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Internal Forces 2D - Theoretical background

 

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

Definition of input value for calculation

The following value are defined as basic forces in centres:

nx(y)

Normal force in node of 2D member in x(y) direction
mx(y)

Bending moment in node of 2D member in x(y) direction
nxy Membrane shear force in node of 2D member
mxy Twisting moment in node of 2D member
h Thickness of 2D element
αx 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…)
αinp,1(2)+ The first (second) inputted direction of calculation at upper surface defined from x axis of LCS of FEM element
αinp,1(2)- The first (second) inputted direction of calculation at lower surface defined from x axis of LCS of FEM element
θα Minimal angle between angle of inputted direction and angle of concrete strut, default value will be 15 °
as+(-)

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

as+(-) = cover+(-) + ds1+(-)
d1(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

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

mα = mx ∙ cos2(α) + my ∙ sin2(α) + mxy ∙ sin(2 ∙ α)

nα = nx ∙ cos2(α) + ny ∙ sin2(α) + nxy ∙ 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

nx-(+) = nx / 2 +(-) mx / z

ny-(+) = ny / 2 +(-) my / z

nxy-(+) = nxy / 2 +(-) mxy / 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

nI∓ = (nx∓ + ny∓) / 2 + 1/2 ∙ √((nx∓ + ny∓) 2 + 4 ∙ nxy,∓)

nII∓ = (nx∓ + ny∓) / 2 - 1/2 ∙ √((nx∓ + ny∓) 2 + 4 ∙ nxy,∓)

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

αI∓ = 0,5 ∙ tan-1((2 ∙ nxy∓) / (nx∓ - ny∓))

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

nEds,i∓ = (nI∓ ∙ sin(αj∓) ∙ sin(αk∓) + nII∓ ∙ cos(αj∓) ∙ cos(αk∓)) / (sin(αj∓ - αi∓) ∙ sin(αk∓ - αi∓))

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 (n1∓ > 0 and n2∓ < 0) the angle of concrete strut αcon∓ 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 (nEds,3∓ < 0 and nEds,3∓ = minimum). The solution can be based on derivation of the following equation.

    nEds,3∓ = (nI∓ ∙ sin(α1∓) ∙ sin(α2∓) + nII∓ ∙ cos(α1∓) ∙ cos(α2∓)) / (sin(α1∓ - x) ∙ sin(α2∓ - x))

    d/dx nEds,3∓(x) = {cos(α1∓ - x) ∙ [nII∓ ∙ cos (α1∓) ∙ cos (α2∓) + nI∓ ∙ sin (α1∓) ∙ sin (α2∓)]} / [sin (α1∓ - x)2 ∙ sin(α2∓ - x)2] + {cos(α2∓ - x) ∙ [nII∓ ∙ cos (α1∓) ∙ cos (α2∓) + nI∓ ∙ sin (α1∓) ∙ sin (α2∓)]} / [sin (α1∓ - x)2 ∙ sin(α2∓ - 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 (nEds,3∓ < 0 and ∑nEds,i∓ = minimum (for nEds,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:

    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

    IF αinp,1- = αinp,1+

    THEN nEds,virt,1- = nEds,1-

    ELSE nEds,virt,1- = (nI- ∙ sin(α2-) ∙ sin(α1+) + nII- ∙ cos(α2-) ∙ cos(α1+)) / (sin(α2- - α1-) ∙ sin(α1+ - α1-))

    (Baumann’s transformation formula where α3- = α1+)

    IF αinp,2- = αinp,2+

    THEN nEds,virt,2- = nEds,2-

    ELSE nEds,virt,2- = (nI- ∙ sin(α2+) ∙ sin(α1-) + nII- ∙ cos(α2+) ∙ cos(α1-)) / (sin(α2+ - α2-) ∙ sin(α1- - α2-))

    (Baumann’s transformation formula where α3- = α2+)

    IF αinp,3- = αinp,3+

    THEN nEds,virt,3- = nEds,3-

    ELSE nEds,virt,3- = (nI- ∙ sin(α2+) ∙ sin(α1-) + nII- ∙ cos(α2-) ∙ cos(α1-)) / (sin(α2- - α3+) ∙ sin(α1- - α3+))

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

    IF αinp,1+ = αinp,1-

    THEN nEds,virt,1+ = nEds,1+

    ELSE nEds,virt,1+ = (nI+ ∙ sin(α2-) ∙ sin(α1+) + nII+ ∙ cos(α2-) ∙ cos(α1+)) / (sin(α2+ - α1+) ∙ sin(α1- - α1+))

    (Baumann’s transformation formula where α3+ = α1-)

    IF αinp,2+ = αinp,2-

    THEN nEds,virt,2-= nEds,2-

    ELSE nEds,virt,2- = (nI- ∙ sin(α2+) ∙ sin(α1-) + nII- ∙ cos(α2+) ∙ cos(α1-)) / (sin(α2+ - α2-) ∙ sin(α1- - α2-))

    (Baumann’s transformation formula where α3+ = α2-)

    IF αinp,3+ = αinp,3-

    THEN nEds,virt,3+ = nsurface,3+

    ELSE nEds,virt,3+ = (nI+ ∙ sin(α2+) ∙ sin(α1+) + nII+ ∙ cos(α2+) ∙ cos(α1+)) / (sin(α2+ - α3-) ∙ sin(α1+ - α3-))

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

     

    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.

    mEd,i = nEds,i- ∙ z- + nEds,virt,i+ ∙ z+

    nEd,i = nEds,i- + nEds,virt,i+

    mEd,i = nEds,virt,i- ∙ z- + nEds,i+ ∙ z+

    nEd,i = nEds,virt,i- + nEds,,i+

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