structural modelling manual by arup

7/25/2019 Structural Modelling Manual by ARUP

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Structural Guidance Note 3.5aStructural Modelling Manual

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STRUCTURAL GUIDANCE NOTE 3.5 REV A

STRUCTURAL MODELLING MANUAL

CONTENTS

1 Introduction

2 Modelling principles2.1 The Aims of Structural Analysis2.2 Modelling procedure2.3 Selecting the Model and its Properties

Scheme design models, Final analysis models2.4 Preliminary Hand Analysis2.5 Computer analysis2.6 Checking the results

2.7 Using the results

3 Modelling Specific Structural Members3.1 General Considerations3.2 Beams and Columns

General Principles, Members of uniform straight cross section, Members ofnon uniform straight section, Perforated members, Curved members

3.3 Joints and connections between beams and columnsGeneral Principles, Joints in Concrete frames, Joints in steel frames, Jointsin other materials

3.4 Slabs and WallsGeneral, Floor slabs in large building models, Suspended floor slabanalysis - flat slabs, Suspended floor slab analysis - beam and slab, Shear

walls in large models , Shear wall analysis, Basement floor slab analysis -flat slab, Basement Floor Slab analysis - beam and slab, Plate and shellstructures

4 Modelling Specific Structure Types4.1 Multistorey buildings4.2 Roof structures4.3 Symmetrical structures4.4 Tunnels and culverts4.5 Cylindrical structures4.6 Bridge decks

5 Member section properties

General, Cross Sectional Area (A) and Shear factors (Ky, Kz), FlexuralSecond Moment of Area (Iyy, Izz), Torsional Second Moment of Area (J)

6 Material properties6.1 General

6.2 Reinforced Concrete

Properties for ultimate strength analysis, Properties for serviceabilityanalysis, Properties for dynamic analysis

6.3 Steel

6.4 Composite

6.5 Timber

6.6 Other materials

Wrought Iron, Cast Iron, Masonry, Aluminium


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7 Modelling Loads7.1 General

7.2 Point of application of load

7.3 Self weight

7.4 Other permanent loads

7.5 Transient loads

7.6 Internal loads

Temperature, Lack of fit, Prestress, Support settlement

8 Modelling restraintsGeneral, Restraint at ground, Raft Foundations, Global restraints, Supportsettlement

9 Modelling for dynamic analysis

10 Interpretation of results10.1 Beams and columns10.2 Slabs and walls

11 Additional Information11.1 Warnings and errors during analysis11.2 Concrete post processing of 2D element results11.3 P-delta effects11.4 Modelling castellated beams, trusses etc. by shear beams11.5 Modelling beams with long slabs11.6 Grillage analogy for slabs11.7 Setting up a model with 2-D elements11.8 Axis systems and element orientation

11.9 Section properties for standard shapes11.10 Multi storey frame models11.11 Modelling bridge decks11.12 Section properties from first principles11.13 Modelling haunched beams11.14 Modelling elements with rigid ends11.15 Modelling beams and slab floors with in plane stresses11.16 Symmetrical & anti-symmetrical loads11.17 Equivalent loads for pre-stressing11.18 Properties of standard materials11.19 Holes in reinforced concrete beams

12 Revision History


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1 INTRODUCTION

This note gives guidance on the representation of a real structure by an idealised model which can

then be analysed.

2 MODELLING PRINCIPLES

2.1 The Aims of Structural Analysis

Analysis is an essential part of the structural design process. Analysis gives us the numbers withwhich to justify the adequacy of the structure in terms of strength, stability and stiffness.

However we do not analyse 'the structure' but a mathematical model whose properties and behaviourwe aim to select so that it adequately represents the real structure. The approximations and

assumptions which must be made to create this model mean there can be no such thing as an exactmodel.

There is a commonly held belief that present day analysis is necessarily more 'accurate' than earliermethods were. However, while results of computer analysis are 'precise', unless the analytical modelis carefully chosen the results may be misleading and could be dangerously wrong.

Before embarking on an analysis it is necessary to consider what analysis is needed:

What needs to be demonstrated, what will dictate the design?Eg strength, deflection, stability

What is the simplest model which can demonstrate this?The simplest model will be the easiest to understand, the quickest to set up anduse and so probably the most cost effective

What effects will be ignored or misrepresented by this model? Are they significant?Eg the analysis will normally assume linear elastic materials and smalldeflections.

What results will be produced? Can they be used to demonstrate the desired effect?Eg 2D element analysis produces stresses which may not be easy to convert intodesign forces and moments.

What will happen to the reactions from this model?Eg Can the foundations resist the forces generated?


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2.2 Modelling procedure

There are a number of steps in the production and analysis of a mathematical model of a structure.

Engineering judgement is needed throughout the process, and the modelling decisions made should

be reviewed as part of the design review. Decide how much of the structure to model. (section 2.3) Decide how much detail is needed in the model. This will depend on whether the model is

for scheme design or final analysis. (section 2.3)

Define geometry of modelDefine member sizes and shapes(section 5)

Decide on appropriate material properties(section 6)Define loading to be applied

Decide how to apply the load(section 7)Analyse the model

Review analysis results(section 2.6)

2.3 Selecting the Model and its Properties

Scheme design models

The model needed for a quick scheme design can usually be very much simpler than the

model used for final detailed design calculations, and in many cases hand calculationswill be sufficient. The form of the structure is still evolving at this stage andfundamental parameters may need to be altered or varied to determine the mostappropriate design.

Issues which are not considered at this stage but which need to be investigated at a laterstage, should be clearly identified for the final designer to pick up. Members derivedfrom such simple models should be sized conservatively to allow for designdevelopment.

Final analysis models

Once the detailed structural geometry has been defined, use of a more complex modelmay be justified. The capacity of the latest generation of computer programs makes itpossible to build very large and complex models but these can be counterproductive,because the volume of data may increase the chances of errors and makes it more

difficult to pick them up. For many structures, analysis of simple sub frames will beadequate for final design.

A structure should only be analysed as a three-dimensional model if the designer issatisfied that simpler models cannot adequately predict the behaviour. This remains trueif a three-dimensional CAD model of the structure has been created. For furtherguidance on when a three-dimensional model is appropriate see multistorey buildings

(section 4.1)

2.4 Preliminary Hand Analysis

The engineer must be able to visualise how he or she expects the structure to work, and must have anidea of the magnitude and form of answers anticipated, before any analysis is done. A sketch of theexpected deflected form and some simple hand calculations should always be done, however

complex the model, to ensure that gross errors in modelling will be picked up.


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2.5 Computer Analysis

Most computer programs for structural analysis will carry out a linear elastic analysis, assume

deflections are not large enough to affect the force distribution and assume that plane sections

remain plane. The engineer must always remember that these assumptions are being made andconsider whether they are appropriate to the structure being modelled.

In structures where deflection would affect the forces in the structure, these effects need to beconsidered (P-delta effectssection 11.3). The assumption that plane sections remain plane is notvalid at complex joints and discontinuities and a more detailed analysis in such areas, using 2Delements, may be necessary. Walls, slabs and plates may be modelled using a grillage of skeletal

elements, or by using 2D elements. Special structures (eg cable nets) or loadings (eg seismic) mayrequire more complex models and/or analyses. Specialist advice should be sought before startingsuch analysis.

Problems may be encountered during analysiswhich either prevent results being produced, or give

warnings that the analysis may not be valid.(section 11.1)Input data should always be checked carefully. Some one who did not develop the model should

check it. The analysis program's facilities for graphical representation of the data should be used tothe full in checking.

2.6 Checking the results

Always confirm that no warning messages were produced by the computer during the analysis. Ifwarnings have been produced, their implications should be evaluated before proceeding further. Seethe program manual for guidance on this topic.

Results should always be checked against the preliminary hand analysis.

Look at the deflected form of the structure under each basic loadcase, looking at themagnitude of peak deflections as well as the form.

Compare the sum of reactions for each basic loadcase with the applied load (to check foranalysis errors like ill conditioning) and the total load on the model with the total load on thestructure (to check for input errors).

If the model and loading is symmetrical, check the results are symmetrical. Check the restraints are restraining the structure in the intended directions.

Compare bending moment, shear force, axial load and reaction plots from the computeranalysis with the predictions.

If the results do not correspond to the predictions the following procedure should be followed:

Check the input data again, graphically and line by line. Check that elements are correctly orientated and relative stiffnesses are sensible.

Check that loading is correctly applied. Seek advice from colleagues and/or specialists. Review the modelling assumptions. Review the hand analysis. Examination of the computer results may show trends which were

not previously recognised, which can in turn lead to a radical reinterpretation of thebehaviour of the structure.

If after this the hand calculations and the computer results cannot be reconciled seek advice. Never

accept the results of a computer analysis without an engineering explanation for the behaviour

predicted by the computer. A number of expensive mistakes have been made in recent years byengineers who have ignored this rule.


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2.7 Using the results

Once the results have been checked and found to be reasonable they can be used for member design.

The resultsfrom the analysis of a skeletal structure can often be used directly for design but mostother results will need to be post-processed to derive forces which can be used in member design.(section 10)

3 MODELLING SPECIFIC STRUCTURAL MEMBERS

3.1 General Considerations

Structures are generally made up of three types of member:

A beam or column, with a lengthwhich is large compared to itscross sectional dimensions.

A slab, wall or plate, with athickness which is smallcompared to its length andbreadth eg a concrete slab.

A solid, with significantdimensions in all three directions,eg a small deep pilecap.

Divisions between these categories are not always clear cut.

Analytical models in a computer are made up of discrete, 'finite' elements, which are only connected

together at their extremities (as opposed to the actual structure with its infinitesimal elementsforming a continuum). Again these finite elements come in three types:

A one-dimensional element,defined by the position of its two

ends and the properties of itscross section, connected at its

ends only (a beam element).

A two dimensional element,defined by the position of its

corners and a thickness,connected at its edges only (a

plate or shell element).

A three dimensional element,

defined by the position of its

corners, connected at its faces

(a brick element). This type of

element is used in specialised

structural analysis only and

will not be considered further

in this note.

The divisions between these types are clear-cut. A one-dimensional element is always represented

mathematically by a line of zero cross section and a two dimensional element by a surface of zerothickness.


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The selection of the appropriate idealised element to model specific members of real structures isdescribed below .

3.2 Beams and Columns

General Principles

Generally these are the easiest members to represent in a model. There will be a simplecorrespondence between the member in the structure and the element in the model. Problems comeatjointswhere the members meet (section 3.3), and where forces in the member are changed bydeflection under load (P-delta effectssection 11.3).

Members of uniform straight cross section

Simple bending theory, assuming plane sections remain plane and neglecting shear deformations,gives an adequate model of beams with a span/depth ratio of 10 or more (for steel and concrete). A

shear beam, which includes the effect of shear deformation, should be used for elements with asmaller span/depth ratio. Using shear beams for all elements will not increase the size of the problemfor the program and so their use is recommended.

It is generally assumed that only part of the cross section of a beam resists shear deformation (eg the

web of an I beam) and so the proportion of the cross section which is assumed to carry shear has tobe defined, usually by a shear factor(section 5.2).

Once the span/depth ratio is 2.5 or less a reinforced concrete beam should be designed as a deepbeam in accordance with CIRIA Guide 2.

Deep steel beams with a span/depth ratio of 1 or less are unlikely to behave as beams and should bestiffened to suit a strut and tie model.


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Members of non uniform straight section

Haunched and tapered beamscan be analysed by choosing a mean stiffness or by modelling differentparts of the beam with different properties. (section 11.13)

Perforated members

Isolated holes or recesses which extend over less than 5% of the length of the beam do notsignificantly affect its stiffness and can be ignored in analysis (but must be considered in design ofthe member). Local effects around holes need to be considered in design, detailed guidance for

dealing with holes in reinforced concrete beamsis given in section 11.19.

A member with a line of holes (for example a castellated beam), a truss or a Vierendeel girder can bemodelled as a single shear beam. (section 11.4)


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Larger holes and long slotsshould be represented in the model by splitting the member into sections.

(section 11.5)

Curved members

A curved member may be modelled as a series of straights. Generally the angle between successiveelements modelling a circular arc should not exceed 15 degrees.

The curve introduces additional forces into the member which are not modelled by the series ofstraights. To include for these effects within the model, each member between intersections with

other members needs to be modelled by at least two elements (or more if necessary to satisfy the 15degree rule given above). The additional node should be placed at the point of maximum deviationof the curve from the straight line joining the ends.

3.3 Joints and connections between beams and columns

General Principles

One-dimensional elements in a model have zero cross section and so they meet at a point. Theconnection at this idealised point in the model can be one of two types:

A rigid joint, in which there is no relative rotation between elements meeting at a point. A pinned joint, in which no moment is transferred between elements meeting at a joint.

A partially fixed joint, where a rotational spring is used to transfer moment across a joint whileallowing some rotation, could be included in a model but the value to be used for the spring stiffnessis uncertain. If the analysis is sensitive to joint stiffness, then analysing for both a fully fixed andfully pinned condition, and designing for the worst case in each element is recommended for

strength design.

Because real members are not of zero thickness, they never meet at a point. True pins are rare, unlessa proprietary bearing is used. Truly rigid joints never occur, although many joints are stiff enoughfor the difference to be neglected. Joints in steel structures(section 3.3.3) even when apparently stiffare usually modelled as pinned which simplifies joint design and ensures that members designed to

resist sway carry all horizontal loads. Where a nominally pinned joint has to be able to rotate (forexample at the end of a long 'simply supported' truss) the capacity of the actual connection toaccommodate those rotations must be checked.


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Joints in Concrete frames

Joints in monolithic concrete frames should normally be modelled as rigid. It is unrealistic to model

any monolithic joint as pinned, and reinforcement must be provided to limit cracking in all suchcases. If in a subframe analysis a beam is assumed to be pinned to the columns then column

moments will need to be calculated separately. A better solution is to model the columns as part ofthe subframe.

Finite Joints

Where the members intersecting at an idealised joint have substantially different flexural stiffnesses,the stiffer members will restrain the flexure of the other members. These flexible members can bemodelled using a single element with a transformed stiffness, or an element with rigid ends.(section11.14)

Large joints can be analysed using 2D elements to determine the distribution of stresses and to give abetter estimate of stiffness of the joint. Note that the dilation of 2D elements under axial load is

restrained by rigid constraints and supports (which are a convenient way to apply loads from a beamanalysis to such models), giving some unexpected and normally spurious stresses local to the

constraint.


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Joints in steel frames

The cost of joints between members in a steel structure is a large proportion of the total cost. It is

therefore desirable, wherever possible, to use simple design methods, assuming pinned connections.

In triangulated structures (trusses) rotations at joints are usually small and so even welded joints canbe considered as pinned. Joint fixity is needed for stability in the design of moment frames andportals and should normally be assumed when a steel member is continuous through a joint.Haunched joints in portals need to be modelled by more than just a beam and column meeting at apoint.

At the end of a steel truss, where a number of steel members meet at 'a point', it would be normal tojoin the diagonal bracing member to the beam before the beam is joined to the column. The beam tocolumn connection will therefore need to be designed for the resultant of the forces in the beam anddiagonal, not just for the end forces in the beam as derived from the analysis.

There are often situations where it is not appropriate for the centrelines of members to meet at a

point and generally the resulting eccentricity needs to be modelled. EC3 annex K.3 allows smalleccentricities to be ignored (the eccentricity has to be less than between 0.25 and .55 times the mainmember depth, depending on the arrangement of the joint)

Joints in other materials

Joints in timber

Achieving a rigid joint in timber construction is very difficult, so joints should generally be modelledas pinned. Connections are often eccentric and the moments generated by this and by partial fixity

need to be considered in design.


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Joints in masonry

Unreinforced masonry joints cannot take tension.

Unreinforced masonry should not generally be analysed in a linear elastic program.

3.4 Slabs and Walls

3.4.1 General

The modelling of slabs and walls is more complicated than the modelling of beams and columns.There are often several different ways in which a given member can be modelled and the best modelto use will depend on a number of factors:

Is it subject to in-plane forces, out of plane forces, or a combination? Is it a uniform isotropic material? (reinforced concrete isnt) If not can the results be

manipulated to give useable design forces?

Is it to be modelled on its own or as part of a larger model? On its own:

Is it a simple, regular structure that can be designed using simple rules, without

computer modelling?Are there holes or discontinuities that will lead to stress concentrations?

As part of a larger model:Are forces within the slab or wall of interest in this model, or just its effect onthe surrounding members?

To illustrate the approach to modelling slabs and walls the following sections show how thesefactors influence the selection of models for some common structural members.

3.4.2 Floor slabs in large building models

Where a large irregular building consists entirely of sway frames, a 3D skeletal model is oftenappropriate (but not forbuildings with stability coressection 4.1). Floor slabs in such buildings willgenerally be very stiff in plane compared to the columns (except at the base of tall buildings wherecolumns are likely to be very stiff ) and axial forces in the slabs are generally neglected in slabdesign. The simplest way to represent the high in plane stiffness of such slabs in the model is to linkthe nodes attached to the slab rigidly together at each level to prevent relative in-plane movement.

If the model is to be used to investigate shrinkage, prestress or temperature effects then a rigid link

cannot be used. A mesh of 2D elements or a grillage of beams could be used to model the slab but

these effects would normally only be investigated in local element models of either a flatslab(section 3.4.3) or abeam and slab deck(section 3.4.4).

3.4.3 Suspended floor slab analysis - flat slabs

Simple methods of analysing regular flat slabs, including coffered slabs, spanning in one or twodirections are described in BS8110 (and EC2) and more details are given in CIRIA Report 110.These methods should generally be used where possible.

In-plane stresses in suspended slabs, for example those generated as horizontal wind loads aretransferred from the facade into the stability cores, are generally neglected in analysis. However propforces from retaining walls onbasement slabs(section 3.4.7) may need to be considered.

If supports are not in a regular pattern then analysis under out of plane loading only can be carriedout by representing the slab as either a grillage of skeletal elements or a mesh of 2D plate bending


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elements. Note that coffered slabs need to be modelled in such analysis asbeam and slab decks(section 3.4.4). Defining the element properties for grillage analysis(section 11.6) is more

difficult but the 2D element mesh (section 11.7) needs to be set up carefully to ensure adequateresults. Interpreting the results of the 2D element analysis is more difficult, unless apost processor

(section 11.2) is available to convert stresses into areas of reinforcement. Guidance on setting up agrillage model for a flat slab is given in CIRIA Report 110.

Cracking of concrete close to supports leads to significant redistribution of moment away from thesupport. This reduction in stiffness (EI) can be modelled by reducing E in the analysis. Iteration torefine the extent of cracking is likely to be necessary. Note that this is effectively a non linearanalysis, and so must be done for the appropriate loadcase, since superposition of loadcases is notapplicable. Sharp peaks of moment/stress at internal supports derived from analysis may beaveraged, as in a hand analysis, over half the column strip width.

If there are large voids in such slabs the modelling round these holes needs to be consideredcarefully.

Checks for punching shear close to supports or concentrated loads should be done by hand, not bytrying to refine the analytical model of the whole slab.

3.4.4 Suspended floor slab analysis - beam and slab

Simple rules for analysing ribbed slabs (one way spanning) and coffered slabs (two way spanning)

are described in BS8110 (and EC2) and these should generally be used where possible.

In-plane stresses are generally neglected in analysis. However prop forces from retaining walls onbasement slabs(section 3.4.8) may need to be considered.

If supports are not in a regular pattern then analysis under out of plane loading only can be carried

out by representing the slab as a grillage(section 11.6) of skeletal elements. 2D element analysis ofribbed or coffered slabs is not recommended.


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If there are large voids in such slabs the modelling round these holes needs to be consideredcarefully.

Checks for punching shear close to supports or concentrated loads should be done by hand, not bytrying to refine the analytical model of the whole slab.

3.4.5 Shear walls in large models

Where a shear wall is to be included in the model of a large building(section 11.10) it will often besufficient to model it as a vertical 1D element. Rigid links, or rigid ended elements(section 11.14),can then be used to join the point where horizontal elements intersect the edge of the wall to the 1Delement representing the wall.

3.4.6 Shear wall analysis

If a shear wall has a complex shape, with voids in different places on different floors, a 2D elementplane stress analysis should be done to identify stress concentrations round the voids.

Out of plane bending of shear walls is normally neglected, because there is other structureperpendicular to the wall which will resist this bending. If this is not the case a general 2D elementmodel, which can carry both in plane and out of plane moments, can be used.

Setting up the 2D element meshneeds to be done carefully to ensure adequate results, particularly atcorners of holes (section 11.7). The mesh illustrated above has been refined in these areas to achievethis.

Interpretation of the results of this analysis into required areas of reinforcement is complicated and

the use of apost processoris recommended (section 11.2).

3.4.7 Basement floor slab analysis - flat slab

Both in plane forces from surrounding retaining walls and out of plane forces from vertical load, areoften significant in the design of basement floor slabs. Where the geometry of such slabs is complex,

a model using general 2D elements able to resist in plane and out of plane forces can be used. If

beam strips across the slab can be identified then a space frame skeletal model (a grillage able toaccept in plane and out of plane loading) can be easier to use.


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Setting up the 2D element meshneeds to be done carefully to ensure adequate results (section 11.7).Interpretation of resultsof this analysis into required areas of reinforcement is complicated, because

the principal stresses in the element will not normally line up with the reinforcement directions(section 11.2).

3.4.8 Basement Floor Slab analysis - beam and slab

Where a basement floor slab has ribs, beams, or coffers it can not be represented adequately by amesh of 2D elements alone. A grillage of beams gives a good model of out of plane effects and isnot very inaccurate in modelling the in plane strains in a regular slab. A mixture of beam and 2Delements gives the best model of the behaviour of this sort of floor but interpreting the results from

such a model is very difficult. More detailsof the options available are given in section 11.15.

3.4.9 Plate and shell structures

General 2D elements can be used to approximate curved surfaces from a series of flat elements.

These structures can then be loaded in any direction. The mesh needs to be fine to limit the anglebetween adjacent elements. Specialist advice on modelling curved shells should generally besought.

Setting up the 2D element meshneeds to be done carefully to ensure adequate results.(section 11.7)

For steel shells the resultant stresses can be used directly for design, provided the effects of bucklingand shear lag (reducing effective widths) are considered in the design.

For concrete shells, interpretation of the results of this analysis into required areas of reinforcementis complicated and the use of apost processoris recommended (section 11.2).

4 MODELLING SPECIFIC STRUCTURE TYPES

4.1 Multistorey buildings

For many structures, where lateral stability is provided by stiff cores, simple sub frame models eachcontaining a single beam and the columns above and below it, will be adequate for the analysis ofconcrete structures and simple hand analysis for steel structures. With very unequal spans thedifference in axial loads in adjacent columns can make a subframe analysis inappropriate (because

the heavily loaded column is compressed more). Columns need to be checked at near the top of thebuilding where bending dominates and axial loads are small. The cores can then be modelled asplane frames resisting the full lateral load in two orthogonal directions.

Using a three dimensional model of the full building in such cases is not recommended. It would notonly be far more complicated to set up and check but would also imply that the whole structure


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forms part of the lateral load resisting system. Changes to the secondary elements during designdevelopment or later refurbishment would require re-analysis.

For structures where a three dimensional analysisis needed, various techniques for generating both

simple and complex models have been developed (section 11.10).

4.2 Roof structures

If a roof structure consists of identical primary frames in one direction, supporting purlins spanningbetween them, there is no advantage in modelling the whole structure. A plane frame model of theprimary frame will be adequate. Sometimes however frames have different stiffnesses or pick up

different loads and support a continuous secondary structure, stiff enough to transfer loads betweenthe frames. Determining the distribution of loads between these frames to apply to a series of planeframes, ensuring compatibility of deflections, would then be complex and iterative. A space framemodel which does this automatically is appropriate.

4.3 Symmetrical structures

If a structure is symmetrical, then only a half or a quarter of the structure needs to be modelled.

However the additional complexity introduced by having to set up different models forantisymmetric and symmetric loads means that this should only be considered if the full model is too

big for the computer to analyse. Detailsof the restraint conditions needed on the axes of symmetryand splitting loads into symmetric and anti-symmetric components are given in section 11.16.

4.4 Tunnels and culverts

Structures with constant cross section over a long distance, without joints and loaded uniformly and

in the plane of the cross section only, can be analysed using 2 dimensional elements in a plane strain

analysis (not to be confused with aplane stress analysissection 3.4.3). However effects close tojoints, local loads and hard points tend to govern such designs, and plane strain analysis is notappropriate in these areas. Soil structure interaction is also likely to dominate design of tunnels andspecialist advice should be obtained on how to model this.


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4.5 Cylindrical structures

Cylindrical tanks and other structures where the loading and structure are symmetrical about an axisof rotation, can be analysed using two dimensional elements in an axisymmetric analysis. Anydeparture from symmetry (a wall attached to the side of the tank for example) will invalidate theanalysis.

4.6 Bridge decks

Bridge decks, unlike buildings, tend to be long span isolated structures. The way these decks aremodelled will depend on the form of the deck. Guidance on the modelling ofbridge decks(section

11.11) is given in Bridge Deck Behaviour by E.C, Hambly (Spon, 1991)

5 MEMBER SECTION PROPERTIES

General

This section describes the derivation of properties for given shapes of member, assuming they are

formed from uniform linear elastic materials. If the material being used is a composite (eg reinforcedconcrete) thesepropertieswill need to be modified (section 6).

Most computer programs will generate the properties required by the analysis from the dimensionsof the cross section for standard sections and this facility should be used where possible. Certain

parameters for unsymmetrical standard sections cannot be generated automatically because theydepend on the end connections. Note that modelling an angle or channel section taking any bending,

torsion, or shear is very complicated because the shear centre is outside the section. Properties forirregular sections may need to be derived by the user.


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Beams and columns have a cross section which can be defined geometrically and this section hasproperties which can be derived from those dimensions. Which properties need to be defined

depends on the type of analysis. The full range of properties (as needed for a 3D space framestructure) are detailed below.

The properties are defined relative to a local axis systemfor the element (section 11.8). Details ofthis system will be given in the computer program manual, generally the local x axis is along the

length of the element and the y and z axes should be the principal axes of the cross section.

Slabs and walls modelled by 2D elements have a thickness only and no further geometric propertiesneed to be defined.

Slabs modelled by 1D elements in a grillageare covered in section 11.6

In a linear elastic analysis the reduction of stiffness in compression of slender members due tobuckling is ignored. It is possible to do an iterative analysis of a structure incorporating tie bars(which can be assumed to take only tension and buckle under compression) by reducing the stiffnessof any tie elements which are found to be in compression, but a non linear analysis may be more

appropriate. Specialist advice should always be sought before carrying out such an analysis.

Cross Sectional Area (A) and Shear factors (Ky, Kz)

The area used for axial stiffness and stress calculations will generally be the cross sectional area (

the concrete section ignoring reinforcement in BS8110). The net area of members perforated by aseries of holes should be used.

The area used for shear stiffness calculations will generally be less than the cross sectional area. It isusually defined by a shear factor, which is the proportion of the total area which is assumed to carryshear (e.g. the web of an I beam). Shear factors for various shapes are derived in Structures Note1992NST_21.

Flexural Second Moment of Area (Iyy, Izz)

The I value used for bending stiffness can be calculated from first principles(section 11.12), and istabulated for most rolled sections in published data sheets, and for simple shapes in publications likethe Steel Designers Handbook.

Note that most sections have significantly different I values in the two orthogonal directions and sothe orientation should always be checked visually in the model.

Torsional Second Moment of Area (J)

Calculation of the torsion constant (J) used for torsional stiffness is an area of modelling where therules to be applied are complex and the appropriate value to use depends on end conditions as wellas the cross section. Methods of deriving Jfor commonly occurring shapes are given in section
http://69.69.9.1/Structures/1992NST/1992NST_21/1992nst_21.cfmhttp://69.69.9.1/Structures/1992NST/1992NST_21/1992nst_21.cfmhttp://69.69.9.1/Structures/1992NST/1992NST_21/1992nst_21.cfm


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11.12. Note that, except for circular cross sections, the torsion constant is not the polar moment ofinertia (which can be calculated from first principles). For a flat slab modelled as a grillage(section

11.6) the torsional stiffness GJ should equal the bending stiffness EI.

In some structures all torsion constants could be set to zero without significantly affecting theresults. This cannot be done in any structure that relies on torsion to carry the load, such as aneccentrically loaded beam. Torsional stiffness must not be neglected in torsionally stiff structuressuch as box sections. The effect of warping, particularly on open ended box girders, cansignificantly reduce the apparent torsional stiffness of such members.

Remember that if torsional stiffness has been included in a model then the torsions derived from theanalysis must be considered in the member design. Simple rules for combining torsions and bendingmoments in a grillage are given in Structures Note 1991NST_16.

6 MATERIAL PROPERTIES

6.1 General

The material properties of a uniform linear elastic material can all be derived from the elastic

modulus, E, (also known as Young's Modulus) and Poisson's Ratio,. Values for these twoparameters are tabulatedalong with the density and coefficient of thermal expansion, for commonconstruction materials in section 11.18

Because real materials are not uniform, or elastic, some modifications to material or sectionproperties need to be made, as described below.

6.2 Reinforced Concrete

Reinforced concrete is a composite, made up of two very different materials. Reinforcing steel can

be assumed to be a linear elastic material for the purposes of analysis but the concrete matrix has avery different stiffness, is subject to shrinkage and creep and cracks under tension. Simple rules

given in design codes allow the use of the concrete section (ignoring reinforcement), the grosssection (including all the concrete and the transformed reinforcement) or the transformed section

(including concrete in the compression zone only plus the transformed reinforcement). Differencesbetween these values can be significant and the designer needs to consider which is appropriate.

Properties for ultimate strength analysis

Most analysis of concrete structures is concerned with deriving forces for ultimate strength designand so is not usually sensitive to the absolute stiffness value assumed. Only if the model contains amixture of steel and concrete members, or members made from concretes with significantly differentstiffnesses (>15%), or the axial stiffness of columns is important, for example in tall buildings, willthe absolute stiffness of the concrete be important.

If absolute stiffness is not critical, the non linear and composite nature of concrete can be ignoredand the standard material properties(section 11.18) can be used with the concrete sectiondimensions. Special rules apply to a flat slab modelled as a grillage(section 11.6). Halving the

torsion constant of any solid concrete member (where this is included in the strength analysissection5.4) to allow for cracking of the section is a reasonable simplification.

Properties for serviceability analysis

In standard construction, deflection under service loads is generally covered by limiting span/depthratios, and cracking is covered by detailing rules. In this case a serviceability analysis is notnormally required.


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If a serviceability analysis is required then more care needs to be taken with the definition ofstiffness. The results obtained from this analysis are unlikely to be accurate and the possibility that

deflections will be significantly larger or smaller than the calculated value must always beremembered. The following matters need to be considered in these exceptional cases to determine

the most appropriate values for material and geometrical properties to use:

Creep

If a concrete member is subject to sustained loading it is observed that the deflection increases overtime. The amount of this increase, known as creep, is sensitive to the time after casting at which theconcrete is loaded, the humidity of the atmosphere, concrete mix, member size and subsequent

loading history. The effects of these factors on the degree of creep is defined in the codes of practicefor concrete and is allowed for by modifying the short term elastic modulus by a creep factor. Notethat E is modified by 1/(creep factor + 1). If a serviceability analysis is being carried out it will notnormally be sufficient to accept default values for long term E.

Shrinkage

Unrestrained concrete shrinks with time. Shrinkage needs to be considered in calculating losses in

prestressed concrete. In very large slabs without movement joints and with stiff external restraintsthe tensile forces generated by restraint to shrinkage may need to be considered. In a frame analysis

program the effect of shrinkage is entered by specifiying an initial strain on the element or, if thisfeature is not available, by specifying a temperature change which will give the equivalent strain.

Reinforcement

Because reinforcing steel is much stiffer than the concrete matrix the uncracked composite materialwill be stiffer than an unreinforced concrete section of equivalent size. It is possible to allow for this

by calculating the geometrical properties of a transformed section, where the area of reinforcement ismultiplied by the modular ratio (Esteel/Econcrete-1). If this is done then the effects of cracking,which reduces stiffness, must also be considered, see below.

Note that axial load in columns increases the area in compression, and so the effect of cracking incolumns will generally be less than in beams.

Note that the value of Econcrete used to determine the transformed section properties will depend onwhether creep effects are being considered. The value used to calculate the section properties shouldalso be used when defining the material properties.

Cracking

Once part of a concrete member goes into tension, it is likely to crack and the stiffness will then bereduced. Because the amount of cracking depends on the loads applied, the stiffness of the member


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will vary with load. This cannot be allowed for in a linear elastic analysis and so, if the mostaccurate estimate of deflection possible is needed, an iterative analysis is required, reducing the

element stiffnesses where an initial analysis of uncracked sections shows that cracking would occur.Stiffness of cracked sections in this case needs to be determined from the moment-curvature

relationship for the section.

Note that it is the stiffness (EI) which needs to be adjusted, and in many cases (including all 2Delement analysis) it is easier to modify E to allow for these effects, instead of changing I (or thesection dimensions from which I is calculated).

Properties for dynamic analysis

Stiffness is important for dynamic analysis and so cracking and reinforcement may need to beconsidered, as for a deflection analysis. If loading is applied rapidly to concrete it is stiffer thanwhen load is applied gradually. The increase in stiffness is of the order of 10% for earthquake orextreme wind loading and so is generally neglected. Larger increases in stiffness are observed with

low amplitude dynamic loads, in vibration analysis of floor slabs for example. Further details aregiven in 'An Arup Introduction to Structural Dynamics' by Mike Willford.

6.3 Steel

Most structural steels can be assumed to behave in a linear, elastic manner up to yield and then todeform indefinitely at that level of stress, forming a plastic hinge. Analysis of models includingplastic hinges is non linear and outside the scope of this note. For further details on the use of plastictheory, seePlastic Design to BS5950by J.M. Davies and B.A. Brown, Blackwell, 1996.

6.4 Composite (steel beam and concrete slab)

Properties for analysis in sagging regions need to be calculated for a transformed section, allowingfor the different modulus of steel and concrete, usually by reducing the effective width of the slab inthe ratio Econcrete/Esteel. Note that in hogging regions the concrete slab will be cracked and will nottherefore contribute significantly to the stiffness of the composite beam. Rules for the proportion ofthe span to be taken as cracked are given in design codes. As with concrete structures the effects ofcreep on the stiffness of the concrete need to be considered.

6.5 Timber

Timber, as a natural material, has a wider variation in stiffness than manufactured materials. Mean

and minimum values of E are given in the codes. Generally the minimum stiffness value should beused in a frame analysis and the mean stiffness should only be used where significant load sharingbetween members is possible (e.g. floor joists). The effect of timber being stiffer than assumed mayneed to be considered.

6.6 Other materials

Typical properties for the materials described below are tabulated(section 11.18) but the following

points need to be considered when modelling these materials:

Wrought Iron

Wrought iron behaves in a similar manner to steel. Further details are given in The Appraisal ofExisting Iron and Steel Structures. (SCI publication 138)


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Cast Iron

Cast iron is not a linear elastic material. Stiffness reduces as load increases and the tension stiffness

is less than the compression stiffness. Brittle failure under tension occurs at comparatively low

stresses. Providing these features are born in mind a linear elastic analysis can give reasonableresults to use in strength design. For deflection calculation a range of stiffnesses would need to beconsidered. . Further details are given in The Appraisal of Existing Iron and Steel Structures. (SCIpublication 138)

Masonry

Masonry is assumed to crack at the joints under tension and so the stiffness of a masonry elementwill reduce once the extreme fibre goes into tension.

Aluminium

Aluminium can be treated as a linear elastic material for analysis. Designers should be aware thatthere will be significant reduction in strength of the material at any welded connection and this lowerstrength will need to be used in design.

7 MODELLING LOADS

7.1 General

It is usually convenient to split the load into a number of loadcases which can then be combined withappropriate load factors.

7.2 Point of application of load

If only part of a structure is modelled, it may not be possible to apply loads to the model at the same

position as they are applied to the structure. Care is needed to ensure that the eccentricity of load isconsidered in this case, particularly for cantilevers.

Sometimes local effects of loading are calculated separately (e.g. wheel load effects on a bridgedeck) and only the global effects are to be determined from the model. In this case it is importantthat loads are applied to the model on the primary structure, with any moments resulting from theeccentricity of loading. Otherwise the results from the global analysis will contain a local componentwhich will therefore be included twice in the total analysis.

7.3 Self weight

It is possible in many programs to have the self weight calculated automatically from the areas of theelements and the density of the materials. A number of issues need to be handled carefully if thisapproach is adopted:

When modelling a slab using a grillage, this approach will apply twice the total weight ofthe slab, unless special precautions are taken (e.g. using two different materials, one with adensity of zero, for longitudinal and transverse elements).

Additional load from stiffeners, connections etc will not be included. Dummy elements with arbitrary large cross section will generate large loads.

Mixing gravity loads and applied loads in a single loadcase is not recommended, because checking

of loading will be complicated and, especially if any of the problems noted above are present, theuse of applied loads to represent the self weight is generally preferable.


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7.4 Other permanent loads

Generally other permanent loads will be present in addition to the self weight of the members and

these loads have to be applied explicitly to the model. It is generally best to apply the load to the

model in a similar manner to the way in which loads are applied to the part of the structure whichhas been modelled (e.g. on a plane frame: apply purlin loads as point loads, slab loads as distributedloads).

7.5 Transient loads

The same principle as for permanent loads should be used. Where it is possible for only part of aload to be applied to the actual structure, (e.g. pattern loading), it is generally preferable to modeleach patch as a separate loadcase and use combinations to create the patterns.

7.6 Internal loads

In addition to applied loads, other effects can change the distribution of forces within the structure,without applying a net force. Many of these effects can be modelled directly but extra care needs tobe taken because the behaviour of the structure under these effects is less easy to check by hand.Checking the direction of the applied effect can often only be done by looking at the results. It is

recommended that these effects are always modelled in separate loadcases from applied loads andfrom each other, to make checking as simple as possible.

Temperature

Unless a structure is free to expand and contract, the effects of a change of temperature need to beconsidered. Even if there is no external restraint to expansion, differential temperatures andtemperature gradients across a member can introduce stresses. Care needs to be taken when using a

uniform temperature change because ina number of circumstances spurious locked in stressescan be generated which can be an order of magnitude higher than the genuine stresses

caused by a restraint to thermal expansion. To avoid this, temperature change must be

applied to all the elements which span directly or partially between restraints (only truly

transverse elements directly orthogonal to the line joining restraints can be omitted).

However if any rigid constraints (includingelements modelled with rigid ends)are in parallel

with beams subject to temperature change (eg a grillage with rigid elements over the column

width) they cannot expand and so will lock in stresses (temperature movements/stresses will

also be underestimated if part of the element is rigid). Also if steel and concrete elements

are in parallel (eg concrete slab members in parallel with primary steel beams) then if they

have different coefficients of expansion, this will lock in stresses (which may or may not be

considered spurious).

Lack of fit

If components are not exactly the right length then forces can be generated in those components and

adjacent ones. This cannot usually be quantified at the design stage, and is deemed to be covered bypartial factors, but may need to be considered if there is a problem on site.

Prestress

Prestressing will shorten the members to which it is applied, with similar effects to lack of fit and if

the prestress is eccentric it will also introduce bending to the members. Unless the prestressedmember is simply supported these distortions will introduce secondary stresses into the structure,

which need to be considered, usually by applying equivalent loadsto the model (section 11.17)


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Support settlement

Support settlementis another special case of lack of fit (section 8.4).

8 MODELLING RESTRAINTS

General

All structures are restrained to prevent global movement. If the whole structure is modelled then themodel will normally be restrained where it touches the ground and the only decisions needed are onthe rigidity of the ground. If only part of the structure is modelled however (e.g. a two dimensionalframe which is stabilised by other structure out of plane), additional restraints(section 8.3) to themodel are needed.

Restraint at ground

It is normal practice to assume initially that the ground is rigid and to allow for the effects of support

settlementseparately (section 8.4). Modelling the joint between the structure and the ground has thesame problems as for otherjoints(section 3.3), but BS5950 states that if full fixity to the support isassumed, then a rotational spring to model flexibility of the ground should be used. A simplerapproach is to analyse the model twice, once assuming pinned supports and once assuming full fixityand to use the moment from the fixed analysis for the design of the base connection only.

Full fixity should only be assumed if both the connection and the foundation can resist the momentsgenerated at the support without significant rotation (less than 10% of the rotation of a pinned

member at that location).

The use of rigid supports can produce very different reactions in adjacent supports which would inpractice be evened out by very slight movements. If examination of the reactions shows this effect, areanalysis using stiff springs (deflecting say 1mm under maximum reaction) is recommended. The

sensitivity of the analysis to variations in this stiffness should be investigated.

Raft Foundations

Where a large building is supported on a ground slab acting as a raft, the stiffness of the soil, thestiffness of the raft and the stiffness of the structure above interact. Soil is not an elastic material andso using springs to represent it can be misleading. An iterative use of a soil displacement programand a structural analysis program where the spring stiffnesses are adjusted to match the predictedground movements is recommended. A check should be made to ensure that any springs which end

up in tension are deleted.

Global restraints

In a general 3 dimensional structure any point has six degrees of freedom, i.e. it can move in 3directions and rotate about 3 axes.To simplify models it is possible to specify that every point is restrained in some of these directions.Such global restraints are selected automatically for some structure types (plane frames and

grillages) where the plane of the structure is defined, but have to be specified by the user for othertypes (plane stress etc) where the structure can be in any plane selected by the user.

Support settlement

Differential settlement between supports can significantly affect the forces in a structure. When thisis the result of different foundation conditions in different places it is not directly related to themagnitude of load and so cannot be modelled by a spring. Generally a maximum magnitude ofdifferential settlement which might happen is defined and this has to be applied to each support inturn, using separate loadcases.


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9 MODELLING FOR DYNAMIC ANALYSIS

The mass of the structure and its contents is fundamental to its dynamic behaviour. Traditionally this

has been modelled by specifying 'lumped masses' at nodes and assuming the model has zero density.It is now possible in some analysis programs to use the areas of the elements to determine the selfmass, (which has the same disadvantages as using gravity to model self weightsection 7.3) and/or tospecify that a particular loadcase should be converted to represent mass.

Because of uncertainties in material and element properties, there is not usually an advantage inusing distributed instead of lumped mass in the analysis and whichever of the options given above is

easiest to input can be used.

The stiffness(section 6.2) of the structure is also fundamental to its dynamic behaviour. Nonstructural members like partition walls can significantly affect the stiffness of a structure under lowamplitude dynamic loading.

Further details are given in 'An Arup Introduction to Structural Dynamics' by Mike Willford.

10 INTERPRETATION OF RESULTS

10.1 Beams and columns

Results from the analysis of the model of a skeletal structure (moment, shear and axial load) canoften be used directly in member design. Exceptions include members where deflectionaffects theforces (section 11.3).

Connection design needs to be considered carefully. In steel structures the capacity of the

connection can often govern the design. To provide a connection capable of transferring the fullmoment capacity of the member might be impossible without haunching beams or stiffening tubeslocal to the connection. Except in the special case of portal frames designed plastically (where it isimportant that plastic hinges are formed away from the connection) the saving in weight achieved byminimising the member size is likely to be outweighed by the extra cost of complex joints.

Stresses derived automatically for beams and columns can give a quick indication of whether

members are grossly oversized or undersized but effects such as buckling mean that they areunlikely to be used directly for final design.

10.2 Slabs and walls

Results from analysis of a slab or wall, either as a grillageof 1D elements (section 11.6) or a mesh of2D elements, will generally need to be processed to derive values to use in design.

Stresses in 2D elements which are modelling a uniform material like steel, can be used directly,provided buckling is not an issue.

Stresses and derived forces in 2D elements modelling reinforced concrete need extensivepostprocessingto derive design forces.(section 11.2)

11 ADDITIONAL INFORMATION

These topics are referred to in the main manual


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11.1 Warnings and errors during analysis

The three most common problems associated with a stiffness analysis are singularity, instability and

ill-conditioning. The first two problems are easily spotted and can be easily rectified; the third is

more difficult to spot, may sometimes be missed completely and is more difficult to cure. Wherethey occur, the model idealisation must be checked and the input data amended accordingly.

Singularity

Singularity occurs when the assembled structural stiffness matrix includes a diagonal term equal tozero. No solution can be obtained from such a matrix. This is normally due to lack of stiffness at ajoint in a given direction or rotation.

In practical terms, this generally does not imply that the structure is a mechanism but it does meanthat some extra restraints must be applied so that the structure may be solved.

Where torsion is likely to occur in the structure, a torsion constant of some value (even if it is verysmall) should be entered to prevent singularity.

Global Instability

Global instability differs from singularity in that all the nodes in the structure have a non-zero

diagonal stiffness term but the structure is not restrained in one or more of its global degrees offreedom.

Ill-conditioning

Ill-conditioning is a numerical problem that arises during the solution of the stiffness equations, as a

result of very large ratios between terms of the stiffness matrix.The problem is shown up in practice by the total applied loads at a node (including the fixed-endcomponents from elements loads) not being equal to the sum of the element end forces of elementsincident at that node. This is immediately apparent at the support nodes. Here the sum of thereactions in the global degrees of freedom not being equal to the sum of the applied loads in thecorresponding direction is a sure indication of ill-conditioning. For non-support nodes, the problemis much more difficult to spot, unless the analysis program itself performs the "out-of-balance" checkand warns the user appropriately.

Large differences in the stiffnesses of elements meeting at a node are generally the cause of ill-

conditioned structures; halving the length of an element by inserting an extra node may provide asolution.

Another general rule is to avoid using relatively very large or small element areas and inertias, and toeither fix or release the nodes attached to the elements instead. Zero areas and inertias will not causeill-conditioning but may cause singularity.Referred to from section 2.5 Computer Analysis

11.2 Concrete Post processing of 2D element results

Because concrete is usually designed as a cracked reinforced section, the stresses calculated from anelastic analysis of 2D elements cannot be used directly for design. Principal stresses are unlikely tobe parallel to the direction of reinforcement.

A method of converting in plane stresses into areas of reinforcement is described in Structures Note1989NST_7and use of a post processor for analysis data which automates this process and can allow

for bending stresses as well is recommended.


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Use of out of plane shear stresses to derive shear reinforcement is not recommended. These stresses

can be used qualitatively to identify concentrations of shear force but punching shear checks shouldbe done by hand at successive shear perimeters as described in the standards.

Referred to from sections 3.4 Slabsand 10 - Results.

11.3 P-delta effects

Normal static analysis calculations assume that displacements do not effect the forces in a structure.If a member carries predominately axial loads then deflection of this member will introduce secondorder, or P-delta, effects. For example, in a sway structure, the vertical loads generate moments.Design codes generally give rules for the design of elements to take account of these effects, such asby applying notional horizontal loads. In special cases, or where stipulated by the design codes, afull P-delta analysis to calculate these second order effects directly is required.

There are no general rules for when a P-delta analysis should be carried out. A P-delta analysisshould be considered if:

deflections are large resulting in P-delta moments greater than 10% of the static moment a structure can sway, and no other methods of allowing for sway have been considered compressive loads are high, so that buckling involving more than one member might be a

problem

If in this last case a P-delta analysis shows significant differences from a static analysis, a bucklinganalysis should also be undertaken to identify the critical buckling load directly.

In a P-delta analysis there is a geometric, or differential, stiffness in addition to the normal structuralstiffness. The geometric stiffness is derived from the forces in the structure, so the solution requirestwo passes. The first pass establishes the forces in the structure allowing the geometric stiffness tobe established for the second pass. In a linear static analysis, provided the model is properlyrestrained, the structure should always be stable, so a solution is always possible. In a P-delta

analysis this is not necessarily the case. If the axial force in an element is too high the elements maybe unstable so that a solution cannot be found. Note too that since this is no longer a linear analysis,

results from different load cases cannot be superposed.


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P-delta analyses can be used to avoid using effective lengths derived from code rules, which do notapply directly to complex structural arrangements. Provided initial imperfections are included in the

model, stresses from the analysis can be used to check an element directly; taking the effectivelength as equal to the element length. If a buckling analysis can be undertaken, the initial

imperfections can be determined from the buckled shapes. Normally the deflection of the mode withthe lowest critical load is scaled to an appropriate amplitude. This may be derived from the initialimperfection ratios used for appropriate elements in the code (see 1996NST_12for advice to Britishand European codes) or be based on construction tolerances. Note that the initial imperfectionsmake allowance for internal stresses as well as lack of straightness.

Referred to from section 2.5 Computer Analysis,3.2 Beams and Columns,and 10 - Results

11.4 Modelling castellated beams, trusses etc by shear beams

The shear stiffness of a castellated beam, or of a truss modelled by a single beam element, issignificantly less than for the equivalent solid beam and so shear deformations always need to be

allowed for, even with slender beams.

Suppliers of castellated and perforated beams supply proprietary software or design charts to use fortheir beams. The user needs to be satisfied before using such products that they are suitable for the

particular application.

An accurate estimate of shear stiffnesses can be calculated by modelling top and bottom flanges andweb members as beam elements in a plane frame. For scheme design the following rules should givean adequate approximation.

In these formulae E is the Youngs Modulus and G is the shear modulus, and for an isotropic

material with Poissons ratio:

Beam Type Bending stiffness Shear area

Castellated BeamsIt=I for T section at holes = pitch of holes

(Isolid + I at hole)/22

192/

s

ItGE

Vierendeel girders where elementsare slender (element span/depth >10)A = pitch of bracing membersb = depth between flangesIf = I for flange memberIb = I for bracing member

(I for top and bottomflanges)

Ib

ba

If

a

GE

+

1224

/2

Vierendeel girders where elementsare stocky (element span/depth d db 312

1 db

!!!

"

#

$$$

%

&

'''

(

)

***

+

,

121

16

36.3

31

4

4

3 bd

b

ddb

or 33db for thin rectangles

Solid Circle,diameter d 4

2d 64

4d 32

4d

Referred to in section 11.12 Section properties

11.10 Multi storey frame models

A number of tools have been developed to assist in the analysis of multi storey frame models, eitherby allowing simple chassis models to be expanded to model a complete building, or to developequivalent simple models during scheming or verification. Papers on these tools are held in R&Dwho can also suggest contact names for further information.

Referred to from section 3.4 Slabs and Wallsand 4.1 Multistorey Buildings

11.11 Modelling bridge decks

Bridge decks generally have clearly identifiable primary members spanning in one direction,supporting secondary members or a deck spanning in the other direction. Because bridges supportmoving concentrated loads, many loadcases are needed and so it is often beneficial to use

comparatively crude skeletal models to model the slab decks. Influence lines can be used to identifyworst locations for the application of load. If a distortion to an element in a skeletal model is

applied, the deflected form of the model illustrates the influence line (or influence surface in 2D).

Local effects of wheel loads on slabs are often calculated from influence charts rather than computeranalysis. In this case loads on an analysis model should be applied to the primary structure to keepglobal and local effects separate.

Details of methods of modelling different deck types are given in 'Bridge Deck Behaviour' by E.C.

Hambly (Spon, 1991) and this book should be consulted for details.Referred to from section 4.6 Bridge Decks


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11.12 Section properties from first principles

Second moment of area (Iyy, Izz)

Complex cross sections can be divided up in to simple shapes. For each shape find the position of itsown centroid, y, the area, A, and I for the shape about its own centroid(section 11.9).

The position of the centroid of the wholesection is then: -

- =

A

yAy

The total second moment of the

section is then:( )2- - += yyAIItot

Torsion Constant, (J)

There is no equivalent set of rules for deriving torsion constants from first principles by addingcomponents. Two examples illustrate this:

A tube with a longitudinal slot has a fraction of the torsion stiffness of an identical closedtube.

Open sections can be considered as a set of simpler shapes but the way a section is split intocomponent parts can have a marked affect.

The following questions need to be considered:

Is it a thin walled closed section? If so use the formula:

( )-

=tds

AJ

24

Is it a thin walled open section? If so treat it as a series of rectangles and use the formula:

( )- = 33/1 tdsJ For thick walled sections more detailed guidance is given in Roark. Open sections should be split

into rectangles, starting with the rectangle with the largest d * t ^3 that can be fitted within thecomplete section.For thick walled closed sections, where the hole dimensions are less than half the dimensions of thesolid rectangle containing it, the hole can be ignored and the section treated as solid.

Referred to from section 5 Section Properties

11.13 Modelling haunched beams

The variation in stiffness along the length of a haunched or tapered beam can have a substantialeffect on the distribution of forces and magnitude of deflections. It should be borne in mind thatminor differences in stiffness due to haunching can have less effect than some of the otherassumptions made during modelling (e.g. using gross uncracked section properties for concrete).Variations in depth of +/- 25% can generally be ignored. With greater variation in depth a haunchedbeam can be modelled by three elements, one representing each haunch, with I taken as the averageof the support and span values (not the properties of an average depth section) and the middle one

with the constant midspan section properties.Referred to from section 3.2 Beams and Columns


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11.14 Modelling elements with rigid ends

Where members of substantially different stiffness are connected, for example where acoupling beam connects parts of a shear wall, an element connecting the centrelines of the

shear walls will be too flexible. Assuming full fixity at the face of the shear wall would

overestimate the stiffness, and so it is generally assumed that full fixity occurs at h/2 from

the face, where h is the depth of the flexible member.

Such a member can be modelled as a single element by increasing its stiffness. Determine

the fraction N of the length which is flexible, then increase the stiffness by 31

Alternatively the element can be modelled with rigid ends if the program allows this.

Modelling the rigid sections by stiff elements is not recommended.Referred to from section 3.3 Joints,and 3.4 Slabs and Walls

11.15 Modelling beam and slab floors with in plane stresses

The 'best' analytical model of the behaviour of a beam and slab floor under combined bending and inplane stress is a model where in plane forces are carried by plane stress elements and out of plane

forces are carried by a grillage of elements with no in plane stiffness and reduced axial stiffness(modelling the part of the beam outside the slab depth only). However a model in this form producesanswers which are very difficult to convert into reinforcement areas. For straightforward shapes asimple grillage(section 11.6), gives in plane deflections which only differ from the 'correct' answersby about 20%. In plane bending moments in the grillage elements generally cancel at nodes and canbe ignored but axial forces need to be considered.

If the slab is a complex shape or has many large holes in it then a plane stress analysis, neglecting inplane stiffness of the beams, (or the combined plane stress and grillage model described above)should be used to establish the behaviour of the slab under in plane loading. Unless this modelidentifies high in plane compressive stresses (greater than 0.2 fcu), or the depth of concrete incompression needed to resist bending in the beam elements exceeds 0.2d, it will be conservative toadd the area of tensile reinforcement calculated from post processing the in plane stresses to the areaof tensile reinforcement needed to resist bending in the beams. If these limits on compression are notsatisfied then specialist help to determine reinforcement areas should be sought.

Referred to from section 3.4.8 Basement Floor Slabs


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11.16 Symmetrical and anti-symmetrical models

Where only part of a symmetrical model can be included in an analysis because of limits on problemsize which can be analysed by a particular program, only a half or a quarter of the structure needs to

be modelled. Two different models, with different support conditions on the axes of symmetry needto be analysed, one used for symmetric loading and one for antisymmetric loading.

Any load to be applied to the model can be split into symmetric and antisymmetric components.Support conditions for axes of symmetry in a plane frame are illustrated below:

Symmetric loading

Anti-symmetric loading

Referred to from section 4.3 Symmetrical Structures


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11.17 Equivalent loads for prestressing

Prestressing a concrete beam imposes a set of loads on the beam which are in equilibrium. If thebeam is simply supported it will distort under these forces without restraint but this distortion isprevented in continuous beams and secondary forces are generated. To determine the magnitude of

these forces, a set of equivalent loads can be applied to the beam to model the effect of prestress. Forthe simplest case of a straight tendon in a constant section beam these forces will be applied at the

ends of the tendon only. If the tendon follows a parabolic curve then in addition to the end forces avertical uniformly distributed load is applied to the beam. If the position of the centroid of the beam

section varies then additional forces are applied. The forces to be applied are summarised in thefigure.

Referred to from section 7.6 Internal Loads

11.18 Properties of standard materials

Basic ranges of properties for commonly used structural materials are given in the table below.These are based on data taken from relevant UK design standards and reference should be made to

the relevant standards for prescribed values to be used when stiffness is critical.

Material Young's Modulus Poisson's Ratio DensityCoefficient ofthermal expansion

kN/mm kg/m strain / degree C

Steel 200-210 0.3 7850 12e-6Concrete (short

term)20 + 0.2 fcu 0.2 2400 10e-6

Timber4 - 20 //0.2-1.3 # grain

N/A (4 - 7) 290 - 1080

Wrought Iron 150-220 0.25 7850 12e-6Cast Iron 60-100 0.25 7850 12e-6

Aluminium 70 0.3 2710 23e-6

Notes


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Concrete properties are modified by creep under sustained loading, reducing the Young's modulusby a factor of 2 to 3. Further details on concrete properties are given in BS8110 Part 2. Properties

for concrete in other countries (eg Hong Kong) are different.Timber properties are defined in EN 338 1995 Table 1.

Referred to from section 6 Material Properties

11.19 Holes in concrete beams (previously SGN 4.6)

INTRODUCTION

This Note describes a simple method for local analysis and detailed design around holes within the

webs of reinforced concrete beams and ribs. It should not be applied to holes in prestressed or deepr.c. beams. (The latter are treated in the CIRIA design guide1 on deep beams.)

No attempt is made to distinguish between holes that are structurally insignificant, and those thatrequire design consideration. The distinction is influenced by a variety of factors including the size

and location of the hole relative to both the depth and the span of the beam. The designer's ownexperience is a more reliable guide than arbitrary limits on size, although it can be said that any hole

of length greater than nominal link spacing or depth greater than one-quarter the beam depth willcertainly need to be investigated. If doubt exists with smaller holes, they should be checked.

BASIS

The presence of a rectangular hole is assumed to change the behaviour of the beam or rib such that itacts locally as a Vierendeel frame. (Obviously if there are multiple holes at close spacing or if thehole dimensions are large in relation to the beam depth or span, the behaviour of the beam will bealtered more radically, and a rigorous overall analysis may then be needed before local effects are

considered.)

Holes of non-rectangular profile may be very conservatively simulated as a rectangular holeenclosing the extreme limits of the actual profile. But where the hole is of a shape that allows thebeam to be simulated locally as a single or double lattice truss, having top and bottom booms and

'diagonals' which can be reinforced to carry loads across the hole, then it is both more realisticmodelling and almost certainly more economical on reinforcement to adopt this truss analogy. Such

an approach can be used when the hole is triangular, circular, or of a shape that can be inscribed


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within an equivalent circle, but it should be limited to cases where the truss members framing thehole can be sized to have coincident centre-lines at 'joints' (thereby avoiding the need for a detailed

analysis of local bending and force transfer at the joint). It must also be possible for diagonal barstrimming the hole to be bent alongside and lapped with the top and bottom reinforcement.

In practice these considerations will restrict the use of the truss analogy method to circular holes

whose diameter is at most no more than one-third, and triangular holes whose height or length is nomore than two-thirds, of the overall beam depth. Even these limits may be too generous for holes inshallow beams or those near the top or bottom of the beam.

The term 'truss' is used to describe the analogy for modelling force transfer across the hole and, inpractice, the analysis may well - for simplicity of working - assume that members are skeletal andpin-jointed. While this will give a 'safe' estimate of the axial forces to be designed for, it must be

remembered that they are, in reality, squat members with monolithic joints, and detailing must takeaccount of this.

Forces are assumed to be transmitted across the hole as follows:

Overall bending of the beam - by equal compression and tension forces having a lever arm which isthe distance between the centroid of the concrete section in compression and the main tensile steelcentroid


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Vertical shear - rectangular hole, Vierendeel frame analogy: by shear and local bending across thehole. The distribution of shear between the concrete boom sections is dependent on the relative

stiffnesses of the sections and on the effective span of these between notional points of fixity, whichshould be at least one-half the overall depth of the boom beyond the extreme end of the hole.

An applied vertical shear acting on a beam is carried across a rectangular hole by local shearing andbending of the boom sections. The distribution of shear between the booms can be determined fromthe assumption that the deflections of the booms across a hole are equal.

An initial distribution can thus be made in proportion to the relative I values of the cross concretesections (assumed uncracked): this will give a safe estimate of shear in the tension boom - which

will probably be cracked - but may underestimate that in the largely or wholly uncrackedcompression boom. This can, conservatively, be designed to carry 100% of the shear; if that leads tounacceptably high stresses, the boom I values should be re-calculated allowing for long-term tensionstiffening in concrete (as recommended in CP110) and an estimated percentage of tension steel.

The design shear on the tension boom should, however, not be reduced from its initially estimatedvalue; the total shear designed for in the two booms will thus exceed the actual value, but this is a


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conservative approach to a difficult analysis which - if it underestimated shears drastically - couldhave equally drastic consequences.

Vertical shear - triangular or circular hole, truss analogy: by components of compression and

tension in the actual or notional diagonal sections adjacent to the hole and by shear across the jointswhere these intersect. The boom section spanning between the further ends of the diagonals may beassumed to carry only a nominal shear;

Local bending (due to loading applied to a boom section above or below the hole) - by bending ofthe section assuming beam action between fixity points (Vierendeel frame) or joints (truss analogy)

adopted when assessing vertical shear transmission. Note that such loads can result in local tensionsat boom ends which require extra tension steel to prevent tearing-out;

Axial force (if present) - by axial forces across the reduced concrete boom sections. The appliedaxial force can usually be taken to act at the centroid of the gross concrete beam cross-section and

can be shared between the top and bottom boom sections by simple statics (analogous to the sharingof load effects from a concentrated load on a simply supported beam, with the 'support' locationstaken as the centroids of the gross concrete boom sections.


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Note, however, that the local behaviour of a beam subject to a large axial force is similar to that of a

deep beam on its side and a detailed investigation of the stresses and strains - e.g. by finite elementanalysis - may, in this case, be essential (but this is outside the scope of this Note);

Torsion (if present) - by shear forces of opposite sign acting as a couple having a lever arm which isthe distance between the centroids of the concrete boom sections, and by local lateral bending of

these sections. Points of fixity are assumed at a distance beyond the extreme end face of the holeequal to one-half of the respective section width: a point of contraflexure will occur at the midspanof the section.

Wi

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