation to the mean value nor monotonic.
Accidental action, which is unlikely to occur with a significant value on a given structure over a given reference period.
2. Classification according to their variation with space
Fixed action, which has a fixed distribution on a structure;
Free action, which may have an arbitrary spatial distribution over the structure within given limits.
3. Classification according to the structural response
Static action, i.e. not causing significant acceleration of the structural or structural elements;
Dynamic action, i.e. causing significant acceleration of the structural or structural elements.

6. Chapter 5 Statistical Analysis of Loads
5.2 Probabilistic Models of Loads

7. 5.2 Probability Models of Loads …1
Stochastic Process Model of Loads
Actually, loads are random variables varying with time in the design reference period i.e. loads are random process.
In general, loads are treated as stationary binomial random process.
Assumptions of Stationary Binomial Random Process
(1) The design reference period is divided into r equal intervals
(2) In each interval, the probability of the load Q occurring is p, while the
probability of not occurring is
(3) In each interval, when the load Q occurs, its magnitude is a non-
negative random variable, and its probability distributions during
different intervals are identical. Let the probability distribution of
the load Q in interval be denoted by

8. 5.2 Probability Models of Loads …2
The function is called an arbitrary point-in-time probability distribution of the load Q.
(4) The magnitudes of the load during different intervals are independent random variables, and they are also independent of the event that the load occurs in these intervals .
Features of Stationary Binomial Random Process
(1) The parameters of the stationary binomial random process are:
The parameters and are determined by statistical surveys or experiential judgments.
The distribution type of should be validated by K-S test.

9. 5.2 Probability Models of Loads …3
(2) The sample function of can be represented by a rectangle wave
function with equal intervals.

10. 5.2 Probability Models of Loads …4
Random Variable Model of Loads
1. Principle of Transformation
Random Process Load Model
Random Variable Load Model
The load Q is represented by the maximum value of the random process load during the design reference period .
Obviously, the value is a random variable.
2. The Probability distribution of
represents the mean times of occurring in the design reference period T .

11. 5.2 Probability Models of Loads …5
being normal distribution

12. 5.2 Probability Models of Loads …6
being Extreme Ⅰ distribution

13. Chapter 5 Statistical Analysis of Loads

14. 5.3 Statistical Analysis of Loads …1
Statistical Analysis of Permanent Load

15. 5.3 Statistical Analysis of Loads …2
Statistical Analysis of Variant Loads
1. Sustained Live Load

16. 5.3 Statistical Analysis of Loads …3
2. Transient Live Load

17. 5.3 Statistical Analysis of Loads …4
Statistical Analysis of Environmental Loads
1. Wind Load
Don’t consider wind direction
Consider wind direction

18. 5.3 Statistical Analysis of Loads …5
2. Snow Load

19. Chapter 5 Statistical Analysis of Loads
5.4 Representative Values of Loads

20. 5.4 Representative Values of Loads …1
Representative Value v.s. Design Value of a Load
1. Representative Value of a Load
The representative value of a load is a value used for the verification of a limit state.
Representative values generally consist of characteristic values, frequent values, quasi-permanent values, combination values.
2. Design Value of a Load
Design value of a load is a value obtained by multiplying the representative value by the partial factor

21. 5.4 Representative Values of Loads …2
Characteristic Value
1. Definition
The characteristic value of a load is the maximum value of the load that acts on the structure during the design reference period.
It is the principal representative value when designing structures. The other representative values are obtained by conversion of the characteristic value.
It is used in both ultimate limit state verification and serviceability limit state verification
It is chosen
either on a statistical basis, so that it can be considered to have a special probability of not being exceeded towards unfavorable values during the design reference period;
or on acquired experience;
or on physical constraints.

22. 5.4 Representative Values of Loads …3
2. Methods
(1) Determined by the return period of a load
where,
is called ( mean ) return period.
where,
is called yearly exceedance probability.

23. 5.4 Representative Values of Loads …4
2. Methods …
(2) Determined by the percentile of
where,
is the probability of not being exceeded during the reference period, it is also called “guarantee probability”.

24. 5.4 Representative Values of Loads …5
Example 5.1
Please refer to the textbook “Structural Reliability” by Professor Ou & Duan.
Turn to Page 15, look at the example 1.3 carefully!

25. 5.4 Representative Values of Loads …6
Frequent Value
1. Definition
The frequent value of a load is the frequent occurring load that acts on the structure during the design reference period.
It is the representative value of variable loads when checking the serviceability limit state by the frequent (short term) load effect combination rule.
2. Method

26. 5.4 Representative Values of Loads …7
Quasi-permanent Value
1. Definition
The quasi-permanent value of a load is the often occurring load that acts on the structure during the design reference period.
It is the representative value of variable loads when checking the serviceability limit state by the combination rules of quasi-permanent value combination and frequent value combination.
2. Method

27. 5.4 Representative Values of Loads …8
Combination Value
1. Definition
When two or more loads act on the structure during the design reference period, the maximum values of these loads cannot occur simultaneously, then the representative values of loads can be taken as its combination values
It is chosen so that the probability that the load effect values caused by the combination will be exceeded is approximately the same as when a single load is considered.
2. Method

28. Chapter 5 Statistical Analysis of Loads
5.5 Combination of Load Effects

29. 5.5 Combination of Load Effects …1
Basic Concepts
The total load Q is a sum of individual load components such as dead load, live load, wind load, snow load, seismic actions, etc. They all vary with time.
When only one kind of the time-dependent loads acts on the structure, its maximum value during the reference period is then used in structural design.
When two or more time-dependent loads act on the structure, their maximum values during the reference period cannot occur simultaneously. Therefore, the load effect combination problem should be considered.

30. 5.5 Combination of Load Effects …2
It is generally assumed that the load effect is linearly related to the load:
where, is called load effect coefficient.
Therefore, the load combination problem is consistent with the load effect combination problem.
The essential point of the load combination problem is to find the probability characteristics of the maximum value of the total load effects :

31. 5.5 Combination of Load Effects …3
JCSS’s Rule for Load Effect Combination
For each load , the design reference period T is divided into equal intervals
all loads are reordered from small to large according to the values of , and let the take integer value.
For an arbitrary load , its maximum load effect during the design reference period is combined with other load effects .
For the load whose numbers of intervals is larger than , the local maximum value during the preceding load interval is taken in the order, and other load effects take their transient values.

32. 5.5 Combination of Load Effects …4
JCSS’s Rule Formula for Load Effect Combination

33. 5.5 Combination of Load Effects …5
Example 5.2
Assume that there are three loads, the interval numbers of each load are:
Give the load effect combination formula of the three loads.
Solution:
Assume that the design reference period
Rank the load according to This has been done.
The intervals of each load are:
The load effect combination results are:

34. 5.5 Combination of Load Effects …6
Example 5.3
The loads acting on the a office building structures generally include:
dead load
sustained live load
transient live load
wind load
The design reference period is
The intervals of these loads are as follows:
Give the load effect combination formula according to JCSS’s Rule.

35. 5.5 Combination of Load Effects …7
Solution:
(1) Solve the interval numbers of loads:
(2) Rank the load effects :
(3) Combine the load effects :

36. 5.5 Combination of Load Effects …8
Turkstra’s Rule for Load Effect Combination
If a load take its maximum load effect during the design reference period, then the other (n-1) loads take their transient values.

37. 5.5 Combination of Load Effects …10
Example 5.4
Please refer to the textbook “Reliability of Structures” by Professor A. S. Nowak.
Turn to Page 170, look at the example 6.3 carefully!

38. 5.5 Combination of Load Effects …9
Simple Rule for Load Effect Combination
The dead load is only combined with the maximum value of a variant load .
Example 5.5
For the problem in example 5.2, the simple rule leads to :
(1) The dead load is combined with a live load:
(2) The dead load is combined with wind load:

39. Homework 5 5.1 Solve the problem 6.1 in text book on P.179 .
Chapter 5: Homework 5
Homework 5
5.1 Solve the problem 6.1 in text book on P.179 .
5.2 Solve the problem 6.2 in text book on P.179 .

40. End of
Chapter 5