Numerical Simulation of the Behavior of Cracked Reinforced Concrete Members

Transcription

1 Materials Sciences and Applications, 2014, 5, Published Online October 2014 in SciRes. Numerical Simulation o the Behavior o Cracked Reinorced Concrete Members Pietro Croce, Paolo Formichi Department o Civil and Industrial Engineering Structural Division, University o Pisa, Pisa, Italy p.croce@ing.unipi.it Received 26 July 2014; revised 6 September 2014; accepted 27 September 2014 Copyright 2014 by authors and Scientiic Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). Abstract Reined non-linear static or dynamic analyses o reinorced concrete structures require the knowledge o the actual orce-displacement or bending moment-rotation curves o each structural member, which depend on the crack widths and on the crack pattern, and ater all on the slip between concrete and reinorcing steel. For this reason the deinition o improved local models taking into account all these local aspects is a undamental prerequisite or advanced assessment o r.c. structures. A numerical procedure which allows to predict the relative displacement between steel reinorcement and the surrounding concrete in a reinorced concrete element, once assigned the stress in the naked steel bar and the bond-slip law is discussed. The method provides as inal outcomes the sequence o crack openings and the individual crack widths, regardless o the particular bond-slip correlation adopted. The proposed procedure is implemented reerring to two relevant experimental case studies, demonstrating that it is able to predict satisactorily actual strain ields and slips along the investigated reinorced concrete elements. Keywords Bond-Slip, Reinorced Concrete, Non-Linear Behavior, Numerical Analysis, Crack Pattern, Crack Opening, Bond Stress 1. Introduction Non-linear static or dynamic analyses are increasingly used to study the behavior o reinorced concrete structures, especially or seismic design [1], but they call or the knowledge o the actual orce-displacement or bending moment-rotation curves o each structural member. These curves represent also the starting inormation or the evaluation o the available ductility o structural elements and or an appropriate estimation o their relevant parameters, required to perorm equivalent elastic How to cite this paper: Croce, P. and Formichi, P. (2014) Numerical Simulation o the Behavior o Cracked Reinorced Concrete Members. Materials Sciences and Applications, 5,

2 analyses, as deined in seismic codes. Evidently, the curves can be obtained experimentally, through sophisticated and typically expensive ad hoc tests, or numerically, resorting to reined theoretical models, suitably validated. Since the precision o the results depends on the type o non-linear analysis which is carried out, the degree o sophistication o the theoretical model can even vary rom case to case. As the non-linear orce-displacement or bending moment-rotation relationships o r.c. members depend on the crack widths and on the crack pattern, and ater all on the slip between concrete and reinorcing steel, advanced models should consider appropriately all these local aspects, usually ignored in classical simpliied approaches, which represent, on the contrary, key issues in reined structural analysis [2]-[7]. Moreover, experimental studies [8] [9] demonstrated that the classical approach to the analysis o reinorced concrete structures, based on the assumption o perect bond between steel rebars and concrete, is unsuitable to predict actual displacements/rotations o structural members, leading to remarkable errors in the evaluation o elements stiness. For these reasons, several studies, also in recent times [10] were devoted to improve the approach, in order to allow to take into account also bond-slip laws. A numerical procedure or the solution o the dierential equation governing the interaction between steel bars and concrete is proposed, which allows to predict the relative displacement between steel reinorcement and the surrounding concrete in a reinorced concrete tie, once assigned the stress in the naked steel bar and the bond-slip law. The proposed approach, which is obviously independent o the particular bond-slip correlation adopted to perorm calculations, provides as inal results the sequence o crack openings and the individual crack widths, or in other words the evolution o the crack pattern in the tie, rom which it can be derived, regarding the tie as a basic sub-element o a r.c. element in bending, the actual non-linear moment-rotation diagram o the element itsel. To validate it, the procedure has been applied to some relevant case studies, aiming at replicating experimental results on r.c. ties, obtained by Bresler and Bertero [11] and by the authors [9]. In the examples, beside the bond-slip models provided in CEB-FIP Model Codes [12] [13], based on ull coninement o concrete members and oten adopted in current practice, an alternative eective bond-slip model, able to take into account indirectly the splitting ailure mode and illustrated in previous papers [8] [9], is adopted. 2. Bond-Slip Modeling in R.C. Elements 2.1. Analytical Modeling As recalled shortly here, the classical equations governing the bond-slip problem can be derived considering equilibrium and compatibility conditions o an ininitesimal r.c. tie. A tie is considered, the length o which is dx (Figure 1), made up by one centered rebar and by the surrounding portion o concrete in tension, subject to a tensile orce F. The equation governing the global equilibrium o the tie is trivially F = F + F, (1) s where F s and F c represent the quotas o the total orce acting on steel and on concrete, respectively. Let φ the diameter o the steel bar and A c, depending on x, the concrete area involved by the bond stress diusion. Dierentiating Equation (1) it results c Figure 1. Equilibrium o an ininitesimal r.c. tie. 884

3 and then 2 πφ df = dfs + dfc = dσs + Ac( x) dσc + σcdac( x) = 0, (2) 4 ( x) ( x) dac dσc = ρ( x) dσs σc, (3) A where σ s and σ c are the stresses in steel and concrete, respectively, and ρ(x) is the geometric reinorcement ratio. Said τ(s) the bond stress, unction o the slip s, the local equilibrium o the rebar over the length dx gives and, inally, the compatibility condition, yields c 2 πφ dσ s = πφτ ( s) dx, (4) 4 ( εs εc)dx= ds, (5) d s x dx ( ) = ε ε, (6) being ε s and ε c the strains in steel and concrete respectively. Combining Equations (2), (4) and (6), the ollowing non-linear dierential equation is obtained ( s) 2 d d d c d = 2 εs + ρ( ) εc + c x φ dσs dσc Ac( x) dx d s c 4τ σ s x A ( x), (7) which can be solved, under appropriate boundary conditions, once input the stress-strain relationships or steel and or the tensile branch o concrete, the bond-slip law and the dependence o A c on x Numerical Solution To solve the non-linear Equation (7), the procedure described hereinater can be implemented, based on the inite dierence method. It is worth to underline that the main diiculty in inding the solution concerns the assignment o the boundary conditions, since, in the present case, the boundary itsel is unknown. The most direct way to ix the boundary conditions is to assign the slip, s 0, and its irst derivative s 0 in a cross section where σ c = 0, typically corresponding to a cracked section or to an end section o the tie. Moreover, since in this section ε c = 0, rom Equation (6), it results s 0 = ε s 0, as soon as the steel strain ε s0 is assigned. Since boundary conditions themselves depend on the strain ield in the concrete tie, solution o Equation (7) requires an iteration process, like the one described hereinater. The proposed procedure is based on an iterative shooting solution o the system o ODEs d F σ sa s s = εs( σs) εc dx Ac ( x) d 4 σs = τ( sx, ), dx φ suitably derived rom Equation (7). The irst phase (phase I), summarized in the low chart shown in Figure 2, allows to deine the boundary conditions up to the irst crack opening in a reinorced concrete tie, the length o which is L *, through the ollowing steps: 1) set the origin o the x-axis in the starting cross section x = x 0 = 0, which coincides with an end section o the tie, where σ c = 0, and subdivide the hal o the tie in N intervals o constant length x; 2) assign the σ-ε constitutive laws or steel and or concrete under tensile stresses; assign the bond-slip law, τ-s; assign the A c (x) unction; (8) 885

4 Figure 2. Flow chart o the proposed numerical procedure up to the irst crack opening. 3) set i = 1, assign tentatively the boundary conditions in x = 0, s 0 and s 0 or equivalently s 0 and ε s0, being s 0 = ε s 0, and, using the bond-slip law and the steel constitutive law, evaluate the bond stress τ 0 (s 0 ) and σ s0 (ε s0 ); 4) starting rom xi 1 integrate (8) using or instance the Runge-Kutta 4 th order method, and determine the slip s i and the steel stress σ s,i at the end o the interval, x i ; 5) using the bond-slip law, evaluate the bond stress τ i (s i ) in x i ; 6) using the equilibrium condition (3) o the tie segment, calculate σ ci, σ = σ A 1 A ρσ ρ σ ( ) ci, ci, 1 ci, ci, 1 i si, i 1 si. 1 A ci,, through the general equation where A c,i is the concrete area at section x i, and ρ i the geometric steel to concrete ratio at the i-th section; 7) stop the iteration when s i = 0, and consequently τ i = 0, or when i = N and check convergence (step 9); 8) set i = i + 1 and iterate the process rom step 4; 9) check convergence: i condition ε s = ε c is satisied at x i, convergence is achieved; otherwise keep unchanged the value o ε s0 and assign a new value or s 0, iterating the process rom step 6 till to convergence; 10) once the convergence is achieved, retain the actual value o s 0 (ε s0 ) and the corresponding real curves ε s (x) and ε c (x) along the tie, together with the actual transer length L a (ε s0 ), as shown in Figure 3. O course, the searching o s 0 can be reined using one o the usual numerical root inding methods, like the secant method, the alse position method or the Newton-Raphson method. Considering that actual values o s 0 (ε s0 ) and ε s0 represent the real boundary conditions at x = 0 when the stress in the naked bar is σ s0 (ε s0 ), the actual boundary conditions can be derived, in terms o ε s0 and s 0, in the whole signiicant range o the loading process, simply increasing ε s0 and iterating again the procedure. Clearly, in the loading process described by the increase o ε s0, the strains ε s and ε c at x = L a and the transer length L a itsel are increasing unctions o ε s0. As soon as the concrete ultimate tensile strain ε ct is attained, or εs0 = εs0, setting La = La( ε s0 ) and s0 = s0 ε s 0 and, provided that the condition or crack ormation, ( ) is ulilled, a irst crack opens at ( 1 ) * a L 2 x= L, satisying the inequality L, (10) a 2 a L L < L. (11) (9) 886

5 Figure 3. Typical distribution o steel and concrete strains in the uncracked tie or a given ε s0. When the crack opens, the strain distributions ε s and ε c on the length L abruptly change and a discontinuity ( 1* ) occurs in the s 0 -ε s0 diagram. The updated strain distributions as well as the updated slip value s 0 can be then derived iterating the procedure or the unique case ε s0, setting the tie length to L, as described aterwards. As known, other cracks can orm or ε s0 values slightly greater than the identiied steel strain level ε s0 until a steady situation is achieved in the whole tie. In this situation consecutive cracks are spaced less than 2 L a, so that the condition σ c < ct is satisied in each point o the tie and the irst crack pattern is ully stabilized. The actual crack locations are generally indeterminate, as the tensile strength o concrete, ct, is a random variable and the unique constraint or L is the inequality (11). In any case, the L distances along the tie, and consequently the irst crack pattern through the Monte Carlo method can be obtained, assigning, or example, a random concrete tensile strength distribution along the tie itsel. Nevertheless, even i the irst phase crack is stabilized, urther cracking is not prevented or increasing values o the external tensile orce F. In act, since the slip s increases as F, and τ depends on s, it can happen that concrete tensile strength ct is reached again or F = F (2) ( 2) ( 2) within a distance L, L L 2, determining a new stabilized cracking pattern. The subsequent phases can be investigated properly modiying the above illustrated procedure. Reerring to the j-th step, the modiied procedure is summarized in the low chart in Figure 4: 1) set the origin o the x-axis in the starting cross section x = 0 o the considered portion o the tie, bounded by ( j 1) adjacent cracks spaced L, and subdivide hal o the interval in M parts o constant length x; 2) set i = 1, assign tentatively the boundary conditions s 0 and s 0, s 0 = ε s 0, at x = 0 and evaluate the bond stress τ 0 (s 0 ) and σ s0 (ε s0 ), bearing in mind that, at the irst step ater the (j 1)-th crack ormation phase, it holds ( j 1) ε the equality εs0 = s0 ; 3) assume the unction s(x) is antisymmetric in the interval L ( j 1) : this implies that, disregarding riction orces, the convergence condition is met i the slip s vanishes at midpoint o the interval, where the concrete stress σ c reaches the local maximum σ cmax, then solve Equation (7) as above (steps 3 to 9); ( j 1) 4) as σ cmax increases with ε s0, a new crack opens at L ( j) 2 as soon as the strain ε s0 attains the value ε s0, ( j) corresponding to σ =. Set ( j) ( j 1) L L cmax ct = 2 and iterate again the process rom 1), until ε s0 attains the ultimate value, ε su. Finally, crack ormation ends and the crack pattern is completely deined when, along the whole tie, it results σ c < ct, whichever the value o ε s0. In the inal stage, experimental evidence conirms that tension stiening is negligible, so that tie behavior tends to it the naked bar behavior. Once completed the procedure, the evaluation o each crack width is trivial, being the sum o the quotas s 0 pertaining to the two parts o the tie delimited by the crack. 887

6 At the end o each iteration step, as soon as convergence conditions are satisied, the couple o boundary conditions, s 0 and ε s0, describing the evolution o the loading process, is inally obtained. In Figure 5 a typical outcome o the procedure s implementation is shown in terms o s 0 -ε s0 curve or the end cross section o a reinorced concrete tie, which has been previously investigated adopting the CEB-MC90 [12] bond-slip relationship illustrated in 3.3 [8]. The example is reerred to a r.c. cylindrical tie reinorced by an axially located 16 mm B450C steel bar. The diameter o the tie, 0.8 m long, was 112 mm. Figure 4. Flow chart o the proposed numerical procedure ater the irst crack opening. Figure 5. s 0 -ε s0 in a r.c. tie (ø 112 mm, 16 mm rebar) according to CEB bondslip model [8]. 888

7 Inspecting the diagram, it is possible to identiy the dierent phases o the loading path, the ormation o ( i) cracks and in particular the discontinuities corresponding to the cracking phases: or the given ε s0 the slip s 0 at the crack location suddenly decreases, relecting the abrupt redistribution o the strain ields in the concrete. 3. Case Studies To validate its eectiveness, the procedure has been to simulate the outcomes o two experimental test campaigns carried out by the authors [9] on ull scale specimens o r.c. ties (case study 1) and by Bresler and Bertero [11] on similar specimens (case study 2). Essentially, the actual (measured) strains on the steel reinorcing bars are compared with those obtained numerically by means o the proposed method. It must be stressed that the classical CEB correlation is valid only when the directly loaded rebar is pulled out rom a ully conined member, where the slip attains values o several millimeters, while in usual structures, like bending beams, the slip rarely exceeds mm. This behavior can be explained considering that, due to tangential tensile stresses around the rebar, the concrete cover splits beore ull activation o bond strength (splitting ailure mode). Obviously, the splitting ailure mode cannot be taken into account directly in one-dimensional bond-slip models, but its eects can be indirectly modeled by using eective bond-slip relationship, suitably determined. On the basis o this consideration, to widen the signiicance o the analysis and urther corroborate the results, beside the classical CEB bond-slip relationship, an alternative eective bond-slip relationship, illustrated in 3.3, has been adopted Test Arrangements The reinorced concrete ties we tested in [9] (case study 1) were made up by a single bar, 16 or 20 mm in diameter, axially located in a concrete cylinder, whose diameter was about 132 mm. More in detail, the results obtained rom short specimens, 235 mm long, are considered here. These specimens were setup to prevent the crack ormation, in such a way to have the opportunity to check the accuracy o the procedure till to the appearance o the irst crack, since the successive phases (cracked specimen) are the reiteration with dierent boundary conditions o phase I itsel. The strains along the steel bar were measured using electrical strain gauges, spaced 25 mm one another and embedded in two diametrically opposite milled grooves (3.5 4 mm) (Figure 6). Moreover, the main objective o tests was the deinition o a suitably improved eective local bond-slip relation law, alternative to the CEB ones, able to take into account indirectly the eects o the splitting ailure mode on the actual non-linear behavior o reinorced concrete members. Bresler and Bertero [11] perormed similar tests on r.c. ties, axially reinorced by one single rebar (case study 2), again instrumented with strain gauges. In this case the strain gauges were located in a milled groove inner to the steel bar, previously cut in two halves and subsequently reassembled, according to a technique previously Figure 6. R.c. tie tested in [9]. 889

8 developed by Mains [14]. The results, which have been considered or the comparison, reer to cylindrical specimens, 406 mm long (16 ), whose diameter was 152 mm (6 ). The diameter o the reinorcing bar, placed axially, was 28.6 mm (1 1/8) (Figure 7). Since the concrete cylinder presented a notch at midspan to predetermine the location o the crack, the analyses reported here reer to the hal-length L/2. Among the available data or case study 2, they have been selected those reerring to the specimen H or the increasing branch o the load cycle nr. 2, which can be assimilated to a monotonic loading phase extending rom 17.8 kn (4 kips) up to kn (30 kips) Basic Assumptions In numerical simulations, constitutive laws or steel and concrete have been modeled according to the Ramberg- Osgood ormulation [15], in the orm n σ a σ ε σ s c s c c c ( ) = + and ε 1 c( σc) m 1 s s n Es Es y m σ a σ = +, (12) E E c c ct being y the yield stress o the steel, ct the tensile strength o concrete, and a s, n, a c and m the curve parameters. The mechanical properties and the curve parameters pertaining to case study 1 were obtained suitably itting the experimental data derived rom ad hoc tests [9], while in case study 2, due to lack o inormation, the limited available experimental data were used, appropriately adapting the parameters evaluated in case 1. The actual yield stress o steel rebars was y 545 MPa in case 1 and y 690 MPa in case 2, while the compressive strength o concrete was c = 29.6 MPa in case 1 and c = 40.8 MPa, in case 2. Furthermore, it has been assumed that the eective concrete area A c (x) varied steeply rom zero to A c within a distance approximately twice the rebar s diameter, according to an exponential law. Anyhow, sensitivity analyses showed that results do not vary signiicantly rom those obtained assuming a constant value or A c Bond-Slip Laws As recalled above, a bond-slip correlation currently adopted in Literature is that proposed by the CEB in Model Code 90 [12], slightly modiied in the inal drat o ib Model Code 2010 [13]. Inspecting the bond-slip laws proposed in Model Codes 90 and in Model Code 2010 or concrete class c 30 MPa (Figure 8), it results that, within the scopes o the present work, the dierences between the two curves are not critical. As discussed in [15], the bond-slip curves proposed by CEB/ib, in particular in the ield where bond deteriorates, are associated with slip values well beyond values actually measured in r.c. members, where rebars are eiciently anchored. Moreover, as said beore, these values would be incompatible with eective crack widths currently measured in real structures, which are directly correlated with actual slip values. Consequently, as clearly demonstrated by the implementation o the procedure too, the adoption o CEB τ-s laws implies that the maximum level o bond stress cannot be attained in real structures, so that the decreasing right branch o τ-s curve, associated with bond deterioration, cannot be covered. On the basis o the reined measurements o steel stress, carried out on the above mentioned r.c. ties during monotonic tensile tests [9], an alternative eective bond-slip relationship can be proposed. The discussion o criteria adopted to derive this law, which is illustrated in Figure 9, again or a c 30 MPa concrete, is out o the Figure 7. R.c. tie tested in [11]. 890

9 Figure 8. CEB-MC 90 and ib MC2010 bond slip models or concrete ( c 30 MPa). Figure 9. Proposed eective bond-slip model or concrete ( c 30 MPa). scope o the present work; nevertheless, however it must be highlighted that the maximum bond stress (which is the same adopted in CEB-FIP models) is associated to a slip value which is signiicantly lower than the corresponding ones in MC90 and in MC2010 models Discussion o Results Using the proposed procedure, the behavior o the two ties tested in case studies 1 and 2 have been simulated, or increasing value o the external orce applied to the rebar. For selected load steps, chosen to appropriately cover those or which experimental data were available, the results are summarized in Figure 10 and Figure 11, pertaining to case study 1 and case study 2, respectively. In the diagrams, or each load step, the curves representing the measured experimentally strain distributions ε-x along the steel bar are compared with those obtained by the implementation, in turn, o the bond-slip correlation suggested by the CEB Model Code 90 (Figure 8) and o the alternative eective bond-slip law suggested in Figure 9. In order to acilitate the comparisons, in the igures the curves obtained adopting CEB model, designated as CEB, are plotted in grey, the curves obtained adopting the alternative model, designated as A, are plotted in black and inally the experimental ones are plotted with piecewise linear curves. Aiming to supplement the inormation, the three amilies o τ-x experimental and numerical curves, describing the bond stress variation along the tie in case 1, are reported in Figure 12. The diagram clearly shows that, as expected, CEB model excites very low bond stresses. Inspecting the diagrams it can be remarked that, in both cases, the proposed procedure allows to perorm reined local investigations and that, even i outside the scope o the present paper, 891

10 P. Croce, P. Formichi Figure 10. Measured and calculated ε-x curves (case 1 [9]). Figure 11. Measured and calculated ε-x curves (case 2 [11]). 892

11 Figure 12. Measured and calculated ε-x curves (case 2 [11]). the use o CEB bond-slip model leads to underestimate the bond stress transer rom steel to concrete; in act, the calculated strains in the rebar are markedly higher than those actually measured and the calculated bond stress are sensibly smaller than the actual ones; on the contrary, the use o the bond-slip model proposed in [9] leads to a good it o bond stresses and o measured steel strains; moreover, the slip values calculated adopting the two alternative τ-s models at end sections o the tie do not ully relect the signiicant dierences registered above; in act in both cases maximum slips are limited to 0.2 mm; but, in case o CEB correlation, these values are one order o magnitude lower than the slip interval associated with the plastic range o the bond-slip CEB curve. This conirms that, as just explained, in r.c. members actual slips are well below those corresponding to the horizontal plateau or to the descending branch o CEB curves. 4. Conclusions A numerical method or the study o the evolution o the crack pattern in r.c. ties, under monotonic loading, simulating the tensile part o a more general r.c. member in bending, has been illustrated. As soon as the stress-strain relationships or steel and concrete in tension are given and a suitable eective bond slip model is adopted, the method allows to calculate the actual distributions o steel and concrete strains and stresses along the tie and, consequently, the actual distributions o bond stresses and slips between the steel reinorcement and the surrounding concrete. A comparative implementation o the procedure is discussed with reerence to the CEB bond-slip model and to an alternative eective bond-slip model, derived rom experimental results obtained in a previous experimental study on concrete ties reinorced with instrumented rebars. The numerical simulations o the ad hoc tests as well as o other similar experimental data available in Literature, conirmed that the proposed algorithm is satisactory and can be suitably adopted or reined non-linear analyses o r.c. ties, whichever the bond-slip model taken into account. 893

12 The numerical procedure can be easily extended, with slight modiications, to more general cases, like beams in bending or beam to column joints, in particular when reined and realistic evaluations o the actual behavior are required. Reerences [1] EN (2004) Eurocode 8: Design o Structures or Earthquake Resistance Part 1: General Rules, Seismic Actions and Rules or Buildings. CEN, Brussels. [2] Ciampi, V., Eligehausen, R. and Popov, E.P. (1982) Analytical Model or Concrete Anchorages o Reinorced Bars under Generalized Excitations. Report. No. UCB/EERC83/23, University o Caliornia, Berkeley. [3] Eligehausen, R. and Bigaj-Van Vliet, A. (1999) Bond Behaviour and Models. Structural Concrete, the Textbook on Behaviour, Design and Perormance. CEB-FIP Bulletins 1, 2, 3. ib, Lausanne. [4] Park, R. and Paulay, T. (2001) Reinorced Concrete Structures. John Wiley & Sons, New York. [5] Tastani, S.P. and Pantazopoulou, S.J. (2002) Experimental Evaluation o the Direct Tension Pullout Bond Test. Proceedings o Symposium on Bond in Concrete From Research to Standards, Budapest, November 2002, [6] Borosnyói, A. and Balázs, G.L. (2005) Models or Flexural Cracking in Concrete: The State o the Art. Structural Concrete, 6, [7] CEB-FIP (2000) Bond o Reinorcement in Concrete. Bullettin nr. 10. ib, Lausanne. [8] Beconcini, M.L., Croce, P. and Formichi, P. (2008) Inluence o Bond-Slip on the Behaviour o Reinorced Concrete Beam to Column Joints. Proceedings o International ib Symposium Taylor Made Concrete Structures: New Solutions or our Society, Amsterdam, May 2008, [9] Formichi, P. (2010) Bond-Slip Laws and Coninement Conditions or Reinorcing Bars in r.c. Elements (in Italian). University o Pisa, Pisa. [10] Casanova, A., Jason, L. and Davenne, L. (2012) Bond Slip Model or the Simulation o Reinorced Concrete Structures. Engineering Structures, 39, [11] Bresler, B. and Bertero, V. (1968) Behavior o Reinorced Concrete under Repeated Load. Journal o the Structural Division ASCE, 94, [12] CEB-FIP (1993) Bullettin d inormation 213/214 CEB-FIP Model Code Thomas Telord, London. [13] Fédération Internationale du Béton (ib) (2012) Bulletin n. 65/66 Model Code Final drat, Volume 1, 2. ib, Lausanne. [14] Mains, R.M. (1951) Measurements o the Distribution o Tensile Bond Stresses along Reinorcing Bars. Journal o the American Concrete Institute, Proceedings, 48, [15] Ramberg, W. and Osgood, W.R. (1943) Description o Stress-Strain Curves by Three Parameters. Technical Note No. 902, National Advisory Committee or Aeronautics, Washington DC. 894

13