Calculation curvature, strain and stiffness caused by shrinkage of 1D element

 

Calculation shrinkage forces

The forces caused by shrinkage are calculated according to formulas below. The forces are calculated for both states: uncracked and cracked cross-section.

Nshr = -ecs(t,ts)·CoefReinfå(Esi·Asi)

Mshr,y = Nshr·eshr,z

Mshr,z = Nshr·eshr,y

where

eshr,y =å(Esi·Asi)/ å(Esi·Asi·ysi) - tiy

eshr,z =å(Esi·Asi)/ å(Esi·Asi·zsi) - tiz

-ecs(t,ts)· - is total shrinkage strain, see "Parameters for calculation of shrinkage strain"

Coefreinf is coefficient increasing amount of reinforcement, see "Coefficient for increasing amount of reinforcement"

Esi - is modulus of elesticity of i-th bar of reinforcement

Asi - is area of reinforcement of i-th bar of reinforcement

ysi – position of i-th bar of reinforcement from centre of gravity of cross-section in y-direction

zsi – position of i-th bar of reinforcement from centre of gravity of cross-section in z-direction

tiy – distance between centre of gravity transformed uncracked/cracked cross-section and centre of gravity of concrete cross-section in y-direction

tiz – distance between centre of gravity transformed uncracked/cracked cross-section and centre of gravity of concrete cross-section in z-direction

 

The forces caused by shrinkage are presented in detailed output in chapter Shrinkage deflection (long-term stiffness). Presented forces are calculated for uncracked cross-section

 

Calculation strain and curvature caused by the shrinkage

Strain and curvature caused by shrinkage are calculated for each 1D elements and these value are calculated for both state (uncracked and cracked cross-section)

Calculation strain caused by shrinkage:

ex =-ecs(t,ts)·CoefReinf·å(Esi·Asi)/(Eceff·Ai)

Calculation curvature around y and z axis caused by shrinkage

(1/ry) = -ecs(t,ts)·CoefReinf·å(Esi·Asi·(tiz-zsi))/(Eceff·Iiy)

(1/rz) = -ecs(t,ts)·CoefReinf·å(Esi·Asi·(tiy-ysi))/(Eceff·Iiz)

 

where

-ecs(t,ts)· - is total shrinkage strain, see "Parameters for calculation of shrinkage strain"

Coefreinf is coefficient increasing amount of reinforcement, see "Coefficient for increasing amount of reinforcement"

Esi - is modulus of elesticity of i-th bar of reinforcement

Asi - is area of reinforcement of i-th bar of reinforcement

ysi – position of i-th bar of reinforcement from centre of gravity of cross-section in y-direction

zsi – position of i-th bar of reinforcement from centre of gravity of cross-section in z-direction

tiy – distance between centre of gravity transformed uncracked/cracked cross-section and centre of gravity of concrete cross-section in y-direction

tiz – distance between centre of gravity transformed uncracked/cracked cross-section and centre of gravity of concrete cross-section in z-direction

Eceff - is effective module elasticity of the concrete calculated according formula Ec = Ec,eff = Ecm/(1+j).

Ecm -secant modulus of elasticity of concrete

j is creep coefficient, see "Parameters for calculation of creep coefficient "

Ai - transformed area uncracked/cracked cross-section

Iiy - transformed second moment of area around y-axis of uncracked/cracked cross-section calculated to centre of gravity transformed uncracked/cracked cross-section

Iiz - transformed second moment of area around z axis of uncracked/cracked cross-section calculated to centre of gravity transformed uncracked/cracked cross-section

 

Calculation stiffnesses for shrinkage

The stiffness of uncracked/cracked cross-section for shrinkage is calculated from strain and curvatures caused by shrinkage with using total level of load (total load combination)