PDF Deformation of Concrete - Seoul National University

Deformation of Concrete

Fall 2010

Dept of Architecture

Seoul National University

2

week

Creep and Shrinkage

Chap. 1 Creep and Shrinkage of concrete and relaxation of steel

1.1 Introduction

1.2 Creep of concrete

1.3 Shrinkage of concrete

1.4 Relaxation of prestressing steel 1.5 Reduced relaxation

1.6 Creep superposition 1.7 Aging Coefficient

1.8 Equation for Aging coefficient 1.9 Relaxation of concrete

1.10 Step-by-step calculation of relaxation function for concrete

1.11 agae-adjusted elasticity modulus

1.1 introduction

1. Subject to change for long time period of time

2. Creep, shrinkage, and relaxation 3. Time function for strain and stress

4. Age of concrete, length of the period after loading, environment, shape of concrete members

1.2 Creep of concrete

1. Instantaneous deformation

2. Under sustained stress

 0 c t



 0 c t



 0

E tc

Slope

 

c t



 0 c t



   , 0 0

c t t c t

  

  

creep

t0

t

     

 ⁰  

0 0

0

1 ,

c

c c c

c

t t t t t

E t

         

1.3 Shrinkage of concrete

1. Drying shrinkage 2. Free shrinkage

 

 

cs t ts cs s t ts

    

1.4 Relaxation of prestressed steel

• Relaxation under constant strain as in a constant-length test





0

log 0.55

10

pr p

p py

t

f

  



 

 

    

 

• The ultimate intrinsic relaxation

 

0 pr 0.4

p

  



 

   ^p0

fptk

  

1.5 Reduced relaxation

• To consider creep and shrinkage of prestressed concrete

pr r pr

  

  

0

ps pr

p

 



    

   

1.6 Creep superposition

     

   

     

 

0

0 0

0 0

1 , 1 ,

,

c t

c c c cs

c c

t t t

t t d t t

E t E

   

    



  

 



1.7 Aging Coefficient

     

   

     

 

0

0 0

0 0

1 , 1 ,

,

c t

c c c cs

c c

t t t

t t d t t

E t E

   

    



  

   

     

     

   

0 0

0 0

0 0

1 , 1 ,

c c c cs ,

c c

t t t t

t t t t t

E t E t

 

  ^   ^ 

1.8 Equation for Aging coefficient

Stress variation

Differentiation

     

     

   

0

0 1

0 0

0

1 , 1 ,

,

t

c c c cs

c t c

t t t d

t t t d t t

E t E d

   

    

 

 

    

   

  ⁰

1

c c

c

t t

  

 

 



  

1 _c /

c

d d

d

d t

  



 ^  ^c  c   ¹

d d

d t d

   

 ^{ } 

Substitution

     

     

   

0 0

0 0

0 0

1 , 1 ,

c c c cs ,

c c

t t t t

t t t t t

E t E t

 

  ^   ^ 

   

   

0  

0 0 1

0

0 0

1 ,

, 1

,

t c

t c

E t t t d

t t d

t t E t d

 

 

 

   

1.9 Relaxation of concrete

Instantaneous stress

Stress at any time

Magnitude of relaxed stress A dimensionless shape function

 0  0

c t cE tc

 

  

c c t

  



 



   , 0 c t cr t t

 

   

c c t

   

   

     0

c c c t

    

  

     ₀

c t c t c t

  

  

     

     

   

0 0

0 0

0 0

1 , 1 ,

c c c cs ,

c c

t t t t

t t t t t

E t E t

 

  ^   ^  ^(1.10)

         

0   0

0 0

0

1 , 1 ,

1 ,

c c c c

c

t t t t

t r t t E t

E t

 

   ^     ^

 0  0

c t cE tc

 

 

 

 

 

1 1

, 1 , / _c ,

t t r t t E t t t

 ^  ^ 

1.10 Step-by-step calculation of relaxation function

     

   

     

 

0

0 0

0 0

1 , 1 ,

,

c t

c c c cs

c c

t t t

t t d t t

E t E

   

    



  

   

     

 ^{1/ 2  }

1/ 2 1/ 2 0

1

1 ,

,

i i j

c i c j cs i

j c j

t t

t t t

E t

 _   ^  _



  

 

  

 

 



Separation of the last term

     

     

  ^{ }

1 1/ 2

1/ 2

1/ 2 1/ 2 0

1

1 ,

1 ,

,

i i j

i i

c i c i c j cs i

c i j c j

t t

t t

t t t

E t E t

 

 _  ^{ }  ^  _



  

  

    

 

 



1.11 Age-adjusted elasticity

     

     

   

0 0

0 0

0 0

1 , 1 ,

c c c cs ,

c c

t t t t

t t t t t

E t E t

 

  ^   ^ 

     

   

   

0

0 0

0 0

1 ,

, ,

c

c c cs

c c

t t t

t t t t

E t E t t

 

  ^  ^ 

   

 ⁰

0

0

, 1 ,

c c

E t t E t

 t t

 

   

 , 0 c c

c

t t

E t t

 ^

 