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006245

Sonja von Bloh, M.Sc.

2026-09-29

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Dlubal Center

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Finite Element Analysis
- Cross-Section Properties

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Program Functions

Interfaces

Result Evaluation

Printout Report

Cross-Section Properties

Cross-Sectional Area

A distinction is made between the geometric and the effective cross-sectional area. The geometric cross-sectional area corresponds to the actual area of the cross-section and is independent of the material properties. On the other hand, the effective cross-sectional area takes into account the different material stiffnesses, for example, in composite cross-sections.

The effective cross-sectional area is related to a reference material with a modulus of elasticity E_ref:

A = \frac{1}{E_{ref}} \int_{A} E ​ d A

Elastizitätsmodul am Flächenpunkt	Modulus of elasticity at the surface point
E_ref	Modulus of elasticity of the reference material

Cross-Sectional Area A

The geometric cross-sectional area is given by:

A_{g e o m} = \int_{A} d A

Geometric Cross-Sectional Area A_geom

Location of Elastic Centroid

The coordinates e_y and e_z describe the location of the elastic centroid relative to the origin of the global coordinate system in the Y and Z directions, respectively.
The elastic centroid is determined by considering the material stiffnesses. Its coordinates are given by the surface integrals weighted by the modulus of elasticity:

e_{y} = \frac{\int_{A} E \cdot Y ​ d A}{\int_{A} E ​ d A}

E_ref	Y-coordinate of the surface point related to the origin
E	Modulus of elasticity at the surface point

Position of Elastic Centroid in y-Direction e_y Relative to Origin

e_{z} = \frac{\int_{A} E \cdot Z ​ d A}{\int_{A} E ​ d A}

y	Z-coordinate of the surface point related to the origin
E	Modulus of elasticity at the surface point

Position of Elastic Centroid in Z-Direction e_z Related to Origin

Second Moments of Area

The second moments of area I_y and I_z refer to the local coordinate system y, z. The origin of the local coordinate system lies at the elastic centroid. Its axes run parallel to the global axes Y and Z.

I_{y} = \frac{1}{E_{ref}} \int_{A} E \cdot {(Z - e_{z})}^{2} ​ d A

E_ref	Modulus of elasticity of the reference material
e_z	Location of the elastic centroid in the z-direction relative to the origin
E	Modulus of elasticity at the surface point
z	Z-coordinate of the surface point related to the origin

Second Moment of Area About Y-Axis I_y

I_{z} = \frac{1}{E_{ref}} \int_{A} E \cdot {(Y - e_{y})}^{2} ​ d A

E_ref	Modulus of elasticity of the reference material
e_y	Location of the elastic centroid in the y-direction related to the origin
E	Modulus of elasticity at the surface point
z	Y-coordinate of the surface point related to the origin

Second Moment of Area About z-Axis I_z

The second moments of area I_u and I_v refer to the principal axes u and v. These axes pass through the elastic centroid and are rotated relative to the local coordinate system y, z by the principal axis angle α.

I_{u} = \frac{I_{y} + I_{z}}{2} + \frac{1}{2} \sqrt{{(I_{y} - I_{z})}^{2} + 4 \cdot I_{y z}^{2}}

I_y, I_z	Second moment of area about the y- or z-axis
I_yz	Centrifugal moment about the y,z axes

Second Moment of Area About u-Axis I_u

I_{v} = \frac{I_{y} + I_{z}}{2} - \frac{1}{2} \sqrt{{(I_{y} - I_{z})}^{2} + 4 \cdot I_{y z}^{2}}

I_y, I_z	Second moment of area about the y- or z-axis
I_yz	Product of inertia about the y,z axes

Second Moment of Area About v-Axis I_v

For cross-sections consisting of several unconnected partial cross-sections, the second moments of area are summed without taking the Steiner component into account.

Centrifugal Moment

The centrifugal moment I_yz refers to the local coordinate system y, z. The origin of the local coordinate system lies at the elastic centroid. Its axes run parallel to the global axes Y and Z.

The centrifugal moment of the ideal cross-section is given by:

I_{y z} = \frac{1}{E_{ref}} \int_{A} E \cdot (Y - e_{y}) \cdot (Z - e_{z}) ​ d A

E_ref	Modulus of elasticity of the reference material
E	Modulus of elasticity at the surface point
e_y, e_z	Coordinates of elastic centroid related to the origin
Y, Z	Coordinates of the surface point related to the origin

Centrifugal Moment About y,z-Axis I_y,z

Polar Second Moments of Area

The polar second moment of area I_p is relative to the centroid and is given by the sum of the two principal second moments of area:

I_{p} = I_{u} + I_{v}

I_u, I_v

Second moment of area about the u- or v-axis

Polar Moment of Inertia I_p

The polar second moment of area I_p,SC is relative to the shear center.

I_{p, S C} = I_{p} + A \cdot (u_{S C}^{2} + v_{S C}^{2})

I_u, I_v	Second moment of area about the u- or v-axis
A	Cross-sectional area
u_SC, v_SC	Shear center coordinate relaed to the elastic centroid in the u- or v-direction

Polar Moment of Inertia Related to Shear Center I_p,SC

Inclination of Principal Axes

The angle α describes the rotation of the principal axis system u, v relative to the local coordinate system y, z.

α = \frac{1}{2} \cdot arctan (\frac{-2 \cdot I_{y z}}{I_{y} - I_{z}})

I_yz	Centrifugal moment about the y,z axes
I_y, I_z	Second moment of area about the y- or z-axis

Inclination of Principal Axes α

Radii of Gyration

The radius of gyration describes the distribution of the cross-sectional area relative to a reference axis. It is calculated as the square root of the ratio of the corresponding second moment of area to the cross-sectional area.

i_{y} = \sqrt{\frac{I_{y}}{A}}

I_y	Second moment of area about the y-axis
A	Cross-sectional area

Radius of Gyration About Y-Axis i_y

i_{z} = \sqrt{\frac{I_{z}}{A}}

I_z	Second moment of area about the Z-axis
A	Cross-sectional area

Radius of Gyration About z-Axis i_z

i_{y z} = \sqrt{\frac{I_{y z}}{A}}

I_yz	Product of inertia about the y,z axes
A	Cross-sectional area

Radius of Gyration About y,z Axes i_yz

i_{u} = \sqrt{\frac{I_{u}}{A}}

I_u	Second moment of area about the u-axis
A	Cross-sectional area

Radius of Gyration About u-Axis i_u

i_{v} = \sqrt{\frac{I_{v}}{A}}

I_v	Second moment of area about the v-axis
A	Cross-sectional area

Radius of Gyration About v-Axis i_v

i_{p} = \sqrt{\frac{I_{p}}{A}}

I_p	Polar moment of inertia
A	Cross-sectional area

Polar Radius of Gyration i_p

i_{p, S C} = \sqrt{\frac{I_{p, S C}}{A}}

I_p,SC	Polar moment of inertia related to the shear center
A	Cross-sectional area

Polar Radius of Gyration About Shear Center i_p,SC

Elastic Section Modulus

The elastic section modulus W_y,min, W_y,max, W_z,min, W_z,max, W_y, and W_z refer to the local coordinate system y, z. The origin of the local coordinate system lies at the elastic centroid. Its axes run parallel to the global Y and Z axes.

W_{y, min} = \frac{I_{y} \cdot I_{z} - I_{y z}^{2}}{I_{z} \cdot z_{min} - I_{y z} \cdot y (z_{min})}

I_y, I_z	Second moment of area about the y-axis or z-axis
I_yz	Product of inertia about the y,z axes
z_min	Minimum ordinate in the local Z-direction
y(z_min)	y-coordinate of the same cross-section point belonging to the ordinate z_min

Elastic Section Modulus About Y-Axis W_y,min

W_{y, m a x} = \frac{I_{y} \cdot I_{z} - I_{y z}^{2}}{I_{z} \cdot z_{m a x} - I_{y z} \cdot y (z_{m a x})}

I_y, I_z	Second moment of area about the y- or z-axis
I_yz	Centrifugal moment about the y,z axes
z_max	Maximum ordinate in the local z-direction
y(z_max)	y-coordinate of the same cross-section point belonging to the ordinate z_max

Elastic Section Modulus About Y-Axis W_y,max

W_{z, min} = \frac{I_{y} \cdot I_{z} - I_{y z}^{2}}{I_{y} \cdot y_{min} - I_{y z} \cdot z (y_{min})}

I_y, Iz	Second moment of area about the y- or z-axis
I_yz	Centrifugal moment about the y,z axes
y_min	Minimum ordinate in the local y-direction
z(y_min)	z-coordinate of the same cross-section point belonging to the ordinate y_min

Elastic Section Modulus About z-Axis W_z,min

W_{z, m a x} = \frac{I_{y} \cdot I_{z} - I_{y z}^{2}}{I_{y} \cdot y_{m a x} - I_{y z} \cdot z (y_{m a x})}

I_y, I_z	Second moment of area about the y- or z-axis
I_yz	Centrifugal moment about the y,z-axes
y_max	Maximum ordinate in the local y-direction
z(y_max)	z-coordinate of the same cross-section point belonging to the ordinate y_max

Elastic Section Modulus About Z-Axis W_z,max

W_{y} = m i n (|W_{y, m i n}|, |W_{y, m a x}|)

W_y,min	Largest negative elastic section modulus about the Y-axis
W_y,max	Smallest positive elastic section modulus about the Y-axis

Elastic Section Modulus About y-Axis W_y

W_{z} = m i n (|W_{z, m i n}|, |W_{z, m a x}|)

W_z,min	Largest negative elastic section modulus about the Z-axis
W_z,max	Smallest positive elastic section modulus about the Z-axis

Elastic Section Modulus About Z-Axis W_z

The elastic coefficients W_u,min, W_u,max, W_v,min, W_v,max, W_u, and W_v refer to the principal axes u and v.

W_{u, m i n} = \frac{I_{u}}{v_{m i n}}

I_u	Second moment of area about the u-axis
v_min	Minimum ordinate in the v-direction

Elastic Section Modulus About u-Axis W_u,min

W_{u, m a x} = \frac{I_{u}}{v_{m a x}}

I_u	Second moment of area about the u-axis
v_max	Maximum ordinate in the v-direction

Elastic Section Modulus About u-Axis W_u,max

W_{v, m i n} = \frac{I_{v}}{u_{m i n}}

I_v	Second moment of area about the v-axis
u_min	Minimum ordinate in the u-direction

Elastic Section Modulus About v-Axis W_v,min

W_{v, m a x} = \frac{I_{v}}{u_{m a x}}

I_v	Second moment of area about the v-axis
u_max	Maximum ordinate in the v-direction

Elastic Section Modulus About v-Axis W_v,max

W_{u} = m i n (|W_{u, m i n}|, |W_{u, m a x}|)

W_u,min	Largest negative elastic section modulus about the u-axis
W_u,max	Smallest positive elastic section modulus about the u-axis

Elastic Section Modulus About u-Axis W_u

W_{v} = m i n (|W_{v, m i n}|, |W_{v, m a x}|)

W_v,min	Largest negative elastic section modulus about the v-axis
W_v,max	Smallest positive elastic section modulus about the v-axis

Elastic Section Modulus About v-Axis W_v

Shear Areas

The shear areas are determined from the FE solution of the Saint-Venant torsion problem. The shear and torsional stiffness matrix D is determined from three calculated unit states of the deplanation function. This matrix relates the warping of the cross-section and the mean shear strains to the primary torsional moment and the shear forces:

\{\begin{matrix} M_{x p} \\ V_{y} \\ V_{z} \end{matrix}\} = D \{\begin{matrix} ϑ \\ γ_{y} \\ γ_{z} \end{matrix}\}, ϑ = \frac{d θ}{d x}

M_xp	Primary torsional moment
V_y, V_z	Shear forces in direction of the y-axis or z-axis
θ	Rotation angle of the cross-section
ϑ	Warping per length unit
γ_y, γ_z	Average shear strains

Shear-Torsion Stiffness Relation

Shear Areas in Local Coordinate System y, z

The shear areas in the local coordinate system are determined from the diagonal entries D₂₂ and D₃₃ of the stiffness matrix D as follows:

A_{y} = \frac{1}{G_{ref}} \cdot D_{22}

G_ref	Shear modulus of the reference material
D₂₂	Shear stiffness coefficient of the shear-torsional stiffness matrix D

Shear Aea in y-Direction

A_{z} = \frac{1}{G_{ref}} \cdot D_{33}

G_ref	Shear modulus of the reference material
D₃₃	Shear stiffness coefficient of the shear-torsional stiffness matrix D

Shear Area in z-Direction

Shear Areas in Principal Axis System u, v

In the principal axes u, v, the shear component of the stiffness matrix is transformed according to the following relation:

A_{s} = S \cdot D_{2 : 3, 2 : 3} \cdot S^{⊤}

S	Rotation matrix
D	Diagonal matrix

Principal Axis Matrix A_s

The rotation matrix S is defined by the principal axis inclination α:

S = [\begin{matrix} \cos α & \sin α \\ - \sin α & \cos α \end{matrix}]

α	Inclination of the principal axes

Rotation Matrix S

The shear areas in the principal axis system are then calculated from the diagonal elements A_s,11 and A_s,22 of the transformed matrix As:

A_{u} = \frac{1}{G_{ref}} \cdot A_{s, 11}

G_ref	Shear modulus of the reference material
A_s,11	Diagonal element of the principal axes matrix A_s

Shear Area in u-Direction A_u

A_{v} = \frac{1}{G_{ref}} \cdot A_{s, 22}

G_ref	Shear modulus of the reference material
A_s,22	Diagonal element of the principal axes matrix A_s

Shear Area in v-Direction A_v

Location of Shear Center

The shear center is the point in the cross-sectional plane through which the line of action of a shear force needs to run so that no additional rotation of the member arises. Conversely, a pure torsional moment applied at the shear center results in warping without additional shear bending. With respect to the shear center, the moment resulting from the shear stresses due to the shear force is zero.
If the shear-torsion stiffness matrix D, derived in the previous section, is related to the shear center, the corresponding coupling elements between shear force and torsion are eliminated based on this mechanical decoupling:

D₁₂=D₂₁=0 and D₁₃=D₃₁=0

Shear Center Coordinates in Local Coordinate System y, z

The coordinates of the shear center, y_SC and z_SC, are determined in such a way that this condition is fulfilled. The calculated coordinates refer to the elastic centroid of the cross-section.

Shear Center Coordinates in Principal Axis System u, v

The transformation of the determined coordinates from the local y, z system to the principal axis system u, v is performed geometrically using the inclination of the principal axis α and the following transformation equations:

u_{S C} = y_{S C} \cdot \cos (α) + z_{S C} \cdot \sin (α)

y_SC, z_SC	Shear center coordinate related to the elastic centroid in the y- or z-direction
α	Inclination of the principal axis

Shear Center Coordinate Related to Elastic Centroid in u-Direction u_SC

v_{S C} = - y_{S C} \cdot \sin (α) + z_{S C} \cdot \cos (α)

y_SC, z_SC	Shear center coordinate related to the elastic centroid in the y- or z-direction
α	Inclination of the principal axis

Shear Center Coordinate Related to Elastic Centroid in v-Direction v_SC

The calculated coordinates refer to the elastic centroid of the cross-section.

Torsional Moment of Inertia

The torsional moment of inertia I_t describes the geometric resistance of the cross-section to pure warping (torsion). The torsional moment of inertia is determined from the FE solution of the Saint-Venant torsion problem. The calculation of the torsional moment of inertia takes into account the cross-section’s deformation (warping) using the following integral relation:

I_{t} = I_{p} - \int_{A} ((Z - e_{z}) \cdot ω_{, y} - (Y - e_{y}) \cdot ω_{, z}) d A

I_p	Polar moment of inertia
ω_,y, ω_,z	Partial derivatives of the deplanation function based on the local coordinates y and z
Y, Z	Coordinates of the surface point related to the origin
A	Cross-sectional area
e_y, e_z	Coordinates of the elastic centroid related to the origin

Torsional Moment of Inertia I_t

Torsional Section Modulus

The torsional section modulus is calculated as the ratio of the primary unit torsional moment $M_xp=1$ to the maximum resulting torsional shear stress due to free torsion:

W_{t} = \frac{M_{x p}}{max τ (M_{x p})}

M_xp	Primary torsional moment
maxτ(M_xp)	Maximum shear stress resulting from the primary torsional moment

Torsional Section Modulus W_t

The calculation uses the maximum of the unsmoothed stress values at the finite element nodes. In the graphical result display, stresses are smoothed by default by averaging the values of adjacent elements at common nodes. This allows for reducing local stress peaks.

The maximum stress from the graphical result display may thus differ from the governing stress used in the calculation of W_t. It is necessary to perform a recalculation based on the displayed maximum stress; however, this may not provide the same value as the one given for W_t.

I_{ω} = \frac{1}{E_{ref}} \cdot \int_{A} E \cdot {ω_{S C}}^{2} d A

E_ref	Modulus of elasticity of the reference material
ω_SC	Warping ordinate at the surface point related to the shear center
A	Cross-sectional area

Warping Resistance Related to Shear Center I_ω

Warping Resistance Related to Shear Center

Warping resistance describes the resistance of the cross-section to warping deformations.
This cross-section property is calculated using the following integral of the squared, material-weighted deplanation function ω related to the shear center:

I_{ω} = \frac{1}{E_{ref}} \cdot \int_{A} E \cdot {ω_{S C}}^{2} d A

E_ref	Modulus of elasticity of the reference material
ω_SC	Warping ordinate at the surface point related to the shear center
A	Cross-sectional area

Warping Resistance Related to Shear Center I_ω

Warping Radius of Gyration Related to Shear Center

The warping radius of gyration is calculated from the warping resistance and the polar second moment of area. It is related to the shear center.

i_{ω} = \sqrt{\frac{I_{ω}}{I_{p, S C}}}

I_ω	Warping resistance related to the shear center
I_p,SC	Polar second moment of area related to the shear center

Warping Radius of Gyration Related to Shear Center i_ω

Warping Modulus Related to Shear Center

The warping modulus is calculated as the quotient of the warping resistance related to the shear center and the maximum value of the corresponding warping ordinate.

W_{ω} = \frac{I_{ω}}{m a x ω}

I_ω	Warping resistance related to the shear center
max ω	Maximum warping ordinate related to the shear center

Warping Modulus Related to Shear Center W_ω

Cross-Section Asymmetry Parameter

The cross-section asymmetry parameter takes into account the influence of a simply symmetric or asymmetric cross-section geometry on the elastic critical lateral-torsional buckling moment and, consequently, on the lateral-torsional buckling behavior.

Cross-Section Asymmetry Parameter Related to Elastic Centroid

r_{y} = \frac{1}{I_{y} \cdot E_{r e f}} \int_{A} E \cdot (Z - e_{z}) \cdot ({(Y - e_{y})}^{2} + {(Z - e_{z})}^{2}) d A

E_ref	Modulus of elasticity of the reference material
I_y	Second moment of area about the y-axis
A	Cross-sectional area
E	Modulus of elasticity at the surface point
Y, Z	Coordinates of the surface point related to the origin
e_y, e_z	Coordinates of the elastic centroid related to the origin

Cross-Section Asymmetry Parameter Related to Elastic Centroid r_y

r_{z} = \frac{1}{I_{z} \cdot E_{r e f}} \int_{A} E \cdot (y - e_{y}) \cdot ({(Y - e_{y})}^{2} + {(Z - e_{z})}^{2}) d A

E_ref	Modulus of elasticity of the reference material
I_y	Second moment of area about the y-axis
A	Cross-sectional area
E	Modulus of elasticity at the surface point
Y, Z	Coordinates of the surface point related to the origin
e_y, e_z	Coordinates of the elastic centroid related to the origin

Cross-Section Asymmetry Parameter Related to Elastic Centroid r_z

r_{u} = \frac{1}{I_{u}} \cdot (r_{y} \cdot I_{y} \cdot cos α - r_{z} \cdot I_{z} \cdot sin α)

I_u	Second moment of area about the u-axis
r_y, r_z	Cross-section asymmetry parameter related to the elastic centroid
I_y, I_z	Second moment of area about the y- or z-axis
α	Principal axis inclination

Cross-Section Asymmetry Parameter Related to Elastic Centroid r_u

r_{v} = \frac{1}{I_{v}} \cdot (r_{y} \cdot I_{y} \cdot sin α + r_{z} \cdot I_{z} \cdot cos α)

I_v	Second moment of area about the v-axis
r_y, r_z	Cross-section asymmetry parameter related to the elastic centroid
I_y, I_z	Second moment of area about the y- or z-axis
α	Principal axis inclination

Cross-Section Asymmetry Parameter Related to Elastic Centroid r_v

Cross-Section Asymmetry Parameter Related to Shear Center

r_{y, S C} = r_{z} - 2 \cdot y_{S C}

r_z	Cross-section asymmetry parameter related to the elastic centroid
y_SC	Shear center coordinate related to the elastic centroid in the y-direction

Cross-Section Asymmetry Parameter Related to Shear Center r_y,SC

r_{z, S C} = r_{y} - 2 \cdot z_{S C}

r_y	Cross-section asymmetry parameter related to the elastic centroid
z_SC	Shear center coordinate relative to the elastic centroid in the z-direction

Cross-Section Asymmetry Parameter Related to Shear Center r_z,SC

r_{u, S C} = r_{v} - 2 \cdot u_{S C}

r_v	Cross-section asymmetry parameter related to the elastic centroid
u_SC	Shear center coordinate related to the elastic centroid in the u-direction

Cross-Section Asymmetry Parameter Related to Shear Center r_u,SC

r_{v, S C} = r_{u} - 2 \cdot v_{S C}

r_u	Cross-section asymmetry parameter related to the elastic centroid
v_SC	Shear center coordinate related to the elastic centroid in the v-direction

Cross-Section Asymmetry Parameter Related Shear Center r_v,SC

Auxiliary Value for Warping Rotation Related to Shear Center

The auxiliary value for warping rotation is a parameter used to describe the warping distribution in a cross-section related to the shear center.

r_{ω, S C} = \frac{1}{I_{ω} \cdot E_{r e f}} \cdot \int_{A} E \cdot ω_{S C} \cdot [{(Y - e_{y} - y_{S C})}^{2} + {(Z - e_{z} - z_{S C})}^{2}] d A

I_ω	Warping resistance related to the shear center
E_ref	Modulus of elasticity of the reference material
E	Modulus of elasticity at the surface point
ω_SC	Warping ordinates at the surface point related to the shear center
Y, Z	Coordinates of the surface point related to the origin
e_y, e_z	Coordinates of the elastic centroid related to the origin
y_SC, z_SC	Shear center coordinates related to the elastic centroid
A	Cross-sectional area

Auxiliary Value for Warping Torsion Related to Shear Center r_ωSC

Plastic Cross-Section Moduli

When determining the plastic cross-section moduli, the magnitudes of the static moments in the tension and compression zones, related to the plastic neutral axis line, are added together.

W_{p l, y} = | S_{t, y} | + | S_{c, y} |

S_t,y	Static moment of the tension zone related to the plastic neutral axis line
S_c,y	Static moment of the compression zone related to the plastic neutral axis line.

Plastic Section Modulus About Y-Axis W_pl,y

W_{p l, z} = | S_{t, z} | + | S_{c, z} |

S_t,z	Static moment of the tension zone related to the plastic neutral axis line
S_c,z	Static moment of the compression zone related to the plastic neutral axis line

Plastic Section Modulus About z-Axis W_pl,z

W_{p l, u} = | S_{t, u} | + | S_{c, u} |

S_t,u	Static moment of the tension zone related to the plastic neutral axis line
S_c,u	Static moment of the compression zone related to the plastic neutral axis line

Plastic Section Modulus About u-Axis W_pl,u

W_{p l, v} = | S_{t, v} | + | S_{c, v} |

S_t,v	Static moment of the tension zone related to the plastic neutral axis line
S_c,v	Static moment of the compression zone related to the plastic neutral axis line

Plastic Section Modulus About v-Axis W_pl,v

Plastic Shape Coefficient

The plastic shape coefficient describes the plastic cross-sectional reserve relative to the elastic load-bearing capacities.

α_{p l, y} = \frac{W_{p l, y}}{W_{e l, y}}

W_pl,y	Plastic section modulus about the y-axis
W_el,y	Elastic section modulus about the y-axis

Plastic Shape Coefficient About Y-Axis α_pl,y

α_{p l, z} = \frac{W_{p l, z}}{W_{e l, z}}

W_pl,z	Plastic section modulus about the z-axis
W_el,z	Elastic section modulus about the z-axis

Plastic Shape Coefficient About Z-Axis α_pl,z

α_{p l, u} = \frac{W_{p l, u}}{W_{e l, u}}

W_pl,u	Plastic section modulus about the u-axis
W_el,u	Elastic section modulus about the u-axis

Plastic Shape Factor About u-Axis α_pl,u

α_{p l, v} = \frac{W_{p l, v}}{W_{e l, v}}

W_pl,v	Plastic section modulus about the v-axis
W_el,v	Elastic section modulus about the v-axis

Plastic Shape Coefficient About v-Axis α_pl,v

Location of Plastic Neutral Axis Line

The location of the plastic neutral axis line is determined by the equilibrium of axial forces. In the theoretically fully plastic state, the resulting tension and compressive forces must be equal in magnitude. They result from integrating the respective yield stresses over the corresponding cross-sectional areas.

\int_{A_{t}} f_{y t} d A = \int_{A_{c}} f_{y c} d A

A_t	Cross-sectional area under tension
A_c	Cross-sectional area under compression
f_yt	Tension yield strength
f_yc	Compression yield strength
A	Cross-sectional area

Position of Plastic Neutral Axis

Location of Plastic Neutral Axis Line in Local y, z Coordinate System

The coordinates y_pl and z_pl describe the location of the plastic neutral axis line relative to the elastic centroid of the cross-section in the local y, z coordinate system.
The coordinate y_pl specifies the location of the plastic neutral axis line that runs parallel to the local z-axis. The coordinate z_pl specifies the location of the plastic neutral axis line that runs parallel to the local y-axis.

Location of Plastic Neutral Axis Lines in Principal Axis System u, v

The coordinates u_pl and v_pl describe the location of the plastic neutral axis lines relative to the elastic centroid of the cross-section in the principal axis system u, v. The plastic neutral axis line parallel to the principal axis v has its distribution through the coordinate u_pl. The plastic neutral axis line parallel to the principal axis u passes through the coordinate v_pl.

Weight

The weight describes the weight of the cross-section per unit length. It is obtained by integrating the specific weight over the cross-sectional area:

G = \int_{A} γ d A

γ	Specific weight
A	Cross-sectional area

Weight G

Location of Center of Gravity

The coordinates of the center of gravity are specified relative to the elastic centroid of the cross-section. They are calculated from the static moments weighted by the respective material density, divided by the total mass per unit length:

y_{C G} = \frac{\int_{A} γ \cdot y d A}{\int_{A} γ d A}

γ	Specific weight of the material of the respective surface element
y, z	y-coordinate of the surface element related to the elastic centroid
A	Cross-sectional area

Coordinate of Center of Gravity Related to Elastic Center of Mass in y-Direction: y _CG

z_{C G} = \frac{\int_{A} γ \cdot z d A}{\int_{A} γ d A}

γ	Specific weight of the material of the respective surface element
y	z-coordinate of the surface element related to the elastic centroid
A	Cross-sectional area

Coordinate of Center of Mass Relative to Elastic Centroid in z-Direction z_CG

Surface Area per Unit Length

The surface area per unit length describes the outer perimeter of the cross-section that is exposed to direct thermal actions from the outside.

Volume

The volume describes the volume of the cross-section related to a unit length. For a unit length, it corresponds to the area of the geometric cross-sectional area A_geom.

Section Factor

The section factor describes the ratio of the component surface exposed to fire to the component volume per unit length. The calculation is performed as the quotient of the surface area per unit length and the volume: A_m / V.

Parent Chapter

Finite Element Analysis