Calculation stiffness of 2D 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 2D 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 of 2D element

• Calculation of principal stresses of 2D element for both surfaces

σ_1± = (σ_x± + σ_y±) / 2 + 1/2 ∙ √[(σ_x± - σ_y±)² + 4 ∙ τ_xy±]

σ_2± = (σ_x± + σ_y±) / 2 - 1/2 ∙ √[(σ_x± - σ_y±)² + 4 ∙ τ_xy±]

• Calculation angle of principal stresses at both surfaces

α_σ1± = 0.5 ∙ tan^-1 [2 ∙ τ_xy± / (σ_x± - σ_y±)]

• Calculation of final value of principal stress

α = α_σ1+ if σ₁₊ ≥ σ_1-

α = α_σ1- otherwise

• Recalculation internal forces to direction of principal stress α

m(α) = m_x ∙ cos²(α) + m_y ∙ sin²(α) + m_xy ∙ sin(2 ∙ α)

n(α) = n_x ∙ cos²(α) + n_y ∙ sin²(α) + n_xy ∙ sin(2 ∙ α)

where n_x, n_y, n_xy, m_x, m_y, m_xy are 2D forces in centre of 2D element

• Recalculation area of reinforcement to direction of of principal stress α

A_s(α) = A_s ∙ cos²(α - α_s)

where A_s, α_s are area and angle of longitudinal reinforcement

• Calculation of non-linear stiffness in first principal direction according to procedure as for 1D element, see "Calculation stiffness of 1D element"
  • for rectangular cross-section (b = 1 m, h = thickness of 2D element in centre of gravity)
  • for internal forces N = n(α), M_y= m(α) and M_z = 0 according procedure as for 1D element
• Calculation of non-linear stiffness in second principal direction according to procedure as for 1D element, see "Calculation stiffness of 1D element"
  • for rectangular cross-section (b = 1 m, h = thickness of 2D element in centre of gravity)
  • for internal forces N = n(α+90) , M_y= m(α+90) and M_z = 0 according procedure as for 1D element

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

Type of stiffness	Respective combination	Direction of principal stress
Short-term stiffness for immediate deflection	Immediate	First (EA1, EIy1, EIz1)
		Second (EA2, EIy2, EIz2)
Short-term stiffness for short-term deflection	Total	First (EA1, EIy1, EIz1)
		Second (EA2, EIy2, EIz2)
Short-term stiffness for creep deflection	Creep	First (EA1, EIy1, EIz1)
		Second (EA2, EIy2, EIz2)
Long-term stiffness for creep deflection	Creep	First (EA1, EIy1, EIz1)
		Second (EA2, EIy2, EIz2)

The following stiffnesses are changes in stiffness matrix for 2D element:

D11 = EI_y1

D22 = EI_y2

D33 = 0.5 ∙ (1 - μ) ∙ (D11 ∙ D22)^0.5

D44 = G ∙ h / 1.2

D55 = G ∙ h / 1.2

D12 = μ ∙ (D11 ∙ D22)^0.5

d11 = EA₁

d22 = EA₂

d33 = G ∙ h

d12 = μ ∙ (d11 ∙ d22)^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.