theory cross section characteristics enu

Theoretical Background

Cross-Section Characteristics

Theoretical Background – Cross-Section Characteristics

ii

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© Copyright 2012 Nemetschek Scia All rights reserved.

Table of contents

iii

Table of contents

Table of contents .............................................................................................................................. iii

Version Information................................................................................................................................. 1

Introduction .............................................................................................................................................. 2

Overview of Cross-section Characteristics .......................................................................................... 3

Axis Systems ..................................................................................................................................... 3 Cross-Section Characteristics ......................................................................................................... 4

Determination of Section Characteristics ............................................................................................. 5

Standardized Cross-section properties .......................................................................................... 5 Overall Procedure ............................................................................................................................. 6 Calculation of Standardized Cross-section properties Part I ....................................................... 7

Basic characteristics ...................................................................................................................... 7

Circumference and Drying surface............................................................................................... 11

Shear Area and Unit Shear Stress ............................................................................................... 12

Radii of Gyration .......................................................................................................................... 13

Elastic Section Moduli .................................................................................................................. 14

Plastic Moments and Section Moduli ........................................................................................... 15

Mono-Symmetry Constants .......................................................................................................... 18

Simplified Torsional Constant ...................................................................................................... 18 Calculation of Standardized Cross-section properties Part II .................................................... 19

Introduction ................................................................................................................................... 19

1D FE Method for Thin-Walled Sections ...................................................................................... 20

2D FE Method for Thick-Walled Sections .................................................................................... 23 Application of Closed-Form Formulae .......................................................................................... 27

Doubly-Symmetric I-section ......................................................................................................... 27

Asymmetric I-section .................................................................................................................... 27

Full Circular section ...................................................................................................................... 28

Full Rectangular Section .............................................................................................................. 29

Polygon with hole ......................................................................................................................... 29

Rectangular Hollow Section ......................................................................................................... 30

Asymmetric Rectangular Hollow Section ..................................................................................... 30

Circular Hollow Section ................................................................................................................ 32

Corrugated Web SIN1 .................................................................................................................. 33

Corrugated Web SIN2 .................................................................................................................. 34 Profile Library Properties ............................................................................................................... 35

References ............................................................................................................................................. 36


1

Version Information

Welcome to the Theoretical Background for Cross-Section Characteristics. This document provides background information regarding the calculation of Cross-section properties according to different methods.

Version info

Document Title Theoretical Background – Cross-Section Characteristics Release 2013.0 Revision 12/2012


2

Introduction

In this Theoretical Background in depth information is given regarding the calculation of cross-section properties. The first chapter gives an overview of the different axis systems as well as a list of all cross-section properties calculated by Scia Engineer. The second chapter details the actual methods for determining cross-section properties. After introducing the standardization of cross-section properties the overall procedure followed by Scia Engineer is explained. The chapter then explains the different numerical methods, both using 1D Finite Elements and 2D Finite Elements, for calculating section characteristics. The chapter concludes with a listing of all closed-form formulae used for standard section shapes.


3

Overview of Cross-section Characteristics

In this chapter the different Axis systems used within Scia Engineer are outlined. The second part of this chapter gives an overview of the properties related to these Axis systems.

Axis Systems

Within Scia Engineer the Cross-section Characteristics are referenced to three distinct Axis systems. a) The UCS or 'Input' Axis system is defined using an arbitrary origin and uses a horizontal Y-axis and a vertical Z-axis. This system serves as a reference from which the center of gravity is calculated. b) The LCS Axis system has its origin in the center of gravity and YLCS and ZLCS axis parallel to the axis of the UCS system. This system serves as a reference from which the rotation of the principal axis is calculated. c) The Principal Axis system has its origin in the center of gravity and principal y- and z-axis rotated according to the angle of rotation between the principal and LCS systems. The following picture illustrates these different Axis Systems:

In case the rotation angle of the Principal Axis system is zero, this system is equal to the LCS Axis system. In this case, only the Principal Axis system is displayed.


4

Cross-Section Characteristics

The following table provides an overview of all Cross-section Characteristics calculated by Scia Engineer:

Property Description

A Area

Ay Shear Area in principal y-direction

Az Shear Area in principal z-direction

AL Circumference per unit length

AD Drying Surface per unit length

cYUCS Centroid coordinate in Y-direction of Input axis system

cZUCS Centroid coordinate in Z-direction of Input axis system

IYLCS Second moment of area about the YLCS axis

IZLCS Second moment of area about the ZLCS axis

IYZLCS Product moment of area in the LCS system

α Rotation Angle of the principal axis system

Iy Second moment of area about the principal y-axis

Iz Second moment of area about the principal z-axis

iy Radius of gyration about the principal y-axis

iz Radius of gyration about the principal z-axis

Wely Elastic section modulus about the principal y-axis

Welz Elastic section modulus about the principal z-axis

Wply Plastic section modulus about the principal y-axis

Wplz Plastic section modulus about the principal z-axis

Mply+ Plastic moment about the principal y-axis for a positive My moment

Mply- Plastic moment about the principal y-axis for a negative My moment

Mplz+ Plastic moment about the principal z-axis for a positive Mz moment

Mplz- Plastic moment about the principal z-axis for a negative Mz moment

dy Shear center coordinate in principal y-axis measured from the centroid

dz Shear center coordinate in principal z-axis measured from the centroid

It Torsional constant

Iw Warping constant

βy Mono-symmetry constant about the principal y-axis

βz Mono-symmetry constant about the principal z-axis

In addition to these properties in each fibre of the cross-section for following unit stress values are calculated:

Fibre stress Description

Shear y Shear stress in principal y-direction caused by a unit shear force Vy

Shear z Shear stress in principal z-direction caused by a unit shear force Vz

Torsion Primary Torsion stress caused by a unit torsion moment Mx

In the following chapters the calculation of these different characteristics is detailed.


5

Determination of Section Characteristics

The first part of this chapter details the general procedure for calculating standardized cross-section properties as well as the procedure used in Scia Engineer. Subsequent subchapters deal with the actual calculation of properties, as well as the applied closed form formulae.

Standardized Cross-section properties

In general the calculation of cross-section properties is divided into 2 parts as shown on the following diagram:

For a detailed background into the calculation of properties according to the above diagram reference is made to Ref.[1]. Applied to Scia Engineer this gives the following differentiation:

As indicated on the above diagram, each part is extended with multiple 'derived' properties i.e. properties which are determined using the base properties calculated in that part.

Cross-section property calculation

Standardized Cross-section properties Part I: Biaxial bending and axial force

Standardized Cross-section properties Part II: Torsion

A, cYUCS, cZUCS, IYLCS, IZLCS, IYZLCS, α, Iy, Iz Extended with: - Ay, Az, AL, AD, iy, iz, Wely, Welz - Wply, Wplz, Mply+, Mply-, Mplz+, Mplz- - Unit stress Shear y, Unit stress Shear z - Initial values for βy, βz - General solid It

dy, dz, It, Iw, Unit warping ω Extended with: - Unit Torsion stress - Final values for βy, βz

Cross-section property calculation

Standardized Cross-section properties Part I: Biaxial bending and axial force

Standardized Cross-section properties Part II: Torsion

- Area - Center of Gravity - Angle of the principal axis system - Principal moments of Inertia

- Shear Center - Torsion Constant - Warping Constant - Standardized Warping Ordinate


6

Overall Procedure

The previous paragraph showed the general principle of calculating cross-section properties using two distinct parts. In addition to these parts, Scia Engineer also takes into account specific overrulings of properties, for example in case the 2D FE Method is used, or in case a cross-section is taken from the Profile Library etc. The following diagram shows the complete calculation procedure as used in Scia Engineer.

By default, for Thick-walled sections the 2D FE Method is activated for Torsional analysis, however this can be de-activated by the user leading to the Simplified Torsion analysis.

In the subsequent paragraphs of this chapter each item of the above diagram is described in detail.


7

Calculation of Standardized Cross-section properties Part I

The first part of the standardised cross-section properties concerns those related to bending and axial force.

Basic characteristics

The basic cross-section characteristics are calculated using the standard formulas from solid mechanics. For detailed information, reference is made to Ref.[3] and Ref.[4].

The cross-section is discretized into n elemental areas dA. First, using the arbitrary origin of the UCS or 'Input' Axis system the following properties are calculated using a horizontal Y-axis and a vertical Z-axis:

Area:

First Moment of Area:


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Using these magnitudes the coordinates of the centroid are determined: Centroid:

The centroid defines the origin of the LCS Axis system with YLCS and ZLCS axis parallel to the axis of the UCS system. According to these axis the second moments of area can be determined: Second Moment of Area:

Product Moment of Area:

Finally, using these magnitudes the Principal Axis system and corresponding characteristics can be determined: Second Moment of Area:


9

Angle of Rotation: in case

in case

and

otherwise

The above determination of the angle of rotation accounts for minor numerical discrepancies. For background information, reference is made to Ref.[1].

In addition, in case the angle of rotation is calculated according to the above formula and exceeds

a tolerance of 3°, the angle is increased by

in case Iz > Iy.

Extension: Multi-Material (Composite) sections

In case of multi-material cross-sections the basic characteristics are determined using the principles given in this paragraph. For background information see Ref.[5]

Centroid

First the area Ai and centroid position of each cross-section part/polygon i are calculated. To determine the location of the centroid (Neutral Axis 'NA') of the whole cross-section the following general equation is used:

Where n represents the number of polygons and Ei the E-modulus of the material of the respective polygon.


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The distances z1, z2, … zn are the distances from the NA to the centroid of each polygon (measured in the UCS Axis system). These distances can be written in function of the centroid distance cZUCS so the above equation can be solved this centroid distance. The above equation illustrates the principle used for cZUCS, in the same way the equation can be written out for cYUCS.

Area

The Area of the multi-material section is calculated using the following general formula:

Where n represents the number of polygons, Ei the E-modulus of the material of the respective polygon and Ai the area of the respective polygon. As indicated by the equation, each polygon of the multi-material cross-section is in fact referenced to the material of the 'first' polygon.

Within Scia Engineer this literally means the 'first' inputted polygon. So the material of this 'first' inputted polygon serves as reference material for the multi-material cross-section. This 'first' material is shown with a cyan background color for easy reference.

Second Moment of Area

The Second Moment of Area of the multi-material section is calculated using the following general formula:

Where n represents the number of polygons, Ei the E-modulus of the material of the respective polygon and Ai the area of the respective polygon. As indicated by the equation, each polygon of the multi-material cross-section is in fact referenced to the material of the 'first' polygon. The above equation is used to determine IYLCS, IZLCS and IYZLCS.


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Circumference and Drying surface

The Circumference per unit length or 'Exterior Surface' AL is calculated as the outer circumference of the cross-section. This calculation accounts for the fact that parts are connected/touching.

For those parts which are not connected the circumference AL is calculated as the summation of the outer circumference of the different unconnected parts:

The drying surface per unit length AD is calculated as the outer circumference AL increased by the circumference of any openings within the cross-section. In case there are no openings AD will thus be equal to AL. An 'opening' in this case concerns any closed in empty area within the cross-section. This calculation method thus accounts for 'constructed' openings for example when creating an RHS from four separate rectangles.


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Shear Area and Unit Shear Stress

The Shear Area Ay and Az in principal directions are determined as follows:

With: b(z) The width of the cross-section at position z from the principal y-axis

b(y) The width of the cross-section at position y from the principal z-axis

Sy(z) The First moment or Area of the 'cut-off' area A', determined according to the principal y-axis

Sz(y) The First moment or Area of the 'cut-off' area A', determined according to the principal z-axis

Iy Second moment of area about the principal y-axis


On the following picture the 'cut-off' area A' is illustrated for the Shear Area Az.


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The Unit Shear stresses in the fibres are calculated as follows: Unit stress Shear y in fibre i:

Unit stress Shear z in fibre i:

With Vy and Vz taken as unity and yi and zi the coordinates of fibre i in the principal axis system.

In case the width b at a given fibre position is zero the Unit Shear stress is taken as zero for that fibre.

For multi-material (Composite) sections reference is made to the 2D FE Method.

Radii of Gyration

The Radii of Gyration iy and iz about the principal axis are determined as follows:


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Elastic Section Moduli

The Elastic Section Moduli Wely and Welz about the principal axis are determined as follows:

The distances z and y according to the principal axis are determined for each fibre of the cross-section. In essence each fibre thus has a different Elastic Section Modulus. The Moduli shown in the cross-section properties are the minimal values taken over all fibres. These minimal values are thus obtained by using the maximal fibre distances as shown in the above formulas. The following picture illustrates the maximal distances for an arbitrary cross-section:

During stress calculations in the fibres (for example in the Steel checks), the stresses are calculated in each fibre separately. These stress calculations thus use the actual Elastic Section Moduli in each fibre and not the minima over the entire cross-section.


15

Plastic Moments and Section Moduli

Basic principle

In this paragraph the basic principle of the plastic property calculation is explained. The principle is illustrated for a general cross-section made out of one material which has equal characteristics in both tension and compression (like for example Steel). As shown on the following picture, this cross-section is loaded by a bending moment M which causes part of the cross-section to be in compression (C) and part of the cross-section to be in tension (T).

All the fibres in this cross-section have yielded as shown by the stress blocks. The Plastic Neutral Axis (PNA) is defined by the axis located between the fibres yielding in compression and those yielding in tension. This axis is off course parallel to the principal axis about which the moment was applied. For a single material cross-section with homogeneous material characteristics the PNA is easily determined as the axis which splits the cross section into two equal areas: the area AC in compression and AT in tension. The Plastic Section Modulus Wpl is calculated as the sum of the First Moments of Area of the part in tension (ST) and the part in compression (SC):

With: AC and AT The areas of the section in compression and tension respectively for a bending moment about the given principal axis.

dC and dT The distances from the centroid of the areas of the section in compression and

tension respectively to the Plastic Neutral Axis, measured perpendicular to the given principal axis.

Using the material strength f of the homogeneous material the Plastic Moment Mpl is calculated as follows:


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General formulation

The basic principle explained in the previous paragraph holds true for a homogeneous uni-strength material. In general however there are several complexities which need to be accounted for:

- The material of the cross-section can have different characteristics in compression and in tension. - The cross-section can be composed out of multiple materials. - The material characteristics depend on the sign of the moment.

Consider the following composite section as an example:

For a positive My bending moment about the principal axis, the concrete will be in compression while the steel will be in tension. In case of a negative My bending moment about the principal axis, the concrete will be in tension while the steel will be in compression. Depending on the position of the Plastic Neutral Axis one of the materials can even be partially in compression and partially in tension. The calculation of the Plastic Moment is therefore split according to axis and according to sign which leads to Mply+, Mply-, Mplz+ and Mplz-. For each of these plastic magnitudes a separate calculation is done. The determination of the Plastic Neutral Axis needs to take into account the material characteristic of each part. In general the following equation is solved which specifies an equilibrium of tensile and compressive forces:

With: n The number of cross-section parts

AC,i The area in compression of part i

fC,i The compressive strength of part i

AT,i The area in tension of part i

fT,i The tensile strength of part i


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With the position of the PNA known, the Plastic Moment can be determined as follows:

In which di signifies the distance from the centroid of the area of part i of the section to the plastic neutral axis, measured perpendicular to the given principal axis.

The above Plastic Moment calculation assumes a 'full bond' between the different materials. The actual Composite checks take into account the effects of partial bond and recalculate the Plastic Moments accordingly.

Since for each part the material strength can be different there is no more straightforward way to obtain the Plastic Section Modulus Wpl. Within Scia Engineer, this value is referenced to the material of the 'first' inputted polygon, see also the paragraph on Multi-Material sections. In addition, since there is both a positive and a negative Plastic Moment for the given axis, the final Plastic Section Modulus is determined using the minimum of both.

With f1 the material strength of the 'first' polygon. This can either be the compressive or tensile strength of this material depending on which stress dominates in this part.

These values for the Plastic Section Moduli are merely used for display in the Cross-Section Manager. The actual Composite checks directly use the Plastic Moments which are thus not referenced to the 'first' material but take into account all material characteristics.

Material Characteristics

As indicated in the above paragraphs the plastic calculation requires the compressive and tensile strength of the respective material. These values are defined as follows for materials with code dependent data:

Material Compressive strength fC Tensile strength fT

Steel fy fy

Aluminium fy fy

Concrete fck Taken as zero

Masonry fck Taken as zero

Timber fc,0,k ft,0,k

Other 240 N/mm^2 240 N/mm^2

Any material which does not have code dependent data is taken as 'Other'.


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Mono-Symmetry Constants

The Mono-Symmetry Constants βy and βz about the principal axis are determined as follows:

With: Iy Second moment of area about the principal y-axis


y & z Coordinates in the principal axis system

y0 Distance between centroid and shear centre, taken as dy

z0 Distance between centroid and shear centre, taken as dz When these parameters are initially calculated the shear centre coordinates dy and dz are not yet determined. The Mono-Symmetry Constants βy and βz are thus initially calculated taking dy and dz equal to zero. After the analysis of Part II the actual shear centre coordinates dy and dz are determined after which the Mono-Symmetry Constants βy and βz are modified accordingly. For more background information regarding these parameters reference is made to Ref.[2]

Simplified Torsional Constant

To finalize the calculation of Part I of the Standardised Cross-section properties the Torsional constant It is calculated using the following simplified formula for a general solid Cross-section:

with

In normal cases this It value will be overwritten by the exact It calculation done in Part II. In case however the Part II calculation is not done the above calculation ensures there is at least an approximate value for It. This approach avoids numerical instabilities during the analysis.


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Calculation of Standardized Cross-section properties Part II

The second part of the standardised cross-section properties concerns those related to torsion.

Introduction

For calculating properties related to torsion the general theory makes a distinction between the following types of cross-sections, see Ref.[1]:

a) Thin-walled, open cross-sections

b) Thin-walled, closed cross-sections

c) Arbitrary, thick-walled cross-sections

A cross-section is defined as thin-walled if, through a reduction to the profile centreline and the application of simplified theories, sufficiently exact calculation results are obtained. Ref.[1]. Within Scia-Engineer a thin-walled section is thus a section for which a centreline is available. To simplify the identification, the Shape Type (thin-walled or thick-walled) is shown in the properties of each cross-section. In literature, for thin-walled, open sections analytical solutions are widely available. For thin-walled, closed (hollow) sections with a single opening analytical solutions are also available Ref.[6] however in case of multiple openings a statically indeterminate problem emerges which requires a large effort to solve analytically. Therefore, within Scia Engineer, a numerical 1D Finite Element Method is used to calculate the torsional properties of any thin-walled section. The main advantages of this method are that it applies to both open and closed sections and can be used for closed sections with any amount of openings. In literature, for thick-walled sections analytical solutions only exist for a few basic shapes such as rectangles, triangles and ellipses. Within Scia Engineer, for thick-walled sections a numerical 2D Finite Element Method is used to provide an exact solution for any shape. In addition, the 2D Finite Element Method can even be applied optionally to thin-walled sections.


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The following table summarizes this approach:

Shape Type Method for Torsional Analysis

Thin-walled open section 1D FEM (Optionally 2D FEM)

Thin-walled closed section 1D FEM (Optionally 2D FEM)

Thick-walled section 2D FEM

The following chapters give an overview of both the 1D and 2D Finite Element Methods.

1D FE Method for Thin-Walled Sections

For thin-walled sections (open or closed or a combination of both) a general One-Dimensional Finite Element approach is applied. For a detailed background regarding this method including calculation examples reference is made to Ref.[1]. Based on the centerline the cross-section is discretised into nodes and elements as schematised on the following picture:

Each element is defined with a begin node a, an end node b and a constant thickness t.


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The Finite Element analysis is carried out using the following steps: Step 1: Calculation of the warping ordinate

Equation system (boundary condition: ):

Element matrices:

,

with:

for D = S

Step 2: Position of the shear centre and standardisation of the warping ordinate:

,

Step 3: Calculation of the cross-section properties Iw and It


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Step 4: Calculation of shear deformations due to shear forces and secondary torsion:

Equation system (boundary condition: ui ):

Element load vector:

with:

Step 5: Calculation of shear stresses:

Shear stresses due to primary torsion:

linearly via t with:

constantly via t:

Shear stresses due to shear forces and secondary torsion:

The above procedure is given here for informative reasons. For a full description of all abbreviations used in this procedure as well as background information and worked out examples, reference is made to Ref.[1]. The main advantage of this method is that it can be used for both open and closed thin-walled sections or combinations of both (sections with openings and outstands). The method is however only valid for sections with a continuous centerline i.e. where all parts are connected by one continuous line.

In case of multiple unconnected parts (like a pair section composed out of two thin-walled sections which do not touch each other) the 1D FE Method cannot be applied since there is no continuous centerline. In such cases the 2D FE Method should be applied.


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2D FE Method for Thick-Walled Sections

For arbitrary thick-walled sections a general Two-Dimensional Finite Element approach is applied.

This is the method used automatically in case of multi-material (composite) sections.

Beside thick-walled sections this method can also be applied to thin-walled sections. As the name indicates, the 2D FE Method discretises the cross-section using two-dimensional elements.

The analysis is split into two separate parts: a Torsion Analysis and a Shear Analysis. The following paragraphs give more information regarding the determination of the default mesh size and both analysis types.

Default Mesh Size

In case no mesh size is inputted the default mesh size is determined as follows:

1. The cross-section is divided into approximately 250 elements:

With A the area of the cross-section

2. In case the area of the circumscribed rectangle around the cross-section exceeds 10 times the area A the mesh size of the previous step is halved:

This correction accounts for thin-walled sections.

3. The mesh size of the previous step is then rounded using a .5 accuracy. This is the mesh size used for the Torsion Analysis.

4. For the Shear Analysis the mesh of the previous step is further refined as follows:

This final step is applied always, also in case a manual input of the mesh size is made.


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As with any Finite Element approach, to obtain accurate results the mesh needs to be sufficiently refined.

Torsion Analysis: Prandtl

The Torsion Analysis determines the Torsional constant It, the Warping Constant Iw and the unit torsion stresses. The analysis is executed according to the Prandtl theory. Within this paragraph the basic principles of the theory are explained. The Prandtl theory (often referred to as the Membrane or Soap-Film Analogy) is based on the similarity of the torsion stress function equation and the equilibrium equation of a membrane subjected to lateral pressure.

Consider an opening in an x-y plane which has the same shape as the cross-section to be investigated.

Cover the opening with a homogeneous membrane.

The pressure against the membrane causes the membrane to bulge out of plane.

The lateral displacement z(x,y) of the membrane and the Prandtl torsion stress function φ(x,y) satisfy the same equation in (x,y)

Prandtl Torsion function:

Elastic Membrane function:

Where z denotes the lateral displacement due to a pressure p and an initial tension S.


25

The theory concludes with the following:

Stress components are proportional to the derivatives of the membrane displacement.

Stresses are proportional to the slope of the membrane.

The twisting moment is proportional to the volume enclosed by the membrane and x-y plane Further elaboration and background information regarding the Prandtl theory and 2D FEM analysis can be found in Ref.[1],[7],[8],[9].

Shear Analysis: Grashof-Jouravski

The Shear Analysis determines the Shear areas Ay & Az and the unit Shear stresses. The analysis is executed according to the Grashof-Jouravski theory. For background information reference is made to Ref.[10]. The following paragraphs describe the theory for the shear Area Az. The same logic can be written out for Ay.

The theory is generally valid in case the following requirements are met:

The cross-section symmetrical about the z-axis

The cross-section is massive, without large holes

Overall the obtained results are better in case the height is bigger than the width

The Shear stresses lead off from the cross-section into one point K.

The area z

z

AA

takes on the shear force Qz.

yT

zT

T

b

z

y

Qz

K

max

xz xz

xyxy


26

The value βz is calculated from the shear stresses in one of the following ways:

1) Only from the vertical components (without influence of )

2

2 2

( )

4 ( )

T

T

y

z

yA

S zAdA

I b z

2) From both components and

2

2

2 2

( )1 tan ( , )

4 ( )

T

T

y

z

yA

S zAz y dA

I b z

In case the cross-section does not meet the requirements of the Grashof-Jouravski theory, the βz

values calculated with the influence of are absolutely incorrect and often unreal. They should

not be used in this case. Depending on the rate of unrealized conditions, the βz values which were calculated only from the

vertical component (without influence of ) are real and can be used in this case.

The user should in all cases evaluate if the values determined by the theory are acceptable or not.

In case of multi-material (heterogeneous) cross-sections the calculated shear areas Ay and Az can be used under the following conditions:

The heterogeneities are symmetrical.

The heterogeneities do not disturb the Grashof-Jouravski stress theory.

The heterogeneity is diffused.

A local heterogeneity consists of less than 10% of the cross-section area.

Openings

As specified, the above theory for shear areas is not valid in case of large openings like for example openings which divide a cross-section into different unconnected parts. A typical example are web openings in steel members. Specifically for such a case a modified procedure is applied: In case:

The cross-section consists of multiple unconnected parts i

The rotation angle α of the cross-section is 0° Then the Shear Analysis of the 2D FE Method is used separately for each part i and the shear area Av,i of each part is stored. The final shear area Av of the cross-section is then calculated as the sum of the shear areas of the different parts:


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Application of Closed-Form Formulae

For many standard cross-section shapes (I-sections, RHS sections, CHS sections …) closed-form formulae exist for the cross-section properties. After the calculation of properties, depending on the shape specific properties are overruled by fixed formulae as indicated in the following paragraphs.

Doubly-Symmetric I-section

For Doubly-Symmetric I-sections (Formcode 1) the Torsional constant It is overruled as follows:

This formula was taken from Ref.[11] In addition the unit torsion stress per fiber is overruled as follows: Torsion_stress = Torsion_stress * (It,old / It,new) With: It,old The original It value It,new The new It value calculated by the above formula These modifications are only done in case the rounding r≠0 i.e. when it concerns a true rolled section shape.

Asymmetric I-section

For Asymmetric I-sections (Formcode 101) the Warping constant Iw is overruled as follows:

This formula was taken from Ref.[12]


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Full Circular section

For a Full circular section (Formcode 11 or geometric 'Circle') with diameter D the Area A is overruled as:

The Second Moments of Area Iy and Iz are overruled as:

The Elastic Section moduli Wely and Welz are overruled as:

The Plastic Section moduli Wply and Wplz are overruled as:

The Torsional constant It is overruled as:

The Shear areas Ay and Az are overruled as:

These formulas were taken from Ref.[4].


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Full Rectangular Section

For a Full Rectangular section (Formcode 7 or geometric 'Rectangle' or 'RECT') with width b and height h the Torsional constant It is overruled as follows:

with

This formula was taken from Ref.[1] The Shear areas Ay and Az are overruled as:

This formula was taken from Ref.[4].

Polygon with hole

For a polygon with hole (geometric 'Polygon with hole') the Torsional constant It is overruled using the second formula of Bredt:

With A' the closed in area, taken as:

With S the circumference of the closed in area, taken as:

With r the radius of the polygon, n the number of corners and t the thickness. The Shear areas Ay and Az are overruled as:


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Rectangular Hollow Section

For a symmetrical RHS (Formcode 2) the Torsional constant It is overruled using the second formula of Bredt:


A' = (H - t) * (B - t) With S the circumference of the closed in area, taken as:

S = 2 * [(H - t) + (B - t)] Where B is the width of the cross-section, H the height and t the thickness. The Warping constant Iw is overruled as follows:

The Shear areas Ay and Az are overruled as:

Ay = A * [ B / (B + H)]

Az = A * [H / (B + H)]

Asymmetric Rectangular Hollow Section

For an asymmetrical RHS (geometric 'O' or geometric 'O asymmetric') the Torsional constant It is overruled using the second formula of Bredt:


A' = Hc * Bc With S the circumference of the closed in area, taken as:

= 2 * (Hc/tha) + (Bc/thb1) + (Bc/thb2)

With Hc and Bc the centerline dimensions:

Hc = H - (thb1 / 2) - (thb2 / 2)

Bc = B - tha


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Where B is the width of the cross-section, H the height, tha the web thickness and thb1 & thb2 the flange thicknesses. Torsional stresses are calculated using an average thickness. The Shear areas Ay and Az are overruled as:

Ay = A * [B / (B + H)] Az = A * [H / (B + H)]


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Circular Hollow Section

For a CHS (Formcode 3 or geometric 'Tube') the Area A is overruled as:

The Moments of inertia Iy and Iz are overruled as:

The Section moduli Wely and Welz are overruled as:

The Plastic section moduli Wply and Wplz are overruled as:

The Torsional constant It is overruled using the second formula of Bredt:


With S the circumference of the closed in area, taken as:

With Dc the centerline dimension, taken as:

Dc = D - t With Di the inner diameter taken as:

Di = D - (2 * t) Where D is the diameter of the cross-section and t the thickness. The Shear areas Ay and Az are overruled as:


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Corrugated Web SIN1

For a corrugated web section SIN1 the Area A is overruled as:

A = 2 * B * t The inertia Iy is overruled as:

In which the distance z1 is determined as follows:

z1 = ( H - t ) / 2 The section modulus Wely is overruled as:

The plastic modulus Wply is calculated by multiplying this Wely value with the ratio of the original Wely and Wply of the (full) section The shear area Az is calculated as:

In these formulas B indicates the width of the cross-section, H the height, Hw the height of the web, t the flange thickness and s the web thickness. The parameters w and sw describe the geometry of the corrugation. These formulas were provided by the company Zeman, Austria.


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Corrugated Web SIN2

For a corrugated web section SIN2 the Area A is overruled as:

A = Bt * tt + Bb * tb The inertia Iy is overruled as:

In which the distances z1 and z2 are determined as follows:

z1= h - (tb / 2) z2 = H - h - (tt / 2)

The distance h is determined as:

h = Sy / A With the modulus Sy calculated as:

The section modulus Wely is overruled as:

The plastic modulus Wply is calculated by multiplying this Wely value with the ratio of the original Wely and Wply of the (full) section The shear area Az is calculated as:

In these formulas Bt and Bb indicate the width of the top and bottom flange, tt and tb the thicknesses of the flanges, H the height of the cross-section, Hw the height of the web and s the web thickness. The parameters w and sw describe the geometry of the corrugation. These formulas were provided by the company Zeman, Austria.


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Profile Library Properties

For those cross-section defined in the Profile Library the properties defined in the Library are used to overrule the calculated properties. As can be seen on the Overall procedure diagram, the properties from the Profile Library are applied after all properties have been calculated. The logic behind this is that the Profile Library might not define all properties but only a few or even none at all. In addition the overruling is done only in case the difference between the calculated property and the property inputted in the Profile Library differs less than 10%. This "10% rule" serves as a safety margin to avoid the application of incorrectly inputted properties in the Profile Library.


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References

[1] Steel Structures: Design using FEM

Kindmann R., Kraus M.

Ernst & Sohn, 2011

[2] The Behaviour and Design of Steel Structures to EC3

Fourth edition

Trahair N.S., Bradford M.A., Nethercot D.A., Gardner L.

Taylor & Francis, 2008

[3] Moments of Area: Introductory Engineering Mechanics

Alexander N.A.

University of Bristol, 2004

[4] Formulas in Solid Mechanics

Dahlberg T.

Linköping University Sweden, 2003

[5] eCourse mechanics

Ch 6. Advanced Beams, Composite Beams

Gramoll K.

http://www.ecourses.ou.edu/

[6] Torsion and Shear Stresses in Ships

Shama M.

Springer-Verlag, 2010

[7] Handbook of engineering mechanics

First edition

W.Flügge

McGraw-Hill, 1962

[8] Berekening van constructies: bouwkunde en civiele techniek

Vandepitte D.

Story-Scientia, 1979

www.berekeningvanconstructies.be

[9] Membrane Analogy for Torsion

Lagace P.A.

MIT, 2001

[10] Grasshof-Žuravského teorie

FEM Consulting

Brno

http://www.ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=me&chap_sec=06.1&page=case_intro


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[11] Sections and Merchant Bars

Sales Programme

Arcelor Mittal, Edition 2011-1

[12] Torsional Section Properties of Steel Shapes

Canadian Institute of Steel Construction, 2002

theory cross section characteristics enu

Documents

basic characteristics

elastic section moduli

prior notice

shear area

photo print

nemetschek scia

version information

prior written permission