Ch3.Basic Principles for the Design Of R.C. Structure --Limit state design based on

probability§3-1. Basic Concept of Limit State Design

3.1.1.Function requirements （功能要求） of structure

1.Safety:In the condition of normal construction and using,the structure can carry various loads and deformation, and be not collapsed.

2.Serviceability: good working property

3.Durability:

reliabilityof structure（可靠性）

3.1.2.Limit state of a structure （极限状态）Definition:When a structure or its a part exceeds a specific state,the function requirements of the structure can not be satisfied,then the specific state is called limit state.

1.The limit state of load carrying capacity( ultimate limit state ):

1).structure slides or overturns:

2).structure collapses as the material strength is exceeded.

3).structure becomes mechanical system.

4).structure loses its stability.

5).structure vibrates resonantly. 2. Serviceability limit state 1).the structure may be prohibitive deformation or deflect.2).the structure may vibrate with too large amplitude.3).the structure may be excessively cracked.

• Action is all reasons what makes structure produce internal force,deformation,stress and strain.

3.1.3.Action on the structure

1.According way of action: 1).directing action:weight2).indirection action:

2.According variation of time• 1).permanent action（永久作用） :

Within the design datum period ,its value does not vary with time. dead mass

Design datum period (T):to relatively fix a datum time during analysis of the structure reliability. 50 years,100

• 2).variable action（可变作用） : wind load, man load• 3).accidental action （偶然作用） : seismic load, explode• 3.According variation of space position

2).movable action:1).fixed action:dead mass

4.According variation of structure response1).static action （静力作用） : 2).dynamic action （动力作用） : seismic load,

man load,wind load

1.4. Action effect S （作用效应） and structural resistance R （结构抗力）

• 1.Action effect S:• the action applying on structure makes the stru

cture produce internal force and deformation. M,N,V,T,f,

• 2.Structural resistance R:• R is capacity that the structure sustains action

effect.

§3-2.Reliability Analysis （可靠度分析）

• 3.2.1.The uncertainty of structure design• 1.the uncertainty of action effect S• 1).variability of load itself.• 2).variation between assumption of calculating

internal fore and practical situation of stressing.• 2.the uncertainty of structural resistance R• 1).the irregularities in the material strength• 2).the variation of section size• 3).the calculating model of resistance

S, R SR

3.2.2.Probability analysis （概率分析）• 1.Random variables Xi and statistical characte

ristics

• The variables to indicate random appearing various result is called random variables.

1).arithmetic mean :

nXn

ii /

1

n

iin X

1

21 2).standard deviation :

3).coefficient of variation （变异系数） : = / •The littler and are , the littler variance is, the more random variable concentrates.

Table3-1 Cubic strength of tested specimens

SpecimenNo.

CubicStrength

(Mpa)

SpecimenNo.

CubicStrength

(Mpa)

SpecimenNo.

CubicStrength

(Mpa)

SpecimenNo.

CubicStrength

(Mpa)

SpecimenNo.

CubicStrength

(Mpa)

1 40.1 8 39.5 15 45.6 22 47.1 29 39.9

2 39.6 9 43.8 16 38.7 23 43.5 30 38.9

3 37.1 10 44.5 17 41.4 24 36.3 31 40.9

4 45.5 11 36.9 18 49.0 25 41.0 32 42.1

5 43.9 12 47.3 19 36.1 26 40.9 33 43.7

6 41.5 13 42.7 20 45.9 27 42.8 34 34.0

7 39.6 14 44.1 21 38.7 28 41.7 35 41.5

=41.59Mpa =3.40Mpa 082.0

2.Normal distribution of probability

• 2).calculating law of random variables:

x

f(x)--density function

F(x)

’’

1).normal curve:N(, )

The litter is, the more the distribution of random variable concentrates.

Distribution function:

dxexFx

2

2

2

2

1

dx

x22

exp2

1

,~,, 21 Nxxx na). nxxxxf 21 nxxxfthen

21:

n

i

xf i

1

2 nxxxxxfb 321).

nxxxxfthen 321

:

n

i

xf i

1

2

• 3.Guaranteed probability （保证率） :

• The probability that the value of random variable is not smaller than(or is not greater than) a certain value u is called as guaranteed probability about the certain value.

x

’u

f(x)

+’=1, u-characteristics

is the guaranteed probability that the value of random variable is not smaller than u.

=(1) , -coefficient of guaranteed probability.u=The probability that x is not greater a certain value u: uxP

ux

P

u

u Gaussian (Standard) normal distribution function

x

y

dx

yyF

2exp

2

1 2

u

uxPuxP 11

uxP

u

2

2: uwhen

%275.22

uxP %725.97221

2.3.Reliability Analysis• 1.Reliability of a structure （结构可靠度） :

• Reliability of a structure is the probability that the structure finishes predetermined function within specified time and in the specified condition. （结构在规定的时间内规定条件下完成预定功能的概率）

• R , S--random variables• define: function （功能函数） : Z=R-S

When Z>0, R>S Structure reliable.Z<0, R<S Structure failureZ=0, R=S Limit state

2.Failure probability:Pf

The probability of Z<0 is called failure probability.Pf

The probability of Z>0 is called reliable probability Ps.

3.Reliability index （可靠指标）

• R and S are normal distribution, then to define:

22sR

SR

Z

Z

• thus: Pf=1-() Tab.3.2

Pf , Object reliability index(table 3.3)

Table 3-3 Object reliability index

Safety class of structures 1st class 2nd class 3rd class

Ductile failure 3.7 3.2 2.7

Brittle failure 4.2 3.7 3.2

Table 3-4 Safety class of structures （结构安全等级）

Safety class of structure

（结构安全等级）

Failure consequence（破坏后果）

Structural type（结构物类型）

1st class（一级）

Very serious（非常严重）

Important structures

（重要结构）2nd class（二级）

Serious（严重）

Ordinary structures（一般结构）

3rd class（三级）

Not serious（不严重）

Secondary structure（次要结构）

§3-3. The Current Chinese Code Practice 3.3.1.Representative Value of load （荷载代表值） 1.The characteristic value of load （荷载标准值） :is basic deterministic value in design. Permanent load Gk:dead weight variable load Qk: Qk= Q+QQ

Design load （设计荷载）G= G Gk

Q=Q Qk

3.Quasi-permanent values of variable loads （荷载准永久值） :q Qk

4.Combined coefficient of variable loads:c 1

q--coefficient of Quasi-permanent values , q1

2.Load factors （荷载分项系数） : factor of permanent loads: G

factor of variable loads: Q

Q

f(Q)

Q Qk

G=1.2,1.35

Q=1.4,1.3

3.3.2. Representative Value of Materials Property:

fk =f-f= f (1-f ) if guaranteed probability is greater than 9

5%, 1.645

fck= f (1-1.645f ) , fyk= f (1-2f ) 2.Design value of material property ( 材料

强度设计值 ): fc,fy f / f -----materials factor design value of concrete strength:fc=fck/c

design value of steel strength :fy=fyk/s

fffk

1.Characteristic value of materials property ( 材料强度标准值 ): fk is the strength value with a 95% degree of confidence on the base of average strength.

c s

3.3.3.Design expression of the ultimate limit state （承载力极限状态设计表达式）

• 1.Expression: oSR• o-the coefficient of structure importance.• Safety class: I II III • o : 1.1 1.0 0.9• (It is important to note that ,various parts of a st

ructure may not be of the same importance.)2.The combined design value of action effect( 作用效应设计表达式 ):

n

iQikciQikQQGkG SSSS

211).1

GkS --the characteristic value of the permanent load effect (M,N,V,T)

QikS --the characteristic value of the variable load effect (M,N,V,T)

3.Structural resistance R=R(fc,fy,a, …)

n

iQikQiGkG SSS

1

9.0

2). for design of ordinary rigid frames( 框架） and bent frames:

3). When the permanent load is dominant:

n

iQikciQiGkG SSS

1

7.0,35.1 ciG

3.4.The design expression of the limit state of serviceability--to check deformation and crack

1.The combined characteristic value of load effect :Sk

Prestressed concrete

the width of maximal cracks:Wmax[W]

2.The Quasi-permanent values of load effect :Sq

•deflection of members: fmax[f ]crack:class I

•class II•class III

n

iQikcikQGkk SSSS

21).1

n

iQikqiGkq SSS

1

n

iQikciGkk SSS

2

).2