chapter 5 statistical analysis of loads. chapter 5: statistical analysis of loads 5.2 probability...

Chapter 5Chapter 5

Statistical Analysis of Loads

Chapter 5: Statistical Analysis of LoadsChapter 5: Statistical Analysis of Loads

5.2 Probability Models of Loads

5.4 Representative Values of Loads

5.1 Loads and Actions

Contents

5.5 Combination of Load Effects

5.3 Statistical Analysis of Loads

5.1 Loads and Actions

Chapter 5Chapter 5 Statistical Analysis of LoadsStatistical Analysis of Loads

5.1 Loads and Actions …15.1 Loads and Actions …1

• 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)

5.1.1 Definitions of Loads and Actions

ActionDirect Action — Load

Indirect Action

5.1.2 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.

5.1 Loads and Actions …25.1 Loads and Actions …2

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.

– 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.

5.2 Probabilistic Models of Loads


5.2 Probability Models of Loads …15.2 Probability Models of Loads …1

(1) The design reference period is divided into r equal intervals

5.2.1 Stochastic Process Model of Loads

Assumptions of Stationary Binomial Random Process

– 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.

( )Q t

/T r T

T

(2) In each interval, the probability of the load Q occurring is p, while the

probability of not occurring is . 1q p (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

( ) [ ( ) , ]QF x P Q t x t ≤


The function is called an arbitrary point-in-time probability distribution of the load Q.

( )QF x

(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:

/T r p

( )QF x

– The parameters and are determined by statistical surveys or experiential judgments.

p

( )QF x– The distribution type of should be validated by K-S test.


( )Q t

( )Q t

t

(2) The sample function of can be represented by a rectangle wave

function with equal intervals.

Tr


5.2.2 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 .( )Q t T

0max ( )Tt T

Q Q t≤ ≤

– Obviously, the value is a random variable.TQ

( ) [ ( )]T

mQ QF x F x

m pr represents the mean times of occurring in the design reference period T .

( )Q t

2. The Probability distribution of TQ( )TQ

F x


3. being normal distribution( )QF x2

1 1( ) exp

22

x QQ

QQ

yF x dy

2

1 1( ) [ ( )] exp

22T

T

TT

x QmQ Q

QQ

yF x F x dy

4

13.5(1 )

TQ Q Qm

4T

QQ

m


4. being Extreme distributionⅠ( )QF x

( ) exp exp QQ

Q

x uF x

( ) [ ( )] exp exp T

T

T

QmQ Q

Q

x uF x F x

TQ Q

lnTQ Q Qu u m

TQ Q

ln

1.2826T

QQ Q

m

0.5772Q Q Qu

1.2826Q

Q

5.3 Statistical Analysis of Loads


5.3 Statistical Analysis of Loads …15.3 Statistical Analysis of Loads …1

5.3.1 Statistical Analysis of Permanent Load

(1.06 ,0.074 )K k kG N G G G

t T0

1, 1,p r T 1.06G kG

0.074G kG

/ 1.06G G kG

/ 0.07G G GV

( ) [ ( )] ( )T

mG G GF x F x F x

1 1T

m prT


5.3.2 Statistical Analysis of Variant Loads

0.257iL kL

1. Sustained Live Load

0.204( ) exp exp

0.092i

kL

k

x LF x

L

0.119iL kL

5 0.352( ) [ ( )] exp exp

0.092iT i

kL L

k

x LF x F x

L

0.406iTL kL

0.119iTL kL

( )iL t

t0

1, 5, 10p r

T


2. Transient Live Load

0.237rL kL

0.164( ) exp exp

0.127r

kL

k

x LF x

L

0.162rL kL

0.441rTL kL

0.368( ) exp exp

0.127rT

kL

k

x LF x

L

0.162rTL kL

( )rL t

t0

1, 1p r

T

( )rL t

t0

1, 5, 10p r

T


5.3.3 Statistical Analysis of Environmental Loads

1. Wind Load( )W t

t0 T

1, 50, 1p r

0.455YW kW

0.359( ) exp exp

0.167Y

kW

k

x WF x

W

0.214YW kW

0.410YW kW

0.323( ) exp exp

0.151Y

kW

k

x WF x

W

0.193YW kW

Don’t consider wind direction

Consider wind direction


2. Snow Load( )S t

t0 T

1, 50, 1p r

00.359YS kS

0

0

0.244( ) exp exp

0.199Y

kS

k

x SF x

S

00.256YS kS

0

0

1.024( ) exp exp

0.199T

kS

k

x SF x

W

01.139TS kS

00.256TS kS

5.4 Representative Values of Loads


5.4 Representative Values of Loads …15.4 Representative Values of Loads …1

5.4.1 Representative Value v.s. Design Value of a Load

– The representative value of a load is a value used for the

verification of a limit state.rQ

1. Representative Value of a Load

– 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 .

dQQ

d Q rQ Q

dQ

r

Q

Q


5.4.2 Characteristic Value

– The characteristic value of a load is the maximum value of the load that acts on the structure during the design reference period.

kQ

1. Definition

– It is the principal representative value when designing structures. The other representative values are obtained by conversion of the characteristic value.

– 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.

– It is used in both ultimate limit state verification and serviceability limit state verification


(1) Determined by the return period of a load

2. Methods

where, rT is called ( mean ) return period.

*

1rT q

* ( )kq P Q Q

where, *q is called yearly exceedance probability.

1( ) 1Q k

r

F QT

1 1(1 )k Q

r

Q FT


(2) Determined by the percentile of

2. Methods …

( )T kF Q

( )T k kF Q p

where, rT is the probability of not being exceeded during the reference period, it is also called “guarantee probability”.

*1(1 ) (1 )T T

kr

p qT

0.95kp 975rT

50T

50rT 0.346kp 50T


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!


5.4.3 Frequent Value

– The frequent value of a load is the frequent occurring load that acts on the structure during the design reference period.

1Q

1. Definition

– 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

10 1k≤ ≤

1 1 kQ Q


5.4.4 Quasi-permanent Value

– The quasi-permanent value of a load is the often occurring load that acts on the structure during the design reference period.

2Q

1. Definition

– 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

20 1k≤ ≤

2 2 kQ Q


5.4.5 Combination Value

– 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 .0Q

1. Definition

– 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

c c kQ Q

5.5 Combination of Load Effects


5.5 Combination of Load Effects …15.5 Combination of Load Effects …1

5.5.1 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.

iQ

– 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.

0max ( )Tt T

Q Q t≤ ≤

1 21

( ) ( ) ( ) ( ) ( ) ( ) ( )n

i G L L W S Ei

Q t Q t Q Q t Q t Q t Q t Q t


– It is generally assumed that the load effect is linearly related to the load:

iC

0 01

max ( ) max ( )n

mT it T t T

i

S S t S t

≤ ≤ ≤ ≤

( ) ( )i i iS t CQ t

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 :mTS

1

( ) ( )n

ii

S t S t

1

( ) ( )n

ii

Q t Q t


5.5.2 JCSS’s Rule for Load Effect Combination

– For an arbitrary load , its maximum load effect during the design reference period is combined with other load effects .

1

1 0 2 0 1 0

10 0 0

( ) ( ) ( )

max ( ) max ( ) max ( )i n

mi i

i i nt T t t

S S t S t S t

S t S t S t

≤ ≤ ≤ ≤ ≤ ≤

– For each load , the design reference period T is divided into equal intervals .

( 1,2, , )iQ i n ir i

– all loads are reordered from small to large according to the values of , and let the take integer value.ir 1/i ir r

– 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.

iQ [0, ]max ( )it T

S t

T

ir


JCSS’s Rule Formula for Load Effect Combination

1 2 11 31 2max ( ) max ( ) max ( ) max ( )

nm n

t T t t tS S t S t S t S t

[0, ]

2 11 0 22 3( ) max ( ) max ( ) max ( )

nm n

t T t tS S t S t S t S t

[0, ]

1 0 2 0 3 0( ) ( ) ( ) max ( )m nnt T

S S t S t S t S t

[0, ]

1

1 0 2 0 1 0

10 0 0

( ) ( ) ( )

max ( ) max ( ) max ( )i n

mi i

i i nt T t t

S S t S t S t

S t S t S t

≤ ≤ ≤ ≤ ≤ ≤


Example 5.2

Assume that there are three loads, the interval numbers of each load are:

1 2 35, 10, 25r r r Give the load effect combination formula of the three loads.

Solution: Assume that the design reference period 50T years

The intervals of each load are:

1 1/ 50 / 5 10,T r 2 35, 2

Rank the load according to . This has been done.ir

The load effect combination results are:

1 1 2 30 50 0 0 5max ( ) max ( ) max ( )mt t t

S S t S t S t ≤ ≤ ≤ ≤10 ≤ ≤

2 1 0 2 30 50 0 5

( ) max ( ) max ( )mt t

S S t S t S t ≤ ≤ ≤ ≤

3 1 0 2 0 30 50

( ) ( ) max ( )mt

S S t S t S t ≤ ≤


The loads acting on the a office building structures generally include:

– dead load – sustained live load– transient live load– wind load

G

1( )L t

2 ( )L t( )W t

The design reference period is .50T yearsThe intervals of these loads are as follows:

50G years

1 210L L years

1W year

Example 5.3

Give the load effect combination formula according to JCSS’s Rule.


(2) Rank the load effects :

1 2, , ,G L L WS S S S

2 1, , ,G L L WS S S S

(3) Combine the load effects :

1 21m G LT L S WSS S S S S

1 23m G L S L T WSS S S S S

1 24m G L S L S WTS S S S S

1 22m G LT L S WSS S S S S

Solution:

1Gr 1 25L Lr r 50Wr

(1) Solve the interval numbers of loads:


5.5.3 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.

1 012

max ( ) ( )n

m jt T

j

S S t S t

[0, ]

0 02 1 23

( ) max ( ) ( )n

m jt T

j

S S t S t S t

[0, ]

1

01

max ( ) ( )n

m n jnt T

j

S S t S t

[0, ]


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!


5.5.4 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:

m G WTS S S

1 21m G LT L SS S S S

1 22m G L S L TS S S S 1 2max( , )m m mS S S

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 .

Chapter 5: Homework 5Chapter 5: Homework 5

End of

Chapter 5Chapter 5

chapter 5 statistical analysis of loads. chapter 5: statistical analysis of loads 5.2 probability...

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probability models of

definitions of loads

time probability distribution

probability distributions

variable action

statistical analysis

accidental action

spacefixed action