DELFT UNIVERSITY OF TECHNOLOGY
Faculty of Civil Engineering
Department of Structural Concrete
Report 5-82-D16
TOWARDS A SIMPLE METHOD OF ANALYSIS
FOR PARTIALLY PRESTRESSED CONCRETE
Prof.dr.ir. A.S.G. Bruggeling
May 1983
RESEARCH REPORT
Tachniscne UnSversiteït t > ^ BibliotheeK Faculieit der Civiele TeèfirtiaK
(Bezoekadres Stevinweg 1) Postbus 5048 2600 GA DELFT
y
REPORT
TOWARDS A SIMPLE METHOD OF ANALYSIS FOR PARTIALLY PRESTRESSED CONCRETE
P r o f . D r . I r . A . S . G . B r u g g e l i n g 1983 Students Ir. D. Boon Ir. W. Slangewal Ir. G.S. Mulder Ir. G.M. Wolsink Ir. J. Zoun Ir. H.H.G. Dijk Scientific collaborators Ir. J. Brakel Ir. J.W. Hofman Ing. H.C. Jager
307-w s^'7
uit^-^ve1 ^:ni 33
d i v e r s c r •^ '^ 0 1 1 7 ^ 0 o-Dutch NOTATION
1 INTRODUCTION
2 DESCRIPTION OF THE RESEARCH PROGRAM
2.1 The development of a model to describe the behaviour of section of a partially prestressed concrete beam
2.2 The calculations 3 METHOD OF ANALYSIS 3.1 Dischinger's approach
3.2 A model to calculate the interaction concrete-steel 3.3 Three models for stress and strain analysis in partially
prestressed concrete
4 BOX GIRDERS FOR BRIDGES CONCRETED IN SITU
4.1 Properties of the materials used in the analysis 4.2 Loads and span
4.3 Cross-section of box girder bridges 4.4 Reinforcement and prestressing steel 4.5 Ultimate bending moment
4.6 Example of calculations for case 8 4.7 Basis of approach
4.8 Results of the calculations
4.9 Example of calculation of stresses 5 FACTORY-MADE DOUBLE T-BEAMS
5.1 Properties of materials used in the calculation 5.2 Loads
5.4 Results of calculations with method described in 3.1 99 5.5 Results of calculations with method described in 3.2 145 5.6 Results of calculations with method described in 3.3 171 5.7 Comparison of results of calculations executed by Prof. 183
Schneider [10], and the results of calculations with
method 3.1 , , , 6 CONCLUSIONS English ... 189
Dutch --—- , . 1 9 5
SUMMARY
This report examines the question whether, and to what extent, it is possible to leave the time-dependent effects out of account in the analysis of partially prestressed concrete, at least in so far as they relate to the redistribution of the stresses over the cross-section. To investigate this possibility it is, first of all, necessary to develop mathematical models which incorporate the time-dependent effects as realistically as possible.
This report discusses three such models, namely, an analytical model based on Dischinger's approach and two models based on the finite dif-ference method. One model is suitable for a small programmable calcu-lator and the other is intended for handling by a larger computer. With the aid of these mathematical models it was investigated what the differences are between the results of simple calculations in which the time-dependent effects with regard to stress redistribution over the cross-section are neglected and the results obtained with the above-mentioned models in which those effects are taken into account.
The check calculations were performed for various degrees of prestres-sing, i.e., for various values of the ratio between the decompression moment and the bending moment due to full loading. Two types of
struc-ture were analysed, namely, a prestressed concrete box girder with posttensioned tendons and a prestressed concrete double T-beam with pretensioned tendons. The effect of variations in creep and shrinkage behaviour of concrete was also investigated in the calculations. The results of the calculations have been plotted in graphs as
func-tions of the degree of prestressing. They include the concrete stress in the compressive zone, the decompression moment, the stresses in the reinforcing steel and the prestressing steel, the reference force, and the curvature.
The investigations show that in many cases the "simple" approach to the analysis is perfectly feasible. It is, however, necessary to
exer-cise caution with regard to the stress in the prestressing steel when the structure is fully loaded. This steel stress may then become too high, particularly with low degrees of prestressing.
SAMENVATTING
In dit rapport wordt ingegaan op de vraag, in hoeverre het mogelijk is om in de berekening van gedeeltelijk voorgespannen beton de tijds-afhankelijke effecten buiten beschouwing te laten, althans in zover deze betrekking hebben op de herverdeling van de spanningsverdeling over de doorsneden.
Teneinde te kunnen onderzoeken of dat mogelijk is, dienen eerst reken-modellen te worden ontwikkeld, waarin de tijdsafhankelijke invloeden
zo realistisch mogelijk zijn ingebouwd.
In dit rapport worden een drietal rekenmodellen besproken, namelijk een analytisch model dat is gebaseerd op de aanpak volgens Dischinger en twee rekenmodellen volgens de differentiemethode. Eén model is ge-schikt voor een klein programmeerbaar rekenapparaat en de andere op een grote computer.
Met behulp van deze rekenmodellen is nagegaan wat de verschillen zijn tussen de resultaten van eenvoudige berekeningen, waarin de tijdsaf-hankelijke effecten met betrekking tot herverdeling van spanningen over de doorsnede zijn verwaarloosd en die met behulp van de bedoelde modellen waarin deze effecten wel worden meegenomen.
De controleberekeningen zijn uitgevoerd voor verschillende voorspan-graden, d.w.z. verhoudingen tussen het ontspan(decompressie)moment en het buigend moment tengevolge van volbelasting. Twee constructie-typen zijn daarin betrokken, namelijk een kokerligger in voorgespannen beton met nagerekt staal en een dubbel T-ligger in voorgespannen beton met voorgerekt staal. In de berekeningen is tevens de invloed van
vari-aties in krimp- en kruipgedrag van beton onderzocht.
De resultaten van de berekeningen zijn grafisch afgebeeld als functie van de voorspangraad. Daarin zijn o.a. betrokken de betonspanning in de drukzone, het ontspanmoment, de staal spanningen in betonstaal en voorspanstaal, de referentiekracht en de kromming.
Uit het onderzoek blijkt dat in veel gevallen de eenvoudige aanpak van de berekening zonder meer mogelijk is. Men dient evenwel
voor-NOTATION
The notation is in accordance with the 1978 CEB-FIP Model Code. Cross-section A Concrete section c A Reinforcing steel s " A Prestressing steel P Modulus of elasticity E Concrete c E Prestressing steel P E Reinforcing steel S ^ Force F Reference force Jv F„ at t = 0 R,o
F„ T at t = 0, with losses of stresses R,rel,o
^R,t "^ ^ = ^ F„ at t = °°
R,"
Moment of inertia I Net concrete section
c . ^20 Degree of prestressing M max M Bending moment M Dead load g
M, Sustained load, including dead load a M- Live load M . Minimum value min M Maximum value max
M„_ Bending moment of decompression M Cracking moment
Compressive force Prestressing force
at t = 0 - initial value
at t = 0 with losses of stress due to relaxation at t = t at t = °° V Width Web of I- or T-beam Top flange Bottom flange Cover
Effective depth - distance from centroid of steel to top fibre of section
Distance of centroid of steel to centroid of concrete section
Prestressing steel Reinforcing steel Strength
Characteristic strength - prestressing steel Characteristic yield strength - reinforcing steel Depth of a concrete section
Top flange Bottom flange
Compressive zone in the case of a cracked tensile zone Kern
Upper kern Under kern
11 c E E s p n = — or n = / c c Equally distributed load q Dead load
g
q. Sustained load (including dead load) Live load
Distance from centroid of concrete section to top or bottom fibre
yi Top fibre 72 Bottom fibre
Internal lever arm
z With respect to prestressing steel z Ultimate state
pu
z With respect to reinforcing steel z Ultimate state su Strain e Concrete e Top fibre ci e Bottom fibre £ Elastic strain at t = 0 CO e Total strain at t = t ct e / ,^ "elastic" part of z ct(el) ^ ct e ^/,N "plastic" part of e ^ Ct((J)) '^ "^ ct 0=° Total strain at t = <»
e Shrinkage strain over the period t = 0 till cs,t
t = t
e Shrinkage strain over the period t = 0 till cs,°°
t = CO
Curvature of a section
Coefficient related to the eccentricity of the steel = A .e^ c p 1 + I c Stress a Concrete c a at t = 0 CO a ^ at t = t ct - J a at t = =° , 's QOO a Top fibre .• ,- • Cl
a Mean value at the centroid of the concrete cm
section
o Concrete stresses at the centroid of the steel ops section a Bottom fibre C2 a Prestressing steel a „ t = 0 - reference value at t = 0 p,R a t = 0 po
o „ t = 0 - reference value with losses of stress p,R,o due to relaxation a ^ at t = t Pt a at t = °o poo CT Reinforcing steel CT at t = 0 so CT at t = t st Creep factor (j) Period from t = 0 to t = t é Period from t = 0 to t = «= A s Relative steel section -r—
A c (jj Prestressing steel P 01 s Reinforcing steel
Change (increase or decrease) ACT Stress in prestressing steel
P
ACT ,„„„ relaxation loss over a period of thousand
p,1000 ^ hours after stressing
ACT Total relaxation loss over the period t = 0
p,co
to t = =°
ACT Total steel stress loss due to shrinkage p,s+c,°°
INTRODUCTION
At the "Betontagung 1979" in Berlin (25-27 April 1979) Prof.Dr.sc.techn. Bachmann (Federal University of Technology, Zurich) and the author of this report were invited to give lectures on the development of partially prestressed concrete in Switzerland and The Netherlands respectively [1, 2 ] .
In his lecture Prof. Bachmann presented his views on partially pre-stressed concrete. Later on he published his "ten theses on partial prestressing" [4, 5 ] .
In the author's lecture special attention was paid to the influence of shrinkage and creep of concrete on the bending moment of cracking of the tensile zone. To illustrate this influence he showed some results of calculations in which time-dependent effects were taken into account.
The model'for statically determinate structures that was used was sub-stantially limited to the case where the centroids of the prestressing steel and the reinforcement coincided and to the assumption that under sustained load the tensile zone was uncracked. Relaxation losses of the prestressing steel were fully taken into account, directly after prestressing. This means that the prestressing force was at that time reduced by the total magnitude of relaxation losses.
After the presentation of their lectures Prof. Bachmann and the author arranged to collaborate with each other in the near future with the object of pooling their views.
In fact Prof. Bachmann neglected in his studies the influence of time-dependent effects on the behaviour of partially prestressed concrete
(crack width for example), whereas the author used models to calculate these influences! The analysis of these influences was interesting for scientific purposes, but presented a real problem to the practical designer.
Commenting on a paper by Naaman and Siriakorn in PCI-Journal, the author wrote on behalf of the complexity of the calculation [3]:
"The second point I want to mention is that in my opinion there is only very slow progress in the use of partially prestressing because the method of analysis developed is too complicated for use in normal practice.
I also developed a design method which I tried to make popular in a report written for practice. We organised a course in which we gave instruction in the practical use of our "simple" calculation method. Some months later I realised in a discussion with engineers engaged in design practice that they felt that our "simple" method did not give them enough possibilities to find out, by the analysis of some alternatives, what is a suitable solution for a given case; and it is not only my simple method that in fact is limited to the design of statically determinate structures. This is in contradiction with actual practice, where most structures are statically indeterminate. So I believe that we really must use our experimental calculation methods ourselves to develop simple design methods to meet practical needs. In this respect our Swiss colleagues have shown us a way. In the Swiss S.I.A. Standard 162, now already ten years old (!), a simple design method for partially prestressed concrete is recommended.
As far as I know, in Switzerland partially prestressed concrete is the solution which is normally chosen. Fully prestressed concrete is the exception there.
In using the experience gained by Swiss engineers we can check the behaviour (deflection, fatigue resistance) of partially prestressed concrete designed in accordance with simple design rules with the calculation methods we have developed. In analysing the results of these calculations we shall find the simple design rules which are needed.
In this respect this paper takes a very important step towards general acceptance of partially prestressed concrete, but not the final one. If the authors agree with my views, I hope that in close cooperation with practitioners we shall together find the way to give them the
(design) tools. I am convinced that with these tools the possibilities of partially prestressed concrete will be recognised and that this construction method will be used on a large scale."
Because the author had already developed in the "Concrete Structures" Group at the Delft University of Technology a model for calculating
time-dependent influences on stress distribution and deformation of cross-sections it was clear that he had to do the work to show the
(im)possibilities of the proposals made by Prof. Bachmann and practical design methods already practised in Switzerland.
In the faculty of Civil Engineering at Delft the students have to carry out studies or do designs to obtain their Master's degrees. The time available for these studies is 35 weeks. Mostly nearly 70% of this time can be spent on specialisation of the study of, for example, concrete structures.
Therefore the author made a proposal in the research program 1978-1979 of the "Concrete Structures" Group to support the development of par-tially prestressed concrete in The Netherlands with studies in which students participated.
The aim of this research project was first to develop the already available model for the calculation of time-dependent effects in
(partially) prestressed concrete structrures into a more realistic one.
The following possibilities had to be incorporated:
- the centroids of the prestressing steel and the reinforcement are not to coincide;
- cracked tensile zone under sustained load;
- real relaxation behaviour of prestressing steel; - tension stiffening of the cracked tensile zone; - statically indeterminate structures;
- influences of difference in bond behaviour of tendons and rein-forcement.
Up till now nine students have collaborated in this research project -not only with enthusiasm, but also with skill, -not evading the problems they encountered. The first meeting of the group took place on
27 November 1978. The first steps with a view to starting the research work as a joint venture of a group of students and scientific
collabo-rators were discussed there.
The students and the scientific collaborators of the "Concrete Struc-tures" Group had a meeting every two months to discuss the progress of the work.
All of them, with exception of one who started in 1982, have already obtained their Master's degrees and they will, it is hoped, introduce their views into concrete practice in The Netherlands.
List of students collaborating in the desk research project "Partially prestressed concrete":
Ir. D. Boon (Scientific collaborator: Ir. J.W. Hofman)
Double T-beams in pretensioned prestressed concrete (with the al-ready existing model)
Ir. W. Slangewal (Sc.coll.: Ir. J.W. Hofman)
Box girder beams in posttensioned prestressed concrete
Ir. G.S. Mulder and Ir. G.M. Wolsink (Sc.coll.: Ir. J. Brakel) Development of a new calculation method to be used on programmable manual calculators; ' ' • " '
introduction of cracked tensile zone under sustained load
Ir. J. Weerheijm and Ir. R. de Hond (Sc.coll.: Ing. H.C. Jager) Models for calculation of time-dependent effects and first approach
to statically indeterminate structures Ir. J. Zoun (Sc.coll.: Ing. H.C. Jager)
Double T-beams in pretensioned prestressed concrete with the cal-culation program developed by Mulder and Wolsink
Ir. H.G.G. Dijk (Sc.coll.: Ing. H.C. Jager)
Development of a model for computer calculation - statically determinate structures;
introduction of tension stiffening and time-dependent relaxation losses of prestressing steel;
with this model, analysis of double T-beams, already prepared by Zoun. In this project close collaboration with Prof.Dr.Ir. J. Blaauwendraad G. de Vos (Sc.coll.: Ing. H.C. Jager)
Further development of the model made by Dijk;
introduction of discrete cracks in the model and calculation of the time-dependent behaviour of a whole beam (instead of a cross-section) as a first step towards the analysis of statically in-determinate structures.
As appears from the list of collaborating students, the project was developed step by step towards the real complexity of statically in-determinate concrete structures in partially prestressed concrete. In fact the study involved concrete structure from ordinary reinforced concrete to fully prestressed concrete.
In the first period six students started together. At that time it was not easy to assign each of them his own particular share of work, because they (and the staff) all had to learn how to proceed in this project.
The first results of this research program were presented at the Delft University of Technology on 10 December 1980.
The meeting - which lasted a whole day - was organized by the "Concrete Structures" Group of the Delft University and the research organisation CUR/VB, Zoetermeer.
Over 250 engineers attended this meeting!
The first six students mentioned in the list presented their work and Ing. H.C. Jager and the author completed the program in such a way as to provide an overall view.
A summary of the lectures was offered to all those present [6]. In March 1982 the first paper was published in the Dutch technical journal "Polytechnisch Tijdschrift - Civiele Techniek".
It presented the possibilities of partially prestressed concrete to concrete designers in The Netherlands and marked the start of a series of papers published, at the rate of one a month, in that jour-nal.
Through this medium the "students" (in fact already qualified engi-neers!) are publishing the results of their studies.
In this research report all the interesting results of the studies carried out up till now will be presented.
It deals with the first analytical model, prepared by the author, and followed by the models developed by Mulder/Wolsink and Dijk. Then the results of parameter studies will be presented. The results of the calculation made by Boon/Slangewal/Zoun and Dijk will be given and analysed.
The aim of this report is therefore to show • ' .
- how it is possible to develop scientific models for the calculation of time-dependent effects;
- whether it is possible to use simple calculation methods - as normally used in present-day practice - for the design and analysis of parti-ally prestressed concrete.
This report will show how it is possible to avoid in practice the use of the complex calculation models which have been developed. It may seem paradoxial, but in fact we tried to develop models with the aim and hope that they will not, or hardly, be used in practical design. The research project has not yet been completed, but we have gained so much information that we believe it is time to present our work and its results in such a way that it can be used in practice. The report is written in English so that we hope to present it at inter-national level as a contribution to promoting the extensive use of partially prestressed concrete.
The answer to the question: "is a simple design method possible?" is in our opinion: "yes". Taking into account some points of importance such as the maximum steel stresses in the prestressing steel, the design of partially prestressed concrete is not more complicated than that of fully prestressed concrete. But the possibilities of partial prestressing are numerous and, in our view, greatly extend the possi-bilities of prestressed concrete.
To put these possibilities into practice the author published a book [7], in Dutch, dealing with the design and calculation of (partially) prestressed concrete.
In preparing this research report Mr. Th. Steyn gave invaluable assis-tance in preparing all the drawings, graphs, etc.
In the research project Ir. J. Brakel, Ir. J.W. Hofman and Ing. H.C. Jager devoted considerable effort to assisting the students in carry-ing out their allotted tasks.
I am especially grateful to Mrs. K.S. Polderman for her excellent lay-out of the text and typing. I am also grateful to Mr. W. Mol and Mr. H.F.S. Spiewakowski for preparing the figures and the printing.
2 DESCRIPTION OF THE RESEARCH PROGRAM
The research work deals with two main items, namely, with the develop-ment of the model and with calculations.
2.1 The development of a model to describe the behaviour of a section of a partially prestressed concrete beam
The tensile zone of the beam may be: a) uncracked under sustained load; b) cracked under maximum load; c) cracked under sustained load.
The centroids of the prestressing steel and the reinforcing steel in the tensile zone may or may not coincide.
The first step was to develop the Dischinger model so that it could be used for the calculations of time-dependent effects due to bending moments.
If the distance between the centroids of the prestressing steel and the reinforcing steel are small, it was assumed that, with some alter-ations, the Dischinger approach could also be used.
In practice, however, the distance between these centroids may be large, especially in the zones of the beams near the supports. Only if the time-dependent effects in these parts can also be investigated will it be possible to develop a model for the time-dependent behaviour of a beam instead of this behaviour for a section.
The Dischinger model that was developed is described in 3.1.
The next step was to develop a model, based on the Dischinger approach, • in which the centroids of the prestressing steel and the reinforcing
steel did not coincide.
The first investigations dealt with the question whether this problem could be solved analytically or whether a differential approach was necessary.
The differential method was therefore at first developed in such a way that a small programmable calculator could be used. In this method already a first approach was made to analysing the behaviour of a
beam with a cracked tensile zone. Because the analysis was in fact concerned with a section, the part of the concrete section in tension was not taken into account. It means that no tension stiffening of the tensile concrete was considered.
Later on it emerged that for the uncracked stage the time-dependent effect could also be calculated analytically.
The third step was to take into account the effect of tension stiffen-ing of the tensile zone of the beam and the relaxation of the stress in the prestressing steel.
In this step, too, a section was considered. It means that the effect of discrete cracks could not be taken into account. In this approach "smeared-out cracks" are introduced. The relaxation loss-time relation-ship was conceived as similar to the creep- (shrinkage) deformation-time relationship. Because the amount of calculation work increased rapidly, it was necessary to use a computer for doing it.
The calculation method now available therefore deals with time-dependent effects also in "sections" with a cracked tensile zone.
Tension stiffening is taken into account by "smeared-out cracks" and also relaxation losses of stress in the prestressing steel are intro-duced as a time-dependent effect.
The program used in the third step is described briefly in 3.3. The development of the computer program has another consequence
affecting Dischinger's analysis. The calculations turn out to be less complicated if the line of reference is located at the centroid of the prestressing steel (and the reinforcing steel). This change is not in fact used in the calculations performed with the Dischinger approach, but this method is nevertheless described in 3.1 because it notably simplifies the analytical calculation.
The calculations
The calculations were performed for two types of statically deter-minate beams, namely, posttensioned prestressed concrete box girder beams and pretensioned prestressed concrete TT-beams.
2.2.1 Simple method of analysis
For each beam the cross-section of the prestressing steel was cal-culated for the case of full prestressing with a simple calculation method. This means that under maximum load the concrete stress in the bottom fibre (tensile zone) in the section at mid-span (maximum value of bending moment) is zero at t = °°.
In other w o r d s , bending moment of decompression (at mid-span) is bending moment under maximum load (at m i d - s p a n ) .
One can write this on the following way: M-_ = M
20 max
Or M^_ = K.M , with K = 1 20 max
If K is the so-called "degree of prestressing" M
max
To determine the cross-section of the prestressing steel one has to calculate the losses of prestress due to shrinkage and creep of con-crete and due to relaxation of the stresses in the prestressing steel. This can, simply, be done by:
1. Calculating the concrete stress CT in the fibre in the _ cp
- centroid of the prestressing steel - under sustained load
- with an assumption of a certain loss of prestress. 2. calculating the shortening of this fibre due to
- shrinkage of concrete £ (negative value) CT
CDS
- creep of concrete é . _ (negative value of CT is compres-«> E ° cps
sive stress).
3. Calculating the loss of prestress due to shortening of the concrete ACT = E .£ + n.d) .CT (negative value!)
4. Calculating the relaxation loss in the prestressing steel
ACT
ACT = 3ACT , „ „ „ (1 + 2 S-tllljZ) p.cc p , 1 0 0 0 CT
ACT .^nn is relaxation loss over a period of 1000 hours p, 1000
for an initial stress CT po
a is initial stress in the prestressing steel. po
5. The tensile stress in the prestressing steel can be calculated with
a = CT - ACT + ACT . , • • . • • •
pa» po p , " p,S+C,o°
In the figures CT is indicated as the steel stress in the design
° pco S _
a p p r o a c h .
If CT is known one can determine the exact place of the centroid of
poo
the prestressing steel in the section, taking into account the rules for sufficient protection (concrete cover) of the prestressing steel and a good construction (enough distance between tendons etc. for easy concreting) and the cross-section of the prestressing steel. K = 1 ; A .CT (e + ki) = M
p p" p max
A = cross-section of prestressing steel
e = distance between centroids of the concrete section P
of the section of prestressing steel
kx = upper "kern" of the concrete section; ki ^
72.A^ Herewith the method to determine the cross-section and place of the prestressing steel is briefly described.
In the calculations also lower degrees of prestressing are introduced. This is done by reducing the area of the cross-section of the prestressing steel so that a whole number of tendons is used.
For example if for K = 1 one needs 14 tendons then one chooses K = 0,86 one needs 12 tendons K = 0.71 one needs 10 tendons K = 0.57 one needs 8 tendons, etc.
Also the place of the tendons in the section of the beam is adapted and the real position of the centroid of the prestressing steel is taken into account for the calculation of the degree of prestressing K.
Therefore K is not chosen primarily but a good design is the starting point. K is derived for that practical case!!
For every value of K the losses of prestress are calculated in the same way as described for K = 1.
The influence of normal reinforcement in the cross-section on the time-dependent effects is neglected in the simple method of analysis.
The section of the normal reinforcement is determined in such a way that the factor of safety of the beam has at least a given minimum value Y- In all our calculations a value y = 1,7 is assumed. There-fore the area of the cross-section of the normal reinforcement can be calculated with the formula:
Y-M - A .f , .z max p pk pu s f .z
sy su In this formula:
z and z are lever arms of prestressing steel and rein-pu su .
forcing steel respectively in the ultimate state.
f , and f are the characteristic strengths of prestressing steel and reinforcing steel respectively in the calculation of the bending moment in the ultimate state.
In the calculation of the bending moment in the ultimate state is assumed that plain sections remain plain.
If the minimum cross-section of the longitudinal reinforcement in the tensile zone the bar diameter must be chosen in such a way that the crack width is limited. In this respect the tensile reinforcement must also be distributed carefully over the tensile zone.
This problem is not discussed in this report but in a separate one [9]. In this report no special attention is paid to the detailing of the reinforcement and its influence on the crack width. Only the steel
stress ACT in the cracks is calculated as a basis for the analysis s,cr
of the crack width.
Posttensioned prestressed concrete box girder beams
This beam with a span of 45 m, a width of 15 m and a height of 3 m (.a relatively stiff box girder!) was prestressed with cables of the V.S.L. system, 12 strands per cable, overall diameter 12.4 mm. Concrete quality B 37.5 (NEN 3861):
characteristic cube strength 37.5 N/mm^ Reinforcement FeB 400 HW:
characteristic yield strength 400 N/mm^ Prestressing steel FeP 1860 - strands - :
characteristic tensile strength 1860 N/mm^.
To study the influence of time-dependent effects it was necessary to decide about the creep and shrinkage behaviour of concrete.
Because the influence of creep and shrinkage may considerably affect the results, it was decided not to adopt just one creep value <)) and one shrinkage value £ , but to study the variation in these values.
° cs,«>
Assuming 75% relative humidity of the air, three combinations of creep and shrinkage were chosen:
mean value é = 2 . 0
CO
dispersion ± 0.5
Therefore three cases were calculated;
creep factor ((> 1.5 2.0 2.5 shrinkage e - 100 x 10~^ - 150 x 10~^ - 200 x 10
° CS,""
These calculations were performed only with the Dischinger approach and are described in Chapter 4.
In this case another study was carried out concerning the influence of cracks on the increase of stress in the prestressing steel of the cables. The results of this study are not described here.
= - 150 X 10 cs,"
2.2.3 Pretensioned prestressed concrete TT-beams
The choice of pretensioned prestressed concrete needs some explanation. In fact we believe that, with some exceptions, partially prestressed concrete is of no interest for products in pretensioned prestressed concrete. Therefore it is questionable whether this type of concrete should be considered.
Our decision to include it in the present treatment of the subject was based on the following considerations:
1) In pretensionsed prestressed concrete the bond of the prestressing steel is of the same magnitude as the bond of the reinforcement. In systems with cables an important difference in bond behaviour
can be expected and is indeed found in research.
To avoid this complication the use of pretensioned prestressed concrete was preferable.
2) There is considerable production of TT-beams in The Netherlands. It is of interest to see if there are as yet unknown possibilities for these beams in cases of more or less abnormal live to dead load ratio in relationship with the span.
Therefore also a TT-beam section is considered which is not of a normally produced type (type TT instead of TTP).
3) The first "student" in this project was an employee of a firm engaged in the production of TT-beams.
The pretensioned prestressed concrete beams have the following charac-teristics:
Two different TTP sections (standard type and TT, see 5.3) Concrete quality B 52.5 (NEN 3861):
characteristic cube strength 52.5 N/mm^ Reinforcement FeB 400 HW
Prestressing steel FeB 1860 - strands Relative humidity of the air 50%
Creep and shrinkage values:
creep factor i>^ K 5 2 2.5
shrinkage e - 200 x 10~^ - 250 x 10~^ - 300 x 10~^
° cs,"
The calculations were performed as follows:
Dischinger approach - 3.1 two types of beams
straight or draped strands
span 14.35 m, 16.2 m, 1 8 m and 20 m Programmable calculator - 3.2 type TTP
straight strands span 16.2 m Computer calculation - 3.3 type TTP
straight strands span 16.2 m.
The first group of calculations yielded much information on various influences affecting the time-dependent behaviour. Therefore it was possible in the two other studies to take a closer look at special effects such as cracking, tension stiffening and detailing of the beams.
The results of these calculations will be given in Chapter 5.
For better representation of the information the results of the cal-culations are merely given in graph form because the trend displayed by the behaviour of sections with different degrees of prestressing is more important than the exact values of stresses, etc. found in the calculations.
Nevertheless all these data are available in our research records and can be studied if necessary.
For comparing the results of the several calculations all the graphs are presented in the same way and drawn to the same scale.
General conclusions from a study of all the results of all the cal-culations are drawn in Chapter 6.
3 M E T H O D S OF A N A L Y S I S 3.1 D i s c h i n g e r ' s a p p r o a c h 3.1.1 C e n t r i c a l l y r e i n f o r c e d c o n c r e t e loaded c o l u m n D i s c h i n g e r d e v e l o p e d a m e t h o d f o r t h e a n a l y s i s of t i m e - d e p e n d e n t e f f e c t s in c e n t r i c a l l y r e i n f o r c e d c o n c r e t e loaded c o l u m n s [ 8 ] . T h i s a p p r o a c h w i l l b e b r i e f l y d e s c r i b e d h e r e . T h e f o l l o w i n g a s s u m p t i o n s a r e m a d e in t h e c a l c u l a t i o n s of t i m e - d e p e n d e n t e f f e c t s . 1) In t h e p a r t o f t h e c o l u m n u n d e r c o n s i d e r a t i o n t h e d e f o r m a t i o n s of the c o n c r e t e a n d steel a r e t h e s a m e .
2) The stresses in the concrete and steel are in the linear elastic part of the stress-strain diagrams of both materials. Therefore the stresses at loading can be calculated assuming linear-elastic behaviour.
3) During the time-dependent process the steel stresses remain in the linear-elastic part of the stress-strain diagram.
4) The modulus of elasticity of the steel (E ) is known.
s
5) The dimensions of the columns and the cross-sectional areas of the concrete (A ) and the steel (A ) are known.
c s
6) The relationship of creep deformations as a function of time is the same as the relationship of shrinkage as a function of time. Therefore the following can be written:
nt,t^)
e ^ - ; . e C S , t é CS,=o E = s h r i n k a g e s t r a i n o v e r t h e p e r i o d t - t c s , t ^ ^ o _. t = time of l o a d i n g of t h e c o l u m n £ = s h r i n k a g e s t r a i n o v e r t h e p e r i o d t - t = oo CS,oo ° ^ o (b = c r e e p f a c t o r t - t = oo. ^oo o7) Creep of concrete is irreversible.
8) Creep strain of concrete can be calculated for a period dt with the creep factor (J)(t+dt) - ^ = —— .
The magnitude of (j) is not influenced by the rate of concrete stress.
N
Fig. 1 Centrically loaded reinforced concrete column
For a period dt and a concrete stress CT ^ at t = t we can calculate:
•^ ct
a) the creep strain of the concrete: ct dc
E dt c
b) the change in magnitude of the concrete stress a over this period dt: do
ct dt
A .do + A .da = O ' s st c ct do , do st ^ _ _L ct dt 0) • dt • A With 0) = -— A c
d) the shortening d£ of a "section" over a period dt; this shortening can be calculated with the equation:
do d* dé d£ ^ ^ {^£t ^ — 1 + £ — 1 E } dt E dt ct • dt cs,<=° •• dt " c c ,, , dCT , dCT d_t ^ Jl_ s 1_ ct dt E • dt ~ wE ' dt c s dCT ^ . d(j)^ E _ ^ (1 ^ - L ) ^ t ^ c ^ . E ) = 0 dt noj dt ct Ó c E g
n = •=— ; CT ^ (compression) and e to introduce as
E ' ct '^ cs,o°
negative values!
The general solution of this differential equation can be written as follows:
e _ ,
ln(CT ^ + ^ 1 ^ . E ) = -rr- • 't'/^ ^ ^ + C c t * c 1 +naj (t, t )
°° o
At t = t the compressive stress CT can be calulated on the following o CO
way:
- N . „ ct CO A (1+nw) ' ^t
c
Therefore the factor C can be found.
£ n c oo
C = ln(CT + , ' . E ) co d) c
^co
The solution of the differential equation leads to the expression for CT as a function of time.
ct e e , CS,"» „ N -Hf- CS,00 _ O ^ = (CT + , ' . E ) . e 't — - ^ . E ct co ó c ó c ^ 0 0 ^00 with n, ^ 1 . ^ • n(jj
If CT is known also o and e, , are known. ct St (t ,t)
1 CT . = — (o^^d+nüj) - CT } st 0) co ct st 1 1 r /. N 1 E/. .^ = -j;— = r=r- • — ICT (1+nw) - CT . } (t ,t) E E no) co ct o s c E/^ ^\ = £ + — (£ + , ) (1 - e i-) (t ,t) o nw o é o oo
The external tensile force F that must act at t = t on the section K, t
if, on application of a compressive force N, the concrete stress is zero can be calculated with
_ £ _
F^ = N.e "^t + -Eli^! .E .A (1+n(o)(1 - e "^t)
R.t i> c c
R,o
The formula shows that the magnitude of the so-called reference force F at t = t is influenced by the following parameters:
R, t
the magnitude of the sustained compressive force N; the relative section of the reinforcement w;
the creep factor (j) ;
the shrinkage of the concrete. 3.1.2 Prestressed concrete column
If a compressive force P at t = 0 is introduced into the reinforced concrete column by stressing steel tendons with an area A at a tensile force P , then, instead of N the value P and instead of
o o . cü = A /A the value u = (A + A )/A , can be substituted into the
s c s p c formula for F .
R, t
The reason is that shrinkage and creep of the concrete will reduce the prestressing force P . This reduc
dependent shortening of the concrete.
the prestressing force P . This reduction is related to the time-N
Instead of CT = -;—7-. s- we can write CO A Cl+nw;
P A (oj = - — ) co A (1+ncü ) s A C S c CT = n.CT so co
In this formula P is the prestressing force (tensile force in the
tendon) directly after transfer of the force to the concrete.
At t = 0 the reference force P (l+no))
F„ = -2 = P (1 +ntü ) . R, o 1 +na) . o p
s
In that case a = a = 0 because, due to the influence of the CO so
reference force, the concrete stress will be zero.
If we introduce the relaxation loss of stress in the steel, that occurs directly after transfer of the prestressing force into the concrete column, we can calculate the reference force "F„ ^
in-R,rel,o
stead of the force F„
R,0
F„ , = (P - A .ACT )(1+nüj ) = F „ - A .ACT (1+nw ) R,rel,o o p p,oo p R,o p p,oo p F_, , = A .CT „
R,rel,o p p,R,o
ACT can be calculated with the formula given by the FIP-CEB Model P,°°
Code.
2ACT ^ ,
ACT = 3ACT ,^„„ (1 + Pil±£^) p,o° p, 1000 CT
^ '^' po
ACT = shortening, to introduce as a negative value.
p,s+c,oo °' °
The tensile force F„ ^ can then be compared with the compressive
R,rel,o '^ ^
force N acting on a concrete column with a steel cross-section (A +A ) .
s p Therefore at t = t _ £ _ . F„ , = F^ , . e '^t + _ £ £ 2 ^ .E .A (1+na))(1 - e '^^) R , t R , r e l , o <{> c c ' CO
If at time t = 0 the column is not only prestressed but also loaded by a sustained tensile force N, the reference force F_ can be
cal-R, t culated with the formula
F„ = F^ , . e-^t + ( N ^ ^ f c s ^ _g ^ (1+nü.)(1-e-''^) R,t R,rel,o A (1+naj) è c"^ c
In the same way the strain over the period t = t can be calculated. We find: . •
a) total strain
e = E - H ^ ( E +-£i2f!)(i - e-^t) ct CO no) CO é
(for the derivation of this formula see 3.1.1)
The stresses of the reinforcement can be calculated with the ex-pression CT = CT .E
st ct s
b) the part of the strain representing the elastic strain associated with the concrete stress at t = t can be calculated with
CT ^ £ £
C t , ^ CS,oo - n ^ CS,oo
E ^ / 1N = -^— = ( e _ + ) . e t _ c t ( e l ) E co (|) ^ é
C 00 ' s o
(for the derivation of this formula see 3.1.1) c) the plastic strain over the same period
^ct((j)) " ^ct ^ct(el) ct((p) no) CO cj)
3.1.3 Section of a prestressed concrete beam subjected to an eccentric pre-stressing force and a bending moment
In the case of an eccentric prestressing force and a bending moment acting at a section we can also use the Dischinger approach to cal-culated time-dependent effects.
Assuming that the centroids of the prestressing steel and the rein-forcement coincide, we can determine the relationship between the steel stress and the concrete stress at the fibre situated at the centroid of the steel.
In the case of prestressing only _ _ P_
A e''
= _ Z_ _
Lie.
= - ^ (1 + c )
cps A I A I c c c c A . e ^ CT = o .5 i f (1 + - ^ ) = ^. c p s cm I cps F i g . 2 The r e l a t i o n s h i p b e t w e e n a and CT cm cpsIn the case where the steel stress changes in time we can also write dCT cps dCT cm Because Therefore dt A . c da cm dt da ps dt dt dCT cm dt = - 0) 1 0)5 • <; dCT A P^ ^s • dt dCT ps dt dCT cps • dt
The solution of the differential equation can then be determined in
the same way as explained for the column in 3.1.1. Instead of nw (column) we must now introduce the value of nto? (beam).
If a sustained bending moment is acting on a section and the tensile zone of the beam is not cracked under this load, we can calculate the reference force F in the same way as for the column:
M G £ A • F„ = F^ - .e-^t + { d- ^ c s ^ _^ ^ c (i+^^5)(i_e-^t) R,t R,rel,o I (l+nca?) ((> c 5 with n ^ 1 . ^ nu 5 = 1 + ^ A .e^ I c
In this case the force F is a tensile force acting on the sections R, t
(concrete and steel) at the centroid of those sections. The force
F produces zero tensile stress at the centroid of the steel sections K, t
in the case of a totally unloaded beam (M, = 0 ) . M , is due to dead d d
load of beam with sustained load on the beam.
1 F acts at the centroid of the section. Therefore the factor — is
R, t t,
introduced into the formula because F is related to the mean value R, t
of the concrete stress at the centroid of the concrete section.
In the case of a section of a beam we can determine the concrete strain £ at the fibre located at the steel centroid and the curvature K
CO o
of the section from the formula
a ) t = 0 r = _ p 2 + M -CO o ' E . A ( l + n w C) • E . 1 ( l + n t o C ) C C S c c e l + n o ) "^o " o • E . 1 ( l + n t o C) "^ • E . 1 ( 1 + n w C ) C C S c c
The derivation of the formula is not given here, but is quite simple. The stresses in the steel and concrete can be found with the formulae:
CT / „ \ = £ . E - K . d . E c i ( M ) CO c o c CT /wN = £ . E - K . e . E c m ( M ) CO c o c = £ . E - K . e . E . 1 c m p ( M ) CO* c o * ' c ' l + n u CT /^x^ = £ . E c p s ( M ) CO c CT ,^. = £ . E + K ( y 2 - e ) . E C 2 ( M ) CO c o -^ c CT ^,,v = £ . E s ( M ) CO s
CT . „ V = £ . E + CT n p(M) c o s p , R so(M) cps(M) co s ''po(M) "^p.R ^ ^so(M) p , R = CT 1+na)5 n . M j . e d p , R , o po * 1+nü) C I (l+ntü O s c s R , r e l , 0 A P (only if the p r e s t r e s s i n g o p e r a t i o n t a k e s p l a c e i n a s t r u c t u r e with a bending moment M, a t t h e s e c t i o n ) d
In these formulae (M) indicates that the prestressing force P and a bending moment M, are taken into account.
yi 'cl r i Ccm / Ccps / ac2
Fig. 3 Stress distribution over the section
b) t = t
1 1 ( M . i z i _ E _ . o
'ct(el) E .A • 1+nuC "" ' e 'R,tc c
The derivation of these formulae is not given here.
The stresses in the steel and concrete can be found as follows; CT ^ = e ^/ 1^-E - K^/ TV.d.E
ci,t ct(el) c t(el) c cm,t ct(el) c t(el) c cmp,t ct(el)' c t(el)* * c' l+nw cps,t ct(el) c CT ^ = e ^,- iN-E + K^^ 1 ^ (y2 - e) .E C2,t ct(el) c t(el) •' c . -F - -F R,t R,rel,o CT = n.CT + '— 7^ ^— st cps,t A + A s p CT - = CT „ + CT pt P,R,o st
The bending moment of decompression (M-p.) and the bending moment at which the tensile zone will crack (M ) can be found as follows:
cr ^20,t = ^ " ^ ^ ^ ^ct(el) ^ "t(el)^y2 - e) = 0 ^ 4n M = M w i t h £ / IN + K ^ / i \ ( y 2 - e ) = £ = — = - ^ — cr c r ( e l ) t(el)'-^^ cr E c
With these formulae the stresses at a section can be calculated at every period after prestressing.
If the sustained load is changed in magnitude after a certain time ti, t2, t3 the creep factor to be taken into account for the extra bending moment is in principle given in the diagram.
Because it is assumed that creep is irreversible, only the creep factor
(<ii ~ <ii^ ) , (<i> - (t)^ ) etc. can be taken into account for the extra
00 t ^ 00 ^ 2
load which is acting on the structure at t = ti, t2 etc. In that case we can calculate the reference force with
F, = F^ , .e-^^t + ! £ 1 ^ . S : ^ (1+na.O(1 - e'^^t) ^
R,t R,rel,o i,^ Z,
The influence of Mi over the period (ti-t) can be determined by sub-stracting from the influence over a period t -t the influence over a period t -ti. o
^(tl.tj
^(t11/^
V
jr
^ 1
«m^-—^^ 1
1 _ ^ - — - " ^ — i '
-b^=^^^
to tlt , t
tx t
Fig. 4 (fi-t-relationship in the case where the magnitude of the load changes with time
Therefore we can writer
influence of Mi = Ml.e \-K c
-n.
A (1 - e'^^t _ 1 + e ^^)-n,
influence of cIn general the influence of M on the reference force F is expressed R, t by: e.A - n ^ - n ,
-n,
m = ^ - 1 {Mi(e ^^ - e ''2) + M2(e ^^ - e '^^^ + c + M (e ^tn _ g n„)j n3.1.4 Calculation of stresses in the case of a cracked tensile zone at t = oo if the steel centroids (A and A ) coincide
s ^p
In that case one can start with the reference force F R,oo'
Assume a curvature K of the section with no influence of concrete X
in tension.
1 . Strains
y . K . E = CT (centroid of compression zone) 2x X c cmx yi «s 1 yi-h, L • — — ^ neutral axi center of g \ m Ap A.
Fig. 5 Linear elastic phase with cracked tensile zone
2. EH = 0 - A .y .K .E + A (d - h ).K .E + F„ = 0 ex -^ 2X X c ps X x s R,oo K {- A .y^ + A (d - h ).n}.E = - F„ X ex 2x ps X C R,oo 3. m = 0 {F„ + E .A (d - h ).K }.z = M R,=o s ps X X z = d - y. + k IX IX
by elimination of K we obtain: ^ X M R,^ A .y (d - y + k ) ex 2X •' IX IX A .y - A (d - h ).n • ex 2X ps X
On the left-hand side of the equation there occurs an internal lever arm and on the right-hand side there occur dimensions of the compres-sive zone with a depth h .
.^ M . . The value of h for- a given -= can be determined by trial and error
X R t
or graphically. '
The equation can be evaluated for various values of h . The relationship between these values and h can be plotted in a graph, from which the
^ . M
depth h associated with the given value of — can be determined.
f X ^ F-,
limit LE
r ' P
M
F i g . 6 Graphical d i s p l a y of r e l a t i o n s h i p between and h
R," ^ Limits in this graphs are:
a) Max. value of d. In that case: M
F„ 5-1 • R,o°
3.2 A model to calculate the interaction concrete-steel (ir. G.S. Mulder, Ir. G. Wolsink and Ir. J. Brakel)
3.2.1 The differential equation for the deformation of the cross-section 3.2.1.1 Starting points
Creep of concrete is introduced by a method based on the "rate of flow". This means that only one creep function is used. It is assumed that:
a) the creep of concrete starts at the instant of prestressing; b) the creep of concrete is dependent on the stress in the
con-crete at a given time.
Shrinkage of concrete is assumed to have a time-dependent behaviour similar to that of creep deformation under constant stress. It is also assumed that shrinkage has the same (mean) value over the whole area of a cross-section. Therefore, in an unreinforced cross-section of a statically determinate member, shrinkage will cause no stress and no curvatures.
^ • 2~,SZ^iSS£^5_°£_£2B£^^£^_^n^_s t eel
The CT-£-diagrams of concrete and steel are assumed to be linearly elastic for every strain of the material under normal service conditions.
^' §£l£^5êS_iïl_£oncrete_and_steel
The stresses in the concrete and the steel have to be uniaxial. The concrete and the steel should have a common vertical axis of symmetry.
3.2.1.2 Definitions
^ • 5^Ë£S£Sïl£S_li2Ë
of symmetry, parallel to the axis of the member (Fig. 7 ) . By virtue of the definition of A, S and I the reference line is a horizontal line in the cross-section itself (see 4 . ) .
Fig. 7 Reference lines
2. Force vector k
The force vector k is a vector of which the first component is a force whose direction coincides with the reference line, while the second component is a moment acting in the plane of symmetry of the member (Fig. 8 ) .
reference line
Ü - I M
-fi-F
Fig. 8 Force vector k
3. Deformation vector v
The deformation vector v is a vector of which the first component is the strain of the reference line, while the second component
is the curvature of the cross-section in the plane of symmetry. In a member with length 1 (unity) in x-direction, the elongation of the reference line is the strain E and the angle between the undeformed and deformed cross-section is the curvature K (Fig. 9)
reference line
- [
X .Fig. 9 Deformation vector v
4. RigiditY_matrix_[_ST2_of_the_cross-section
The rigidity matrix [ST] of the cross-section represents the relation between the force vector k and the deformation vector v, according
to:
with and
[ST].v = k
If the strains E and K as shown in Fig. 9 are defined as positive, it follows from considerations of equilibrium that:
[ST] = EA -ES -ES EI in which:
E = modulus of elasticity; A = cross-sectional area;
S = static moment of the cross-section with respect to the reference line;
I = moment of inertia of the cross-section with respect to the reference line.
5. Risidit2_matrix_of_the_conerete_and_steel The rigidity matrix of the concrete:
[C] = E
A -S c c -S I
c c The rigidity matrix of the steel:
A -S [S] = E s s
-S I s s
For a reinforced concrete section: [ST] = [C] + [S]
3.2.1.3 Prestressing
Prestressing can be interpreted as a k vector. However, there is a difference between pretensioned and posttensioned steel.
In the case of pretensioned steel the vector k , caused by the pre-stressing force, acts on a cross-section in which the prepre-stressing steel is incorporated in the rigidity matrix [ST].
In the case of posttensioned steel the vector k , caused by the post--n
tensioning force, acts on a cross-section in which the posttensioned steel has not been incorporated in the rigidity matrix (Fig. 10 and 11). ly reference line pretensioned steel
7 ^*
k = -VF 1
P I
-F .ejP J
V = V - -Vk -n F P -F .e P V -n If k = k , then v ^ v -V -n -V -n
Fig. 11 Force vector posttensioned P.C.
Since the time-dependent deformation appears after the posttensioned steel has been stressed, some kind of transition has to be made from post- to pretensioned steel.
In other words, a fictitious vector k has to be'determined, which gives the section of the member in Fig. 10 the same deformation as it gives the section in Fig. 11.
This idea will be worked out for the general case of a cross-section with mild steel, pretensioned steel and posttensioned steel. A starting point will be that the moment caused by dead weight acts on the cross-section during the stressing of the posttensioned steel.
Therefore the point of departure is a cross-section with mild steel, pretensioned steel and posttensioned steel. Three k-vectors are now acting on the cross-section:
k = k-vector caused by the dead weight of the member; k = k-vector caused by the pretensioned steel;
k = k-vector caused by the posttensioned steel. n
-Furthermore, the following notation will be used:
[S+] = rigidity matrix of steel, including posttensioned steel; [S-] = rigidity matrix of steel, excluding posttensioned steel. Now the deformation vector v follows from:
V = ([C] + [S-]) ^ (k + k + k ) _g _v _n
We have now to define the fictitious k-vector, k ^, which will give -sub
the same deformation v to a cross-section in which the posttensioned steel has been replaced by pretensioned steel:
v = ([C] . [S.])-^k^^, (2)
Substituting (2) into (1) and multiplying both members of the equation by ([C] + [S+]) results in:
k , = ([C] + [S+])([C] + [S-]) ^ (k^ + k + k )
-sub -g -V -n (3)
From here on k . will be used instead of (k + k + k ) , so that all -sub -g -V -n
problems dealing with posttensioned steel can be handled as if the posttensioned steel were pretensioned steel.
All is now ready for the time-dependent calculation. 3.2.1.4 Time-dependent deformation
It is assumed that a cross-section is loaded at time t = 0 and that after a certain period this cross-section has undergone a direct and a time-dependent deformation v(t); Fig. 12.
v(t)
[^(t)
[^(t)
Fig 12 Deformation vector v(t) at t = t
Now the reinforcement is assumed to be free from the concrete, so that the concrete can creep and shrink freely during a short period dt = (t + dt) - t.
The deformation vectors due to creep and shrinkage during this period dt are dv, and dv ; Fig. 13.
reference line dx -du^ dv, = dv = -r dE I dK dE dK
Fig. 13 Change in deformation vector
However, in reality the concrete and the steel have the same defor-mation. It can be shown that the increase of the deformation of
the composite section (steel + concrete) due to creep and shrinkage during the period dt is:
dv(t) = ([C] + [S]) ^[C](dv^ + dv )
- • • -(p -r
(4)
It remains to calculate dv. and dv . -d) -r - dv, :
At time t the deformation vector is v(t); Fig. 12. Because the steel behaves in a linearly elastic manner the part of the k-vector carried by the steel (k ) can be calculated as:
-s k = [S].v(t)
-s
The rest of the k-vector, k , has to be carried by the concrete; so: •
k = k - [S].v(t). c
-Now the elastic deformation of the concrete, v ,, is defined -eel
as:
V . = [C]"''.k or V , = [C] ^(k - [S].v(t))
-eel -c -eel - -
(5)
Then the increase of the deformation vector of the concrete, due to creep, is:
dv^ = 4J .V ,.dt • • (6)
-41 dt -eel or (with (2)): dv^ = 4$ {[C]"^(k - [S].v(t))}.dt (7) (p dt -dv : ' -rShrinkage is only time-dependent. If r(t) is the shrinkage function, the increase of the deformation vector of the concrete due to
shrinkage is:
A dr
dv =
-T--r dt .dt (see also 3.2.1.1, sub 1.) (8) Substitution of (7) and (8) into (4) results in:
: ^ = -([s] + [c])"^[s]. ^7 .v(t) + ([s] + [c])"^ 4$ -k +
dt dt • dt
-(9) + ([S] + [C]) ^[C]. ^
The starting values v of this set of two differential equations can be calculated from:
V = ([C] + [S])'^k. o -Remark
The idea of the "elastic deformation" of the concrete, v ., , will -eel
also be used in the non-linear version of the cross-section analysis However, in this case v . has to be determined iteratively.
-eel
Nevertheless, it will be assumed that (6) remains valid also in the case of non-linear behaviour of the concrete. This will be the main postulate of the time-dependent non-linear calculation of the cross-section (see also 3.2.2.4).
§2iH£l22_2£_.Èil£_dif f erential_e2uation__(9)_
differential equation (9) can be written in its most comprehensive form as:
dv
-^ = f(t)[A].v + f(t).a (10)
[A] and a are respectively a matrix and a vector, which can be derived from (9).
Let [ E ] be the matrix whose columns are formed by the "eigen" values of [ A ] . Now [A] is written as:
[A] = [E] ^ [ E ] [ A ] [ E ] ^ [ E ]
It is known from linear algebra that [ E ] [ A ] [ E ] is a diagonal matrix whose diagonal is formed by the "eigen" values of
[A] : \i and
X2-Multiplication of (10) by [ E ] results in: dv
[E] . ^ = f ( t )
[o'
0 [E].V.+ [E].f(t).a k 1 m n The elements of [E] are assumed to beIntroduce a new vector z = [E].V.
The original simultaneous set of differential equations (10) can now be reduced to two independent first-order differential equations in z1 and z 2:
dzi dt
^ = f(t).Xi.zi + f(t)(kai + laz)
(11) dZ2
-T-2- = f(t).X2.Z2 + f(t)(mai + na2) Here ai and a2 are the components of a.
When zi and Z2 have beem solved from (11), the time-dependent deformation v(t) can be determined from:
V = [E] ^ .z
The general analytic solution has the following form: / s -Si.(t)(t) -S2.(J)(t)
E(t) = Yl + Y2.e ^ f^ ' + Y3.e ^ ^^ r . ^ -Si.(f)(t) ^ ^ -S2.<}>(t) K(t) = Yl* + Y5 .e + Ys .e
When the reinforcement is placed at only one level of the cross-section, another solution is valid.
The general solution was programmed on a TI 59. 3. Stresses
When the deformations £(t) and K(t) have been determined, the stresses in the mild steel, the pretensionsed steel and post-tensioned steel can be calculated, as well as those in the con-crete.
If the Y-coordinate of the reinforcing bar is y , the stresses are:
o ., , ^ T = E {E(t) - y .K(t)} mild steel s •'o
pre t. steel s o o ^ , = E {E(t) - y .K:(t)} + CT
where: CT = stress in the steel before the tendons have been o
released from their anchor block;
CT ^ ^ , = E { £ ( t ) - £ ( 0 ) - y ( K ( t ) - K ( 0 ) ) } + CT
p o s t , s t e e l s o o
where: CT = stress in the posttensioned steel just after grouting; = deformation vector just after grouting of the
post-tensioned steel.
If the Y-coordinate of the concrete fibre is y , the stresses are;
o ^ = E (E , - y .K T) concrete c eel o cel with 'eel
"eel
= V , = [C] ^{k , - [S].v(t)} (see also (5) in —eel sub
--3.2.1.4, sub 1.) 3.2.2 Non-linear stress-strain relations'nip - Statically indeterminate
beam
3.2.2.1 Introduction
be introduced, based on an arbitrary stress-strain relation for the concrete.
The essential part of this method will be illustrated by an example with one degree of freedom: a centrically loaded and reinforced column
(3.2.2.2).
Here the time-dependent deformation will be neglected.
After this the method will be extended to an arbitrarily reinforced cross-section, loaded by a k-vector (3.2.2.3).
By making a simplified assumption concerning the creep of a non-linear stress-strain relationship the model can be extended again to a non-linear time-dependent method of analysis (3.2.2.4).
Finally, a statically indeterminate beam will be designed on the basis of this theory (3.2.2.5).
3.2.2.2 Model with one degree of freedom
If cracking occurs there is no linear relation between F and E , even if time-dependent effects are neglected.
Consider a reinforced concrete tensile member with central loading and reinforcement. Suppose it is possible to determine the tension-stiffness ST = F / E , for any given strain E , where F is the force act-ing on the column and E is the strain (Fig. 14).
F
A
;,£a
^>^//y>/>/y
Fig. 14 Tension bar of reinforced concrete
Fo
^1
^(l 1 / ^ ^ ^ - — • El ^2 ^3 ^o ^ STi = arctan I ST2 - arctan II etc,Fig. 15 Relationship F-E
In this diagram also a particular force F is given. The problem now is to detemiine the strain £ corresponding to F . With regard to this
0 0
it should be realised that in reality the relation between F and E is not known, only the relation between ST and E.
Since ST is known for every E , ST is also known for E = 0; say STi. A first approximation of E , say EI is then obtained with:
El =
STi
Now ST2, corresponding to E I , can be determined. A second, better approximation of E , say E2, is obtained with:
£2 =
ST2 and so on.
Model with two degrees of freedom
Similarly to 3.2.2.2 an iterative process will be developed to calculate the non-linear defoirmation vector v , when an arbitrary cross-section, stress-strain diagram and force vector k are given.
-o
It is assumed that at time t = 0 the creep and shrinkage are also zero. Further a non-linear relation between stress and strain of the concrete is given. In a first approximation a linear stress-strain relation . . of the concrete is assumed.
With this assumption and the known position of the steel in the cross-section it is possible to obtain a first approximation of the deformation vector V , say vi, according to:
vi = [STj.k^.
Here [STi] = [S] + [Ci], in which [S] is the rigidity matrix of the steel and [Ci] the rigidity matrix of the concrete with a linear stress-strain relation.
When vi has been determined, a first approximation of the stress dis-tribution in the concrete can be obtained, making use of vi and the given stress-strain relation of the concrete.
Generally speaking, this stress distribution will differ from the linear stress distribution (Fig. 16, 17 and 18).
^ y
• • •
Fig. 16 Cross-section + k-vector ->• v-vector
L.
reference line^2
7 f f / / r^.
I>
^'^'
H
'-'reduced
° unreduced
Fig. 18 Reduced and unreduced stresses
A second approximation of the rigidity matrix of the concrete can now be calculated "as follows:
CT^^^(y).b(y).dy a,,^(y).b(y).y.dy
'^^='c{ 7-r:öö '^^-"'cf/ CT_,(y)
[C2].= yi unred C = C 21 12 C22 = E ƒ yi unred y2 <^red^y^-^^y^-y'-'^y c CT ,(y) yi unredHere CT j(y) is the stress in the concrete, corresponding to vi, at a distance y from the reference line (Fig. 18).
When [C2] has been determined, a second approximation of the rigidity matrix is obtained from:
[ST2] = [C2] + [S]
With this a second approximation of the deformation vector can be ob-tained from:
V2 = [ST2].k .
- -o
With V2 a third approximation of the stress distribution can be made, and so on.
Generally six or seven iterations proved to be sufficient.
In this chapter this iteration process will be further referred to as:
3.2.2.4 Time-dependent analysis . . '
It is assumed that the cross-section has undergone a certain creep and shrinkage deformation during the period from t = 0 to t = t. At time t = t the force vector k is assumed to be removed and the
-o
steel to be free from the concrete (Fig. 19).
reference line
n
\ ^IJ
*—L -CSl X, CSlFig. 19 Force vector k is removed
Now there cannot be any stress in the concrete. For this reason the deformation of the concrete under these conditions is called the
"stressless" deformation, v ,. , -csl
A part of the removed force-vector is now used to produce in the steel the same deformation as in the concrete. This part k follows from:
k = [S].v , .
-csl
The rest of the force-vector k , namely k - k, is used as input for the function ITER. The result of ITER(k - k ) , is called the "elastic" deformation of the concrete, v ,. This name has been chosen because
-eel
the function ITER is a non-linear elastic calculation. The total deformation of the cross-section becomes:
V = V 1 + V T .
- -csl -eel
The idea of an elastic deformation was already introduced in 3.2.1.4, sub 1 .
This idea is similar to that just mentioned. The only difference is
while here v ., can be a non-linear elastic deformation as well. -eel
According to 3.2.1.4, sub 1 the following proposition is made: dt dt ' -eel"
In 3.2.1.4, sub 1 this proposition could directly be derived from the definition of the creep function; in the case of non-linear behav-iour of the concrete it is only a postulate.
It is now advantageous to define first the stressless deformation v ^
° -csl and then the total deformation v.
The stressless deformation is caused by shrinkage and creep.
Shrinkage only depends on time, creep depends also on stress (or direct elastic deformation). The following differential equation can be es-tablished: dv -, ,, d£ , -csl _ d4_ £ ,1, dt " -eel • dt dt ^0-^ V , = I T E R ( K - [S].V ., -eel -o -csl
so
% i =
at -o -csl dt dt 0 ITER(<- [S].v^ ,)
. ^ . ^ d )
At time t = 0 ^ v , = 0. -cslWith these boundary values the differential equation can be solved. A solution based on a first-order numerical method (Euler) can be given
in a block flowchart (Fig. 20).
Actually, however, a second-order method (Heun) was used, which is about 20 times faster.
3.2.2.5 Statically indeterminate beam
Based on the theory explained in 3.2.2.3 and 3.2.2.4, a computer program was written for the calculation of some statically indeter-minate beams.
(^ START^
1
V T = 0 a t t = 0 - c s l V . = I T E R ( K ) - c e l3
time = t + At A (see 3 . 2 . 2 . 5 ) V , - c s l V , .+ V A<i>it) - (t)(t-At)) + ( r ( t ) - r ( t - A t ) ( „ ) - c s l - c e l O V , = I T E R ( K - [S].V J - c e l - c s l no B (see 3 . 2 . 2 . 5 ) y e s V = V 1 + V , - c s l - c e l (END^ F i g . 20 Flowchart A~2i
77777" B A•~7i
B F i g . 21 H y p e r s t a t i c beamThe loading i s uniformly d i s t r i b u t e d over the l e n g t h of the beam and may be time-dependent.
