Calculation stiffness of 1D element

 

Members which are not expected to be loaded above the level which would cause the tensile strength of the concrete to be exceeded anywhere within the member should be considered to be uncracked. Members which are expected to crack, but may not be fully cracked, will behave in a manner intermediate between the uncracked and fully cracked conditions. New stiffness (stiffness with taking into account cracking) is calculated in centre of each 1D element.

Two types of stiffness are calculated:

Short-term stiffness - is calculated using 28 days modulus of elasticity E_c = E_cm, it follows that value of stiffness is loaded directly from properties of the concrete material

Long-term stiffness - is calculated using effective E modulus based on creep coefficient for acting load, it follows E_c = E_c,eff = E_cm / (1 + φ).

Calculation effective modulus of elasticity is based on equation 5.27 in EN 1992-1-1, but instead of effective creep coefficient φ_ef, only creep coefficient φ is used.

The following procedure is used for calculation stiffens for

• calculation transformed CSS characteristics of uncracked section (A_i, I_i, t_i...)

• calculation stiffness’s of uncracked cross-section ((EI_y)_I,( EI_z)_I, (EA)_I) to centre of uncracked transformed cross-section

• calculation maximum value of tensile stress of uncracked cross-section (value σ_ct,res) for respective characteristic combination (N_char,res, M_char,res,y, M_char,res,z)

• calculation maximum value of tensile stress of uncracked cross-section (value σ_ct,imm) for immediate characteristic combination (N_char,im, M_char,im,y, M_char,im,z)
• check condition σ_ct ≥ σ_ct,imm
  • If condition is fulfilled, then respective characteristic combination will be used for calculation, N_char = N_char,res, M_char,y = M_char,res,y, M_char,z = M_char,res,z, σ_ct = σ_ct,res
  • otherwise immediate characteristic combination will be used N_char = N_char,im, M_char,y = M_char,im,y, M_char,z = M_charim,z, σ_ct = σ_ct,im

• check condition(σ_ct ≤ σ_cr)

  • If condition is fulfilled, then cross-section is uncracked nad

    • bending stiffness around y-axis EI_y = (EI_y)_I

    • bending stiffness around z axis EI_z = (EI_y)_I

    • axial stiffness EA = EA_I,

  • Else, cross-section is cracked and average stiffness will be calculated

• calculation transformed CSS characteristics of cracked section (A_ir, I_ir, t_ir...)
• calculation stiffness’s of fully cracked cross-section ((EI_y)_II,( EI_z)_II, (EA)_II) to centre of cracked transformed cross-section
• calculation stress in tensile reinforcement of fully cracked cross-section (value σ_sr) for characteristic combination (N_char, M_char, M_char,z)
• calculation stress in tensile reinforcement of fully cracked cross-section (value σ_s) for respective combination (N, M_y, M_z

• calculation the distribution coefficient ξ according equation 7.19 in EN 1992-1-1

  ξ = 1 - β ∙ (σ_sr / σ_s)

  where

  β	coefficient taking account of the influence of the duration of the loading or of repeated loading on the average strain β = 1 for calculation short-term stiffness β = 0,5 for calculation long-term stiffness

• calculation averages value of stiffness’s based on equation 7.18 in EN 1992-1-1

  • bending stiffness around y-axis (EI_y) = 1 / [ξ / (EI_y)_II + (1 - ξ) / (EI_y)_I]

  • bending stiffness around z-axis (EI_z) = 1 / [ξ / (EI_z)_II + (1 - ξ) / (EI_z)_I]

  • axial stiffness (EA) = 1 / [(ξ / (EA)_II + (1 - ξ) / ( EA)_I]

Stiffness is recalculated to principal axis for unsymmetrical cross-section.

The four type of stiffnesses is calculated for each 1D element and each dangerous combination:

Type of stiffness	Respective combination
Short-term stiffness for immediate deflection	Immediate
Short-term stiffness for short-term deflection	Total
Short-term stiffness for creep deflection	Creep
Long-term stiffness for creep deflection	Creep

 

The following stiffnesses are changes in stiffness matrix for 1D element:

EA_x = EA

GA_y = GA_z = G ∙ EA_x / (1.2 ∙ E_c)

EI_y = EI_y

EI_z =EI_z

GI_x = 0.5 ∙ 1 - μ) ∙ (EI_y ∙ EI_z)^0.5

where

G	shear modulus of the concrete calculated according to formula G = 0.5 ∙ Ec / (1 + μ)
μ	Poisson's coefficient of the concrete loaded from material properties of the concrete

Eccentricity of stiffness (distance between centre of gravity of concrete cross-section and centre of gravity of cracked transformed cross-section) is not taken into account in version SCIA Engineer 21.