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Calculation of dimensional forces for 2D member

The forces from FEM analysis (n_x,n_y, n_xy,m_x, m_y, m_xy) calculated in direction of LCS has to be recalculated to dimensional forces, it means forces calculated in direction of reinforcement at upper and lower surface.

Procedure for calculation

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

The dimensional internal forces is calculated only in two inputted directions. The different directions for lower and upper surface can be used.
The dimensional forces without influence of additional tensile force caused by shear is calculated
Baumann’s transformation formulas [1] is used in calculation
The angle of concrete strut is optimized to force in direction of concrete strut was the smallest. In second step there will be more method for optimization

The following procedure is used for calculation:

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

For recalculation dimensional forces the following values are necessary:

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	The angle of x axis of LCS of FEM element. The angle of x axis will be depends on selected LCS system to which forces is calculated (LCS of FEM element is used)
α_inp,₁₍₂₎₊	The first (second) inputted direction of calculation at upper surface defined from x axis of LCS of FEM element defined in Concrete setup
α_inp,_1(2)-	The first (second) inputted direction of calculation at lower surface defined from x axis of LCS of FEM element defined in Concrete setup
Δα	Minimal angle between angle of inputted direction and angle of concrete strut, default value is 15 degrees. It is internal value, it follows user can not change this value.
a_s+(-)	Distance of the centroid of the upper(lower) defined longitudinal reinforcement from upper (lower) surface of 2D element. For surface with two layer of reinforcement a_s+(-) =cover_+(-) + d_s1+(-)
d₁₍₂_)+(-)	Diameter of longitudinal reinforcement in first(second) direction at upper(lower) surface defined in Concrete setup
Cover_+(-)	Cover of longitudinal reinforcement at upper(lower) surface calculated to surface of the closer bar to the surface. Value is defined in defined in Concrete setup.
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 of inner lever arm for 2D element

The inner lever arm is necessary to know for calculation surface forces. Inner lever arm is calculated in direction of angle of first principal moment (the forces is recalculated to this direction).

The forces for calculation inner lever arm is calculated according to formulas below

where
n_x(y)	normal force in x(y) direction
m_x(y)	bending moment in x(y) direction
n_xy	membrane shear force
m_xy	twisting moment
α	angle of first principal moment

If inner lever arm is not calculated (forces are zero or equilibrium is not found) ,then inner lever arm is calculated according to formula

z = 0,9*d

where
d	is effective height , which is calculated according to formula. If m_α >=0 then d = d_loelse d = d_up
d_lo	the effective height for lower surface. It is distance the reinforcement at lower surface to upper surface of the element d_lo = h-a_s-
d_up	the effective height for upper surface. It is distance the reinforcement at upper surface to lower surface of the element d_lo = h-a_s+

The parts of inner lever arm is calculated according to formulas below:

z₊

the part of lever arm for upper surface (above centre of cross-section)

If m_α >=0 then z₊ is distance of center of compressive concrete to centre of CSS.

If m_α <0 then z₊ is distance of center of tensile reinforcement to centre of CSS.

If value z is not calculated (forces are zero or equilibrium is not found) then z₊ =0.45d

z_-

the part of lever arm for lower surface (under centre of cross-section)

If m_α >=0 then z_- is distance of center of tensile reinforcement to centre of CSS

If m_α <0 then z_- is distance of center of compressive concrete to centre of CSS .

If value z is not calculated (forces are zero or equilibrium is not found) then z_- =0.45d.

The formulas z = z₊+z- has to be fulfilled for parts of inner lever arm too.

Calculation of normal forces at both surfaces of 2D element

The inputted internal forces is recalculated to both surfaces according the following formulas:

Calculation of principal forces at both surfaces of 2D element

The principal forces at both surfaces are calculated from normal forces at both surfaces according to

and direction of first principal force is calculated according to formula

Recalculation of principal forces at both surfaces to inputted directions

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

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 α_2- (i=2,j=3,k=1_-) the formula will be following
	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_∓= α_inp,₁_∓ – α_n1_∓ The second inputted direction: α₂_∓= α_inp,₂_∓ – α_n1_∓ Direction of concrete strut: α₃_∓= α_con_∓ – α_n1_∓
α_con_∓	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: α_con_∓= 0,5·( α_inp,₁_∓ α_inp,₂_∓). For hyperbolic state of stress (n₁_∓> 0 and n₂_∓< 0) the angle of concrete strut should be found by optimization method , see next chapter. For the angle of concrete strut, the following conditions have to be fulfilled: (α_inp,₁_∓+n·180deg) -Δα ≥α_con_∓ ≥ (α_inp,₁_∓+n·180deg) +Δα for n=0,1,2 (α_inp,₂_∓+n·180deg) -Δα ≥α_con_∓ ≥ (α_inp,₂_∓+n·180deg) +Δα for n=0,1,2

Direction of reinforcement for recalculation at lower surface

Optimization of angle of concrete strut

The direction of concrete strut at lower (upper) surface can be calculated for all states of stresses excepted of hyperbolic state of stress (n₁_∓> 0 and n₂_∓< 0) according to formula: α_con_∓= 0,5·( α_inp,₁_∓ α_inp,₂_∓). For hyperbolic state of stress (n₁_∓> 0 and n₂_∓< 0) the angle of concrete strut should be found by optimization methods. The following rules for optimization angle of concrete strut is used:

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

the angle of concrete strut has to fulfil the following conditions:

(αinp,1∓ +n·180deg) -Δα ≥αcon∓ ≥ (αinp,1∓ +n·180deg) +Δα , n=0,1,2

(αinp,2∓ +n·180deg) -Δα ≥αcon∓ ≥ (αinp,2∓ +n·180deg) +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 is calculated according to Baumann’s transformation formula, where for virtual forces at lower surface direction of reinforcements and direction of concrete strut for upper surface and vice versa

	If α_inp,₁_- = α_inp,₁₊, then else
	If α_inp,₂_- = α_inp,₂₊, then else
	If α_inp,₃_- = α_inp,₃₊, then else
	If α_inp,₁₊ = α_inp,₁_-, then else
	If α_inp,₂₊ = α_inp,₂_-, then else
	If α_inp,₃₊ = α_inp,₃_-, then else