Calculation curvature, strain and stiffness caused by shrinkage of 2D element

 

Calculation shrinkage forces

The forces are calculated for in centre of gravity of the each element and there are calculated in two directions:

• First direction of principal stress a
• Second direction of principal stress a+90°

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

n_shr = -e_cs(t,t_s)·Coef_Reinfå(E_si·A_si(a))

m_shr = n_shr·e_shr,z

where

e_shr,z =å(E_si·A_si(a))/ å(E_si·A_si(a)·z_si) - t_iz(a)

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

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

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

A_si(a) - is area of reinforcement of i-th bar of reinforcement in first (angle a)/second direction (angle a+90°)of principal stress

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

t_iz(a) – distance between centre of gravity transformed uncracked/cracked cross-section and centre of gravity of concrete cross-section in z-direction and in n first (angle a)/second direction (angle a+90°)of principal stress

 

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 2D elements and these value are calculated for both state (uncracked and cracked cross-section). The values are calculated in both direction of principal stresses

Calculation strain caused by shrinkage:

e_x =-e_cs(t,t_s)·Coef_Reinf·å(E_si·A_si(a))/(E_ceff·A_i(a))

Calculation curvature around y and z axis caused by shrinkage

(1/r) = -e_cs(t,t_s)·Coef_Reinf·å(E_si·A_si(a)·(t_iz(a)-z_si))/(E_ceff·I_iy(a))

 

where

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

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

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

A_si(a) - is area of reinforcement of i-th bar of reinforcement in first (angle a)/second direction (angle a+90°)of principal stress

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

t_iz(a) – distance between centre of gravity transformed uncracked/cracked cross-section and centre of gravity of concrete cross-section in z-direction and in n first (angle a)/second direction (angle a+90°)of principal stress

E_ceff - is effective module elasticity of the concrete calculated according formula E_c = E_c,eff = E_cm/(1+j).

E_cm -secant modulus of elasticity of concrete

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

A_i(a) - transformed area uncracked/cracked cross-section in the first (angle a)/second direction (angle a+90°)of principal stress

I_iy(a) - transformed second moment of area around y axis of uncracked/cracked cross-section calculated to centre of gravity transformed uncracked/cracked cross-section in the first (angle a)/second direction (angle a+90°)of principal stress

 

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)

• bending stiffness in direction of first principal axis EIy₁ =m_tot(a)/(1/r)₁

• bending stiffness in direction of second principal axsi EIy₂ =m_tot(a+90)/(1/r)₂

• axial stiffness in direction of first principal axis EA₁ = n_tot(a)/e_x,1

• axial stiffness in direction of second principal axis EA₂ = n_tot(a+90)/e_x,2

where

n_tot(a), n_tot(a+90 - are axial force from total combination in 2D element recalculated to direction of first and second principal axis, see "Calculation stiffness of 2D element"

m_tot(a), m_tot(a+90 - are bending moments from total combination in 2D element recalculated to direction of first and second principal axis, see "Calculation stiffness of 2D element"

e_x,1(2) - is strain caused by shrinkage calculated in direction of first (second ) principal axis

(1/r)₁₍₂₎ - is curvature caused by shrinkage calculated in direction of first (second ) principal axis

 

 

 

Deflection for shrinkage is calculated in FEM analysis for total combination, therefore the stiffness are calculated with using internal forces for total combination