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AQUA

General

Materials and

Cross Sections

Version 11.00

SOFiSTiK AG, Oberschleissheim, 2001


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AQUA General Cross Sections

This manual is protected by copyright laws. No part of it may be translated, copied or

reproduced, in any form or by any means, without written permission from SOFiSTiK

AG. SOFiSTiK reserves the right to modify or to release new editions of this manual.

The manual and the program have been thoroughly checked for errors. However,

SOFiSTiK does not claim that either one is completely error free. Errors and omissions

are corrected as soon as they are detected.

The user of the program is solely responsible for the applications. We stronglyencourage the user to test the correctness of all calculations at least by random

sampling.


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AQUAGeneral Cross Sections

i

1 General 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1. Task Description 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Theoretical Principles 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1. Materials 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2. Coordinate System 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3. Normal Stresses 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.4. Effective Width 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.5. Warping and Shear Stresses 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.6. Torsional Moment of Inertia 25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7. Shear Stresses in Solid Sections 26. . . . . . . . . . . . . . . . . . . . . . . . . . .2.8. Program Limits 212. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.9. Bibliography 213. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Input Description 31. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1. ECHO Extent of Output 36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2. CTRL Contol of Computation 38. . . . . . . . . . . . . . . . . . . . . . . . . .3.3. Materials 311. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4. NORM Default Design Code 313. . . . . . . . . . . . . . . . . . . . . . . . . . .3.5. MAT General Material Properties 314. . . . . . . . . . . . . . . . . . . . . . .3.6. MATE Material Properties 315. . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.7. MLAY Layered Material 318. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.8. BMAT Elastic Support / Interface 319. . . . . . . . . . . . . . . . . . . . . . .3.9. NMAT Nonlinear Material 322. . . . . . . . . . . . . . . . . . . . . . . . . . .3.10. MEXT Extra Materialconstants 336. . . . . . . . . . . . . . . . . . . . . . . . .3.11. CONC Properties of Concrete 337. . . . . . . . . . . . . . . . . . . . . . . . . .3.12. STEE Properties of Metals 344. . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.13. TIMB Properties of Timber 352. . . . . . . . . . . . . . . . . . . . . . . . . . . .3.14. MASO Masonry / Brickwork 354. . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.15. SSLA StressStrain Curves 356. . . . . . . . . . . . . . . . . . . . . . . . . . . .3.16. SVAL Crosssection values 358. . . . . . . . . . . . . . . . . . . . . . . . . . . .3.17. SREC Rectangle, Tbeam, Plate 361. . . . . . . . . . . . . . . . . . . . . . .3.18. SCIR Circular and Annular Sections 364. . . . . . . . . . . . . . . . . . . . .3.19. TUBE Circular and Annular Steel Crosssections 365. . . . . . . . .3.20. CABL Cable sections 366. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.21. SECT Freely defined CrossSections 369. . . . . . . . . . . . . . . . . . . .3.22. CS Constuction Stages 374. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.23. SV Additional CrossSection Properties 375. . . . . . . . . . . . . . . . .

3.24. POLY Polygonal CrossSection Element / Blockout 377. . . . . . .3.25. VERT Polygon Vertices in Absolute Coordinates 379. . . . . . . . . .


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3.26. DVER Polygon Vertices in Relative Coordinates 382. . . . . . . . . .

3.27. CIRC Circular CrossSection Elements 383. . . . . . . . . . . . . . . . . .3.28. CUT Shear Sections 384. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.29. PANE ThinWalled CrossSection Element 389. . . . . . . . . . . . .3.30. PLAT ThinWalled CrossSection Element 391. . . . . . . . . . . . . .3.31. WELD Welded Shear Connection 393. . . . . . . . . . . . . . . . . . . . . . .3.32. PROF Rolled Steel Shapes 395. . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.33. SPT Points for Stresses 3102. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.34. SFLA CrossSection StressStrain Curves 3104. . . . . . . . . . . . . . .3.35. WPAR Parameters for Wind Loading 3106. . . . . . . . . . . . . . . . . . . .

3.36. WIND Coefficients for Wind Loading 3107. . . . . . . . . . . . . . . . . . . .3.37. Reinforcement 3109. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.38. RF Single Reinforcement 3112. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.39. LRF Linear Reinforcement 3113. . . . . . . . . . . . . . . . . . . . . . . . . . . .3.40. CRF Circular Reinforcement 3115. . . . . . . . . . . . . . . . . . . . . . . . . . .3.41. CURF Perimetric Reinforcement 3117. . . . . . . . . . . . . . . . . . . . . . .3.42. INTE Interpolation of sections 3119. . . . . . . . . . . . . . . . . . . . . . . . . .

4 Description of Output 41. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1. CrossSection Overview 41. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2. Material Properties 41. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3. CrossSection Properties 42. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.4. CrossSection Elements 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Examples 51. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1. Polygonal Column CrossSection 51. . . . . . . . . . . . . . . . . . . . . . . . .5.2. Torsion of Reinforced Concrete CrossSections 59. . . . . . . . . . . . .5.3. TBeam with Effective Width 510. . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4. Thinwalled Steel Box 513. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.5. Polygonal CrossSection with inner perimeter 519. . . . . . . . . . . . . .5.6. Composite Section 525. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.7. Examples in the Internet 527. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


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1

General

1.1. Task Description

AQUA calculates the properties of crosssections of any shape and made ofany material. The crosssection properties for a static analysis are determined, as well as characteristic magnitudes for the calculation of normal andshear stresses. Crosssections need to be defined before input of the staticsystem or dimensioning with AQB.

According to their complexity there are four types of sections:

Static Properties of Crosssections

All static properties of crosssections are directly specified. This includesshear deformation areas and stress resistance values. The values may betaken from other cross sections with a multiplication factor. These crosssections are mainly used in the static calculations. Their usage in AQB isstrongly restricted.

Standard Crosssections

(Rectangle, Tbeam, annular etc.)A standard crosssection must always be input by means of a single command. All static properties of the crosssection, including the torsional moment of inertia, are available. The section moduli for the maximum stressesare known, yet a detailed calculation of the shear stress at all points is notperformed. Shear deformation areas and center of shear are not estimated.

Combined action with other cross section parts is not possible.Freely Defined Thinwalled Crosssections

A freely defined thinwalled crosssection may contain any number of thinelements. A thin element assumes that the variation of the normal stress andmost shear stresses over the thickness is neglectable. This has the consequence that the inertia about the weak axis also vanishes. Available elementsare panels, standard steel shapes and welded joints, as well as reinforce

ments.Resistance moduli for all stresses are available at all points of the crosssection.


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Torsional moment of inertia and warping resistance, as well as center of shearand shear deformation areas, are determined for open or closed shapes, butthey can be specified explicity for special cases as well. Composite crosssections can be defined.

Freely Defined Solid Crosssections

A freely defined solid crosssection consists of any number of outer and innerperimeteres in the form of circles or polygons, as well as of reinforcement elements. Structural steel shapes can be integrated.

Resistance moduli for all stresses are only available at distinct points of thecrosssection.

The torsional moment of inertia, the center of shear and the shear deformation areas can be calculated, or they can be input separately. The warping resistance can not be determined. Composite sections or effective widths of thepolygons can be defined.

After definition with AQUA, the cross sections can be represented graphically with AQUP.

In most cases sections are somehow regular. AQUA allows to use variant definition possibilities to deal with those instances.

You may describe the section via CADINP variables within a block,which is then used multiple times.

You may interpolate between two sections linearely

You may define a section with a hierarchical sequence of references asa template based on a few construction points and generate sections bychanging these points.

You may describe the position of those construction points by a 3D modell with curved reference lines.

It is also posssible for AQUA to redefine all interpolated or otherwise generated sections with a simple command (INTE).


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2

Theoretical Principles

2.1. Materials

Properties of materials must be distinguished according to whether they areto be kept as close as possible to real values (e.g. for dynamic calculations) orto be used with a safety coefficient for calculating an ultimate loadbearingcapacity.

Whereas the safety factors were formerly assigned moreorless at random,sometimes to the load and sometimes to the material, more recent regulations (Eurocode) provide a clearer separation between safety factors for theloads and factors for the material.

Since the material safety factors still depend on the nature of the load or thetype of design, AQUA generates and stores only the genuine properties of thematerial. However, AQUA accounts for some safety factors which are independent of the particular loading case, such as long term reduction factors.

Nevertheless, a safety coefficient can be entered in AQUA for each material;this is used in AQUA for calculating the full plastic section forces and moments, and can be used in AQB by strain checks.

2.2. Coordinate System

Crosssections are described according to DIN 1080 in the local yz coordinate system of the bar. Here the xaxis points in the longitudinal direction

of the bar. The observer is looking at the positive boundary of the section (fromthe end of the bar to the beginning).

For simple beams, the exact position of the coordinate system is not important. For more complex systems however especially in bridge design, the origin of the coordinate system is always on the beam axis, thus defining any excentricities and unsymmetrical haunched beams with ease.


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Coordinate system

x, y, z Local beam coordinate system, freely selectable, is defined inGENF relative to the global coordinate system(will be heavily used in GEOS).

y, z sectional coordinate system for minimum of inertia (=coordinatesystem shifted to the gravity center)

2.3. Normal Stresses

The loadbearing behavior of a generic bar without foundation according to1st order theory, yet with warping, can be described with a differential equation matrix:

| Fx Fy Fz Fw | | vxII | | px |

| | | | | |

| Fy Fyy Fyz Fyw | | vyIV | | py |

E | | | | = | | (1)

| Fz Fyz Fzz Fzw | | vzIV | | pz |

| | | | | | | Fw Fyw Fzw Fww| | xIV | | mx+GItxI |

with the following definitions:


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vx, vy, vz displacements parallel and perpendicular to the barx rotation about the axis of the barpx, py, pz loads parallel and perpendicular to the barmx torsional load

The rest of the parameters are static properties of the crosssection (geometrical area moments). Since it is impractical to incorporate all of the staticproperties into the calculation, certain standardisations are normallyadopted:

The axial force refers to the gravity center of the bar

Fy= Fz= 0 (2)

Bending takes place about the principal axes

Fyz= 0 (3)

Warping can occur freely in the crosssection

Fw= 0 (4)

The torsional moment and the shear forces refer to the center of shear.

Fyw= Fzw= 0 (5)

Conversely, the conditions in (2) through (5) can be used in determining thegravity center, the orientation of the principal axes, the free moduli of warping and the center of shear.

The determination of the area moments is simple, and is not described inmore detail. The next paragraph will deal with the more complex calculationof the unit warping w.

The normal stresses of a bar crosssection can be described by means ofSwains expression and the unit warping:

xNAMyIzMzIyzIyIzI2yz

zMzIyMyIyzIyIzI2yz

yMbCM

w (6)

In stress analysis, there is no problem in dealing with rotated principal axes.

When analysing a frame structure, however, one must start, as a rule, by rotating the crosssection coordinate system to coincide with the principalaxes. STAR2 does this on its own in case of three dimensional structures. In


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special cases, AQUA can also rotate the crosssection to coincide with theprincipal axes. In such case, the local yaxis must then be rotated accoridingly in GENF too.

2.4. Effective Width

The socalled effective widths are used in literature for modelling the effectswhich derive from the disk loadcarrying action of the plate of a Tbeam ora box crosssection. The concept of an equivalent substitute width with constant normal stress naturally demands different approaches depending onthe task at hand (statics, design).

AQUA is able to define the noneffective areas directly by means of polygonalelements. AQUA then stores the crosssection values for the total crosssection as well as for the effective crosssection. Static analysis usually refersto the effective parts, whereas prestressing refers to the total crosssection.In STAR2, however, this can be explicitly switched one way or the other. Also,when dimensioning with AQB, the user can refer to the total crosssection atvarious locations.

The effective widths are not taken into consideration during shear stress

computations due to many consistency reasons.

2.5. Warping and Shear Stresses

In case of warping as well as shear stressing due to torsion and shear force,the crosssection no longer remains plane. A deflection w occuring at thecrosssection in the longitudinal direction of the bar causes shear stresses.

All the problems of theory of elasticity can be analysed by use of the forcemethod or the displacement method. While the force method is frequently

used in calculations by hand and for nonlinear problems, the displacementmethod is better suited for processing with the computer. Both proceduresare implemented in AQUA for solid cross sections. Certain simplificationsof the following equations can be made in case of thinwalled sections, whichfacilitate a quick solution for all tasks. These sections are therefore alwaysanalysed by the matrix displacement method.

A general formulation for the crosssection warping w according to the displacement method conforms to the equilibrium condition

G2wy2

2wz2

xx

(7)


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and the boundary condition

xynyxznz 0 (8)where the shear stresses are given by

xyG wyzxx (9)

xzG wzyxx (10)

The right side of (7) can be computed, for example, by (6). Assuming constant

normal force and constant crosssection properties, one gets:

xx

QzIzQyIyzIyIzI2yz

zQyIyQzIyzIyIzI2yz

yMt2CM

w (11)

The following boundary condition applies to Saint Venants torsion problem(x/x=1):

w

nzny

ynz (12)

The right side of (7) is identical to zero.

2.6. Torsional Moment of Inertia

The torsional moment of inertia according to the displacement method Itisderived by

Ity2z2wy2wz2dF (13)As long as AQUA does not solve the differential equation (7), only an estimateof the torsional moment of inertia is possible. The last equation shows thatthe polar moment of inertia can be substituted for Itin case of warpfreecrosssections.

ItIpIyIz (14)

For all cross sections (14) provides an upper limit, which e.g. is about 10%above the exact value for a square.


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A better approximation was given by Saint Venant:

IT A442 IyIz (15)

This value is exact for circular and elliptical crosssections. For compact solidcrosssections this value provides a good approximation.

In case of open sections, however, it is sensible to consider a correction afterWienecke /2/ in Consideration of the Crosssection Perimeter, which has beenimplemented in AQUA.

Deviations in rectangular crosssections:

a/b 1/1 2/1 10/1

exact 0.140 0.458 3.13 b4

Saint Venant 0.152 0.486 3.01

Wienecke 0.124 0.418 3.24

By hollow crosssections with more than 30 percent inner perimeteres, an

equivalent hollow crosssection based on the external and internal perimeters is used for a more refined estimate. By composite sections this formula (15) is used for each partial crosssection and the components areadded.

2.7. Shear Stresses in Solid Sections

The calculation of shear stresses for solid sections in AQUA requires that theuser specifies the method to be used and the checking locations. The problem

is extremely complicated and can be solved with a variety of methods. Thisis controlled by the CTRL option STYP:

CTRL STYP 0 force method

CTRL STYP 1 displacement method only for Itand locationof shear center (default for concrete)

CTRL STYP 2 displacement method for torsion

force method for shear

CTRL STYP 3 displacement method for torsion and shear


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shear areas are determined (default for steel, wood)

In postcracking (state II) analysis, AQB always employs the force methodwith proportional axial force. In case of composite crosssections options 2and 3 should be used with caution. The input of explicit shear sections is required as a rule.

2.7.1. Equivalent Hollow Crosssections

While DIN 1045 still allows the calculation of torsional stresses according to

state I, both DIN 4227 and EC2 allow for their calculation on an equivalenthollow crosssection. As long as AQUA does not use the integral equationmethod, the force method is used in conjunction with the definition of anequivalent hollow crosssection.

2.7.2. Shear Cuts

The user normally uses the command CUTto define a so called cut throughthe sectional geometry where a check of the shear stresses should take place.

Each cut is assigned an identification with three characters. The cut can bedefined as parallel to an axis or as an freeform polygon line. Every segmenthas its own material number and it will cut only through crosssection elements with the same material number. Gaps between the segments will beclosed by means of virtual connections. A width of the substitute torsionalcrosssection is available as a special option for the description of equivalenthollow crosssections of reinforced and prestressed concrete. Two partial cutsare generated for each section in this case.

If the user does not supply any input, one or two axisparallel cuts will becreated through the gravity center. This is generally not sufficient even fora simple TBeam, nor for composite sections, where the reference materialnumber of the section is not necessarily represented at that location. The userwill see a warning for general sections therefore.

CTRLSTYP allows the user to control how many of these standard cuts willbe generated (0/1/2).

The cut can hit the crosssection several times creating partial cuts. Eachpartial cut has a direction s and three defined points of interest: beginning(A), middle (M) and end (E):


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Shear section

The internal forces perpendicular to the cut M and N act in such way thatpositive axial forces cause tensile stresses, and positive moments cause tensile stresses at the EndPoint.

The shear stressing is described primarly by the resistance moduli of theshear stresses at the three points. Additional values are calculated for the

proportioning of reinforced concrete structural elements: A mean torsional shear stress which, after being multiplied by the

width of the partial section, must be covered by reinforcement. Thiscorresponds to a section modulus for the shear flow.

The total cut width, by which the shear stresses due to the shear forcemust be multiplied in order to obtain the shear flow from shear force.

These distinctions are very significant in the definition of equivalent hollow

crosssections.

2.7.3. Force MethodThe force method is implemented in AQUA only for "statically determined"crosssections, i.e. simply connected ones. For multiply connected crosssections, the user must either know the location of zero shear stress or specifythe distribution of the shear to the multiple segments of the cut. Since incracked estate the force method is much more simpler, the distribution valuesare needed in any case for reinforced concrete sections.

Analysis of the torsion stressing is not elementary even by the force method(stress function with soap film analogy). The resistance areas for the tor


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sional shear stresses are therefore prescribed by two values per section. Thefirst value defines the shear at midarea (Bredts equivalent caisson). Thesecond value defines the increase along the section:

mMtWTm MtWTd

The default is one of the following two values, depending on whether thecrosssection is a hollow one or an equivalent hollow one:

WTm 12Akb0 WTd 1It

minb,d

The sign of the shear stresses is determined based on the orientation of thesection relative to the shear center.

The shear force components are computed by the classic formula

VI Sb

V is only valid for prismatic beams with constant normal force

I has to be generalized with Swains formula

For S the separated part of the cross section is not known for multipleconnected sections.

shear stress does not need to be constant across the width b

The separated part of the crosssection is the one to the left of the cuts direction on the positive side. During this computation any missing partial sections are automatically filled in. It is therefore extremely important to inputthe sections correctly, and especially to maintain their sequence.

For special cases, such as dowel outline joints, deductible areas, equivalenthollow crosssections, multiply connected crosssections etc., the componentof the shear force for each partial cut can be provided by a factor.

Multiply connected crosssection types require special considerations:


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Shear sections in hollow crosssection

Similarly difficult is the processing of crosssections consisting of severalpolygons, either inner perimeters or composite crosssections, not hitted bythe polygonal shear cut. In such cases AQUA examines all points of the polygon to see whether they are inside or on the boundary of the already evaluatedpartial section. Openings must therefore always be defined after the polygonsthat surround them. In case of composite crosssections it may be helpful topay attention to the cut direction or the sequence of the polygons.

In the definition of cuts across several materials the user must take care thateach segment of the cut has the correct material number, because a cut willhit only parts with the same material number. It makes a difference for thehorizontal shear in a composite flange if a dowel is before or behind the cut.

Cuts through crosssections with "open air" between their parts can not beanalysed as the section does not hit any elements. A similar problem occursif a cut has the wrong material number. This may happen especially with thestandard cuts through the gravity center.


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Problem case

Some additional advice applies to oblique cuts. Since the shear force at an oblique cut does not vary significantly compared to to the straight cut, the widthof the cut does. The selection of an inappropriate cut direction can thereforeresult in the computation of too small shear stresses.

The stress evaluation with the displacement method uses always the grosssection while the force method may use only the effective part of the section.The latter is the default behaviour. But with CTRL SCUT +8 you may switchto the full section if needed.

2.7.4. Displacement MethodThe analysis by the displacement method employs the integral equationmethod developed by Katz. The crosssection contour is discretized into multiple socalled "boundary elements." A linear formulation of the warping ismade for each element, and the boundary condition is satisfied by a Galerkinweighted residual.

The number of elements determines the accuracy of the solution. In case of

a square, for instance, the unit lateral warping on all the axes of symmetryis zero. A nonvanishing solution therefore can be obtained only by definingat least four elements per side. AQUA uses each polygon edge as one element,which can be further subdivided depending on its size. Duplicate edges areautomatically removed.

Since a finer subdivision increases the computational time with a power ofthree, the subdivision should not be made too fine. The user can control themesh size be CTRLSDIV. This indicates the maximum size of a boundary el

ement compared to the largest dimension of the crosssection.CTRL SDIV 0 No subdivisionCTRL SDIV 1 maximum 1/2CTRL SDIV 2 maximum 1/4CTRL SDIV 3 maximum 1/8 (default)CTRL SDIV 4 maximum 1/16

The method computes the shear stresses due to shear force and torsion at allstress points and shear sections. The program also computes the torsional

moment of inertia and the shear deformation areas. The description of innerperimeteres of any shape and at any location is automatically taken into account.


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Under no circumstances are the results of this method to be accepted uncritically. It is a numerical approximate method. Local singularities of the shearstresses, such as those at reentrant corners for example, can generate veryhigh stresses.

The following table shows the convergence of the method using the exampleof a square with a side length of 6 m. The torsional moment of inertia and theshear stresses at a centerline near the boundary are shown. Due to the constant formulation of the linearly varying boundary condition, results that arevery close to the boundary are relatively inaccurate, while values on theboundary are much better.

SDIV 1 2 3 4 5 exact

Mom.of iner. IT

Warping

tau Boundary

tau 3.000 tau 2.999

tau 2.990

tau 2.900

216.0 188.2 183.3 182.4 182.2

0.000 1.542 1.308 1.330 1.319

0.03139 0.0214 0.0220 0.0222 0.0222

0.0456 0.0372 0.0297 0.0258 0.02390.0303 0.0285 0.0252 0.0235 0.0228

0.0252 0.0255 0.0236 0.0227 0.0223

0.0197 0.0219 0.0214 0.0212 0.0212

182.2

1.312

0.0222

The associated shear problem due to shear force can be solved exactly, evenwith coarser element subdivision, and it yields resistance 0.04167 and sheardeformation area 0.8333 A = 30.0 m2.

2.8. Program Limits

The following program limits hold:

Materials 999Crosssections 999Reinforcement layers 10Polygon vertices per polygon 255Shear sections per crosssection 255


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2.9. Bibliography

[1] Katz,C. (1986)SelfAdaptive Boundary Elements for the Shear Stress in BeamsBETECH 86, Boundary Element Technology Conference 1986Massachusetts Institute of Technology, Cambridge U.S.A.

[2] Wienecke, U.J. (1985)Zur wirklichkeitsnahen Berechnung von Stahlbeton und

Spannbetonstben nach einer konsequenten Theorie II.Ordnungunter allgemeiner Belastung.Dissertation Technische Hochschule Darmstadt 1985

[3] Werner,H. (1974)Schiefe Biegung polygonal umrandeter StahlbetonQuerschnitteBeton und Stahlbetonbau 1974 S 9297

[4] Roik,Carl,Lindner (1972)

Biegetorsionsprobleme gerader dnnwandiger StbeWilhelm Ernst & Sohn, Berlin Mnchen Dsseldorf 1972

[5] Roth/Griesshaber (1966)Praktische Berechnung auf Biegung und Torsion beanspruchterStbe mit dnnwandigen QuerschnittenTeubner, Leipzig.

[6] Bornscheuer, F.W. (1952)Systematische Darstellung des Biege und Verdrehungvorganges unter besonderer Bercksichtigung der WlbkrafttorsionDer Stahlbau 21 (1952), S 19

[7] Schade, D. (1969)Zur Wlbkrafttorsion von Stben mit dnnwandigem QuerschnittIngenieurArchiv, 38, S 2534


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31Version 11.00

3

Input Description

AQUA let the user define general cross sections with arbitrary geometry andmaterials. For simple sections and materials you do not need a special license,but for all sections starting with record SECT you need a license for AQUA.

Before defining a section you have to specify the materials. Materials are addressed by an arbitrary number. Please remind that in keeping track of construction phases in AQBS, it is assumed that materials with higher materialnumbers will be added at a later time.

A classified section is defined by just one input record. All sectional values willbe calculated including torsional and shear properties. The maximum components for all stresses are known, but a detailed analysis at different locationswithin the section will not take place.

SVAL Sections without geometry

RECT Rectangular sections, plates, Tbeams and joistsCIRC Circular and annular sectionsTUBE Tubular sections (AQUA only)PROF Rolled Steel shapes (AQUA only)CABL Cable sections (AQUA only)SECT General section (AQUA only)

With AQUA crosssections can be redefined at any time during the processing of the project without affecting other defined sections. However if any

material definition is made, all existing crosssections are deleted. The distributions of reinforcements and stresses are deleted too, unless CTRLRESThas been specified otherwise.

Freely defined crosssections always start with the command SECT,whichspecifies the crosssection number. All subsequent input commands describethis one crosssection, which may consist of several partial crosssections(external perimeter, inner perimeter, reinforcement layout etc.). The inputfor a crosssection is concluded either by the next SECTcommand or by two

END commands.

Input is made in free format according to the input language CADINP.


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Records Items

ECHOCTRL

OPT VALOPT VAL

NORM

MATE

MLAY

BMAT

NMAT

MEXT

CONC

STEE

TIMB

BRWO

SSLA

DC NDC COUN

NO E MUE G K GAM GAMA ALFA E90M90 OAL OAF SPM FY FT TITL

NO T0 NR0 T1 NR1 T2 NR2 T3 NR3T4 NR4 T5 NR5 T6 NR6 T7 NR7 T8NR8 T9 NR9 TITL

NO C CT CRAC YIEL MUE COH DIL GAMBTYPE MREF H

NO TYPE P1 P2 P3 P4 P5 P6 P7P8 P9 P10

NR ART VAL VAL1 VAL2 VAL3 VAL4 VAL5

NO TYPE FCN FC FCT FCTK EC QC GAMALFA SCM ESLA FCK GC GF MUEC TITL

NO TYPE CLAS FY FT FP ES QS GAMALFA EPSY EPST FDYN REL1 REL2 R K1 SCM

TITLNO TYPE CLAS EP G E90 QH QH90 GAMSCM FM FT0 FT90 FC0 FC90 FV FVR OALOAF ALFA TITL

NO STYP SCLA MCLA E G MUE GAM ALFASCM FC FCK FHS FTB TITL

EPS SIG TYPE TEMP


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33Version 11.00

Records Items

SVAL

SREC

SCIR

TUBE

CABL

SFLA

INTE

NO MNO A AY AZ IT IY IZ IYZCM YSC ZSC YMIN YMAX ZMIN ZMAX WT WVYWVZ NPL VYPL VZPL MTPL MYPL MZPL BCYZBTYP TITL

NO H B HO BO SO SU ASO ASUMNO MRF ITF SAY SAZ DASO DASU REF TITL

NO RA RI SA SI ASA ASI MNO MRFITF DAS TITL

NO D T MNO BC TITL

NO D TYPE INL MNO F K W KETITL

EPS F VERT TYPE LEV

NO NS0 NS1 S NREFY1 Z1 Y2 Z2 Y3 ... Y12 Z12


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Records Items

SECT

CS

SV

POLY

VERT

DVER

CIRC CUT

PANE

PLAT

WELD

PROF

RF

LRF

CRF

CURF

SPT

WPAR

WIND

NO MNO MRF ALPH YM ZM FSYM BTYPBCY BCZ KTZ TITL

NO TITL

IT AK YSC ZSC CM CMS AY AZ AYZLEVY LEVZ

TYPE MNO YM ZM SMAX REFP REFD

NO Y Z R PHI TYPE YEFF ZEFF REFP/D

NO DY DZ R PHI TYPE YEFF ZEFF

NO Y Z R MNO REFP REFD REFRNO YB ZB YE ZE NS MS WTM WTDMNO MRF LAY ASUP OUT TYPE VYFK VZFKINCL BMAX BRED REFA RFDA REFB RFDB

NO YB ZB YE ZE T MNOREFA RFDA REFB RFDB R PHI OUT FIXA FIXEAS ASMA LAY MRF TORS DAS

NO YB ZB YE ZE T MNOREFA RFDA REFB RFDB R PHI OUT FIXB FIXE

NO YB ZB YE ZE T MNOREFA RFDA REFB RFDB

NO TYPE Z1 Z2 Z3 MNOALPH YM ZM REFP DTYP SYM REF MREFVD VB VS VT VR1 VR2 CW

NO Y Z AS ASMA LAY MRF TORS DAR SIG REFP REFD

NO YB ZB YE ZE AS ASMA LAY MRFTORS D AR R PHI REFA RFDA REFB RFDB

NO Y Z R PHI AS ASMA LAY MRFTORS D AR REFP REFD REFR

A DE AS ASMA LAY MRFTORS D AR CENT

NO Y Z WTY WTZ WVY WVZ SIGY TEFFSIGC TAUC MNO REFP REFD

CS KR ICE TRAF YMIN YMAX ZMIN ZMAX

ALPH CWY CWZ CWT CLAT S AG


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Version 11.0036

3.1. ECHO Extent of Output

ECHOItem Description Dimension Default

OPT A literal from the following list:MAT Material parametersSECT Crosssection elementsREFP References of elementsSDEF Crosssection values restartSYST System statistic

PICT Properties of PicturesIEQ Integral equation methodFULL Select all options

LIT FULL

VAL The extent of the outputOFF nothing computed / outputNO no output

YES regular outputFULL extensive output

EXTR extreme output

LIT FULL

In case of no ECHO input all options are set to YES. The input of the optionalone is therefore sufficient for increasing the value to FULL. The commandname ECHO must be input by every command.

MATYES Material parameters in abbreviationFULL Stressstrain curves of materials added

SECTYES Overview of crosssection values onlyFULL The most important values for each crosssectionEXTR The individual elements of the crosssection added

REFPNO No printoutFULL For section templates all references of coordinates

are added to the printout

SDEFYES The crosssections input in this run only


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37Version 11.00

FULL plus the unmodified crosssections in the databaseEXTR plus all interpolated sections

SYSTYES Statistics of total usage of sections and masses in the

system (only available for restart)

PICTNO No pictures to be included

YES Nice pictures with shadingFULL Contours including basis static elements

EXTR Detailed picture including labelsIEQ

NO No additional printoutFULL Detailed print of the analyzed topology of the section

for the intgral equation system.


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3.2. CTRL Contol of Computation

CTRLItem Description Dimension Default

OPT A literal from the following list:

RFCS Minimum reinforcementfor computingideal crosssection values

STYP Method of shear in solid

sectionsSDIV Subdivision for intergral

equation methodSCUT Number of standard shear

sectionsFIXL Max. factor for thickness

step at buckling paneldetection

REST Deletion of data at restart

LIT FULL

VAL The value of the option *

CTRL RFCScontrols whether minimum reinforcement should be consideredin the calculation of the crosssection values:

0 do not consider1 consider for composite sections (default)2 consider for all sections

3 consider also effect on dead load+4 do not assign reinforcement to any partial section

CTRL RESTcontrols what to do with existing data in the database. As default AQUA will erase all if materials are defined, and the minimum reinforcements, limit stresses and beam stiffnesses if only sections are defined.This is usually meaningful in order to avoid unforeseeable results. In somecases though it is desirable to process these results further. This can functionwithout problems only if the assignment of the layers and the use of the material numbers in the individual cross sections is not changed.

0 delete old values in the database (default)1 keep old values in the database


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39Version 11.00

2 Keep all values, even if material is changed implies a reanalysis of the sections.

The meanings of STYP, SDIVand SCUTare explained in more detail inparagraph 2.7.2..

CTRL STYPcontrols the computation of shear stresses in solid sections:

0 force method

1 displacement method only for Itand location of shear center(default for concrete)

2 displacement method for torsion,force method for shear

3 displacement method for torsion and shear, shear deformationareas are determined (default for steel, timber)

Options 2 and 3 should be used for composite sections, but with caution.

SCUTcontrols the generation of the two axisparallel standard sections.

When SCUT 1 is input, only the shear section corresponding to the mainbending direction is generated. (Output extent by unaxial bending!). If avalue added by 8 is defined, the shear according to the force method will beevaluated for the gross section instead of the effective section.

The fineness of the subdivision for the integral equation method is controlledby the input value CTRL SDIV. This indicates how large an element may becompared to the largest dimension of the crosssection.

0 No subdivision1 maximum 1/22 maximum 1/43 maximum 1/84 maximum 1/16 (default)

A fang measure for the detection of crosssection parts combined with eachto other can be defined additionally. The value SDIV 4.001 defines 1 mm, thevalue of 4.005 e.g. 5 mm as fang measure.


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311Version 11.00

3.3. Materials

SOFiSTiK supports a large number of different material descriptions. All willbe addressed by a unique material number and should be usable everywherein general. The default for the material type is dependant on the selected design code.

The basic properties are input via the records:

NORM Selection of a design code familyMAT General Materialdefinition (obsolete)MATE General Materialdefinition including strength

CONC Concrete MaterialSTEE Steel and other metallic materialsTIMB Timber/lumberMASO Masonry / BrickworkMLAY Layered composite material for QUADElements

These records are mutually exclusive but may be enhanced by other records:

BMAT Elastic support

NMAT Nonlinear material properties for MAT/MATE(to be used in ASE/TALPA for QUAD and BRIC elements)

SSLA uniaxial strainstress law for materials CONC/STEE/TIMB/BRWO

MEXT Special material properties

Input of material is possible in all parts of the program system. However it

is selfevident that not all parameters are used for all types of analysis or system. Each material has a standard name given by its classification, whichmight be extended by the user. If the user wants to replace the standard completely, he has to start his own text with an equal sign (e.g. =my own Text).

Properties of materials must be distinguished according to whether they areto be kept as close as possible to real values (e.g. for dynamic calculations) orto be used with a safety coefficient for calculating an ultimate loadbearingcapacity. Whereas the safety factors were formerly assigned moreorless at

random, sometimes to the load and sometimes to the material, more recentregulations (Eurocode) provide a clearer separation between safety factorsfor the loads and factors for the material. However, since the material safety


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factors still depend on the nature of the load or the type of design, it will notbe possible to define all safety factors with the material itself.

SOFiSTiK distinguishes therefore:

Properties and safety factors for the standard Design

Mean values or calculatoric values and safety factors for serviceabilityand deformation analysis

If some design codes (DIN 18800, DIN 10451) apply additional safetyfactors to the mean values, this may be defined with the stressstrain relation

via SSLA. The safety factor defined with the material will thus be used onlyfor the full plastic forces in AQUA.


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313Version 11.00

3.4. NORM Default Design Code

NORMItem Description Dimension Default

DC Design code familyEC EurocodesDIN Deutsche NormenOEN sterreichische NormenSIA Schweizer NormenBS British Standard

US US Standards (ACI etc.)JS Japanese StandardGBJ Chinese Building CodesIS Indian Standards

LIT EC

NDC Number of a specific design code Lit16

COUN Countrycode for boxed values within EC30 = Greece

31 = Netherlands32 = Belgium33 = France34 = Spain39 = Italy41 = Switzerland43 = Austria44 = Great Britain45 = Danmark

46 = Sweden47 = Norway49 = Germany351 = Portugal352 = Luxembourg353 = Ireland358 = Suomi/Finland

*

Some properties of Eurocode are dependant on national variants (boxed va

lues). The country code may be used to select those values.


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3.5. MAT General Material

Properties

MATItem Description Dimension Default

NO

EMUEGK

GAMGAMA

ALFA

Material number

Elastic modulusPoissons ratio (between 0 and 0.49)Shear modulusBulk modulus

Specific weightSpecific weight under buoyancyThermal expansion coefficient

kN/m2

kN/m2

kN/m2

kN/m3kN/m3

1

*0.2**

25*

E5

EYMXYOAL

OAF

SPM

TITL

Anisotropic elastic modulus EyAnisotropic poissons ratio mxyMeridian angle of anisotropyabout the local xaxisDescent angle of anisotropy

about the local xaxisMaterial safety factor

Material name

kN/m2

deg

deg

Lit32

EMUE

0

0

1.0

The record MAT can define general materials that can be used for sections orQUAD and BRICelements. The material number must be unique for everymaterial.

This record has been superseeded by MATE, supplying input of strength andelasticity constants in MPa. Further comments are available there.


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315Version 11.00

3.6. MATE Material Properties

MATEItem Description Dimension Default

NO

EMUEGK

GAMGAMA

ALFA

Material number

Elastic modulusPoissons ratio (between 0 and 0.49)Shear modulusBulk modulus

Specific weightSpecific weight under buoyancyThermal expansion doefficient

MPa

MPaMPa

kN/m3kN/m3

1

****

25*

E5

E90M90OAL

OAF

SPMFYFT

TITL

Anisotropic elastic modulusAnisotropic poissons ratioMeridian angle of anisotropyabout the local xaxisDescent angle of anisotropy

about the local xaxisMaterial safety factorDesign strength of materialultimate strength of material

Material name

kN/m3

deg

deg

MPaMPa

Lit32

EMUE

0

0

1.0

Sometimes it is more convenient to define the elastic constants by other values than the Elasticity modulus and the Poisson ratio. You may transform

your values by the following formulas:E Elastic modulusEs subgrade modulus (horizontally cosntrained)K Bulk modulusG Shear modulus Poissons ratio

K E3(1

2)

G E2(1

)

E 9KG(3KG)

3K 2G6K 2G


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Version 11.00316

EsE1

(1)(12)

G 3KE9KE

G 3K(1 2)2(1)

If not specified, missing values will be calculated according to these formulas.It is however possibel to define non consistent constants. If no values aregiven, E will default to 30000 MPa and MUE to 0.2.

Orthotropy may be defined via material and thickness of QUADElements.

(confer record GRP in GENF and remarks in manuals to ASE, SEPP andTALPA).

The description of a transversal orthotropie according to Lechnitzky has onedirection that has different properties, while the description in the plane perpendicular to this remains isotropic. This covers most practical problems liketimber and rock. If this special direction is z it holds:

xxEyE90

zE

90

yyEx

E90

zE90

z zE9090

(xy)E90

Please mark, that the poisson ratio M90 is no longer bound to 0.5 and isstrongly connected to the Elasticity modulus.

For beams the main value of the fibres is the xaxis, perpendicular values yand z will be E90 and M90.

For planar systems (TALPA) the basic values are in the xz plane. The valueOAF is the angle between the local xdirection and the global xdirection. Youhave to exchange the indices y and z in the above formula.

For shells and plates (ex. plywood) we assume that the fibres are in both x andy direction. The anisotropy effects reduce to different shear moduli for in

plane shear and the transverse directions.For three dimensional continua, the orientation is given by the meridina anddecent angle, known from geology. They describe the deviation of the constant


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317Version 11.00

height lines to the north direction and the inclination of the layers. They areequivalent to first and third of the Euler angles. The transformation is defined by two rotations to be selected by the gravity direction. North is thecyclic permutation of the gravity. (ie. xaxis for GDIR ZZ or NEGZ, yaxisfor GDIR XX or NEGX and zaxis for GDIR YY or NEGY). First the northaxis will be rotated about the vertical axis by the amount of OAF, then therotated xy plane will be rotated about the xaxis by the amount of OALagainst the vertical.


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Version 11.00318

3.7. MLAY Layered Material

MLAYItem Description Dimension Default

NOT0NO0T1NO1...

T9NO9TITL

Number of composite materialThickness of first layerMaterial number of first layerThickness of second layerMaterial number of second layer

Thickness of 9thlayerMaterial number of 9thlayerMaterial Designation

**

*

Lit32

1!!!!

With MLAY you may define for QUAD elements a composite layered materialof up to 10 layers. Each layer may be defined with an positive absolute thickness or a negative relative one. The total thickness of the element will be calibrated to the sum of the thicknesses of the material definition. If some layershave negative thickness only these layers will be adopted. Otherwise a uni

form scaling will take place.

If you have a sandwich element with two outer laminates with a given thickness:

MLAY 1 0.02 1 $$ upper laminate 1.00 2 $$ interior laminate 0.02 1 $$ lower laminate

then this data will be applied to match two QUAD elements with a total thickness of 0.10 or 0.15 as follows:

MLAY 1 0.02 1 $$ upper laminate 0.06 2 $$ interior laminate if 0.10 total thickness 0.02 1 $$ untere Deckschicht

MLAY 1 0.02 1 $$ upper laminate 0.11 2 $$ interior laminate if 0.15 total thickness 0.02 1 $$ lower laminate

A standard material definition will also be generated with mean values.


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319Version 11.00

3.8. BMAT Elastic Support /

Interface

BMATItem Description Dimension Default

NO Material number 1

CCTCRAC

YIEL

MUECOHDILGAMB

Elastic constant normal to surfaceElastic constant tangential to surfaceMaximum tensile stress of interfaceMaximum stress of interface

Friction coefficient of interfaceCohesion of interfaceDilatancy coefficientEquivalent mass distribution

kN/m3

kN/m3

kN/m2

kN/m2

kN/m2

t/m2

0.0.0.

0.0

REF

MREFH

ReferencePESS Plain stress conditionPAIN Plain strain conditionHALF circular disk on halfspace

CIRC circular hole in infinite diskSPHE sperical hole in infinite spaceNumber of a reference materialReference dimension (thickness/radius)

LIT

m

NO!

The bedding approach works according to the subgrade modulus theory(Winkler, Zimmermann/Pasternak). It facilitates the definition of elastic supports by an engineering trick which, among others, ignores the shear deformations of the supporting medium. The bedding effect may be attached to

beam or plate elements, but in general it will be used as an independant singleor distributed element. (see SPRI, BOUN, BEAM or QUAD element and themore general description of BOREProfiles)

The determination of a reasonable value for the foundation modulus oftenpresents considerable difficulty, since this value depends not only on the material parameters but also on the geometry and the loading. One must alwayskeep this dependance in mind, when assessing the accuracy of the results ofan analysis using this theory.

The subgrade parameters C and CT will be used for bedding of QUADElements or or the description of support or interface conditions. A QUAD el


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Version 11.00320

ement of a slab foundation will thus have a concrete material and via BMATthe soil properties attached to the same material number.

If subgrade parameters are assigned to the material of a geometric edge(GLN), spring elements will be generated along that edge based on the widthand the distance of the support nodes.

Instead of a direct value you may select a material and a reference dimensionfor some cases with constant pressure [1]:

Flat layer with horizontal constraints e.g. for elastic support by columns and supporting walls (plane stress condition):

CsEH 1

(1)(1) CtEH

12(1)

Flat layer with horizontal constraints for sttlements of soil strata(plane strain condition):

CsEH (1)

(1)(12) CtEH

1(1)

Equivalent circular disk with radius R on unlimited halfspace:

CsER 2(1)(1)

Circular hole in unlimited disk with plane strain conditions

CsER 1

(1)(12) CtCs

Spherical hole in infinite 3D elsatic continua

CsER 2

(1) CtCs

Including a dilatancy factor describing the normal strain induced by shear deformations, we have for the stresses:

Cs (unDILut)Ctut

Nonlinear effects are controlled by CRAC, YIEL, MUE and COH:


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321Version 11.00

Cracking: Upon reaching the failure load the interface fails inboth the axial and the lateral direction. The failureload is always a tensile stress.

Yield load: Upon reaching the yield stress, the deformation component of the interface in its direction increases without a corresponding increase of the stress.

Friction coefficient:If a friction and/or a cohesion coefficient are input, thelateral shear can not sustain forces greater than:

Friction coefficient * normal stress + CohesionIf the axial interface has failed (CRAC), the lateral shear acts only if 0.0 hasbeen input for both friction coefficient and cohesion.

The nonlinear effects can be taken into account only by a nonlinear analysis.The friction is an effect of the lateral interface, while all other effects act uponthe normal stress.

[1] Katz, C., Werner, H. (1982)

Implementation of nonlinear boundary conditions in Finite ElementAnalysisComputers & Structures Vol. 15 No. 3 pp. 299304


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Version 11.00322

3.9. NMAT Nonlinear Material

NMATItem Description Dimension Default

NOTYPE

Material numberKind of material law

LINE Linear materialMISE Mise / Drucker Prager lawMOHR Mohr Coulomb lawGUDE Gudehus law

ROCK Rock materialFAUL Faults in rock materialLADE Lade lawDUNC DuncanChang lawHYPO Schad lawSWEL SwellingMEMB Textile membrane

LIT

1!

P1

P2P3P4...P10

1st parameter of material law

2ndparameter of material law3rdparameter of material law4thparameter of material law

...10thparameter of material law

*

***

*

The types of the implemented material laws and the meaning of their parameters can be found in the following pages.

In a linear analysis the yield function for the nonlinear material is merelyevaluated and output. This enables an estimation of the nonlinear regionsfor a subsequent nonlinear analysis.

If TYPE LINE is given, the material remains linear.

3.9.1. Invariants of the stress tensorFor the present chapter, as long as not specified differently, the following conventions hold:

I1xyz

Deviatoric stress tensor:


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323Version 11.00

sxxI13

syyI13

szzI13

J2 12(sx2sy2sz2)xy2yz2xz2

J3sxsysz 2xyyzxzsxyz2

syxz2

szxy2

13

sin1 3 3

J32J2

32

; 6

6

3.9.2. Material Law MISEElastoplastic material after MISE or DRUCKERPRAGER with associatedflow rule.

fp2 I1 J2 p1

3 0

Application range:

Metals and other materials without friction

Parameters:

P1 = Comparison stress [kN/m2]P2 = Friction parameter []P3 = Hardening module [kN/m2]P4 = Tensile strength z [kN/m2]P5 = Compressive strength (cap) c [kN/m2]

Several substitutes for P1 and P2 can be used for the calculation of commonparameters in soil mechanics. Commonly used e.g. is the compression cone:

P16ccos

3 sin P22sin

3 (3 sin)

The values for the internal cone are better suited for plane strain conditions:


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Version 11.00324

P16ccos

3 sin

P22sin

3 (3 sin)By specification of parameter P5 the model can optionally be extended by aspherical cap (in principal stress space) that limits the volumetric compressive stress to a maximum value. This can be meaningful in particular formainly hydrostatic straining. The cap is defined by:

f 12 2

2 32 P52 P52 P52 0

Reference:

M.A.ChrisfieldNonlinear Finite Element Analysis of Solids and Structures. Vol. I.Essentials. Chapter 14. Wiley & Sons (1991)

M.A.ChrisfieldNonlinear Finite Element Analysis of Solids and Structures. Advanced Topics. Vol. II. Chapter 6. Wiley & Sons (1997)

3.9.3. Material Law MOHRElastoplastic material with a prismatic yield surface and a non associatedflow rule after MOHRCOULOMB. The model is extended by means of aspherical compression cap and plane tension limits. Formulation of yieldcondition and plastic potential using stress invariants:

f 13I1 sin J2 (cos

sin sin

3)ccos 0

g 13I1 sin J2 (cos sin sin3 )

with:

Application range: soils with friction and cohesion

Parameters: Default values:

P1 = Friction angle [degrees] (0.)P2 = Cohesion c [kN/m2] (0.)P3 = Tensile strength z [kN/m2] (0.)


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325Version 11.00

P4 = Dilatation angle [degrees] (0.)P5 = Compressive strength (cap) c [kN/m2] ()P6 = plastic ultimate strain u[o/oo] (0.)P7 = ultimate friction angle u[grad] (P1)P8 = ultimate cohesion cu[kN/m2] (P2)

Special comments:

The following expressions are better suited for checking the yield criterion:

f = I m III b 0

m 1 sin1 sin

b 2ccos1 sin

By specification of parameter P5 the model can optionally be extended by aspherical cap (in principal stress space) that limits the volumetric compressive stress to a maximum value. This can be meaningful in particular formainly hydrostatic straining. The cap is defined by:

f 12

22

32

P52

P52

P52

0Reference:

M.A.ChrisfieldNonlinear Finite Element Analysis of Solids and Structures. Advanced Topics. Vol. II. Chapter 14. Wiley & Sons (1997)

O.C.Zienkiewicz,G.N.PandeSome Useful Forms of Isotropic Yield Surfaces for Soil and Rock

Mechanics. Chapter 5 in Finite Elements in Geomechanics(G.Gudehus ed.) Wiley & Sons (1977)

3.9.4. Material Law GUDEElastoplastic material in its extended form after Gudehus with non associated flow rule.

f = q2 c7p2+ c6p c5 < 0

g = q2 c9p2+ c8p

with:


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Version 11.00326

p = (x+ y+ z)/3 = (3sin)/(3+sin)

q 121 J2 1

3 3 J32J2

c5 = (12c2cos2)/A ; A = (3sin)2

c6 = (24c cos sin)/A

c7 = (12 sin2)/A

c8 = (24c cos sin)/B ; B = (3sin)(3sin)

c9 = (12 sinsin)/B

Application range: soils with friction and cohesion


P1 = Friction angle [degrees] (0.)

P2 = Cohesion c [kN/m2

] (0.)P3 = Tensile strength z [kN/m2] (0.)P4 = Dilatation angle [degrees] (0.)P5 = Compressive strength (cap) c [kN/m2] ()P6 = plastic ultimate strain u[o/oo] (0.)P7 = ultimate friction angle u[grad] (P1)P8 = ultimate cohesion cu[kN/m2] (P2)

Special comments:

This law is capable of describing a multitude of plane or curved yield surfaces.For g=1 a circle in the deviatoric plane is obtained. The dilatation angle isusually set to zero or equal to the friction angle.

By specification of parameter P5 the model can optionally be extended by aspherical cap (in principal stress space) that limits the volumetric compressive stress to a maximum value. This can be meaningful in particular formainly hydrostatic straining. The cap is defined by:

f 12 22 32 P52 P52 P52 0Reference:


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327Version 11.00

W.Wunderlich, H.Cramer, H.K.Kutter, W.RahnFinite Element Modelle fr die Beschreibung von Fels Mitteilung8110 des Instituts fr konstr.Ingenieurbau der Ruhr UniversittBochum, 1981

3.9.5. Material Law ROCKElastoplastic material with marked shear surfaces

f1 = tan (p1) p2 + < 0

g1 = tan (p4) +

f2 = p3 < 0

g2 = f2 (Kluftflche/Fault)

f3 = tan (p6) p7 + < 0

g3 = tan (p9) +

f4 = I p8 < 0

g4 = f4 (Felsmaterial/Rock)Application range:

Plane strain conditions and anisotropic material


P1 = Crevasse friction angle [degrees] (0.)P2 = Crevasse cohesion c [kN/m2] (0.)

P3 = Crevasse tensile strength z [kN/m2

] (0.)P4 = Crevasse dilatation angle [degrees] (0.)P5 = Angle of crevasse direction [degrees] (*)

with respect to xaxis (0180)P6 = Rock friction angle [degrees] (0.)P7 = Rock cohesion c [kN/m2] (0.)P8 = Rock tensile strength z [kN/m2] (0.)P9 = Rock dilatation angle [degrees] (0.)

Special comments:

This law ignores the effect of the third principal stress acting perpendicularlyto the model. One can, however, specify the strength of the rock as well as the


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strength of the slide surfaces, which are defined by the angle P5 (default valueis that of an anisotropic material). The flow rule of the shear failure is nonassociated if P4 is different from P1.

Any of the two limits can be deactivated in special occasions by specifying = c = 0.0.

Reference:

W.Wunderlich,H.Cramer,H.K.Kutter,W.RahnFinite Element Modelle fr die Beschreibung von Fels MitteilungNr. 8110 des Instituts fr konstruktiven Ingenieurbau der Ruhr

Universitt Bochum, 1981.

3.9.6. Material Law FAULDiscrete faults in materials

f1 = tan c + < 0

g1 = tan+

f2 = z < 0g2 = f2

Application range:

Additional discrete faults to a given rock material


P1 = Crevasse friction angle [degrees] (0.)

P2 = Crevasse cohesion c [kN/m2

] (0.)P3 = Crevasse tensile strength z [kN/m2] (0.)P4 = Crevasse dilatation angle [degrees] (0.)P5 = Meridian angle of crevasse plane [degrees] (*)P6 = Descent angle of crevasse plane [degrees] (*)P7 = plastic ultimate strain u[o/oo] (0.)P8 = ultimate friction angle u[grad] (P1)P9 = ultimate cohesion cu[kN/m2] (P2)

Special comments:

This material law may be specified up to three times in addition to any othernonlinear material to allow for the description of multiple faults.


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3.9.7. Material Law LADE

Elastoplastic material after LADE with non associated flow rule.

fI13

27p1paI1

m

I3 0

gI13

27p4paI1

m

I3

withpa= 103.32 kN/m2 = atmospheric air pressure

I1 = (I p3) ( p3) (III p3)

I3 = (I p3) (II p3) (III p3)

Application range: all materials with friction including rock and concrete


P1 = Parameter "" ()P2 = Exponent "m" ()P3 = Relative uniaxial tensile strength [kN/m2] (0.)P4 = Parameter "" for flow rule ()P5 = compressive strength (cap) c[kN/m2] ()P6 = plastic ultimate strain u[o/oo] (0.)P7 = ultimate Parameter "" (P1)P8 = ultimate Exponent "m" (P2)

Special comments:

Material LADE has shown very good accordance between analytical and experimental results. In practice therefore, the parameters can be taken fromexperiments on the materials strength. The law at hand can also describeconcrete or ceramics. A simple comparison with the material parameters ofthe MohrCoulomb law can be made only if the invariant I1 is known.

By specification of parameter P5 the model can optionally be extended by a

spherical cap (in principal stress space) that limits the volumetric compressive stress to a maximum value. This can be meaningful in particular formainly hydrostatic straining. The cap is defined by:


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f 12 2

2 32 P52 P52 P52 0

Reference:

P.V.LadeFailure Criterion for Frictional Materials in Mechanics ofEngineering Materials, Chap 20 (C.s.Desai,R.H.Gallagher ed.)

Wiley & Sons (1984)

3.9.8. Material Law DUNC

Hypoelastic material based on DuncanChang.Loading:

Et1

p7 1 sinp1 IIII2p2 cosp1 2I sinp1

2

p4maxp3I, 0

pa

p6

Unloading and reloading:

Etp5maxp3I, 0

pa

p6

pa= 103.32 kN/m2 = atmospheric air pressure

Application range:Deformation analyses with little plastification and with stress paths not verydifferent from a triaxial test.

Parameters: Default values:P1 = Friction angle [degrees] (0.)P2 = Cohesion c [kN/m2] (0.)P3 = Tensile strength z [kN/m2] (0.)P4 = Reference elastic modulus [kN/m2] ()

during loadingP5 = Reference elastic modulus [kN/m2] ()

during unloadingP6 = Exponent (

0) [] ()

P7 = Calibration factor ( 0) [] ()

Special comments:


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The model distinguishes between primary loading, unloading and reloading different moduli for loading and un/reloading can be specified.

Loading is defined as an increase of the stress level S:

S

1 sinp1 IIII2p2 cosp12I sinp1

The initial state should be calculated linearly doing so, parameters definingthe loading history are initialized and the resulting stress state is interpretedas loading".

In case of unloading, after having passed a deviatoric stress minimum, a primary loading branch is traced again > simulation of cyclic loading behavioris possible.

The original law according to DUNCAN/CHANG has been modified in orderto allow for a better simulation of the plastic flow in soil materials. Poissonsratio is not kept constant but is defined as a function of the tangential modulus of elasticity and the bulk modulus. The bulk modulus is kept constant in

this case.By P6=P7=0 one can define a law with each a constant elastic modulus forloading and unloading.

In order to avoid numerical difficulties, the elastic modulus in the MAT record should not be chosen smaller than the initial elastic modulus.

Anisotropic materials are not possible with this model.

Reference:J.M.Duncan, C.Y.ChangNonlinear Analysis of Stress and Strains in Soils

J.Soil.Mech.Found.Div. ASCE Vol 96 SM 5 (1970) ,16291653

C.S.Desai, J.T.ChristianNumerical Methods in Geotechnical Engineering, 8188McGrawHill Book Company

3.9.9. Material Law HYPOHypoelastic material after Schad.


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Bulk and shear moduli during loading:

K = p1 p7p p8qmaxG = p2 p5(I+III) p6q

Bulk and shear moduli during unloading:

K = p3

G = p4where:

p = (x+ y+ z)/3q = I III

Application range: isotropic materials


P1 = Initial bulk modulus [kN/m2] ()P2 = Initial shear modulus [kN/m2] ()P3 = Bulk modulus for unloading [kN/m2] ()

P4 = Shear modulus for unloading [kN/m2] ()P5 = Parameter [] ()P6 = Parameter [] ()P7 = Parameter [] ()P8 = Parameter [] ()P9 = Tensile strength [kN/m2] (0)

Special comments:

This law must have a vanishing shear modulus at failure by MohrCoulomb,thus the following expressions are obtained:

p2 = p62 c cos

p5 = p6sin

Anisotropic Material constants are not possible with this model.

Reference:

H.Schad

Nichtlineare Stoffgleichungen fr Bden und ihre Verwendung beider numerischen Analyse von Grundbauaufgaben. MitteilungenHeft 10 des BaugrundInstituts Stuttgart (1979)


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3.9.10. Material law SWEL

Additional Parameters for swelling of materialsApplication range: Selling of soils in case of unloading

Relationship between stress and final state swelling strains:

qi

p1

0 i0i

logi0i 0iip2

logc0i p2 ii 1..3

i = principal normal stresses0i = equilibrium state of stress wrt swelling (initial condition),

transformed to the direction of principal normal stresses i


P1 = modulus of swelling [] (0.0033)P2 = magnitude of the smallest compressive stress below which

no more increase of swelling occurs [kN/m2] (10.0)

Special comments:

Swelling of soils is a complex phemomenon that is influenced by various factors. There are two swelling mechanisms of practical importance that can bedistingushed for which the presence of (pore) water is a common prerequisite. The first mechanism is termed as the osmotic swelling" of clay minerals,which basically is initiated by unloading of clayey sedimentary rock. The second mechanism takes place in sulfateladen rock with anhydrite content. Inthis case the swelling effects are due to the chemical transformation of anhydrite to gypsum which goes along with a large increase in volume (61%).

For both described mechanisms a principal dependency between the swellingcaused increase in volume and the state of stress was observed both inlaboratory and in insitu experiments. The formula employed here, repre


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sents a generalization of the 1dimensional stressstrain relationship, whichHUDER and AMBERG derived from oedometer tests.

The equilibrium state wrt swelling 0is defined by means of the GRPrecord.Doing so, we use the option PLQ in order to reference a (previously calculated)load case as primary state for swelling". This state is in equilibrium wrtswelling (normally insitu soil prior to construction work). In the course ofconstruction work occuring unloading related to this primary state causesswelling strains according to the formula above.

The SWEL record is specified in addition to a linear elastic or elastoplasticbasic material.

Anisotropy is not possible with this model.

Reference:

P.WittkeGattermannVerfahren zur Berechnung von Tunnels in quellfhigem Gebirgeund Kalibrierung an einem Versuchsbauwerk.Dissertation RWTHAachen, Verlag Glckauf 1998

W.WittkeGrundlagen fr die Bemessung und Ausfhrung von Tunnels inquellendem Gebirge und ihre Anwendung beim Bau der

Wendeschleife der SBahn Stuttgart.Verffentlichungen des Institutes fr Grundbau, Bodenmechanik,Felsmechanik und Verkehrswasserbau der RWTHAachen 1978

W.Wittke, P.RisslerBemessung der Auskleidung von Hohlrumen in quellendemGebirge nach der Finite Element Methode.

Verffentlichungen des Institutes fr Grundbau, Bodenmechanik,Felsmechanik und Verkehrswasserbau der RWTHAachen 1976,Heft 2, 746

Nichtlineare Stoffgleichungen fr Bden und ihre Verwendung beider numerischen Analyse von Grundbauaufgaben. MitteilungenHeft 10 des BaugrundInstituts Stuttgart (1979)

3.9.11. Material law MEMB

Parameters for textile membranesP1 Factor for Stress change

(only in special cases, cnf. ASE GRP FACS)


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P2 Factor for compression stiffness0.0 no compressive stress possible1.0 full compressive stress possible0.1 intermediate values for scaling the elasticity modulus


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3.10. MEXT Extra

Materialconstants

MEXTItem Description Dimension Default

NOTYPE

VALVAL1VAL2VAL3

VAL4VAL5

Number of materialType of constant

Value of material constantFirst additional material valueSecond additional material valueThird additional material value

4thadditional material value5thadditional material value

LIT

****

**

1!

With MEXT you may define special material values for any type of material.The definition of TYPE selects one of the following possibilities:

With KRVAL defines the equivalent roughness according to Table 10.8.1 ofEC 1 part 24, needed especially for wind loads on circular sections:

Surface Roughnessk [mm]

Surface Roughnessk [mm]

glas 0.0015 galvanised steel 0.2

polished metall 0.002 spinning concrete 0.2

smooth painting 0.006 cast in situ concrete

1.0

spray painting 0.02 rust 2.0blasted steel 0.05 masonry 3.0

cast iron 0.2

Hint: In table 4 of DIN 1055 part 4 slightly larger values are defined .


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3.11. CONC Properties of Concrete

CONCItem Description Dimension Default

NOTYPE

Material number (1999)Type of concrete:

C regular concreteLC lightweight concreteB,LB,SB concrete DIN 1045/4227SIA,LSIA concrete SIA 162

BS concrete (BS 8110)ACI American StandardCBC Chinese Building CodesIS,IRC Indian StandardsCE with constant E modulus

LIT

1*

FCN Strength class f ck/fcwk(nominal strength) N/mm2 *

FC

FCTFCTKECQCGAM

ALFASCMTYPR

FCRGCGFMUECTITL

Design value of concrete strength

Tensile strength of concreteLower fractile strength valueElastic modulusPoissons ratio or shear modulusUnit weightThermal expansion coefficientTypical material safety factorType of service state line

LINE = constant elastic modulus

A,B = shorttime lines (Eurocode2)R = calc. mean values (DIN)RS = R with k=1.3 (SLWAC)

Strength for nonlinear analysisEnergy at break for compressive failureEnergy at break for tensile failureFriction in cracksMaterial name

N/mm2

N/mm2

N/mm2

N/mm2

*kN/m3

LIT

N/mm2

N/mmN/mmN/mmLit32

*

***

0.225

1E5**

******

3.11.1. Eurocode 2

According to Eurocode 2 the following types are available:


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C = regular concreteLC = lightweight concrete

The cylindrical strength is to be input for FCN. The default value is 20.

Some properties are dependant on national variants. The definition of NORMCOUN is used to switch between those values. As EC2 and DIN 10451 differconsiderably, you should use NORM to select the proper design code family,but you may also append the characters :DIN" or .EC" to the given class toset the given code explicitly for that material.

The default values for design strength and elastic are derived as follows:

FC = 0.85 fck

FCT = 0.3 fck2/3 (fck< 55)

FCT = 2.12 ln((fck+8)/10+1) (f ck> 50)

EC = 9500 ( fck+ 8 ) 1/3

By lightweight concrete (LC) according to EC24, value EC must be defined

explicitly or by means of GAM. The raw unit weight class can be input forGAM too, GAM and EC will then be defined appropriately. For the raw weight in kg/m3we have = (1.5)100

EC = 9500 ( fck+ 8 ) 1/3 ( /2200 ) 2

For detailed analysis of concrete according to appendix 1 you need the kindof cement. You may specify this by appending a Literal to the class of concrete

N,R normal or rapid hardening cement (= 0.0)S slowly hardening cement (= 1.0)RS with high strength cement (= +1.0)

The usual stressstrain curve of the C types is the parabolic rectangularstressstrain diagram of Eurocode 2 / DIN 1045 / OeNORM B 4200 / SIA 162.For nonlinear analysis or deformation analysis, there are other types A/B/Rfollowing the expression:

fc knn2

1 (k2)n

with


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n = / c1

k = (1.1EC) c1/fc

For fc we have for the curves A and B the value fck+8, for R or RS the value0.85fckaccording to DIN 10451. The maximum strain is limited accordingto the strength. The B line does not possess a falling branch, and it is thuseventually more stable numerically.

The safety factors SCM are preset to 1.5. In AQB, however, they must be selected explicitly, because they are dependent on the loading combinations.

For concrete with high strength the factor will be increased by , which willalso be incorporated in the strainstress laws, to allow a global safety factorto be used for the design.

For nonlinear analysis with a constant safety factor according to DIN 10451the strength of the concrete will be reduced, while those of the steel will beraised. This servicability work law is selected with the literal R at positionTYPR. As DIN 10451 distinguishes between normal and light sand, thelatter may be adressed with the literal RS.

3.11.2. DIN 1045 old / DIN 4227 / DIN 18806:The new DIN 1045 will be addressed by the national variant of Eurocode EC2. The old DIN can be addressed with the old literals.

B = regular concrete (DIN)LB = lightweight concrete (DIN)SB = prestressed concrete (DIN)

The default FCN is 25 for B and LB, and 45 for SB. FCT is defined by:

FCT = 0.25 FCN 2/3

Defaults in accordance with old DIN 1045 / DIN 4227:

FCN 10 15 25 35 45 55

FC

DIN 1045 (B)

DIN 4227 (SB)

EC

7

22000

10.5

26000

17.5

15.0

30000

23

21

34000

27

27

37000

30

33

39000


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as well as the following highstrength concretes:

FCN 65 75 85 95 105 115

FC

EC

40.0

40500

45.0

42000

50.0

43000

55.0

44000

60.0

44500

64.0

45000

The elastic modulus or the weight has to be specified in case of lightweightconcrete. However, the raw unit weight class according to DIN 4219 (1.0 2.0)may be input for item GAM. The default for GAM and EC then complies withDIN 1055. A bilinear stressstrain curve is usually employed for light

weight concrete.For standard concrete a parabolarectangular diagram will be selected according to Eurocode EC2 / DIN 1045 / NORM B4700 / SIA 162. SCM will default to 1.00. If you analyse composite sections you might want to change thevalue. High strength concrete will have lesser ultimate strains.

3.11.3. NORM B 4700 / B 4750Although the OENORM B 4700 calls itself close to Eurocode, it deviates justwith the classification of concrete based on the cubic stregth instead of the cylindrical strength. As the designation is C resp. LC the user has to select theoption NORM OEN or append those literals to the class value.

C = regular concrete (NORM 4700)LC = lightweight concrete (NORM 4700)

The default FCN is 25.

Defaults in accordance with OeNORM B 4700:

FCN 20.0 25.0 30.0 40.0 50.0 60.0

FC

FCTEC

15.0

1.927500

18.8

2.229000

22.5

2.630500

30.0

3.032500

27.5

3.535000

45.0

4.137000

SCM is preset to 1.5, FCTK to 0.7FCT.

The standard choice for regular concrete is the parabolicrectangular stressstrain diagram in accordance with Eurocode 2 / DIN 1045 / OeNORM B 4700

/ SIA 162. The value of SCM is preset to 1.5.


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3.11.4. Swiss Standard SIA 162 (1989)

As type we haveSIA = regular concrete (SIA 162)LSIA = lightweight concrete (SIA 162)

The nominal strength FCN is the mean cubical strength. The first value ofthe concrete class must thus be used (e.g. B 35/25 should be input as SIA 35).The elastic moduli are the mean values from Figure 31 in Section 5.18 derSIA. Half of the EC values are assigned to lightweight concrete.

FCN 20.0 25.0 30.0 35.0 40.0 45.0FCNmin

FC

FCT

EC

10.0

6.5

2.0

29000

15.0

10.0

2.0

31000

20.0

13.0

2.0

33500

25.0

16.0

2.5

35000

30.0

19.5

2.5

36000

30.0

23.0

2.5

37000

The default stressstrain diagram is the parabolicrectangular one in accordance with Eurocode 2 / DIN 1045 / OeNORM B 4200 / SIA 162. SCM willbe preset with 1.2.

3.11.5. British Standard BS 8110As type we have:

BS = normal weight concrete BS 8110

The nominal strength FCN is the cubical strength. The design strength is obtained by

FC = 0.67 FCN

British Standards employ a parabolic rectangle curve, starting from a designcube strength = FC/0.67 with 0.24 strain at full plasticity and an initialstiffness of 5.5 vaccording to Figure 2.1. The safety factor SCM is preset to1.5.

3.11.6. American concrete institute ACI 318M99As type we have the specified compressive strength in MPa:

ACI = normal weight concrete ACI 318M

As the value of fc should not exceed the value of 25/3 MPa in general anddifferent reductions have to be applied for lightweight concrete, we use the


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tensile stress to define the value of fc . The modulus of rupture fris the

upper fractile value of the tension strength. ACI 9.5.2.3 defines:fr 0.75* fc 0.75*253

or for lightweight concrete:

fr 0.70 * min( fc ,1.8*fctm)fr 0.70*0.75* fc

The ratio of the fractiles is thus 1.26. The mean value fctmwill be preset to0.5* fc . All other values will be derived from this value by a factor. If neededthe lower fractile may be given, which will then set the upper value. But thisvalue is only used for those cases where explicitly the value fris used withina formula.

3.11.7. Chinese StandardsAs type we have:

SGBJ = Standard TB 10002.399 (Railway Bridge)

The nominal strength FCN (15 to 60) and the the design strength are takenfrom table 3.1.3. Youngs modulus is derived from 3.1.4.

3.11.8. Indian Standards IS 456 / IRC 21As type we have:

IS = Indian Standards IS 456 (10 bis 80)IRC = Indian Roads Congress IRC 21 (15 bis 60)

The nominal strength FCN is the cubical strength. The design strength is obtained by

FC = 0.67 FCN

Indish Standards employ either allowed stresses (IRC resp. Annex B of IS456) or a parabolic rectangular curve with 2 and 2.5 o/oo strain. The allowedstresses will be converted to a serviceability stress strain law. The elasticitymodulus is preset according to IS to 5000* f

ck, for IRC according to table 9.

The tensile strength is preset to 0.7* fck . The safety factor SCM is preset to1.5.


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3.11.9. Linear Elastic Concete

A linear elastic material without tensile stresses is specified for CE. This canbe used for servicability analysis, older design codes or stresses of foundations.


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3.12. STEE Properties of Metals

STEEItem Description Dimension Default

NRTYPE

CLAS

Material number (1999)Type of the material

S / PS reinforc./prestress. steelBST/PST Reinf./prestr. steel DINFE / S / ST Structural steel EC/DINGU Cast iron

AL,AC,AW Aluminium alloymore types see comments

Steel class or quality

LIT

*

1*

*

FYFTFPESQS

GAMALFA

SCM

Yield strength (f0.02)Tensile strengthElastic limit (f0.01)Elastic modulusPoissons ratio or shear modulus

Unit weightThermal expansion coefficient

Default for AL:Typical material safety factor

N/mm2

N/mm2

N/mm2

N/mm2

*

kN/m3

****

0.3

*1.2E52.38E5

*

EPSYEPSTREL1REL2

RK1FDYNTITL

Permanent strain at yield strengthUltimate strainCoefficient of relaxation (0.70 )Coefficient of relaxation (0.55 )

Bond coefficient by DIN 4227 Table 8.1Bond coefficient per EC 2 / Vol. 400Allowed stress rangeMaterial name

o/ooo/oo%%

N/mm2

Lit32

***0

150/2002.0/0.8**

The steel types S (partial), FE, ST, GU, BS, A and AL, AC, AW can be usedfor crosssections. All other designations can be used only as reinforcementand prestressing tendons. The safety factors are considered by AQB first, because they depend on the loading combination.


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Defaults for structural steel:

FY FT EPST FP EPSY ES GAM

Eurocode:

* FE 360

FE 430

FE 510

FE 275FE 355

235 360 210000 78.5

275 430 210000 78.5

355 510 210000 78.5

275 430 210000 78.5355 510 210000 78.5

DIN:ST 33

ST 37

ST 52

S 235

S 275

S 355

* GU 52

GU 17

GU 20GU 200

GU 240

GU 400

190 330 210000 78.5

240 370 210000 78.5

360 520 210000 78.5

240 360 210000 7

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