fbpx

In Consteel, the calculation of cross sectional interaction resistance for Class 3 and 4 sections is executed with the modified Formula 6.2 of EN 1993-1-1 with the consideration of warping and altering signs of component resistances. Let’s see how…

Application of EN 1993-1-1 formula 6.2

For calculation of the resistance of a cross section subjected to combination of internal forces and bending moments, EN 1993-1-1 allows the usage -as a conservative approximation- a linear summation of the utilization ratios for each stress resultant, specified in formula 6.2.  

$$\frac{N_{Ed}}{N_{Rd}}+\frac{M_{y,Ed}}{M_{y,Rd}}+\frac{M_{z,Ed}}{M_{z,Rd}}\leq 1$$

As Consteel uses the 7DOF finite element and so it is capable of calulcating bimoment, an extended form of the formula is used for interaction resistance calculation to consider the additional effect.

$$\frac{N_{Ed}}{N_{Rd}}+\frac{M_{y,Ed}}{M_{y,Rd}}+\frac{M_{z,Ed}}{M_{z,Rd}}+\frac{B_{Ed}}{B_{Rd}}\leq 1$$

Formula 6.2 ignores the fact that not every component results the highest stress at the same critical point of the cross-section.

In order to moderate this conservatism of the formula, Consteel applies a modified method for class 3 and 4 sections. Instead of calculating the maximal ratio for every force component using the minimal section moduli (W), Consteel finds the most critical point of the cross-section first (based on the sum of different normal stress components) and calculates the component ratios using the W values determined for this critical point. Summation is done with considering the sign of the stresses caused by the components corresponding to the sign of the dominant stress in the critical point.

(For class 1 and 2 sections, the complex plastic stress distribution cannot be determined by the software. The Formula 6.2 is used with the extension of bimoments to calculate interaction resistance, but no modification with altering signs is applied)

Example

Let’s see an example for better explanation.

GATE

Did you know that you could use Consteel to perform dual analysis with 7DOF beam and/or shell elements?

Download the example model and try it!

Download model

If you haven’t tried Consteel yet, request a trial for free!

Try Consteel for free
dual_superbeam_analysis
dual_superbeam_analysis
dual_superbeam_analysis
dual_superbeam_analysis
dual_superbeam_analysis
dual_superbeam_analysis

Did you know that you could use Consteel to calculate effective cross-section properties for Class 4 sections?

Download the example model and try it!

Download model

If you haven’t tried Consteel yet, request a trial for free!

Try Consteel for free

Introduction

This verification example studies a simple fork supported beam member with welded section (flanges: 200-12 and 100-12; web: 400-8) subjected to bending about major axis. Constant bending moment due to concentrated end moments and triangular moment distribution from concentrated transverse force is examined for both orientations of the I-section. Critical moment and force of the member is calculated by hand and by the Consteel software using both 7 DOF beam finite element model and Superbeam function.

Geometry

Normal orientation – wide flange in compression

Constant bending moment distribution

Triangular bending moment distribution – load on upper flange

Triangular bending moment distribution – load on bottom flange

Reverse orientation – narrow flange in compression

Constant bending moment distribution

Triangular bending moment distribution – load on upper flange

Triangular bending moment distribution – load on bottom flange

Calculation by hand

Factors to be used for analitical approximation formulae of elastic critical moment are taken from G. Sedlacek, J. Naumes: Excerpt from the Background Document to EN 1993-1-1 Flexural buckling and lateral buckling on a common basis: Stability assessments according to Eurocode 3 CEN / TC250 / SC3 / N1639E – rev2

Normal orientation – wide flange in compression

Constant bending moment distribution

Reverse orientation – narrow flange in compression

Computation by Consteel

Version nr: Consteel 15 build 1722

Normal orientation – wide flange in compression

Constant bending moment distribution

First buckling eigenvalue of the member which was computed by the Consteel software using the 7 DOF beam finite element model (n=25). The eigenshape shows lateral torsional buckling.

Superbeam

First buckling eigenvalue of the member which was computed by the Consteel software using the Superbeam function (δ=25).

gate

Introduction

This verification example studies a simple fork supported beam member with welded section (flanges: 200-12; web: 400-8) subjected to bending about major axis. Constant bending moment due to concentrated end moments and triangular moment dsitribution from concentrated transverse force is examined. Critical moment and force of the member is calculated by hand and by the Consteel software using both 7 DOF beam finite element model and Superbeam function.

Geometry

Constant bending moment distribution

Triangular bending moment distribution – load on upper flange

Triangular bending moment distribution – load on bottom flange

Calculation by hand

Constant bending moment distribution

Triangular bending moment distribution

Computation by Consteel

Version nr: Consteel 15 build 1722

Constant bending moment distribution

7 DOF beam element

First buckling eigenvalue of the member which was computed by the Consteel software using the 7 DOF beam finite element model (n=16). The eigenshape shows lateral torsional buckling.

Superbeam

First buckling eigenvalue of the member which was computed by the Consteel software using the Superbeam function (δ=25).

Triangular bending moment distribution – load on upper flange

7 DOF beam element

First buckling eigenvalue of the member which was computed by the Consteel software using the 7 DOF beam finite element model (n=16).

Superbeam

First buckling eigenvalue of the member which was computed by the Consteel software using the Superbeam function (δ=25).

Triangular bending moment distribution – load on bottom flange

(more…)

Introduction

This verification example studies a simple fork supported beam member with welded section equivalent to IPE360 (flanges: 170-12,7; web: 347-8) subjected to biaxial bending due to concentrated end moments and compression due to axial force. Second order deformations of the middle cross-section of the member are calculated by hand and by the ConSteel software using both 7DOF beam and shell finite elements and Superbeam function. In addition to the verification, the difference between modelling with 6DOF and 7DOF elements is demonstrated.

Geometry

Calculation by hand

The first order and the simple amplified (P-δ) deformations can be analitically calculated by the well known formulas. The calculation of the second order deformations considering true, three-dimensional behaviour of the beam is however so complicated that there are only approximate analitical formulas available for hand calculation. The formula below can be found in Chen, W. and Atsuta, T.: Theory of Beam-Columns, Vol. 2: Space behavior and design, McGRAW-HILL 1977, p. 192  

Computation by Consteel

Version nr: Consteel 15 build 1722

First order

Second order – 6DOF beam element

The second order deformation of the member which was computed by the ConSteel software. It is visible that there is no torsion, only increments of the lateral displacements due to P-δ effect:

Second order – 7DOF beam element

The second order deformation of the member which was computed by the ConSteel software using the 7DOF beam finite element model (n=16). It is visible that there is torsion and further increment in the lateral displacement (Dy):

Second order – Shell finite element

The second order deformation of the member which was computed by the ConSteel software using the shell finite element model (δ=25mm):

Second order – Superbeam

The second order deformation of the member which was computed by the ConSteel software using the Superbeam model (δ=25mm):

(more…)

Introduction

This verification example studies a simple fork supported beam member with IPE 360 section subjected to axial force and bending about the minor axis due to lateral, distributed force. The second order bending moment and the maximum axial compressive stress of the member is calculated by hand and by the Consteel software using the 7DOF beam finite elements.

Geometry

Calculation by hand

Computation by Consteel

Version nr: Consteel 15 build 1488

7DOF beam element The second order bending moment diagram of the member which was computed by the Consteel software using the 7DOF beam finite element model:

Normal stress in the middle cross-section:

(more…)

Introduction

Our verification examples are created to be able to compare hand calculation results with Consteel anaysis results with using either 7DOF beam or shell finite elements. This example is a member of mono-symmetric I- section loaded with transverse concentrated load.

Geometry

Calculation by hand

Computation by Consteel

Version nr: Consteel 15 build 1488

Deformation of the member with the numerical value of the maximum rotation (self-weight is neglected):

GATE

Introduction

Our verification examples are created to be able to compare hand calculation results with Consteel anaysis results with using either 7DOF beam or shell finite elements including Superbeam function. This example is a member in torsion loaded with concentrated torque.

Geometry

Calculation by hand

Computation by Consteel

Version nr: Consteel 15 build 1488

Deformation of the member due to concentrated twist moment:

Bimoment of the member due to concentrated twist moment:

Warping normal stress in the middle cross-section:

Maximum deformation of the middle cross-section:

gate

Civil engineering software in general use the traditional beam-type deformation representation where the section is shown on the deformation of the reference line. In Consteel 15 we use an advanced method for deformation representation which makes it smooth and realistic. The analysis results are the same, but with the improved visualisation the real 3D behavior of the structure can be better seen.

gate