Random Time Dependent Resistance Analysis on Reinforced Concrete Structures 3

GUAN Chang2sheng1) WU Ling2)

1)  Wuha n  University of  Technology 2) Wuha n University of Science a nd Technology

( Receive d :Aug. 20 ,2001)

Abstract :  The  analysis  method  on  random  time  dependence  of reinf orced  concrete  material  is  introduced , the effect mechanism on reinf orced concrete are discussed , and the random time dependence resistance of reinf orced concrete is studied. Furthermore , the corrosion of steel bar in reinf orced concrete structures is analyzed. A practical statistical method of evaluating the random time dependent resistance , which includes material , structural size and calculation  influence , is  also  established.  In  addition , an  example  of  predicting  random  time  dependent resistance of reinf orced concrete structural element is given.

Key words : random analysis ; time dependent resistance ; reinf orced concrete

1 Introduction

Reinforced concrete is one of the most useful materi2 als in civil engineering for its suitability and inexpensive price. Therefore , a large number of this kind material is used. However , in traditional design of  construction structure and material  research  of  reinforced  concrete , litte attention to the time dependent strength of reinforced concrete is paid , especially the random properties of the various effects on the materials are hardly studied[1 ,2 ] .

Until recent year there are a few researches  that  mention the problem about the random time dependence of rein2 forced concrete in engineering structures. Li and Melch2 ers[3 ] have studied the failure probability of reinforced concrete columns under stochastic loads ,and given a tech2 nique to calculate the time2variant failure probability. Li and Liu[4 ] have  studied the durability of  concrete structure

considering the effect of reinforced corrosion. Leng[5 ] et al have discussed the status and problem of study  on  the stress corrosion of concrete. Also Lu[6 ] et al have dis2 cussed the state of reinforcement corrosion. Generally speaking , reinforced concrete resistance obviously de2 crease , which depends on different factors[7 ] . To evaluate

the safety of concrete structure ,a basic theory for analyz2 ing durability of  reinforced  concrete  should  be  proposed. It is necessory to study  random time  dependent  properties of reinforced concrete structures.

2 Time Dependent Effect Mechanisms  of Reinforced Concrete

GUAN Chang2sheng ( 管昌生) :Born in 1957 ; Prof . ; Institute of Architecture and Civil Engineering ,Wuhan University of Technol2 ogy ,Wuhan 430070 ,China

3 Funded by Hubei Province Construction Bureau( G200219)

There are many factors affecting the resistance of re2 inforced concrete. More than 50 kinds of chemical corro2 sion factors exist only in the area of water basin , water works and water conditioning. The practical method to ob2 tain a time dependent model of reinforced concrete is a multi2factor comprehension technique. In general ,for sin2 gle2factor ,many results have been shown that if only con2 crete carbonization is considered , a formula of predicting carbonization thick can be written as :

D ( t)  = K t (1)

where D ( t ) , K and t are  thick , speed  coefficient  and time of carbonization ,respectively.

Up to now many predicting models are used in steel corrosion , fatigue damage , and chemical erosion , but no common recognized results unanimously exist . Generally speaking , the decreasing resistance function of reinforced concrete includes some variables that can express as steel

bar ,concrete  ,geometric  size  ,environment  condition near2

by , and properties of random and time dependence of re2 sistance[1 ,5 ]. Obviously , reinforced  concrete  resistance is a function of random process or variable of series of mate2 rials and structures. Reinforced concrete carbonization in  air is called neutralization ,which is a complex and slow neutralization process of CO2 in the air and alkaline mate2 rial in reinforced concrete. In air it  will  take  several decades to completely carbonize the protective coating of steel bar in dense concrete , but only take a few years in non2dense concrete. If fineness of carbonized materials is higher , its strength is lower and carbonization process is faster in decreasing cross section of structure. Carboniza2 tion makes alkalinity lower and  steel  bar  corrosion[2 ] . Steel  bar corrosion  is a process that  iron in  steel  bar  sur2

face losses electron  continually and  dissolves in water  un2

der the condition of oxygen and water interaction. There2 fore , the volume of eroded materials is expended several

times , which can make the protective coating of reinforced concrete crack , drop , and cohesion damage between steel bar and concrete along steel bar direction so that  rein2 forced concrete structure losses bearing capability. Recent researches show that the steel bar gets into a serious cor2

rosion state when non2carbonized protective coating is only 15mm thick.  On the other  hand ,  the  start  time  of corro2

sion may be in advance due to incomplete  coating  and crack in concrete , also corrosion velocity may increase greatly. When steel stress is less than yielding point , the corrosion velocity is stable , but increases several  times if the corrosion velocity passes over the yielding point . The latter is called stress corrosion , which is of brittleness and more dangerous. Steel corrosion can make bearing area of steel bar decrease , cohesion damage , and lose structure effectiveness. Erosion  air  and liquid  all  have  strong effect

on  reinforced  concrete ,  which  can  make   reinforced con2

crete erosion ,decrease protective coating of concrete , and finally accelerate steel corrosion. When bearing dynamic load , reinforced concrete structure can undergo fatigue damage ,leading strength and rigidity to decreasing , form2 ing crack and expansion. Usually fatigue damage can be divided into determine and random models , the  former deals with determine period loading , and the latter is rel2 ative to the random of material and load.

Besides all mentioned above , there are many other factors , such as temperature ( including high or low , and cyclic temperature ,freezing damage) ,humidity , (including cyclic of dry and humid) ,and the amplitude of loading , which can result in resistance decreasing of reinforced concrete structure.

3 Random Time Dependence Analysis

Under the condition that every random variable of structure is independent on each other , and considering time dependence characteristic of  material , the  random time dependent function of reinforced concrete material

variable Kf ,respectively.

By the same way the similar random analytical model may be set up for studying the randomness and time de2 pendence to geometric parameters and calculation methods of reinforced concrete structure.

The resistance of reinforced concrete can be ex2 pressed by resistance function  R  ( x1 ,  x2 ,  ,  xn )  , in which xi denotes the  parameters  of  materials , geometry and calculation procedure of structural material . If random and time dependence are applied simultaneously to resis2

tance  and relative  variables , xi can be  dealt with as a ran2

dom process. Moreover , considering fatigue mechanics , calculation model and initial conditions , time dependent resistance model can be written as :

R ( R0 , t)  =ηR0 KpR ( x1 , x2 , , xn) (5)

where R ( R0 , t) is resistance random process , R0 is ini2 tial resistance ,η is a random process of fatigue damage , and Kp is an undeterminable  parameter  of  calculation model , xi denotes a random process of material  parame2 ters . Many experiments on steel corrosion indicate  that  steel erosion process accords with normal random process. By  separation  procedure ,  random  time   dependent  resis2

tance R ( R0 , t) can simply be expressed as :

R ( R0 , t)  = R ( t) R0 (6)

where R ( t ) is a determining time  dependent  function , and R0 is random initial resistance.

In addition , some other undetermined conditions should be employed , such as undetermined resistance of actual material , geometric size and  calculation  model . As to a simple structure , resistance function of random pro2 cess also can be simplified as :

R ( t)  = g ( t)  Kp ( y) Rp ( t) (7) where g ( t) is time dependent function , Kp ( t ) is unde2 terminable parameter of calculation model ,and Rp ( t ) is structure resistance determined by calculating model .

To a single structural element equation (7) can be written as :

R ( t) =  Kp ( t) KM ( t) KA ( t)  R K ( t) (8)

characteristic can be written as[8 ] :

where K ( t ) , K ( t ) ,and K

( t ) are respectively time

KM ( t) = k - 1 K ( t) K

(2)

p M A

0 0 f

where K0 ( t) is a random variable indicating property dif2 ference of structural material and testing material ; k0 is a coefficient of property difference of structural and testing material ,and Kf is a random variable of testing material .

As statistics theory , the mean value and standard

dependent parameter of calculation ,geometry and material characteristic of reinforced concrete structural  element . R K ( t) is standard value of material resistance.

The mean value and standard deviation of equation

(8) are respectively as follows :

μR  ( t) = μK    ( t) μK  ( t) μK    ( t) R K ( t) (9)

P M A

deviation coefficient of KM ( t) are as follows respectively :

σR ( t)  = [σ2

( t)  +σ2

( t)  +σ2

( t) ]1/ 2 (10)

μ  ( ) = - 1μ ( ) μ

(3)

KP KM KA

t k t

M 0 f

In engineering application , decreasing function random

δ ( t) =

M

(4)

process of concrete structural resistance can  be  expressed by :

where μk   ( t)  and δK  KM  ( t)  respectively are mean value

0 0 R ( t)  =ζ<( t) (11)

and  standard  deviation  coefficient  of   random  process. K0

( t) ,μ and δ are the mean value and standard devia2

f f

where R ( t ) is random process of concrete structural re2 sistance ζ,   is  relative  random variable ,and  < ( t )   is  time

tion  coefficient   of   testing  material   property and  random dependent determining function. Based on durability grade

60 Journal  of  Wuhan  University of  Technology -  Mater. Sci. Ed. Dec. 2002

of designed concrete structure , equation (11) can be es2 tablished as :

μcu ( t)  = μcu (1 - α1 t) (12)

δcu ( t)  =δcu (13)

μy ( t)  = μy (1 - α2 t) (14)

δy ( t)  =δy (15)

where μcu ( t) and μy ( t) are respectively resistance mean value of concrete and steel bar when reinforced concrete is used up  to time  t ;δcu ( t)  and δy ( t)  are  standard devia2 tion coefficients of concrete and steel bar of reinforced concrete at time  t  respectively ;μcu ,μy  ,δcu ,δy  are initial values of  various  variables ;α1 and α2 are  resistance  de2 crease coefficients of concrete and steel respectively ,

which relate to structural material and can  be  obtained from material experiment .

4 Engineering Application

A structural element of reinforced concrete bearing axial pressure is tested , some statistical data of structural

resistance is calculated. Consider initial values : C30 con2 crete ,the mean value μcu = 1. 41 and standard deviation σcu  = 0. 19 ;20MnSi  steel  bar ,mean value μy  = 1. 14 and standard deviation σy  = 0. 07 ;  initial  section  size is 1m2 .

There are 0. 02 variation on each side of the structural el2

ement. Resistance decrease coefficients of concrete and steel obtained from Ref . [ 4 ] are respectivelyα1 = 8 × 10 - 7 and α2 = 2. 2 ×10 - 6. Under these conditions the random time dependent resistance of the reinforced con2 crete structure bearing axial load can be obtained.

According to design theory of reinforced concrete structures , resistance function can be formed by R =  Acu Rcu + AyRy ,in which Rcu and Ry are initial resistance of concrete and steel bar respectively , and then using the

theory  suggested  in  this  paper ,  the  random  time depen2

dent resistance can  be  calculated.  Some  statistical  data are listed in Table 1. From the results it is seen that the resistance change of reinforced concrete is obvious ( de2 creasing with time) . This phenomenon  is significant  and not to be ignored for evaluating the safety of reinforced concrete.

Table 1 Random Time Dependent Resistance of Reinforced Concrete Structural Element Bearing Axial Load

t/ year	0	10	20	30	40	50
Mean value/ KN	7625. 20	7323. 95	7136. 56	6885. 75	6517. 44	5987. 32
Standard deviation/ KN	1248. 63	1193. 80	1164. 73	1129. 49	1080. 28	1011. 05
Variation coefficient	0. 1637	0. 1629	0. 1632	0. 1640	0. 1657	0. 1688