13 Fitzroy Street London W1T 4BQ Telephone: +44 (0) Facsimile: +44 (0)

Transcription

1 Oasys AdSe Theory

2 13 Fitzroy Street London W1T 4BQ Telephone: +44 (0) Fasimile: +44 (0) Central Square Forth Street Newastle Upon Tyne NE1 3PL Telephone: +44 (0) Fasimile: +44 (0) oasys@arup.om Website: oasys-sotware.om

3 Oasys AdSe Oasys Ltd All rights reserved. No parts o this work may be reprodued in any orm or by any means - graphi, eletroni, or mehanial, inluding photoopying, reording, taping, or inormation storage and retrieval systems - without the written permission o the publisher. Produts that are reerred to in this doument may be either trademarks and/or registered trademarks o the respetive owners. The publisher and the author make no laim to these trademarks. While every preaution has been taken in the preparation o this doument, the publisher and the author assume no responsibility or errors or omissions, or or damages resulting rom the use o inormation ontained in this doument or rom the use o programs and soure ode that may aompany it. In no event shall the publisher and the author be liable or any loss o proit or any other ommerial damage aused or alleged to have been aused diretly or indiretly by this doument.

4 Oasys AdSe Contents Introdution 6 Sign Convention 6 Analysis options 8 Loading 8 Solution 9 Solver Solution Spae 10 Solver Searh Proess 10 SLS and ULS loads analysis 11 ULS strength analysis 1 Long, short and long & short term analysis 1 Strength redution or material partial ators or ULS 14 Tension in Conrete 14 EC tension stiening 15 Stiness 16 Craking 18 Craking Moment 18 Crak-width 18 Conrete material models Conrete material models or dierent odes Parabola-retangle 3 EC Conined 5 Retangle 5 Bilinear 6 FIB 7 Popovis 7 Mander & Mander onined urve 9 BS8110- tension urve 30 TR59 31 Interpolated 3 Conrete properties 33 Notation 33 ACI 34 Oasys Ltd 016

5 Oasys AdSe AS 35 BS BS CSA A3.3 / CSA S6 38 EN HK CP 41 HK SDM 4 IRC IRS Bridge 45 IRC Rebar material models 48 Symbols 48 Rebar material models or dierent odes 48 Elasti-plasti 49 Elasti-hardening 49 BS Pre-stress 51 Progressive yield 51 Park 5 Linear 53 No-ompression 53 ASTM strand 53 Crak alulation 54 CSA S6 54 EN Appendix 58 Alternative stress bloks 58 Reerenes 60 Steel materials urve 61 Oasys Ltd 016

6 Oasys AdSe Introdution AdSe is a general setion analysis program, giving setion properties or irregular setions. While oussed on onrete setions, other materials an be speiied. It will: Calulate the ultimate resistane o irregular setions (reinored & pre-stressed) Provide N/M & M yy/m zz harts Servieability alulations inluding rak widths / raking moment Provide moment urvature and moment stiness harts The basi assumption in AdSe is that plane setions remain plane. Sign Convention In AdSe the setion is onsidered to be in the y-z plane as illustrated below z y x This means that moments are deined as ollows: -M MR M MYY MZZ Oasys Ltd 015 6

7 Oasys AdSe The ollowing table illustrate moment, ompression and urvature onventions M yy M zz Moment angle, Neutral Axis angle, NA yy zz + M ve 0 C NA + M + M ve + ve C NA C 0 + M ve NA NA - M +M ve + ve C - M (180 ) -180 (180 ) - ve 0 NA C - M - M ve - ve C NA 0 - M ve C NA + M - M ve - ve C NA Key: C Region o Compression NA Neutral Axis Oasys Ltd 015 7

8 Oasys AdSe M Any partiular moment Analysis options For the Ultimate Limit State, the options available inlude: Ultimate moment apaity o the setion. Stresses and strains rom the ultimate applied load and stresses at the ultimate limit state. Ultimate resistane N/M interation hart. Ultimate resistane M yy/m zz moment interation hart. For the Servieability Limit State, the program alulates: Craking moment. Stresses, stiness and rak widths or eah analysis ase. Moment-stiness and moment-urvature harts. The ollowing load types an be simulated: Pre-stress using unbonded tendons Shrinkage and temperature eets AdSe will ind the ultimate apaity o a simple or ompound setion. It will ind the state o stress & strain in the setion under a variety o loading onditions or servieability and ultimate material properties. Servieability analysis will generate a plot o neutral axis position and rak widths around the setion as well as ull numerial output The ultimate apaity harts have developed signiiantly. The user an speiy a table o additional points (N/M or Myy/Mzz), with labels, whih will be plotted onto the graph. Also the user an speiy a number o values o axial ore and Myy/Mzz plots will be drawn or eah value o axial ore on the same graph. User input strain planes an applied to see the impat on the ultimate apaity harts. Servieability harts will plot moment versus urvature, seant stiness or tangent stiness or a given value or range o axial ore and moment angle. Loading The reerene point or loading and strain planes is taken by deault as the Geometri Centroid, but this an be overridden by speiying a user speiied point. Oasys Ltd 015 8

9 Oasys AdSe Solution The basi idea behind AdSe is that the state o strain aross a setion varies linearly and an be deined by a strain plane. As the variation is linear the strain plane an be deine by a salar axial strain (applied at a speiied point) and a urvature vetor.,, x y z Then the axial ore and moments in the setion are then deined as N M M x y z A A A da z da y da This an be haraterized in the same way as the strain plane as N N, M, M x y z There are many dierent alulations in AdSe but they are all deined as solution to a set o equations whih are untions o the strain plane. When heking the strength o a setion given the axial ore and a moment vetor, the solution is to ind the maximum moment possible beore the setion ails. There are several riteria to be heked but the primary riteria is Where N x N x M max, app M, app u N x is the axial ore, M is the moment angle, max is the maximum strain in the setion, u is the ultimate (ailure) strain o the material and the subsript app is or applied. The basi solution proedure is selet an initial trial strain and alulate the target values. N x M max 1 3 The solution is then iterative until a onverged solution or the riteria above is ahieved. One this strain plane is ound the ultimate moment M is the moment rom integrating the inal strain plane. u Oasys Ltd 015 9

10 Oasys AdSe Solver Solution Spae Any partiular load ondition an be onsidered as a point in the N M yy, M zz, spae. It is them possible to onstrut a ailure envelope this an be or ULS strength or or a servieability riteria, suh as raking. The ailure envelope is then typially an onion shape with axial ore as the vertial axis. Any point inside the spae represent a valid ore state, but a point outside the surae has ailed. The analysis is then onerned with determining the ore position with respet to the ailure surae. In a ULS analysis the solution holds the axial ore onstant and ind the ratio o the applied moment to the moment orresponding to the projetion on to the ailure surae. Similarly the hart options N/M orresponds to a vertial slie through the onion and a horizontal slie orresponds to an M yy/m zz hart. Solver Searh Proess The solution is iterative, and assumes plane setions remain plane. The iteration searhes through possible strain planes. Strain at point y, zis y z x z y. For eah strain plane: loked in strains are added, stresses are alulated rom the non-linear material urves, ores and moments are alulated by integration o stresses over the setion The three searh onditions are then heked. Analysis Condition 1 Condition Condition 3 ULS : Strength Axial ore Moment angle Ultimate strain Oasys Ltd

11 Oasys AdSe ULS : N/M harts Moment angle Strain ondition Ultimate strain SLS & ULS : Loads Axial ore Moment angle Applied moment SLS : Craking moment Axial ore Moment angle Craking strain The iteration ontinues until the three variables that satisies the (up to) three onditions.,, are ound with orm a strain plane ˆ x z y SLS and ULS loads analysis For a loads analysis AdSe searhes or a strain plane whih satisies: axial ore = applied axial ore moment = applied moment moment angle = applied moment angle When looking at servieability it is useul to be able to dierentiate between short and long term onditions. Long term analysis takes aount o reep, while short term analysis assumes that no reep takes plae. Oasys Ltd

12 Oasys AdSe ULS strength analysis For a strength analysis there are several riteria that may govern the strength o the setion. For a onrete (ompression) governed setion axial ore = applied axial ore moment angle = applied moment angle onrete strain : onrete ailure strain = 1 For a reinorement (tension) governed setion axial ore = applied axial ore moment angle = applied moment angle rebar strain : rebar ailure strain = 1 Some odes implement a third ondition whih limits the ompressive strain on the setion when the whole setion is in ompression. Long, short and long & short term analysis When looking at servieability it is useul to be able to dierentiate between short and long term onditions. Long term analysis takes aount o reep, while short term analysis assumes that no reep takes plae. The reep is deined by a reep oeiient or onrete. It is assumed that the other materials are unaeted by reep. This oeiient is used to modiy the material stress-strain urves. In a long term analysis the total strain is assumed to inlude the strain due to load plus an additional strain due to reep. In the linear ase this an be written as load reep E E Rearranging this gives E 1 AdSe takes these reep eets into aount by modiying the eetive elasti modulus E reep E 1 Resulting in stress-strain urves strethed along the strain axis. 'Long and short' term analysis is an option in AdSe to give a more detailed understand the servieability behaviour o setions. Note: this is not available or all design odes. Oasys Ltd 015 1

13 Oasys AdSe Loading is deined in two stages. Firstly long term loading, ombined with a reep ator and then an additional short term loading. In some irumstanes, the long term loading is a permanent or quasi permanent loading, and the short term loading is an extreme event that happens ater an extended period o time. However in many ases short term loading will our intermittently throughout the lie o the setion. The long and short term analysis option in AdSe will model the seond ase. Firstly the raking moment is alulated assuming the total long & short term axial load and moment diretion, and short-term material properties. Seondly the strains and stresses are ound or the long term loads, and long term material properties. These strains are used to alulate the reep eets o long term loads where reep strain is: reep long 1. In this analysis the BS8110 Pt. tension urve will use 0.55N/mm as the maximum stress, i the setion was deemed raked under the total load. This will model the onservative assumption that, i raked, this happened at an early stage o the setions lie. The raking moment is then realulated, or the total load inluding the reep strain in the onrete alulated above. This will have the eet o slightly reduing the raking moment i a ompressive ore has been ating on the setion or a long time. This is the ase, beause the stress in the onrete will have redued as the onrete reeps and more stress is transerred to the reinorement. Finally a short term analysis is perormed or the total loads, using short term material properties and the alulated reep strain to inlude or the long term eets. Note that i the same proess is ollowed manually using sequential AdSe analyses the initial raking moment will be alulated rom the long term load only. This will give dierent results than the automated AdSe 'long and short' term analysis in a small number o ases. The ases aeted are where the BS8110 Pt. tension urve is seleted, and the setion is raked under total load, but unraked under the long term load, and the stress under long term load is between 0.55 and 1.0 N/mm at the entroid o tension steel. Some odes allow an intermediate term analysis, depending on the ratio M M In this ase E E E q g inter short long E E short E 1 1 M q M g 1 Elong M M 1 M M E q g q g Oasys Ltd

14 Oasys AdSe Strength redution or material partial ators or ULS There are two main approahes appliable to setion analysis: strength redution and material partial ators. The strength redution approah is use in the ACI (Amerian) and AS (Australian) odes. In this ase the material strengths are used unatored, the strength o the setion is alulated and then redued using a strength redution ator ϕ. The partiular value o ϕ depends on the strain plae at ailure. In most other odes ators are applied to the material in the ULS. Giving a redued strength or the material in design d m The CSA (Canadian) odes use a similar approah but use a ator m. d m The values o m and m are speiied in the ode but typial values are m m Conrete Reinorement Pre-stress tendons Strutural steel 0.9 Tension in Conrete Conrete exhibits a 4 phase behaviour in response to tension stresses. Oasys Ltd

15 Oasys AdSe Low tension stress onrete tension stiness similar to ompression Craking starts stiness drops o as raks orm Craks ormed, raks open up stiness drops o more rapidly as raks open up Fully raked no residual stiness let This behaviour is omplex as it is ontrolled by the reinorement. The simpliied means presribed to deal with these phenomena vary rom ode to ode. All odes state that ultimate analysis and design should ignore the tension stiening rom the onrete. All odes will aept ully raked setion properties as a lower bound on stiness. Servieability analysis is usually perormed or stiness, stress/strain heks, or rak width heks. Some odes imply a dierent tension stiening method or rak width as opposed to the other heks. This may lead to a disparity in AdSe results between the 'raking moment' and the moment at whih the rak width beomes > zero. The ode rules are developed or a retangular setion with uniaxial bending and one row o tension steel. However the rules are not extended to setions made up o various zones o onrete, some with loked in strain planes. Beause the tension stiening is a untion o the amount o 'damage' / raking in the setion, adjoining tensile zones need to be onsidered in evaluating the tension strength o a zone, as these may ontain steel whih will ontrol the raking. BS8110 Pt presents a stress/strain 'envelope' whih provide means o alulating an eetive tensile Young's modulus or a linear tension stress/strain urve. ICE Tehnial note 37 presents a more sophistiated envelope approah than BS8110 and is oered as an option in ADSEC. BS5400 presents the same approah as BS8110 in Appendix A or stiness als. But this is rarely used. Instead the main body o the ode gives a rak width ormula based on strains rom an analysis with no tension stiening. The rak width ormula itsel inludes some terms to add bak in an estimate o the ontribution rom tension stiening. Re BS equation 5. EC proposes analyses, one with ull tension stiness and one with none. The inal results are an interpolation between these results. Reent researh about the raked stiness o onrete has shown that the tension stiness measured in the laboratory an only be retained or a very short time. This means that both the tension stiening given in BS8110 and TN 37 is un-onservative or most building and bridge loadings. AdSe inludes these indings or BS8110 and will give a smaller tension stiness than previous versions. EC tension stiening EC tension stiening is desribed in Euroode setion equation EC does not have a speii tension stiening relationship used in analysis. Instead, two analyses are arried out assuming raked and unraked stiness values, and the atual urvature & stiness is an interpolation between the results based on the amount o raking predited. Oasys Ltd

16 Oasys AdSe The raking moment, M r element o an unraked onrete setion has reahed, is deined as the moment when the stress in the outer most tensile tm. The tension stiening options oered or EC in AdSe are zero tension, linear tension, and interpolated. 'Zero-tension stiness' will give a onservative, ully raked lower bound. The 'linear tension stiening' uses the Elasti modulus o the onrete to produe a linear stress strain relationship. This is or heking o the other results only and it is not appropriate to use this beyond the raking moment. Note that the values in EC or servieability are based on mean onrete properties rather than the harateristi values used or ultimate analysis and design. The interpolation depends on the amount o damage sustained by the setion. This is alulated by AdSe based on the proximity o the applied loading to the raking moment. But or setions whih have been raked in a previous load event the minimum value o or use in equation 7.18 an be input. The deault value o is 0. To take aount o the ast drop in min tension stiening ollowing raking, the value o in equation 7.19 deaults to 0.5. AdSe does not use equation 7.19 to alulate the damage parameter. Instead is alulated rom the raking strain tm Em and the most tensile strain unr in the setion under an unraked analysis under ull applied load. The E used or to determine unr is short term (not modiied or reep). For omposite setions is alulated or eah omponent i using the omponent material properties or i, and the most tensile strain on the omponent or unr. The highest value o will be used or i in stiness & raking alulations. Note engineering judgement should be used to assess i this approah its the partiular situation. 1 min unr unr unr I the EC interpolation is seleted or the tension stiness at servieability, the properties whih depend on the average behaviour along the element (eg stiness, urvature and rak widths) are based on the interpolated strain plane. However or moments greater than stresses output by AdSe or the interpolated tension stiness are rom the ully raked analysis, beause these represent the maximum stresses whih our at rak positions. Stiness AdSe operates on strain, using non-linear materials. AdSe will show how the stiness o the setion hanges with load and the eet o non-linear material behaviour. There are a number o M r, the Oasys Ltd

17 Oasys AdSe ways in whih the stiness o a reinored onrete setion an be approximated. These are show in the diagram below. This diagram plots AdSe results along with the approximate stiness values or omparison For a symmetri setion, symmetrially loaded, stiness an be expressed as M EI I there is an axial ore, loked in strain plane, or pre-stress, there will be a residual urvature at zero moment. This urvature an be alled 0 so AdSe uses EI M 0 The urvature at zero moment may not be in the same diretion as the applied moment angle. To allow or this, the ormula is urther modiied to give EI M 0 appl NA Where alulation. appl is the angle o applied moment and NA is the neutral axis angle rom the 0 Oasys Ltd

18 Oasys AdSe Craking Craking Moment The programme alulates the total axial ore and moment rom the loads in the analysis ase deinition. It uses the axial ore and the angle o the applied moment to deine the raking moment analysis task. The programme then searhes or a strain plane that gives the raking strain as shown below. Integrating the stresses rom this strain plane over the setion will give an axial ore equal to the applied ore, and a moment whih is parallel to the applied moment. The value o this moment is the raking moment. Short-term material properties are always used or the raking moment alulation. Crak-width Crak width ormulae to BS8110 and BS5400 are based on a weighted interpolation between two eets. Close to a bar, rak width is a untion o the bar over, untion o the depth o tension zone, h x. min. Between bars, it is a Oasys Ltd

19 Oasys AdSe BS8110 & Hong King Code o Pratie Crak widths an be alulated to BS8110 using any o the three available tension stiening options. Ater alulating the raking moment, AdSe will searh or a strain plane whih gives ores and moments within tolerane o the applied ores and moments. The resulting strain distribution is used to alulate the rak width. The maximum rak width output is related to the given resultant moment orientation. This is partiularly important or irular setions, as the maximum strain may not our between two bars. This would give a lower rak width value than may our in reality. The sides o the setion are divided into small segments and the rak width alulated or eah segment. The rak width ormula 3ar m w ar 1 h x min is given in BS8110, setion 3.8, equation 1. Crak width alulations involve a large amount o engineering interpretation or aeted setions, setions with voids, setions with re-entrant orners, and multi-zone setions. Depending on the situation, a dierent deinition o 'over' is required. The programme stores the minimum over to eah bar and uses this in the alulation o rak width. This means that the over used in the alulation may not relate to the side being heked (it will always give a onservative result). The reason or this approah is that urved setions are analysed as a multiaeted polygon, and there may be no bars present parallel to a small aet. This is beause the number o aets may be greater than the number o bars. Oasys Ltd

20 Oasys AdSe The rak width alulation is done on a zone by zone basis using the zonal strain plane (resulting strain plane + omponent strain plane + onrete only omponent strain plane). This omponent strain plane applied to the whole setion is used to alulate neutral axis depth and setion height ( x and h ) relative to the whole setion using all the setion oordinates. The rak width inludes the term inreased. a r min. I min is smaller than a r the rak width is For eah division on the onrete outline the losest bar is ound (minimum a r ). For a reentrant orner, and a bar whih is on the 'outside' o the setion with re to the side being heked a warning lag is generated, A onservative rak width an still be alulated using the minimum over to the bar. I the over is greater than hal the depth o the tension zone, the rak width in both odes in invalid. The term or onrete only raking should be used instead. This is m in AdSe. 1.5 h x. This is inluded These are warning in the rak width alulation. They do not neessarily mean that the answer is wrong. But do mean that the graphial results should be heked or engineering interpretation min < ontrolling bar diameter rak width not valid Controlling bar is remote rom rak loation Controlling bar and rak are loated on either side o re-entrant orner Cover to ontrolling bar measured to dierent side rom rak loation BS540, Hong Kong Struutres Dsign Manual & IRS Bridge Code The BS8110 speiiation above is valid or BS5400 plus some additional points. The BS5400 inludes a udge or eetive tension stiening. So or rak widths to be orret using the BS5400 ormula, SLS analysis must use no tension stiening in alulation o the strains around the setion. This udge requires alulation o 'the level o tension steel'. This is re-alulated on a omponent by omponent basis using a similar method to BS8110- tension stiening. The tension steel is identiied as the steel whih is in the tension zone when the zonal strain plane is extended aross the whole setion (the zonal strain plane is the resulting strain plane + zonal loked in plane + zonal onrete only plane). From the steel bars identiied, the entroid o steel ore is alulated using the atual stress in the bars ignoring prestress, and ignoring any bars in ompression. One the level o entroid o tension steel is ound, the width term b t needs to be alulated. This inludes interrogation o all the setion oordinates (as or x and h ). I more than sides ross the level o entroid o tension steel, the width is taken as the distane between the two extreme dimensions. This needs to be heked or setions with large voids, or hannels with Oasys Ltd 015 0

21 Oasys AdSe thin legs, as the term b t h is used to make an approximation or the ore in the tension zone and assess the area o tension steel versus the area o tension onrete. It may be appropriate to substitute a smaller value o b t. BS5400 inludes a notional surae a distane ' nom ' rom the bars. AdSe will look at all bars to deine this surae exluding any with 'negative over'. This should be reviewed, partiular or setions with sharp aute angles and re-entrant orners. In the example below the adjustment to sides A and side C may not be the adjustment that would be hosen by engineering judgement. In this situation, the raking parameters output or the relevant sides an be extrated rom the output, and the results realulated, substituting the orreted values. The rak width equations in BS are either equation 4 w 1 a h d r 3a r m nom where the strain m is given by equation 5 m a d M q 9 1 h d M g 3.8bt h 1 10 s As or the alternative equation 6 w 3 a r m are oered by AdSe Oasys Ltd 015 1

22 Oasys AdSe Conrete material models Symbols u onrete stress onrete strength Conrete strain Strain at whih onrete stress is maximum Strain at whih onrete ails Units The deault units are: Stress, strength Elasti modulus MPa (psi) GPa (psi) Conrete material models or dierent odes Dierent material models are available or dierent design odes. These are summarised below: ACI 318 AS 3600 BS 5400 BS 8110 CSA A3.3 CSA S6 EN 199 HK CP HK SDM IRC:11 IRS Bridge IS 456 Compression Parabolaretangle Retangle Bilinear Linear FIB Popovis Oasys Ltd 015

23 Oasys AdSe EC Conined AISC 360 illed tube Expliit Tension No-tension Linear Interpolated BS TR 59 PD 6687 Expliit Expliit envelope inerred rom retangular blok PD 6687 variant o EN 199 only Parabola-retangle Parabola-retangles are ommonly uses or onrete stress-strain urves. Oasys Ltd 015 3

24 Oasys AdSe Oasys Ltd The paraboli urve an be haraterised as d b a At strains above the stress remains onstant. For most design odes the parabola is taken as having zero slope where it meets the horizontal portion o the stress-strain urve. 1 1 d The Hong Kong Code o Pratie (supported by the Hong Kong Institution o Engineers) interpret the urve so that the initial slope is the elasti modulus (meaning that the parabola is not tangent to the horizontal portion o the urve). s s d E E E E 1 where the seant modulus is d s E In Euroode the parabola is modiied n d 1 1 and n 50MPa

25 Oasys AdSe n > 50MPa EC Conined The EC onined model is a variant on the parabola-retangle. In this ase the onining stress inreases the ompressive strength and the plateau and ailure strains.,, u, Retangle u , The retangular stress blok has zero stress up to a strain o onstant stress o. d (ontrolled by ) and then a α β ACI ( - 30)/7 [0.65:0.85] AS ( - 8) [0.65:0.85] AS [0.67:0.85] Oasys Ltd 015 5

26 Oasys AdSe BS / BS CSA A3.3 1 max(0.67, ) CSA S6 1 max(0.67, ) EN MPa 1 - ( - 50)/00 > 50MPa MPa ( - 50)/400 > 50MPa HK CP > HK CP 007 > MPa MPa MPa HK SDM 0.6/ IRC: MPa 1 - ( - 60)/50 > 60MPa MPa ( - 60)/500 > 60MPa IRS Bridge 0.6/ IS Bilinear The bilinear urve is linear to the point and then onstant to ailure., d Oasys Ltd 015 6

27 Oasys AdSe FIB The FIB model ode deines a shemati stress-strain urve. This is used in BS 8110-, EN199-1 and IRC:11. This has a peak stress FIB This is deined as FIB k 1 k with E k Where the ator is ode dependent. Code BS EN IRC:11 FIB MPa MPa 1.05 Popovis There are a series o urves based on the work o Popovis. Oasys Ltd 015 7

28 Oasys AdSe These have been adjusted and are based on the Thoreneldt base urve. In the Canadian oshore ode (CAN/CSA S474-04) this is haraterised by n k3 n 1 nk with (in MPa) 10 k3 0.6 n E n n 1 1 k The peak strain is reerred to elsewhere as pop. pop All the onrete models require a strength value and a pair o strains: the strain at peak stress or transition strain and the ailure strain. Oasys Ltd 015 8

29 Oasys AdSe Mander & Mander onined urve The Mander 1 urve is available or both strength and servieability analysis and the Mander onined urve or strength analysis. For unonined onrete, the peak o the stress-strain urve ours at a stress equal to the unonined ylinder strength and strain generally taken to be Curve onstants are alulated rom E se and r E E E se Then or strains 0 the stress an be alulated rom: r r r 1 where 1 Mander J, Priestly M, and Park R. Theoretial stress-strain model or onined onrete. Journal o Strutural Engineering, 114(8), pp , Oasys Ltd 015 9

30 Oasys AdSe The urve alls linearly rom as To generate the onined urve the onined strength eo to the spalling strain u. The spalling strain an be taken, must irst be alulated. This will depend on the level o oninement that an be ahieved by the reinorement. The maximum strain u, also needs to be estimated. This is an iterative alulation, limited by hoop rupture, with possible values ranging rom 0.01 to An estimate o the strain ould be made rom EC ormula (3.7) above with an upper limit o The peak strain or the onined urve, is given by:,, Curve onstants are alulated rom and E se,, r E E E se as beore. E is the tangent modulus o the unonined urve, given above. Then or strains 0 u, the stress an be alulated rom:, r r r 1 where, BS8110- tension urve BS8110- deine a tension urve or servieability Oasys Ltd

31 Oasys AdSe TR59 Tehnial report 59 deines an envelope or use with onrete in tension or servieability. The material is assumed to behave in a linearly elasti manner, with the elasti modulus redued beyond the peak stress/strain point based on the envelope in the igures below Oasys Ltd

32 Oasys AdSe Interpolated Interpolated strain plains to ACI318 and similar odes ACI318 and several other odes give a method to ompute a value o the seond moment o area intermediate between that o the unraked, I g, and ully raked, I r, values, using the ollowing expression: I e M M r a 3 I g M 1 M r a 3 I r where M r is the raking moment and M a is the applied moment. AdSe SLS analyses determine a strain plane intermediate to the unraked and ully raked strain planes. The program determines a value or, the proportion o the ully raked strain plane to add to the proportion 1 o the unraked plane so that the resulting plane is ompatible with ACI318 s approah. Unortunately, sine ACI318 s expression is an interpolation o the inverse o the urvatures, rather than the urvatures themselves, there is no diret onversion. It should also be noted that although I is deined as the value o seond moment o area ignoring the reinorement, it is assumed that this deinition was made or simpliity, and AdSe inludes the reinorement. M r M a, the unraked urvature be I Let 3 To ACI318, the interpolated urvature 1 1 I II and the aim is to make this equivalent to, g and the ully raked urvature be II. Oasys Ltd 015 3

33 Oasys AdSe II I 1, Equating these two expressions gives II I I II whih an be re-arranged to give The ratio 1 II I 1 1/ 1 II I is appropriate or uniaxial bending. For applied loads N M y, M z unraked and ully raked strain planes I, yi, zi and II yii, zii is replaed by the ratio N M M N M M II y yii z zii I,, and, respetively, II I y yi z zi, whih is independent o the loation hosen or the reerene point. In the absene o axial loads, this ratio ensures that the urvature about the same axis as the applied moment will omply with ACI318; in the absene o moments, the axial strain will ollow a relationship equivalent to that in ACI318 but using axial stiness as imposed to lexural stiness. The ratio M r M a is also inappropriate or general loading. For the general ase, it is replaed by the ratio t ti, where t is the tensile strength o the onrete and ti is the maximum onrete tensile stress on the unraked setion under applied loads. Summary: 1 N M M N M M II y yii z zii 1 3 I 1 ti t Sine is larger or short-term loading, all urvatures and strains are alulated based on shortterm properties regardless o whether is subsequently used in a long-term servieability alulation. Conrete properties Notation y yi z zi onrete strength d onrete design strength t E onrete tensile strength elasti modulus Poisson s ratio (0.) oeiient o thermal expansion (varies but / C assumed) u strain at ailure (ULS) Oasys Ltd

34 Oasys AdSe ax ompressive strain at ailure (ULS) plas strain at whih maximum stress is reahed (ULS) max assumed maximum strain (SLS) peak strain orresponding to (irst) peak stress (SLS) pop strain orresponding to peak stress in Popovis urve (SLS) 1 u ACI The density o normal weight onrete is assumed to be 00kg/m 3. The design strength is given in by d The tensile strength is given in Equation 9-10 by 0. 6 t 7. 5 (US units) t The elasti modulus is given in as E 4. 7 E (US units) The strains are deined as ε u ε ax ε plas ε max ε peak Parabolaretangle ε u (1-3β) ε u Retangle ε u ε β Bilinear ε u (1 - β) ε u Linear ε max FIB Oasys Ltd

35 Oasys AdSe Popovis ε pop EC Conined AISC illed tube Expliit ε u AS The density o normal weight onrete is taken as 400kg/m 3 (3.1.3). The design strength is given in (b) by with d and limits o [0.67:0.85] The tensile strength is given in by 0. 6 t The elasti modulus is given (in MPa) in 3.1. as E E mi mi 40MPa mi 0.1 mi 40MPa This tabulated in Table (MPa) E (GPa) Oasys Ltd

36 Oasys AdSe The strains are deined as ε u ε ax ε plas ε max ε peak Parabolaretangle Retangle ε β ε β Bilinear Linear ε max FIB Popovis ε max ε pop EC Conined AISC illed tube Expliit BS 5400 The density o normal weight onrete is given in Appendix B as 300kg/m 3. The design strength is given in Figure 6.1 by 0. 6 d The tensile strength is given in as t but A.. implies a value o 1MPa should be used at the position o tensile reinorement. The elasti modulus tabulated in Table 3 (MPa) E (GPa) Oasys Ltd

37 Oasys AdSe The strains are deined as ε u ε ax ε plas ε max ε peak Parabolaretangle ε u ε RP Retangle Bilinear Linear ε max FIB Popovis EC Conined AISC illed tube Expliit ε u RP BS 8110 The density o normal weight onrete is given in setion 7. o BS as 400kg/m 3. The design strength is given in Figure 3.3 by d Oasys Ltd

38 Oasys AdSe The tensile strength is given in as t but Figure 3.1 in BS implies a value o 1MPa should be used at the position o tensile reinorement. The elasti modulus is given in Equation 17 E 0 0. The strains are deined as ε u ε ax ε plas ε max ε peak Parabolaretangle ε u ε u ε RP * ε RP Retangle ε u ε u ε β Bilinear Linear ε u ε max FIB ε u 0.00 Popovis EC Conined AISC illed tube Expliit ε u ε u ε u u MPa RP CSA A3.3 / CSA S6 Oasys Ltd

39 Oasys AdSe The density o normal weight onrete is assumed to be 300 kg/m 3 ; see (A3.3) and (S6). The design strength is given in by d max 0.67, The tensile strength is given in Equation 8.3 (A3.3) and in (S6) 0. 6 (or CSA A3.3) t 0. 4 (or CSA S6) t For normal weight onrete the modulus is given in A3.3 Equation 8.. E 4. 5 and in CSA S E The strains are deined as ε u ε ax ε plas ε max ε peak Parabolaretangle ε u (1-3β) ε u Retangle ε u ε β Bilinear ε u (1 - β) ε u Linear ε max FIB Popovis ε pop EC Conined AISC illed tube Expliit ε u Oasys Ltd

40 Oasys AdSe EN 199 The density o normal weight onrete is speiied in as 00 kg/m 3. The design strength is given in by d For the retangular stress blok this is modiied to 50MPa d MPa d The tensile strength is given in Table 3.1 as t MPa.1ln MPa t The modulus is deined in Table E The strains are deined as ε u ε ax ε plas ε max ε peak Parabolaretangle ε u ε ε Retangle ε u3 ε 3 ε β Bilinear ε u3 ε 3 ε 3 ε u3 ε 3 Linear ε u ε FIB ε u1 ε 1 Popovis EC Conined ε u, ε, ε, AISC illed tube Oasys Ltd

41 Oasys AdSe Expliit ε u ε u? ε u m u1 u MPa MPa k MPa MPa k u MPa HK CP The density o normal weight onrete is assumed to be 400kg/m 3. The design strength is given in Figure 6.1 by d The tensile strength is given in as t but implies a value o 1MPa should be used at the position o tensile reinorement. The elasti modulus is deined in E The strains are deined as ε u ε ax ε plas ε max ε peak Oasys Ltd

42 Oasys AdSe Parabolaretangle ε u ε u ε RP Retangle ε u ε u ε β Bilinear Linear ε u ε u FIB ε u 0.00 Popovis EC Conined AISC illed tube Expliit ε u ε u ε u u MPa E d RP GPa E d HK SDM The density o normal weight onrete is assumed to be 400kg/m 3. The design strength is given in (b) o BS by 0. 6 d The tensile strength is given in as t but rom BS5400 a value o 1MPa should be used at the position o tensile reinorement. The elasti modulus is tabulated in Table 1 Oasys Ltd 015 4

43 Oasys AdSe (MPa) E (GPa) The strains are deined as ε u ε ax ε plas ε max ε peak Parabolaretangle ε u ε RP Retangle Bilinear Linear ε max FIB Popovis EC Conined AISC illed tube Expliit ε u RP IRC 11 Oasys Ltd

44 Oasys AdSe The density o normal eight onrete is assume to be 00kg/m 3. The design strength is given in d In A.9() the strength is modiied or the retangular stress blok to MPa d d MPa The tensile strength is given in by A.() by t MPa.7ln MPa t The elasti modulus is given in A.3, equation A-5 10 E 1.5 The strains are deined as 0.3 ε u ε ax ε plas ε max ε peak Parabolaretangle ε u ε ε ε u ε Retangle ε u3 ε 3 ε β Bilinear ε u3 ε 3 ε 3 ε u3 ε 3 Linear ε u ε FIB ε u1 ε 1 Popovis EC Conined ε u, ε, ε, AISC illed tube Expliit ε u ε u? ε u Oasys Ltd

45 Oasys AdSe u1 u MPa MPa k MPa MPa k u MPa IRS Bridge The density is assumed to be 300kg/m 3. The design strength is given in (b) by 0. 6 d The tensile strength is given in as t The elasti modulus is tabulated in (MPa) E (GPa) Oasys Ltd

46 Oasys AdSe The strains are deined as ε u ε ax ε plas ε max ε peak Parabolaretangle ε u ε RP Retangle Bilinear Linear ε max FIB Popovis EC Conined AISC illed tube Expliit ε u RP IRC 456 The density is assumed to be 00 kg/m 3. The design strength is given in Figure 1by d The tensile strength is inerred rom 6.. as 0. 7 t The elasti modulus is deined in Oasys Ltd

47 Oasys AdSe E 5 The strains are deined as ε u ε ax ε plas ε max ε peak Parabolaretangle Retangle ε β Bilinear Linear ε max FIB Popovis EC Conined AISC illed tube Expliit Oasys Ltd

48 Oasys AdSe Rebar material models Symbols y rebar stress rebar strength u p rebar ultimate strength rebar strain strain at whih rebar stress is maximum u strain at whih rebar ails Rebar material models or dierent odes Dierent material models are available or dierent design odes. These are summarised below: ACI 318 AS 3600 BS 5400 BS 8110 CSA A3.3 CSA S6 EN 199 HK CP HK SDM IRC:11 IRS Bridge IS 456 BS 5400 Pre-stress Progressive yield Park Linear Elastiplasti Elastihardening Noompression Oasys Ltd

49 Oasys AdSe ASTM strand Expliit Elasti-plasti The initial slope is deined by the elasti modulus, E. Post-yield the stress remains onstant until the ailure strain,, is reahed. u For some odes (CAN/CSA) the initial slope is redued to E. Elasti-hardening The initial slope is deined by the elasti modulus, E, ater yield the hardening modulus governs as stress rises rom y, yd to,. For EN 199 the hardening modulus is deined u u in terms o a hardening oeiient k and the inal point is redued to ud (typially 0.9 uk ). uk, k yd The relationship between hardening modulus and hardening oeiient is: E h k 1 uk Eh uk k y y y E y E 1 Eh where the ailure strain is Oasys Ltd

50 Oasys AdSe The material ails at ud where ud uk. This is deined in Euroode and related odes. BS 5400 In tension the initial slope is deined by the elasti modulus, E, until the stress reahes The slope then redues until the material is ully plasti, stress remains onstant until the ailure strain, o u yd k e yd., at o E. Post-yield the yd, is reahed. For BS5400 k 0. 8 e and Oasys Ltd

51 Oasys AdSe In ompression the initial slope is deined by the elasti modulus, E, until the stress reahes ke yd or a strain o o. It then ollows the slope o the tension urve post-yield and when the strain reahes o the stress remain onstant until ailure Pre-stress The initial slope is deined by the elasti modulus, E, until the stress reahes then redues until the material is ully plasti, remains onstant until the ailure strain, and o u yd k e yd. The slope, at o E. Post-yield the stress yd, is reahed. For BS8110 and related odes 0. 8 k e Progressive yield The initial slope is deined by the elasti modulus, E, until the stress reahes k e yd. The slope then redues in a series o steps until the material is ully plasti, ater whih the stress remain onstant. The points deining the progressive yield are ode dependent. Oasys Ltd

52 Oasys AdSe Park The initial slope is deined by the elasti modulus, E, until the stress reahes then zero or a short strain range, then rising to a peak stress beore ailure. yd. The slope is ud ud yd u u p p u p p E ud yd Oasys Ltd 015 5

53 Oasys AdSe Linear The initial slope is deined by the elasti modulus, E, until the ailure strain is reahed. No-ompression This is a linear model when in tension whih has no ompressive strength. ASTM strand The ASTM A 416 deines a stress-strain urve doe seven-wire strands. This has an initial linear relationship up to a strain o with progressive yield till ailure. The stress strain urves are deined or speii strengths. For Grade 50 (175 MPa) the stress-strain urve is deined as For Grade 70 (1860 MPa) the stress-strain urve is deined as Bridge Engineering Handbook, Ed. Wah-Fah Chen, Lian Duan, CRC Press 1999 Oasys Ltd

54 Oasys AdSe In the Commentary to the Canadian Bridge 3 ode a similar stress-strain relationship is deined. For Grade 1749 strand E p For Grade 1860 strand E p A more detailed disussion o modelling strands an be ound in the paper 4 by Devalapura and Tadros Crak alulation CSA S6 Code Approah The equation or the rak width is given in setion as w k b s The parameters srm in mm as rm sm kb and depend on the setion and the ause o raking. The ode deines s rm k db 3 Commentary on CSA S6-14, Canadian Highway Bridge Design Code, CSA Group, Stress-Strain Modeling o 70 ksi Low-Relaxation Prestressing Strands, Devalapura R K & Tadros M K, PCI Journal, Marh April 199 Oasys Ltd

55 Oasys AdSe Where As A t and At is the onrete area exluding the reinorement. The ode gives values or bending and 1.0 or pure tension. In AdSe we interpolate between these values using k max min max max min,0 max, But limited to the range [0.5:1]. The k as 0.5 or db term is taken as the average bar diameter in the tension zone, and the area o steel in is taken as the weighted area o the bar in the tension zone A s d i 4 i And the weighing is based on the stress in the bar ompared with the stress in the extreme bar. i extreme The area o onrete in is A min A.5h, A h / 3 t b Oasys Ltd

56 Oasys AdSe where is the entroid o reinorement in tension. The strain term is given in the ode as sm E s s 1 w s Where sis the stress in the reinorement at the servieability limit state and the reinorement at initial raking. In AdSe this is implemented as w is the stress in sm s E s 1, rup t Where alulation, s is the stress in the most tensile reinorement at the servieability or a ully raked, rup is the rupture strength o the onrete and t is the maximum tensile stress in the onrete at the extreme ibre assuming an unraked material. Loal Approah The above approah beomes diiult to justiy when the setion is not in uniaxial bending. In these ases the alternative loal approah an be used. This assumes there is a loal relationship between a bar and the surrounding onrete. The irst stage is to identiy the most tensile bar and determine the over to this bar. We then deine ht as the over plus hal the bar diameter. db ht Oasys Ltd

57 Oasys AdSe The depth rom the neutral axis the most tensile bar b is alulated, and hb is then deined as h b db b Then the onrete area is based on a dimension h min.5h, h / 3, h The width assoiated with this is So that w min 5h, w t t b 1 w t s v A h w EN199-1 The equation or the rak width is equation 7.8 In this w k s r, max sm s r, max is given by m s r, max k3 k1kk4 p, e The ode gives values or k 1 as 0.8 or high bond bars and 1.6 or plain bars or pre-stressing tendon. The ode gives values or k as 0.5 or bending and 1.0 or pure tension. In AdSe we interpolate between these values using k max min max max min,0 max, But limited to the range [0.5:1]. k 3 and k4 are nationally determined parameters whih deault to 3.4 and 0.45 respetively. Where the spaing o bar is large, 5 p,e deined as s r, max 1. 3 h x, then is the ratio o reinorement to onrete in the raking zone where the area o onrete is h d, A h x 3, A A, e min A.5 h Oasys Ltd

58 Oasys AdSe Appendix Alternative stress bloks General stress bloks Parabola-retangles are ommonly uses or onrete stress-strain urves. The paraboli urve an be haraterised as a p b p Deine & p d I the urve is taken to be tangent to the plateau then at 1, 1and 0 d Solving or the oeiients gives a 1and b so Oasys Ltd

59 Oasys AdSe The area under the urve is given by A p d For bi-linear urve with the strain transition at b the area under the urve to p is A b b b 1 b 1 Equating the areas or so 1 b b 3 3 3, b, p For a retangular stress blok with the strain transition at r the area under the urve to p is Oasys Ltd

60 Oasys AdSe Ab 1 r Equating the areas or so 1 r r , r, p Reerenes Popvis S (1973) A numerial approah to the omplete stress-strain urve o onrete, Cement and Conrete Researh 3(5) pp Thoreneldt E, Tomaszewiz A and Jensen J J, (1987) Mehanial Properties o high-strength onrete and appliation in design, Proeedings o the symposium on Utilization o High-Strength Conrete, Stavanger, pp Kent, D.C., and Park, R. (1971). "Flexural members with onined onrete." Journal o the Strutural Division, Pro. o the Amerian Soiety o Civil Engineers, 97(ST7), Oasys Ltd

61 Oasys AdSe Steel materials urve The steel stress-strain urve is haraterised a liner response to yield, ollowed by a ully plasti zone, beore hardening until ailure. The hardening zone an be approximated by a parabola yd a b Deining the peretly plasti strain limit as p and assuming zero slope at u then 1 a p b p ud yd a u b u b a u The dierene between the irst two gives ud yd 1 a b u p u p And substituting the third into this gives or ud yd 1 a u p u u p Oasys Ltd

62 Oasys AdSe 1 ud a u b a u p p yd 1 b a p Oasys Ltd 015 6