Theoretical background

 

The following procedure will be used for calculation of dimensional forces 2D:

• Definition of input value for calculation

• Calculation of inner lever arm

• Calculation of normal forces at both surfaces of 2D element

• Calculation of principal forces at both surfaces of 2D element

• Recalculation of principal forces at both surfaces to inputted direction (Baumann’s transformation formula)

• Optimization of angle of concrete strut

• Calculation of virtual forces at lower(upper) surface for inputted direction and direction of concrete strut for upper(lower surface)

• Recalculation of forces at surfaces to centre of gravity of cross-section of 2D member

Definition of input value for calculation

The following value are defined as basic forces in centres:

• n_x(y) - normal force in node of 2D member in x(y) direction
• m_x(y) - bending moment in node of 2D member in x(y) direction
• n_xy - membrane shear force in node of 2D member
• m_xy - twisting moment in node of 2D member
• h - thickness of 2D element
• α_x - angle of x axis of LCS of FEM element. The angle of x axis will be depends on selected LCS system to which forces will be calculated (LCS of FEM element, LCS of 2D element, UCS…)
• α_inp,1(2)+ - the first (second) inputted direction of calculation at upper surface defined from x axis of LCS of FEM element
• α_inp,1(2)- - the first (second) inputted direction of calculation at lower surface defined from x axis of LCS of FEM element
• θ_α Minimal angle between angle of inputted direction and angle of concrete strut, default value will be 15 °
• a_s+(-) Distance of the centroid of the upper(lower) defined longitudinal reinforcement from upper (lower) edge of 2D element. For surface with two layer of reinforcement

  a_s+(-) = cover_+(-) + d_s1+(-)

• d_1(2)+(-) Diameter of longitudinal reinforcement in first(second) direction at upper(lower) surface
• cover_+(-) Cover of longitudinal reinforcement at upper(lower) surface calculated to surface of the closer bar to the surface
• z - Inner lever arm for upper (lower) surface.
• z_+(-) - The position for recalculation forces at surfaces to centre of gravity of cross-section of 2D member.

Calculation lever arm for 2D element

The lever arm is necessary to know for calculation surface forces. The a special cross-section set has to be created for calculation of lever arm (value z). Value z will be calculated in direction of angle of first principal moment (the forces will be recalculated to this direction and cross-section set will be created in this direction).The reinforcement will be designed for recalculated forces and from designed reinforcement inner lever arm will be calculated

If value z is not calculated (forces are zero or equilibrium is not found) the value z will be calculated according to formula

z = 0,9 ∙ d

where

d	effective height , which is calculated according to formula IF mα ≥ 0 then d = dlo else d = dup
dlo	the effective height for lower surface is distance of centroid of inputted element at lower surface for whole element from upper edge of the element dlo = h - as-
dup	the effective height for upper surface is distance of centroid of inputted element at upper surface for whole element from lower edge of the element dup = h - as+

The distance of centre of compressive concrete and centre of tensile reinforcement to centre of cross-section has to be calculated too

z = z₊ + z_-

where

z+	the part of lever arm for upper surface (above centre of cross-section) IF mα ≥ 0 then z+ is distance of centre of compressive concrete to centre of CSS IF mα < 0 then z+ is distance of centre of tensile reinforcement to centre of CSS IF value z is not calculated (forces are zero or equilibrium is not found) then z = 0,45 ∙ d
z-	the part of lever arm for lower surface (under centre of cross-section) IF mα ≥ 0 then z- is distance of centre of tensile reinforcement to centre of CSS IF mα < 0 then z- is distance of centre of compressive concrete to centre of CSS IF value z is not calculated (forces are zero or equilibrium is not found) then z = 0,45 ∙ d

The forces for calculation of z will be will be calculated according to formulas below

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)

Calculation of normal forces at both surfaces of 2D element

The inputted internal forces will be recalculated to both surfaces according the following formulas

n_x-(+) = n_x / 2 +(-) m_x / z

n_y-(+) = n_y / 2 +(-) m_y / z

n_xy-(+) = n_xy / 2 +(-) m_xy / z

Calculation of principal forces at both surfaces of 2D element

The principal forces at both surfaces will be calculated according to the following formulas

n_I∓ = (n_x∓ + n_y∓) / 2 + 1/2 ∙ √((n_x∓ + n_y∓) ² + 4 ∙ n_xy,∓²)

n_II∓ = (n_x∓ + n_y∓) / 2 - 1/2 ∙ √((n_x∓ + n_y∓) ² + 4 ∙ n_xy,∓²)

and direction of first principal force will be calculated according to formula

α_I∓ = 0,5 ∙ tan^-1((2 ∙ n_xy∓) / (n_x∓ - n_y∓))

Recalculation of principal forces at both surfaces to inputted (Baumann’s transformation formula)

The recalculation the principal forces to inputted direction will be done separately for both surfaces with using Baumann’s transformation formula

n_Eds,i∓ = (n_I∓ ∙ sin(α_j∓) ∙ sin(α_k∓) + n_II∓ ∙ cos(α_j∓) ∙ cos(α_k∓)) / (sin(α_j∓ - α_i∓) ∙ sin(α_k∓ - α_i∓))

where

i, j, k, i	index of direction (direction for recalculation of forces) i, j, k, i = 1,2,3,1. For example for lower surface and for calculation of forces in second direction α2- (i = 2, j = 3, k = 1) the formula will be following nEds,2- = (nI- ∙ sin(α3-) ∙ sin(α1-) + nII- ∙ cos(α3-) ∙ cos(α1-)) / (sin(α3- - α2-) ∙ sin(α1- - α2-))
αi,j,k±	angle between inputted direction or direction of concrete strut and direction of first principal forces at lower or upper surface The first inputted direction: α1∓ = αinp,1∓ - αI∓ The second inputted direction: α2∓ = αinp,2∓ - αI∓ Direction of concrete strut: α3∓ = αcon∓ - αI∓
αcon∓	the direction of concrete strut at lower (upper) surface. This value can be calculated for all states of stresses excepted of hyperbolic state of stress (n1∓ > 0 and n2∓ < 0) according to formula: αcon∓ = 0,5 ∙ (αinp,1∓ + αinp,2∓). For hyperbolic state of stress (n1∓ > 0 and n2∓ < 0) the angle of concrete strut should be find by optimization method , see next chapter. For the angle of concrete strut, the following conditions have to be fulfilled: (αinp,1∓ + n ∙ 180°) - Δα ≥ αcon∓ ≥ (αinp,1∓ + n ∙ 180°) + Δα; n = 0,1,2 (αinp,2∓ + n ∙ 180°) - Δα ≥ αcon∓ ≥ (αinp,2∓ + n ∙ 180°) + Δα; n = 0,1,2

Optimization of angle of concrete strut

For hyperbolic state of stress (n_1∓ > 0 and n_2∓ < 0) the angle of concrete strut α_con∓ should be find by optimization methods. The two methods for optimization of angle of concrete strut will be available:

The force in concrete strut will be compressive and the smallest (n_Eds,3∓ < 0 and n_Eds,3∓ = minimum). The solution can be based on derivation of the following equation.

n_Eds,3∓ = (n_I∓ ∙ sin(α_1∓) ∙ sin(α_2∓) + n_II∓ ∙ cos(α_1∓) ∙ cos(α_2∓)) / (sin(α_1∓ - x) ∙ sin(α_2∓ - x))

d/dx n_Eds,3∓(x) = {cos(α_1∓ - x) ∙ [n_II∓ ∙ cos (α_1∓) ∙ cos (α_2∓) + n_I∓ ∙ sin (α_1∓) ∙ sin (α_2∓)]} / [sin (α_1∓ - x)² ∙ sin(α_2∓ - x)²] + {cos(α_2∓ - x) ∙ [n_II∓ ∙ cos (α_1∓) ∙ cos (α_2∓) + n_I∓ ∙ sin (α_1∓) ∙ sin (α_2∓)]} / [sin (α_1∓ - x)² ∙ sin(α_2∓ - x)²] = 0

The force in concrete strut will be compressive and the sum of all compressive forces in all direction at one surface will be the smallest (n_Eds,3∓ < 0 and ∑n_Eds,i∓ = minimum (for n_Eds,i∓ ≤ 0)). The optimization method for finding this angle of concrete strut is not developed.

For the angle of concrete strut, the following conditions have to be fulfilled:

• (α_inp,1∓ + n ∙ 180°) - Δα ≥ α_con∓ ≥ (α_inp,1∓ + n ∙ 180°) + Δα; n = 0,1,2
• (α_inp,2∓ + n ∙ 180°) - Δα ≥ α_con∓ ≥ (α_inp,2∓ + n ∙ 180°) + Δα; n = 0,1,2

Calculation of virtual forces at lower(upper) surface for inputted direction and direction of concrete strut for upper (lower surface)

The virtual forces will be calculated according to Baumann’s transformation formula, where instead of angle of concrete strut angle of direction for upper or lower surface will be added

• n_Eds,virt,1-

IF α_inp,1- = α_inp,1+

THEN n_Eds,virt,1- = n_Eds,1-

ELSE n_Eds,virt,1- = (n_I- ∙ sin(α_2-) ∙ sin(α₁₊) + n_II- ∙ cos(α_2-) ∙ cos(α₁₊)) / (sin(α_2- - α_1-) ∙ sin(α₁₊ - α_1-))

(Baumann’s transformation formula where α_3- = α₁₊)

n_Eds,virt,2-

IF α_inp,2- = α_inp,2+

THEN n_Eds,virt,2- = n_Eds,2-

ELSE n_Eds,virt,2- = (n_I- ∙ sin(α₂₊) ∙ sin(α_1-) + n_II- ∙ cos(α₂₊) ∙ cos(α_1-)) / (sin(α₂₊ - α_2-) ∙ sin(α_1- - α_2-))

(Baumann’s transformation formula where α_3- = α₂₊)

• n_Eds,virt,3-

IF α_inp,3- = α_inp,3+

THEN n_Eds,virt,3- = n_Eds,3-

ELSE n_Eds,virt,3- = (n_I- ∙ sin(α₂₊) ∙ sin(α_1-) + n_II- ∙ cos(α_2-) ∙ cos(α_1-)) / (sin(α_2- - α₃₊) ∙ sin(α_1- - α₃₊))

(Baumann’s transformation formula where α_3- = α₃₊)

• n_Eds,virt,1+

IF α_inp,1+ = α_inp,1-

THEN n_Eds,virt,1+ = n_Eds,1+

ELSE n_Eds,virt,1+ = (n_I+ ∙ sin(α_2-) ∙ sin(α₁₊) + n_II+ ∙ cos(α_2-) ∙ cos(α₁₊)) / (sin(α₂₊ - α₁₊) ∙ sin(α_1- - α₁₊))

(Baumann’s transformation formula where α₃₊ = α_1-)

• n_Eds,virt,2+

IF α_inp,2+ = α_inp,2-

THEN n_Eds,virt,2-= n_Eds,2-

ELSE n_Eds,virt,2- = (n_I- ∙ sin(α₂₊) ∙ sin(α_1-) + n_II- ∙ cos(α₂₊) ∙ cos(α_1-)) / (sin(α₂₊ - α_2-) ∙ sin(α_1- - α_2-))

(Baumann’s transformation formula where α₃₊ = α_2-)

n_Eds,virt,3+

IF α_inp,3+ = α_inp,3-

THEN n_Eds,virt,3+ = n_surface,3+

ELSE n_Eds,virt,3+ = (n_I+ ∙ sin(α₂₊) ∙ sin(α₁₊) + n_II+ ∙ cos(α₂₊) ∙ cos(α₁₊)) / (sin(α₂₊ - α_3-) ∙ sin(α₁₊ - α_3-))

(Baumann’s transformation formula where α₃₊ = α_3-)

 

Recalculation of forces at surfaces to centre of gravity of cross-section of 2D member

The forces will be recalculated to centre of gravity of cross-section of 2D member.

• Forces at centroid for direction inputted for upper surface

m_Ed,i = n_Eds,i- ∙ z_- + n_Eds,virt,i+ ∙ z₊

n_Ed,i = n_Eds,i- + n_Eds,virt,i+

• Forces at centroid for direction inputted for lower surface

m_Ed,i = n_Eds,virt,i- ∙ z_- + n_Eds,i+ ∙ z₊

n_Ed,i = n_Eds,virt,i- + n_Eds,,i+