Behaviour of a single crack in reinforced and plain concrete subjected to sustained shear loading: Empirical formulae for the crack displacements

and reinforced concrete subjected

to sustained shear loading

Empirical formulae for the crack displacements

jr. J . W . Frénay

1 1

T U Delft

Faculty of Civil Engineering

Division of Mechanics and Structures

Section of Concrete Structures Stevin Laboratory Delft University of Technology

Behaviour of a single crack in reinforced and plain concrete subjected to sustained shear loading

Empirical formulae for the crack displacements

-b y

J.W. F r e n a y

a

S T E V I I V J

Mailing address:

Delft University of Technology Concrete Structures Group Stevinweg 1

2628 CN Delft The Netherlands

L A B O R A T O H Y

CONCRETE STRUCTURES

No p a r t of t h i s r e p o r t may be published without w r i t t e n permission of t h e author © 1 9 8 6 DUT, Delft, Holland.

Technische Universiteit Delft Bibliotheek Faculteit der Civiele Techniek

(Bezoekadres Stevinweg 1) S f^ Postbus 5048 S-- P^. // 2600 GA DELFT

CONTKNTS page

Preface 4

Summary 5

1. Introduction "^ 6

2. Static shear strength Tu 10

3. Experiments and handling of test data 12

4. Analysis of the test results 14

5. Determination of formulae for the directly occurring and the

time-dependent crack displacements under sustained shear loading 18

5.1 Push-off specimens with embedded reinforcing bars 18 5.2 Push-off specimens with free restraining rods 20

6. Development of creep coefficients (pw and (ps 22 6.1 Reinforced concrete push-off specimens 22 6.2 Plain concrete push-off specimens 34 6.3 Non-linearity of creep coefficients 43

7. Shrinkage and creep of concrete 52

8. Conclusions 57

9. References 58

10. Notation 59

PREFACE

The work carried out is part of the activities within the CUR committee A26

Concrete Mechanics (CUR= Netherlands Centre for Civil Engineering Research, Codes and Recommendations). It is being supervised by Prof.Dr.-Ing. H.W. Reinhardt and Prof.Dr.Ir. J.C. Walraven of Darmstadt University of Technology (Germany).

SIM4ARY

Empirical formulae for the directly occurring crack displacements wei and sei as well as for the time-dependent displacements wc(t) and Sc(t) have been derived for cracked concrete specimens subjected to sustained shear loading. Steel bars perpendicularly crossing the crack-plane provided its restraint stiffness. Two cases have been distinguished:

- embedded reinforcing bars; - free restraining rods;

Both circumstances were evaluated by means of a regression analysis. The

complete displacement behaviour of all the tests is presented by a set of eight formulae, being functions of the experimental variables fee; pfsy; t/Tu and (Tco. The proposed ranges of the variables are indicated. No significant

contribution of the initial crack width Wo to the displacement behaviour could be demonstrated due to its minor variation. The calculated displacements are in good agreement with measured test data.

Next, the development of creep coefficients <pw =wc (t)/we i and <Ps=Sc (t)/se i is presented graphically. The formulae allow fast and accurate computation of the time-dependent displacements. Moreover, experimental results can now easily be compared with the results of the theoretical model to be developed for

1. INTRODUCTION

As a result of the increase in scale and complexity of new structures,

sophisticated methods are being used for structural analysis. These methods take account of the non-linear behaviour of cracked reinforced concrete. Whereas in the case of bending the behaviour of reinforced concrete has been extensively investigated and the physical model is generally accepted, there is still a lack of knowledge and of modelling in the case of shear forces, especially when the concrete is cracked. In chapters 1-4 of this report, the results and analysis of previous sustained shear tests will be briefly summarized. The experiments were carried out in the Stevin Laboratory on cracked push-off specimens. Detailed information has been presented in [l]-[4]. Next, in chapters 5-7 empirical

formulae have been determined describing the observed time dependent displacement behaviour of the crack under shear loading.

Fig. 1 Types of push-off specimens used: (a) with embedded bars; (b) with

free restraining rods , Dimensions in iran.

The 400x600 mm^ push-off specimen has a central crack-plane with a 120x300 mm^ cross-sectional area Ac, which is formed before the start of the actual shear test. The initial crack width is Wo. Both types of specimens are presented in fig. 1.

As for the reinforced specimens, the restraint stiffness in the direction perpendicular to the crack plane is achieved by means of 8 mm diameter

deformed bars embedded in the concrete. For the unreinforced specimens, four smooth steel rods were used, which were placed in cylindrical tubes running

perpendicularly through the crack plane. The rods are attached to the short faces of the specimen by means of bolts. The axial steel stresses were measured by means of strain gauges glued on the surface of each rod. Further information is presented in [1,2].

The push-off specimen is loaded by a force V. To improve the gradual and centric introduction of the external shear force into the crack-plane, the cantilevers at both top and bottom of the specimen have been post-tensioned transversely. Each specimen was subjected to sustained shear loading of at least 90 days' duration. The displacements of the crack-plane were recorded periodically on the front and

the rear face of the specimen. The shear stress T=V/Ac is applied at a loading rate of approximately 0.01-0.03 N/mm^ per second.

The directly occurring displacements wei and sei of both crack halves are

measured when the sustained shear stress T has just been reached. See fig. 2. The adjusted shear stress level is defined as T/Tu in which Tu is the static shear strength of a push-off specimen. Also, these displacements correspond to a time duration t=0 hrs. With increasing duration of the load, the displacements w(t) and s(t) increase. As appears from fig. 2, it follows that:

t^O hrs : w(t) = wei + Wc(t) [mm]

s(t) = sei + Sc(t) [mm] ..(1)

in which wc(t) and Sc(t) represent the displacement increments from the instant t=0 hrs onwards. 1 0 (N/mm^l 1:0 1 A . I ! Welti 1

l \

i

tlN/mm^i Wltrml Simml

In summary, the experimental variables are: -cube compressive strength of the concrete

fee=51 or 70 N/nmi2. The two mixes contained Portland cement (medium rapid-hardening) and glacial river aggregates having a maximum diameter Dmax^lömm and a grading curve according to Fuller (table 1). High-strength concrete was chosen in view of offshore structures;

Table 1. Mix proportions and 28-day strength.

_ _ _ - - - — ^ ^ ^ cement content w/c ratio aggr./cem. ratio superplasticizer compaction index air content fresh density

cube compr. strength coeff. of variation mix A 325 0.50 5.98 — 1.15 1.2 2430 51 6.7 Mix B 420 0.38 4.47 2.50 1.10 2.0 2453 70 6.1 [kg/m^] .[-] [-] [% of cem.]

[-]

w

[kg/m3] [N/mm2] [%]

- normal restraint stiffness

* embedded bars; reinforcement ratio p=0.011, 0.017 or 0.022 i.e. 4, 6 or 8 stirrups respectively, all perpendicularly crossing the crack plane. The yield strength of the 8 mm diameter deformed bars was fsy=460 and 550 N/mm^

respectively. The relative rib areas were fR=0.050 and 0.059 respectively; * restraining rods}four 10 ram or 16 mm diameter rods with fsy=335 N/mm^ were used for each specimen.

- sustained shear stress level T/Vu

% embedded bars: the shear stress varied between 1=5.7-11.5 N/nini^ , i.e. 45%-89% of the static shear strength Tu;

* restraining rods;the shear stress varied between r=4.0-6.5 N/mm^, i.e. 49%-84% of Tu.

- initial concrete compressive stress Ceo

The initial restraint normal compressive stress on the cracked concrete surface was (7c 0 = 1. 0 or 2.0 N/min^ ;

- initial crack width wo

The initial crack width varied between Wo=0.01-0.05 mm.

Average values of Wo were 0.03 and 0.02 mm for the reinforced and the plain conrete specimens respectively. The corresponding numbers of tests carried out were 32 and 8.

The specimens were cured in a fog chamber (19°C:99% RH) for 22 days. Next, they were placed in the laboratory (20°C: 50% RH). Tests started when the concrete had reached an age of 28 days.

2. STATIC SHEAR STRENGTH Tu

The sustained shear stress of the long-term tests is referred to the static shear strength of the push-off specimens of Walraven [5]. An empirical formula has been derived for the case of embedded reinforcing bars [L]. The data of four different test series were used:

P

Tu = oc(pfsy) [N/mm2] .. (2)

in which: a. = 0.822.fccO " o ^ and P = 0.159. fcc° 3°^ , for which

fee represents the 150 mm cube compressive strength; p the reinforcement ratio and fsy the steel yield strength. From 88 experiments it was found for the average ratio Tu(exp.)/ru(calc.)=1.001 with a coefficient of variation of 10.9% [4].

Static shear tests on plain concrete performed by Walraven [5] and Daschner [2] yield another relation for Tu:

Tu = 1.647.(rcoO-427.f^^o.32i [N/mm2] ,.(3)

with a coefficient of correlation of r=0.99. Note that the diameter of the restraint rods was also an experimental variable having no significant

influence on Tu. The equations (2) and (3) are presented graphically in figs. 3 and 4 respectively.

shaor stress T^j [N/mm^l

p-lsylN/mm'l

Fig. 3 Shear strength Tu according to equation (2) for reinforced

/• •

2.A

\l\°' '[N/mm*]

20 1.6 1.2 0 8 0.4 «^ ,

>;r^

^ > ^ ^ o B -^ •

1 1

X TH Dtlft « TU MUnch 1 1 O rn 1 0 2 0.4 0 5 0.8 1.0 U 1.4 1.6 18 ff/t, ass [N/mm J

Fig. 4 Shear s t r e n g t h Tu according t o equation (3) for p l a i n c o n c r e t e push-off specimens [ 9 ] .

3. EXPERIMENTS AND HANDLING OF TEST DATA

The development of the average crack width w and the parallel displacement s has been graphically presented as a function of the loading period. The measured displacements needed correction in view of the normal and shear deformations of

the concrete between the reference points near the crack. Regarding the specimens

with internal restraint rods, the development of the normal concrete compressive stress Cc(t) was also measured. Examples of the measured time dependent

displacements are shown in figs. 5 and 6.

Welt) [«"'mm) ScM)l10-'mml

10 • 10' tlhrsl

Fig. 5 Measured displacements wc and sc for a specimen with embedded reinforcing bars T=11.4 N/mm^; T/Tu=0.70; wei

and Se1=0.063 mm; fsy=550 N/mm2 [6].

Sc(tll10-'mml

0.128 mm

tthrsl

F i g . 6 Measured d i s p l a c e m e n t s Wc and se f o r a s p e c i m e n w i t h i n t e r n a l

r e s t r a i n t r o d s ; T=6.5 N/mmS;T/ru=0.74; Wei=0.044 mm and s e i = 0 . 0 2 4 mm [ 6 ] .

Functions We(t), Se(t) and s(w) were determined for each specimen by means of a statistical analysis. The displacement behaviour has been expressed by so-called "power-functions". For each specimen three equations have been computed:

[mm] ••(4) [mm] • -(5)

[mm] ..(6)

in which cti, a2...ai2 are coefficients obtained from a non-linear regression analysis. The average difference between measured and calculated displacements was less than 0.007 mm, which is almost equal to the accuracy of the measuring device used. The boundary conditions of eq. (4), (5) and (6) are:

s(w=Wo) = 0.0 [mm] We (t = 0 hrs) = 0.0 [mm]

se(t=0 hrs) = 0.0 [mm] ••(7)

The computed displacements deviated by less than 0.002 mm from the theoretical values in eq. (7). Figs. 5 and 6 show examples of the calculated curves. The 90% confidence intervals of the displacements have been indicated. Note that t=105 hrs correspond to approximately 11.4 years. The mathematical treatment used for the measured data has some advantages:

- all the measurements are transformed to their logarithmic values, so that the small displacements as well as the short periods of observation also contribute significantly to the definite equations of the curves;

- the behaviour of the displacements is analysed in an objective way; - the scatter of the measurements and of the displacement behaviour is

quantified;

- the computation allows extrapolation and interpolation to loading periods which have not been investigated experimentally.

The regression equations of the individual experiments could nqt be combined mathematically, nor could they be expressed as functions of the experimental variables. This is the reason for a further analysis of the test data.

s(w) = tti+tta. (w+ffa)^ Wc(t) = a5+a6.(t+a7)^8

a

S e ( t ) = ttg+aio.(t+aii) 12

4. ANALYSIS OF THE TEST RESULTS

There is a need to develop analytical formulae describing the displacement behaviour of the push-off specimens investigated:

- the effect of each experimental variable can be studied; moreover, it allows a so-called sensitivity analysis;

- non-tested circumstances can be simulated;

- formulae are convenient for comparing test data with the results of a theoretical model for shear transfer in cracked concrete.

It should be noted that by means of analytical formulae the displacement behaviour will be unique for a chosen combination of the test variables. This fact is in contrast with experimental reality; measurements on two or more "identical" specimens will differ and display scatter caused by measuring errors and by the heterogeneity of concrete which gives rise to variations of the complicated time-dependent effects (shrinkage, creep, bond between concrete and steel). The further analysis of the test results is composed of two

consecutive phases:

1) Description of the directly occuring and the time-dependent displacements In chapter 3 it is concluded that the regression equations (4),(5) and (6) of the individual specimens could not be combined mathematically. Next, four

boundary conditions were formulated for each equation (eq. (4), (5) and (6)) of one specimen. These boundary conditions could be combined and expressed as regression functions of the experimental variables, by means of a statistical treatment. The last step is the computation of new equations (4), (5) and (6) which satisfy the boundary values. This approach demands an implicit

calculation method by a micro-computer and is reported in [2,3]. Similar

calculations can be executed for the static curves (T-w, T-s) which are related to the application of the sustained shear loading. The mathematical method is summarized in fig. 7. An example of the calculation of an expression for we(t) according to eq. (5) will now be given:

wc(t=0 hrs) =0.0[mm] dwc(t=100 hrs)/dt =gi [mm/hrs] wc(t=2000 hrs) =g2 [mm]

in which gi, gz and g3 are functions of the variables. Next, the new coefficients o(5-«8 can be computed implicitly.

boundary conditions for llmc-dtpL curvti ^ . ' l l • • • • 3»

vtrificaiian o( additional conditiont for t(wl

find « i . a , - a » for «titl via Nrvlon-Rophion

find 0 ( , « : . - « 4 J for ic») and s4w| via N.-R. Isom* proctduri as •n^\i\

Wj(ll,«t(fl and »<wl art d«ltrinin«d

Fig. 7 Overview of the calculations carried out in [2,3]

Comparisons of measurements and calculations for the reinforced and for the unreinforced push-off specimens are presented in tables 2-5. It should be noted that the magnitudes of the variables incorporated into the computer program have been limited to values within the experimental range.

Table 2. Comparisons for the instantaneous displacements Wei and sei;

the results refer to the push-off specimens with embedded reinforcing bars [3]. quantity number of observations X = calc./meas. coeff. of variation Wel Se 1 30 30 1.07 0.96 0.22 0.33

Table 3. Comparative overview of experimentally and theoretically determined boundary conditions for the internally

reinforced push-off specimens [3].

boundary condition dwc/dt (lOOh) wc (2000h) dwc/dt (2000h) dsc/dt (lOOh) sc (2000h) dse/dt (2000h) ds/dw (w=0.05mm) ds/dw (w=0.20mm) s (w=0.20ram) number of x observations 24 24 24 24 24 24 24 24 24 ~ calc./meas. 1.04 1.04 1.01 1.03 1.05 1.10 1.02 •1.02 0.97 coeff. of variation 0.30 0.24 0.27 0.20 0.23 0.25 0.18 0.20 0.23

Table 4 Comparison of measured and calculated values of We 1 and se 1 for 10 push-off specimens with free restraining rods , including two static tests [2]. quantity We 1 Sel number of observations 10 10 x=meas./calc. 0-.88 0.80 coeff. of 1 variation 0.20 0.34

Table 5 Similar to table 4; results refer to push-off specimens with free restraining rods [2].

1 bound.condition 1 dwe/dt(t= 100) We (t=2000) dwc/dt(t=2000) dsc/dt(t= 100) sc (t=2000) dsc/dt(t=2000) ds/dw (w=0.05) ds/dw (w=0.20) s (w=0.20) number of observations 8 8 8 8 8 8 8 8 8 x=meas./calc. 1.05 0.92 1.16 0.90 1.05 0.84 1.07 0.82 0.95 coeff. of variation 0.10 0.09 0.30 0.13 0.09 0.21 0.13 0.17 0.08

2) Empirical formulae for the displacements.

The computational results of 1) can easily be fitted by fairly simple formulae for the displacement behaviour of the complete test series. The formulae will be presented in chapter 5 and in appendices I and II.

5. DETERMINATION OF FORMULAE FOR THE DIRECTLY OCCURING AND THE TIME-DEPENDENT CRACK DISPLACEMENTS UNDER SUSTAINED SHEAR LOADING

5.1 Push-off specijBeDS with embedded reinforcing bars

The range of the variables investigated is as follows:

30 < fee < 75 N/mm2 4 4 pfsy < 12 N/mm2

0.60 < T/Tu < 0.90 provided that T^ 3 N/mm2 ..(9)

Note that for fee the width of the proposed interval is wider than is given in section 5.2 for the sustained shear tests. The reason is that the calculated values agree very well with experimental results of Walraven [5] who carried out tests on specimens made of relatively low grade concrete. The displacements will approximately be reduced to the measuring accuracy if the variables are smaller than the lower limits of the intervals proposed in eq. (9). The initial crack width wo was always less than 0.05 mm and its value had no significant influence on the calculated displacements. All the calculated displacements refer to an average value wo=0.03 mm.

Formulae fi, f2, f3 and f4 are presented in appendix I:

Wel = f l ( f c c , p f s y , T / T u ) [mm] S e l = f2 ( f e c , p f s y , T / T u ) [mm] Wc = f3 ( f e e , p f s y , T / T u , t ) [mm] Sc = f4 ( f c c , p f s y , T / T u , t ) [mm]

w i t h f i = ( X i . ( T / T u ) ^ « - + 0 3 . ( T / T u ) * * . p f s y . . . ( 1 0 )

in which ^1 , 0.2, 0.3^ 0.4, are functions of fee and t (i= 1,2,3 and 4 ) ; t refers to the loading period in [hrs], which is limited to 10® hours in the computer program.

Table 6 shows that the formulae satisfactorily describe the measured

displacement behaviour. Due to the small values of Wc and se at t=10° hrs, minor differences in displacements may already contribute to the rather high coefficients of variation sx/J?. Note, that for t=10^ hrs the "measured" displacements are extrapolated values according to eq. (5) and (6).

Table 6. Comparison of measured and calculated displacements; each result refers to 26 tests on push-off specimens with embedded reinforcing bars. j q u a n t i t y 1 c [mm] We 1 S e l Wc ( t = 10Ohrs)^ Wc ( t = 1 0 3 h r s ) w c ( t = 1 0 5 h r s ) s c ( t = 1 0 0 h r s ) ^ s e ( t = 1 0 3 h r s ) 1 s c ( t = 1 0 5 h r s ) : a l c . mean c [mm] 0 . 1 2 6 0 . 0 9 1 0 . 0 2 5 0 . 1 0 0 0 . 1 9 6 0 . 0 1 9 0 . 0 8 7 0.182 x = m e a s . / c a l c . 1.07 1.05 1.09 0 . 9 8 1.05 1.20 1.08 1.07 c o e f f . of v a r i a t i o n 0 . 1 5 0 . 2 8 0 . 3 5 0 . 2 6 0 . 2 6 0 . 3 4 0 . 2 4

0.33

1

*for 24 tests

5.2 Push-off specimens with free restraining rods

Two differences occur on comparing the proposed range of the variables with that of equation (9) in section 5.1:

- now equal variables have a smaller width of the allowed interval (fee ; T/Tu);

- due to the measurements of the axial steel stresses of the restraining rods, Ceo is an additional variable.

The range of the variables investigated is as follows:

50 < fee < 75 N/mm2 1.0 ^Cco .^2.0 N/ram2

0.60 < T/Tu ^ 0.90 provided that T > 3 N/mmZ ..(11)

All the computed displacements refer to an average value wo~0.02 mm. The diameter d of the internal rods had no significant influence on the measured displacement during the observation period. This is probably due to the rather limited number of only eight sustained experiments carried out. The normal

stiffnesses corresponding to d are: " d=10 mm : pfsy = TT . 102.335/(120.300) = 2.92 N/mm2

d=16 mm : pfsy = 2.92 . 2.56 = 7.48 N/mm2

Formulae fs, f6, f7, fs have been derived for the directly occurring and for the time-dependent displacements of the crack-plane subjected to a sustained shear loading:

Wel = fs (T/Tu, Ceo) [mm] Sel = f 6 (T/Tu, CTco) [mm] Wc = f7 (T/Tu, Ceo) [mm] Wc = fs (T/Tu, Ceo) [mm]

with: fi= 61+62.(T/Tu)+63.(ln(T/Tu)) ..(12)

in which 61,62,63 are functions of Ceo and t (i= 5,6,7 and 8 ) ; ,t corresponds to the loading period in [hrs], limited to 10® hours for

calculations. All the formulae are presented in appendix II. Table 7 presents a comparison of measured and calculated displacements. Note, that for t=10^ hrs the "measured" displacements are extrapolated values according to eq. (5) and (6).

Table 7. Comparison of measured and calculated displacements. The results refer to 8 push-off specimens with free restraining rods .

q u a n t i t y c [mm] Wel S« 1 "~ wc ( 1 = 1 0 ° h r s ) w c ( t = 1 0 3 h r s ) Wc(1=10»hrs) s c ( t = 1 0 o h r s ) s c { t = 1 0 3 h r s ) s c ( t = 1 0 5 h r s ) c a l c . mean c [mm] 0 . 0 7 5 0 . 0 8 3 0 . 0 4 8 0 . 1 1 3 0 . 2 1 8 0 . 0 4 2 0 . 1 0 7 0 . 1 7 9 x - m e a s . / c a l c . 1 . 0 4 1 . 0 6 1 . 0 1 1 . 0 9 1 . 1 2 0 . 9 5 1.11 1 . 0 6 c o e f f . of v a r i a t i o n 0 . 3 0 0 . 3 5 0 . 4 6 0 . 3 0 0 . 2 8 0 . 3 8 0 . 3 0 0 . 4 6

From table 7 it can be concluded that the coefficients of variation have rather high values. One of the reasons is that only eight sustained shear tests were

conducted. Finally, it should be stated that formulae fs and fs deviate from other formulae derived in appendix V of [2]. Improved empirical expressions

with increased confidence were determined by also involving some experimental

data of Walraven [5]. Only experiments for fcc> 30 N/mm2 and wo» 0.02mm were analysed; six push-off tests were used namely those of mix 1 (Ceo=0.70 N/mm, fee=38.5 N/mm2 and Ceo=1.05 N/mm2; fee=36.7 N/mm2) and of mix 3

(Ceo=0.70 N/mm2; fee=57.4 or 60.8 N/mm2). The formulae fs..fa indirectly

show the influence of fee on the displacement behaviour by means of eq. (3)

for Tu.

In summary, a comparison has been made between the results reported in chapters 4 and 5; see table 8. The data presented include a combination of the

instantaneous crack displacements wei and sei and the time dependent increments Wc and sc. The implicit calculation (see page 14) of the

displacement behaviour of cracked concrete under sustained shear loading is more complicated than the explicit method (eq. (10) and (12)), However, the

first method generally gives a more accurate description of the experimental

results.

Table 8. Comparison of the calculation methods used.

Method of c a l c u l n l i o n i m p l i c i t ( t a b l e s 2 - 5 ) e x p l i c i t ( t a b l e s 6 - 7 ) r e i n f o r c e d s p e c . n , -i? s j / X 24 1.04 0 . 2 5 26 1.07 0 . 2 8 p l a i n c o n c r . s p e c . n 7f sy A 8 0 . 9 5 0 . 1 9 8 1.06 0 . 3 4

6. DEVELOPMENT OF CREEP COEFFICIENTS fw AND (ps 6.1 Heinforced concrete push-off specinens

For to= 28 days, the total time-dependent displacements can be expressed as

functions of defined creep coefficients (pw and (ps ;

w(t)=Wel+We(t)=Wel (l+(pw(t)) [mm] ..(13) s(t)=sei+sc(t)=sei(l+(ps(t)) [mm] ..(14)

These creep coefficients have been calculated with the aid of the formulae fi, f2, fa and f4 developed in chapter 5. From eq. (1),(5) it follows for the reinforced push-off specimens; '

crack width : (pw = We(t)/wei = fa/fi [-] ..(15) shear displacements: (ps = sc(t)/sei = f4/f2 [-] ..(16)

in which (pw=(ps=0.0 for t=0 and to =28 days at the start of the experiment. Table 9 gives a survey of the plots for (pw and (ps presented as functions of the experimental variables. See also figs. 8-23.

Table 9. Survey of the plots figs. 8-23 representing the development of (pw and <ps for the reinforced push-off specimens. x - y fee-(p t-(p pfsy-(p T/Tu T pfsy t [-] [N/mm2] [N/mm2] [ h r s ] 0 . 6 0 - 4 / 8 / 1 2 * 10O/103/105 0 . 7 0 - " " 0 . 8 0 - " " 0 . 9 0 - " " 5 . 5 4 . 0 f e e = 4 0 - 7 0 N/mm2* 7 . 4 8 . 0 8 . 8 1 2 . 0 " 0 . 8 0 - - 10°./103/105 f e e = 5 1 / 7 0 N/mm2* f i g s . n o s . 8+9 10+11 12+13 14+15 16^17 18+19 20+21 22+23

Figs. 8-15 show the development of (pw and (ps for constant values of the normal restraint stiffness pfsy. It can be concluded that the curves become steeper as pfsy decreases and/or as the sustaining period t is longer.

Moreover, (ps-values are systematically larger than ipw-values. A first reason is that Se1<we1 for instantaneous displacements smaller than approximately 0.20mm. Moreover, generally for the time dependent displacement increments; se>we. As a consequence the crack opening curves show an increasing value of its first derative ds/dw for a longer sustaining period.

Figs. 16-21 show a fairly considerable increase of (pw and (ps for low concrete grades; minor inaccuracies occur for t>10^ hrs= 1.14 year, which are probably due to the lack of test data in this area. In fact the formulae fi, f2, f3, f4 have been extrapolated to non-tested circumstances including low values of fee. The development of the creep coefficients for a constant shear stress level is presented in figs. 22-23.

Another point to realize is that the instantaneous displacements are rather small, so that a small deviation may have a great effect upon the calculated creep coefficients. Table 10 summarizes the instantaneous displacements Wei

Table 10. Values of We i / sei in 10-^ nnn, calculated according to eq. (10) for fi and fa respectively. pfsy fee [N/mm2] [N/mm2] 4.0 30 51 70 6.0 30 51 70 8.0 30 51 70 10.0 30 51 70 12.0 30 51 70 T/Tu [-] 0.60 0.70 0.80 0.90 35/27 58/47 95/93 152/190 52/35 83/63 127/107 184/172 70/36 106/62 150/100 205/153 46/36 73/58 114/105 175/203 59/41 94/73 142/120 204/190 76/39 114/68 163/111 223/172 57/45 88/68 133/117 199/216 66/48 104/82 156/133 223/207 81/42 122/74 175/122 240/190 68/54 102/79 152/129 223/230 73/54 115/71 171/146 243/225 86/45 131/80 187/133 257/209 79/63 117/89 171/141 247/243 80/60 125/101 186/160 263/243 92/48 139/86 200/144 275/228 •

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4) > OJ c cn > — 1 0 1 i - H üT to UI V -o . •O 5 * j V . ü O

o

co'

o

_CM

o

ö

» U) u «ƒ) II M

9-O O

co'

o

CN'

o

ö

c o • 5 II

9-o cn i j -o c •r- 'i-c 3) E C c i—H i l > il 0 uo r H T r—l Ui « Q -T3 ^ co ..-< V i U O o co" CM' o

o

ö

10^ 10^ 10^ 10^

• t [hrs]

F i g s . 1 6 - 1 7 . Curves (pw(t) and (pt. ( t ) f o r T- 5 . 5 N/min2 and

pfEy=4.0 N,/mm2 ; v a r i a t i o n of f c c .

o

I

10° 10^ 10^ 10^ 10^ 10^

t [hrs]

0; o «/) II IA

9-1

'E'E

?

- £ £ ^ z z ?r -'.o e f-oo .g

"3 ^

ö o< ^ ^ n o n i n 3 ö o ' - o o ^ j i n t o p~ O O • •

? ^

^

K

\ \

\V

- j i n ' t ^ o lO < (/l in O

" o

O CM O o

o

O O

co

o

CM"

o

ö

i n O " '—

" o

* " tn O * ( N O (fl u. JZ ^—' • < - ' B z ^ o w i~ 0 VH .—^ 4-1 1.1 9 -T3 C C3 u S-. 0 c 0 •r4 .I-I CO •H ^ tO > . .N <s. in 0) > i-D ^> oi "-^ 00 l—l w bO 00 II >, (T-I Q .

F i g s . 2 0 - 2 1 . Curves (p„(t) and (p^(t) f o r T= 8 . 8 N/mm^ and pfsy=12.O N/mm2; v a r i a t i o n of f c c . I I

10^ 10^ 10^

t [hrs]

0.0 I A ^

3.0

-2.0

1.0

0.0

pf^y [N/mm1

pfsy [ N / m m ^ ]

Figs. 22-23. Development of (fw and ips for T/Tu =0.80; variation of t and fc c .

6.2 Plain concrete push-off specimens

Similar to eq. (13)-(16) creep coefficients (pw and (ps have been defined;

crack width : (pw=wc (t)/we i =f7/ff [-] ••(17) shear displacement : (ps=sc (t)/se i =f8/f6 [-] ••(18)

The results of the calculations are presented in figs. 24-37. The variables used for these plots are presented in table 11. The instantaneous displacements

are given in tables 12-13. It should be emphasized that the formulae fs, f0,

f? and fa are based on only eight sustained shear tests. Because of this the curves for (pw and (ps have a rather low confidence, especially for T/Tu >0.85

and Cc o >1.5 N/mm^.

Table 11. Survey of the plots figs. 24-37 representing the development of (pw and ips for the plain concrete push-off specimens. x-y fcc-(p t-(p O"co-(p T/Tu T (Teo t [-] [N/ram2] [N/mra2] [hrs] 4.0 1.0 10O/103/105* 5.0 1.0 5.0 2.0 0 . 6 - 0 . 8 * - - 1.0 " " — • 1.5 0.80 - - 100/103/105^ 0.90 -figs.nos. 24+25 26+27 28+29 30+31 32+33 34+35 36+37

Table 12. Values of We 1 and se1 in 10-3 juj^ calculated

from equation (12) for fs and fe respectively.

CTco [N/mm2] 1.0 1.5 2.0 0.60 115/90 48/44 17/2 T/Tu 0.70 65/37 35/23 23/8 ^''^ 0.80 61/37 50/28 43/30 0.90 91/78 85/71 73/75

Table 13. Similar to table 12. (Tco [N/nmi2] 1.0 1.5 2 . 0 T 4 . 0 68/39(0.69)* 101/74(0.62) 55/56(0.58) 84/106(0.52) 27/26(0.51) 40/55(0.46) [N/mm2] 5 . 0 75/57(0.86) 58/33(0.78) 37/16(0.73) 38/24(0.65) 18/10(0.64) 18/18(0.58) 6 . 0 - / - ( - . - ) 106/96(0.93) 73/55(0.87) 45/23(0.78) • 36/20(0.77) 22/3 (0.69) * T/Tu-values

91 O {/) II IA

9-" o 1—1 E E t ; z z ^

3 3 -E

II II E o -— c n i n i N m i£> lo o" o" o"

o o o in lO t^ ^ ^ ^ o IA k . JZ

A

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en O

/// 1

' / \ / / °o < IN

£

o

U3 i n i n

o

i n u o

o

cd

o

U3 O O O Ö z o I I l—l 1 CM <N • _ ! E E E £ t; Z Z ^

5S -^i

II II E • i-o° i C7> i n < N l O l O lO o' o" o" o o o i n lO rv y " o .c / / / / o o

/// 1

/ / / / / / °o < IN

£

i n i n

o

U3 i n i n

o

i n o u L3 § O •ö a

i.

o 0< o l—i il > iJ C l CM IA ta O

cd

03 O

o

cl

o

ö

6.0

A.O

2,0

Oeo= 1.0 N/mm^ fee (N/mm^l 50 60 70 X/X^ [ - ] 0.86 0.82 0.78

ao I \ r

50 55 60 65 70

fee [N/mm^]

6.0

^.0

2.0

0.0

Oeo= 1.0 HlmtT? fee [N/mm ] 50 60 70 X/X^ [ - ] 0.86 0.82 0.78

Ar

Ll'O'hrs

50 55 60 65 70

fee [N/mm^]

Figs. 26-27. Development of f» and (PL for r=5.0 and C"co=1.0 N/mm^; variation of t.

6.0

A.0

2.0

X = 5 0N/mm^ Oeo= ZON/mm^ 1 fee (N/mm^l 1 X/X„ [-] 50 60 70 0.64 0.61 058

1 A

^ • " ~ - ^ • ^ - ^ .^ U t =

^i235i^

J o ^

loVs

10°hrs

6.0

A.0

2.0

0.0

50 55 60 65 70

fee [N/mm^]

Figs. 28-29. Development of (pw and (ps for T=5.0 and C"CÜ=2.0 N,/mm?; variation of t. X = 5 0 Oco= 1.0 fee [N/mm^l 50 60 70 N/mm^ N/mm^ x/x, [-1 0.64 0.61 0.58

1 A

V

" ~ ^ ^ t.

r^-^

.^^ t= t =

^^D?;^

-JSlïs.^^

As^

loV

Co 00 I

50 55 60 65 70

fee [N/mm^]

10° 10^ 10^ 10^ 10^ 10^

t [hrs]

Figs. 30-31. Curves (Pw(t) and (ps(t) for crco=1.0 N/mn*;

variation of T/Tu.

0.0

X/Xu 0.80 0.70 0.60

V

1 ^ r> ^ ^ , / ^ -/ / /

1

/

10° 10^ 10^ 10^ 10^ 10^

t [hrs]

I co I

q„=1.5N/mm2 • X/X, 0.80 0.70 0.60

A

V

n . ^

^y

A

V/

-i

V'

_V

y

y

o

\ll

II

A

1 c

10° 10^ 10^ 10^ 10' 10^

t [hrs]

F i g s . 3 2 - 3 3 . Curves (pw(t) and (p£,(t) f o r Cc o = 1 . 5 N/niii|2; v a r i a t i o n of T/Tu.

10° irf 10^ 10^ 10' 10^

t [hrs]

6.0

40

2;0

0.0

C. JN/mm']

co

Figs. 34-35. Development of (pw and (ps for T/Tu =0.80; variation of t.

6.0

A.O

2.0

0.0

Cco[N/mm^]

6.0

A.O

2.0

0,0

Cco[N/mm ]

Figs. 36-37. Development of (Pw and fs for T/Tu =0.90;

variation of t.

6.0

^.0

2,0

0.0

I

Cco[N/mm^]

8.3 Non-linearity of creep coefficients

According to the CEB-FIP model code [8] the creep coefficient of plain concrete is independent of the induced stress level. As a consequence the creep

deformation Ec =(pc .Eei =(pc .CTc/Ec is proportional to the concrete stress. This proportionality is related to the linear elastic material behaviour of concrete in structures under service conditions.

This observation is in contrast with the creep coefficients (pw and (ps of cracked concrete under sustained shear loading. Due to the high stress level

(T/TU=0.70-0.90) these coefficients correspond more or less to limit state conditions of the material for which non-linear behaviour can be expected of the direct as well as the time-dependent displacements. See for example figs. 32-33. Two cases have to be distinghuished;

-cracked reinforced concrete

For constant values of pfsy and fee the development of (pw and (ps is

presented as a function of the time of load application t. The corresponding shear stress levels are T/Tu=0.60; 0.70; 0.80 and 0.90. See figs. 38-49. A variation of both creep coefficients is found for t=ti hours which means that non-linearity occurs;

M<P(ti)I |(p(T/Tu=0.90) - (p(T/Tu=0.60)|

0,5. = . 100S5 > 0% ..(19) ?(ti) (p(T/Tu=0.90) + (p(T/Tu=0.60)

If the ratio of eq. (19) equals zero, then the induced shear stress level has no influence upon the development of the creep coefficients. If the ratio of eq. (19) increases, the greater is the non-linearity of the creep coefficients. For ti=102 hrs and ti=105 hrs average values of eq. (19) are 19.7% and

19.6%. See table 14. Note, that the ratios are mostly smaller the Jiigher the restraint stiffness pfsy of the reinforcement. Moreover, the instantaneous displacements wei and Sei are both also non-linear functions of the shear stress level T/Tu. See table 10.

Table 14. |A(p('*p-values in % according to equation (19) for T/Tu=0.60-0.90; reinforced concrete specimens.

fee pfsy [N/mm2] [N/nmi2] 5 1 . 0 4 . 0 8 . 0 1 2 . 0 7 0 . 0 4 . 0 8 . 0 1 2 . 0 a v e r a g e v a l u e s : •M<Pw|/2.(pw t=102 h r s 105 h r s 2 6 . 3 2 6 . 3 1 9 . 1 2 3 . 9 1 3 . 8 2 2 . 7 2 7 . 5 2 0 . 0 2 2 . 4 2 1 . 3 1 8 . 8 2 1 . 9 2 1 . 3 2 2 . 7 l^(psl/2.(ps t U 0 2 h r s 105 h r s 1 2 1 . 1 1 9 . 2 9 . 1 1 3 . 4 0 . 7 9 . 6 2 7 . 1 1 5 . 1 2 5 . 6 1 9 . 0

1

2 4 . 1 2 1 . 9 1 8 . 0 1 6 . 4

-cracked plain concrete

Ratios according to eq. (19) for ti=10^ hrs and for ti=105 hrs are

presented in table 15. Each value refers to a constant normal compressive stress Ceo. See also figs. 30-33. Note, that the concrete strength has no direct influence on the displacements (eq. (13)). Relatively high average ratios lid(pK2.(p were fotmd; 43.6% and 68.9% for ti =102 ^rs and ti=105 hrs respectively which are probably partly caused by the limited number of experiments carried out.

Table 15. M(rt/4P-values in % according to equation (19) for T/Tu=0.60-0.90; plain concrete specimens.

C"c 0 [N/mm2] 1.0 1.5 2.0 av.values Ul(pwl/2.(pw t = 102 hrs 105 hj-s 25.8 73.7 1.0 41.6 42.7 43.7 23.2 53.0

M(psl/2.(ps 1

t=102 hrs 105 hrs 66.7 95.6 50.8 7.6 74.5 82.5 64.0 84.7

Xu s 8 .A N/mm' Pftys 4.0N/mm' fcc =51 .ON/mm'

10° 10^ 10^ 10^ 10* 10^

t [hrs]

F i g s . 3 8 - 3 9 . Development of (pw and (ps f o r fee =51 N/

v a r i a t i o n of t .

^.0

3.0

2.0

1.0

0.0

and pfsy=4 N/a

10° 10' 10^ 10^ 10' 10^

t [hrs]

10* 10^

\ [hrs]

F i g s . 4 0 - 4 1 . Development of (pw and (pE. f o r fcc=51 N'/mmZ and pfey=8 N/mm^; v a r i a t i o n of t .

t(hrs)

F i g s . 4 2 - 4 3 . Development of (pw and f s f o r fee =51 N/mm^ and pfEy = 12 N/mm''; v a r i a t i o n of t .

10^ 10^

t(hrs)

Pf^ • 4.0 N/mm fee -70.0 N/mm

10^ 10^ 10^ 10^

i [hrs]

3.0

2.0

1.0

0.0

Pisy = 4 . 0 N/mm2 fee =70.0 N/mm2 I CD I

Figs. 44-45. Development of (pw and (pr. for fee =70 N/mm^ and pf8y=4 N/mm^;

10^ 10^

t(hrs)

ao

2.0

1.0

0.0 ^

10^ 10^ 10^ 10^

t [hrs]

Figs. 46-47. Development of (pw and (ps for fee =70 N/mm^ and pf£.y=8 N/mm2;

10° 10^ 10^ 10^ 10^ 10^

\ [hrs]

3.0

2.0

1.0

0,0

10^ 10' 10^

t [hrs]

Figs. 48-49. Development of ipw and (Ps for fee =70 N./mm^ and pfsy = 12 N/mm^;

7. SHRINKAGE AND CREEP OF CONCRETE

In this chapter additional information will be given on time-dependent properties of the two concrete grades as used for the sustained shear tests. Shrinkage tests were carried out for both mix A and mix B; see [1], page 28. Measurements were performed on 150x150x600 mm prisms placed in a vertical position at 20°C and 50* R.H. The longitudinal deformations were recorded periodically on two opposing surfaces over a length of 400 mm. Measurements started at an age to=23-26 days immediately after removal of the specimens

from the fog room. Shrinkage has been calculated according to:

ecs(to,t)= t/(a+p.t) * (20)

and: Ecs(to,t)= a+p.ln(t) (21)

in which:

t = duration of drying in days;

to = age of concrete in days at start of test; ff,p = regression coefficients.

It was found for 10<t<460 days:

mix A: with r=.98 £cs(to,t)= t/(116.29+2.54.t) .10-3 (22) with r=.98 ECS(to,t)= -154+90.ln(t) .lO-^ (23) mix B: with r=.97 ecs(to,t)= t/(91.7+2.95.t) .10-3 (24)

with r=.98 ecs(to,t)= -123+82.ln(t) .lO"© " (25)

The equations (22)-(25) have been plotted in fig. 50, Test data have been presented in [1]. If to+t exceeds approximately 110 days then both eq. (23) and (25) overestimate the measured shrinkage.

AOO

300

EcslIO J

10^ 10^

t [hrs]

Z,00

300

200

Ecs 110-6]

10^

10-t[hrs]

Next, shrinkage has been calculated according to the rules of the CEB-FIP model code [8]. Assuming an equivalent size ho=90 mm for the prisms, it is found by means of a regression analysis on the data of the model code for to=7-60 days and 20°C, SCó RH:

ecs(to,t)= [2770.exp(-16.96/ln(to+t))-204].10-6 ( 2 G )

in which to+t is the present age of the concrete in hours. In table 16 formula (26) has been compared with the measurements on prisms. Although the initial difference is large the ultimate values agree closely. It should be stated that when quantifying shrinkage the CEB-rules do not take account of the differences

in concrete strengths. They only consider the consistency of the mix. In this respect the same deformations are found for mix A and mix B.

Table 16. Comparison of measurements and shrinkage according to equations (20) and (26).

^t[days] 10 50 100 , 200 00 mix A 70 198 258 303 394 ecs[10-e] mix B 79 195 239 269 339 eq. (2^) 28 88 131 182 320

The development of creep coefficient (pc has not been investigated experimentally. From the CEB model code [8] (Pc has been calculated [1]:

(pc= pa(t)+(pd.pd(t-to)+(pf .[Pf (t)-pf (to)] • (27)

For to=28 days it follows for a period of load-application t in days:

mix A: (pc(to,t)= -2.075+0.0361n(t)+0.7071n(t+28) (28) mix B: (pc (to , t)= -2.127+0.0361n( t)+0.7071n(t+28) (29)

The results are shown graphically in fig. 51. The (pc-values can be compared with creep coefficients (Pw and (ps of the sustained shear tests on plain

cracked concrete, presented in sections 6.2-6.3. The results and discussions of these comparisons will be subject of further study.

8.0

fc [-]

6.0

kO

2.0

0.0

10^ 10^

t [hrs]

Fig. 51. Development of (pc(to,t) calculated for both mix A and mix B according to eq. (28) and (29).

8. CONCLUSIONS

- By means of empirical formulae developed in this report the displacement behaviour of a single crack under static and sustained shear loading can satisfactorily be described as functions of fee, pfsy, X/Xu, Ceo

and the period of load application t. These variables should satisfy the

ranges of the indicated intervals. Restraint stiffness of the crack was achieved by means of internal steel rods or by means of embedded rein-forcement. Formulae have been developed for both cases;

- The formulae allow an analysis of the effect of each experimental variable on the displacement of the crack-plane; moreover, non-tested circumstances can be simulated;

- The formulae enable an accurate and a fast comparison between measured displacements and those computed by means of a theoretical model;

- The instantaneous displacements calculated according to fi, f2, fs and fe have been compared with measurements by Walraven [5] on statically loaded push-off specimens; the results show that there is close agreement;

- It should be realized that because of the small number of shear tests on cracked plain concrete, the confidence of the relevant formulae developed may be limited;

- The calculated creep coefficients (pw and (ps develop in accordance with the physical expectation. For rather high shear stress levels X/tu these coefficients display non-linearity. Variation of fee has a

considerable effect on these coefficients;

- The calculations show that for a sustained loading period of t=10S hrs= 114 years, no final values of (pw and (ps are attained.

- In extrapolating the measured displacements by means of empirical formulae based on regression analysis of the test data, it is more particularly necessary to take account of the internal structure of concrete and time dependent changes. That is why a theoretical model [7] shall be extended to time effects.

9. REFERENCES

1. Frenay, J.W., Shear transfer across a single crack in reinforced concrete under sustained loading. Part I, Experiments, Stevin report 5-85-5,

Delft University of Technology, 1985, 114 pp.

2. Frénay, J.W., Shear transfer across a single crack in plain concrete under sustained loading. Experiments and analysis, Stevin report 5-85-13, Delft University of Technology, 1985, 116 pp.

3. Frénay, J.W., Shear transfer across a single crack in reinforced concrete under sustained loading. Part III, Analysis of experiments, Stevin

report 5-85-7, Delft University of Technology, 1985, 114 pp.

4. Walraven, J . C , Frénay, J.W., Pruijssers,. A.F., Influence of concrete strength and load history on shear friction capacity (to be reported in 1988 in PCI-Journal).

5. Walraven, J . C , Vos, E., Reinhardt, H.W., Experiments on shear transfer in cracks in concrete, Part I, Description of results, Stevin report 5-79-3, Delft University of Technology, 1979, 89 pp.

6. Reinhardt, H.W., Frénay, J.W., Creep analysis of structures, 4th Rilem Symposium, Evanston, USA, August 1986, 12 pp.

7. Walraven, J . C , Aggregate interlock: a theoretical and experimental analysis, Thesis, Delft University of Technology, 1980, 197 pp.

8. CEB-FIP model code for concrete structures, Paris, 1978, pp. 55-60

9. Walraven, J . C , The behaviour of cracks in plain and reinforced

1 0 . NOTATION

Unless otherwise stated, the dimensions a r e N , mm or N/min^

d - diameter o f restraint b a r

fi - indicates formula for displacement (i= 1,2...8) fee - cube compressive strength of concrete at to days f«y - yield strength o f steel

fa - relative rib coefficient [-]

gj - indicates formula for b o u n d a r y condition,

i,j - subscripts for formulae for regression coefficients [-] r - coefficient of correlation [-]

s - parallel displacement (or:slip)

Sc - slip increment due to sustained loading s«i - instantaneous slip

t - duration of load application [hours or days] to - age of concrete at start of test [days] w - crack width (or; separation)

Wc - separation increment due to sustained loading Wel - instantaneous separation

Wo - initial crack width

Ac - cross-sectional area of shear plane [mm^] As - cross-sectional area of reinforcing bar [mm^] V - shear force

a,P,y' - regression coefficients [-] 5,e - regression coefficients [-]

If - regression coefficient [-] p - reinforcement ratio (= A s / A c ) [-] Ceo - initial concrete compressive stress r - shear stress

Tu - static shear s t r e n g t h

(ps - slip c r e e p coefficient (=sc(t)/sei) [-] (pw - separation creep coefficient (=wc(t)/wei) [-]

ll. APPENDICES

page

I. Formulae fi, f2, f3 and f4 ^2

Appendix I Formulae f i , f 2 , f3 and f4 ( e q . ( l O ) )

G e n e r a l f o r m u l a : fi = <ti(x/Xuf'^ + tta (x/Xuf''^. pfsy [mm] f o r i = 1, 2 , 3 and 4 . D i s p l a c e m e n t Wel S e l Wc Sc C o e f f i c i e n t s ttj f o r j = 1 , 2 P i j ( f c c ) ° ' 2 i P 3 j ( f c c ) * ^ ' ' J [ Y i j + c C 2 j . l n ( t ) + Y 3 j . ( l n ( t ) ) 2 [ Y 7 j + Y 8 J . l n ( t ) + Y 9 j . ( l n ( t ) ) 2 3 a n d 4 " Y 4 j + Y 5 j . l n ( t ) + Y 6 j . ( l n ( t ) ^ f c c -^ [ Y i o j + Y i i j • I n ( t ) + Y i 2 j . ( l n ( t ) ) * ] f c c C o e f f i c i e n t s P f o r We1 and s e 1 : ^ j = l : 1 j=2: j = 3 : j = 4 :

Pu

+0.0714 80.1235 0.0340 0.3470 P2J 0.2693 -0.8078 -0.2488" 0.5015 p3 j 7.3663 173.7260 0.00043 0.00165 p4 j -0.8944 -0.9315 0.8366 1.8657

C o e f f i c i e n t s Y for Wc and sc^ / Yu Y 2 j Y 3 j Y 4 j Y s j Y6J Y 7 j Y s j Y9J Y i o j Y i i j Y12J j = l 8 . 5 6 2 2 + 2 . 7 8 0 8 - 2 . 9 4 0 - 1 . 3 3 7 0 - 0 . 0 3 0 1 0 . 0 0 9 7 5 9 1 . 7 8 4 8 - 6 5 . 8 1 8 4 1.4323 - 2 . 3 4 1 5 +0.0514 +0.0057 j = 2 7 0 5 . 1 1 8 3 - 7 3 . 4 8 9 1.1068 - 1 . 4 2 7 4 - 0 . 0 6 1 1 0 . 0 1 3 7 1 1 6 . 6 3 7 3 4 . 8 5 6 8 - 1 . 2 6 6 2 - 0 . 7 8 1 4 - 0 . 1 0 5 9 +0.0142 j = 3 0 . 7 5 6 3 - 0 . 0 6 4 6 0 . 0 0 2 3 - 1 . 2 7 6 3 0 . 0 6 9 2 0 . 0 0 5 0 0 . 6 4 4 5 5 . 7 6 9 2 - 0 . 5 0 3 7 - 1 . 1 7 1 3 - 0 . 2 6 1 7 + 0 . 0 2 3 8 j = 4 1 4 . 6 0 6 1 - 0 . 9 8 3 1 0 . 1 4 8 4 - 0 . 3 1 9 1 0 . 0 3 0 7 0 . 0 0 5 4 7 0 . 6 0 0 5 6 2 . 8 6 8 7 - 5 . 1 3 1 4 - 0 . 6 9 4 1 - 0 . 0 7 6 6 + 0 . 0 0 4 8

Computer-program for fi-f4: 4 p r l n t " " ; p r i n t " c a l c u l a t i o n o-f w e l , s e l , w c c t J ' a n d s c i , t : 6 p r I n t " a c c o r d i n g t o - f o r m u l a e .f 1 , + 2 , t 3 a n d t 4 " : p r i n t 5 p r i n t " c r a c k e d c o n c r e t e w i t h emibedded r e i n f o r c i n g b a r s ' l e p l = . e 7 l 4 : p 2 = . 2 0 9 3 12 p 5 = 8 t i . 1 2 3 5 : p 4 = - . S 0 7 S 14 p 5 = . e 3 4 ö : p ó = - . 2 4 8 S l ó p 7 = . 3 4 7 ö : pS= . 5 0 1 5 18 q 1 = 7 . 3 ó ó 3 : q 2 = - . 8 9 4 4 20 q 3 = 1 7 3 . 7 2 ó : q 4 = - . 9 3 1 5 22 q 5 = . 0 ö Ö 4 3 : q ó = . 8 3 6 6 q 7 = . 0 0 1 0 5 : q 8 = l . 8 ó 5 7 t c = 70 : + s = l 2 : p r i n t f c , - f s " ^ p r I n t " 1 o a d i n g p e r i o d i n h r s . ; n o r m a l p r i n t ; p r i n t " a n d c o n c r e t e c o m p r e s s i v e ' sprint :pr 1 n t 24 25 26 28 30 31 32 33 34 3ó 33 40 42 50 52 54 56 53 8 8 90 91 92 93 94 95 9i 98 99 100 1 10 115 120 125 130 135 140 190 195. 200 205 250 2 5 5 265 275 290 295 30 0 310 400 410 500 510 520 530 540 5ó0 570 575 576 5 7 7 s t r e s s level al = a 2= a 3= a 4= a 5= a 6= a 7= a 3= b l = b2=-b3= b4= b5= b6= b7= 08= - 6 5 . 8 1 8 4 : - 0 . 0 5 1 4 : 4 . 8 5 Ó 8 : . 1 0 5 9 : 5 . 7 Ó 9 2 : . 2 6 1 7 : 6 2 . 8 6 8 7 : f)766 : c l = l . 4 3 2 3 c 2 = - . 0 0 5 7 c 3 = - l . 2 6 6 2 c 4 = - . 0 1 4 2 c 5 = - . 5 0 3 7 c 6 = - . 0 2 3 3 c 7 = - 5 . 1 3 1 4 c 8 = - . 0 0 4 8 pr i n t"the shear r l = p l * ( + c * p 2 ) r2=p3»(-f c'p4) r3=p5«< tc * p ó ) r4=p7*< t'c * p 8 ) s l = q l * < f c ' q 2 ) s2=q3*<+"c*q4) s3=q5*< fc"qó) S4=q7*<-fc"q8) + or i = 1to5 read t e we< i ) = r l » C t e * r 2 ) + - f s « r 3 * < t e V 4 ) sei;i> = £ l « ( t e * s 2 ) + -fs«3 3 * < t e ' s 4 ) n e X t j = l : t(j)=.10 5 9 1 . 7 8 4 3 2 . 3 4 1 5 1 1 6 . 6 3 7 3 .7314 .6445 1.1713 70 ..1005 Ó941 s t=logt t<J)> dl = al+bl*t + c l » ( f 2 ) d 2 = a 2 + b 2 » t + c 2 « < t * 2 ) d 3 = a 3 + b 3 « t + c 3 * C t * 2 > d 4 = a 4 + b4»t + c4»< t*2) d 5 = a 5 + b 5 » t + c 5 « ( 1 * 2 ) d 6 = a i + b 6 « t + c6»< t'2> d7=a7+b7«t: + c7«< t'2) d 8 = a S + b 3 * t + c8*( t *2> d2=d2*<-l> , d 4 = d 4 * < - l ) d 6 = d ó * ( - 1 ) d8=dS«(-n e l = d l * < t c ' d 2 ) e2=d3*(-fc-d4) e3=d5*C-fc"dó) e 4 = d 7 « < f c - d 8 ) r e s t o r e •for i = l t o 5 ; readte scCi ,j)= el*<te*e2)+e3*<te*e4)«-fs next if j = 7 then500 j = j + l : t < j ) = 1 0 « t < j - n : 9 o t o 9 9 r e s t r a i n t s t i f f n e = s r hi o « f: s t r e n g t h in n'rrim2" : pr i n t indicated by te"sprint I fl n . rr.ffi- ' al=8. i22 2=-l.3370 bl=2.7808 b2=-0.0301 a3= 705.U83:b3=-73.4419 a4=-1.4274 a5= .7563 ' i a6=-1.2763 a7=14.6061 aS=-.3191 j = l :t(j >=.10 t=log< t(J)> b4=-.06l1 b5=-.0ó46 b6= .0692 b7=-.9831 b8=.0307 cl = -. c2= . :c3=l c4= c5= c6=-. c7= . 2940 00 97 .1068 .0137 .0023 0 0 50 1484 cS=-.0054

577 t=l09(t<j)) 580 dl=al+bl*t+cl«<t*2) 590 d2=a2fb2*t+c2«<t*2) 600 d3=a3+b3«t+c3*<t*2> 610 d4=a4+b4*t+c4*<t'2) 620 d5=a5+b5*t + c 5 * ( f 2) 630 d6=a6+b6»t+c6*(t*2> 640 d7=a7+b7*t+c7»(t*2) 650 dS=aS+bS»t+c3*(t'2> 700 el=dl»(fc'd2) 710 e2=d3*(fc'd4) 720 e3=d5*<fc-d6) 730 e4=d7«cfc-d8) 732 restore 735 for i=1to5:readte

740 wc(i,.j)= el*< te*e2)+e3«< te*e4)«f s 7 4 5 next 746 if j=7 then750 747 j=j+l:t<j)=10*t<j-1):9oto577 750 restore 800 open4,4:open2,4,1:open3,4,2 8 0 2 p r i n t « 4 , " f u n c t i o n s f l - f 4 " 8 0 3 p r i n t « 4 , " c r e e p c o e f f i c i e n t s wc/wel and s c / s e l " 305 p r i n t » 4 , " f c = "fc 810 p r i n t « 4 , " f s = "fs 8 1 5 tu=^ .822*c f c * ,406) )*cf s" •: . 159*<f c .303) ) ) 320 printil4,"tu= "tu 825 p r i n t « 4 , 830 for 1=1to5:readte 835 pr I n t»t4 , " " 3 4 5 p r i n t » 4 , " t / t u wel s e l " 855 printtt2,te,we<i),se<i) 860 pr int»4, " " 8 6 5 f o r j = l t o 7 870 fw<j)=wc<i ,j)/we<I) 8 7 5 f5<j)=sc<i ,j)/se<i) 376 fw<.j) = int( 1000*fw< j) )/1000 S77 f si; j>=int< 1000«fs< j ) ) / i e 0 0 890 p r i n t « 2 , t < j ) , f w ( j ) , f s < j ) 895 nextj :nexti 900 c l o s e 4 ; c 1 o s e 2 ; c l o s e 3 950 I ist25 1000 end 2000 data . 5 , .6,.7,.8 , .9 60000 ï =ive " ï'O : r ec^aas " , 9

Appendix I I Formulae f 5 , f e , f? and fa ( e q . ( 1 2 ) ) G e n e r a l f o r m u l a e : f5 , fe o r f7 o r fe = 5i t-52 (r/Tu )+63 . l n ( r / T u ) [mm] f o r fs and f e : f5 = f 5 . v fe = f e . v w i t h v = 0 . 4 7 5 + 0 . 782(l/crco ) D i s p l a c e m e n t We 1 S e l Wc Sc C o e f f i c i e n t s 6i ( i = i , 2 , 3) 6i = E i i + e 2 i .C"co+e3i . l n ( c r c o ) 1 6 i = £ 4 i + £ 5 i . C c o + E s i . l n ( C c o ) 1 6i=(pii+(p2i . l n ( t ) + ( p 3 i ( I n t ) 2 j [ + ( p 4 i + ( p s i . l n ( t ) + ( p 6 i . ( l n ( t ) ) 2 ] . c - c o 5 i = ( p 7 i + ( p 8 i . l n ( t ) + ( p 9 i . ( l n ( t ) ) 2 [+ (pioi+(piii • l n ( t ) + ( p i 2 i . ( l n Ct))2j.crco C o e f f i c i e n t s £ f o r wei and s e i : i = l : i = 2: i = 3 : E l i £ 2 1 £ 3 i - 7 . 1 5 6 2 +4.9537 - 6 . 0 2 1 0 +7.5180 - 5 . 1 9 7 0 + 6 . 4 4 0 9 - 5 . 4 8 3 1 + 3 . 7 1 7 8 - 4 . 2 1 3 2 £4 1 E s i £ 6 j j - 1 0 . 2 3 9 0 + 7 . 6 2 7 7 - 1 0 . 8 1 5 7 1 0 . 3 8 7 4 - 7 . 6 5 6 5 +10.9718 - 8 . 5 1 5 1 + 6 . 4 7 1 3 - 8 . 8 5 8 3

Coeff i c i e n t s (p for wc and sc :

\x

"pli <P2i (p3i (p4 1 "P5i •Pei •pTi (psi (P9i ( p i o i ( p i l 1 <Pl2i i = l -1.4991 1.3824 -0.1590 0.7480 -0.6992 . 0.0795 -0.4128 0.6279 -0.1618 0.2506 -0.3684 0.0861 i=2 1.9444 -1.5919 0.1864 -0.9544 0.8051 -0.0925 0.6063 -0.7068 0.1886 -0.3359 0.4121 -0.0992 i = 3 -0.7714 0.7485 -0.0861 0.3848 -0.3794 0.0491 -0.2041 0.3466 -0.0887 0.1269 -0.2066 0.0475

a 2= a 5= b2= b5= c2= c5= 1 .3824 - . 6 9 9 2 - 1 . 5 9 1 9 .8051 .7485 - . 3 7 9 4 :a3= ; a6= :b3= :b6= :c3= :c6= -.1590 .0 795 .1364 - . 0 9 2 5 - . 0361 .0431 Computer-program for f s - f a : 200 pr I n t " •'; p r i n t " c a l cu 1 a t i on o f w e l , s e l , wc ( t ; a n d 5c •• t ' " : p r i n t 2 0 5 p r i n t " a c c o r d i n g t o f o r m u l a e f 5 , f é , f 7 a n d f 6 " : p r i n t 210 p r i n t " p l a i n c r a c k e d c o n c r e t e w i t h i n t e r n a l r e s t r a i n t b a r = " : p r i n t 2 1 5 s l = - 7 . 1 5 6 2 : a2= 4 . 9 5 3 7 : a3= - ^ . . 0 2 1 0 216 b l = 7 . 5 1 8 0 : b 2 = - 5 . 1 9 7 0 : b 3 = 6 . 4 4 ó v 2 1 3 c l = - 5 . 4 8 3 1 : c 2 = 3 . 7 1 7 8 : c 3 = - 4 . 2 1 3 2 220 d l = - l 0 . 2 3 9 0 : d 2 = 7 . 6 2 7 7 : d 3 = - 1 0 . 3 1 5 7 2 2 2 e l = 1 0 . 3 8 7 4 ! e 2 = - 7 . 6 5 6 5 : e 3 = 1 0 . 9 7 1 8 2 2 5 f l = - S . 5 1 5 1 : f 2 = 6 . 4 7 1 3 : f 3 = - 8 . 3 5 3 3 2 3 5 d = l 6 : fc= 5 5 : 5=2 : v = . 4 7 5 + . 7 3 2 / < s * £ ) 240 p r I n t " 1 o a d i n g p e r i o d i n h r s . ; b a r d i a . i n m m . j c o n c r e t e s t r e n ' j t h i n 2 4 5 p r I n t : p r i n t " a n d i n i t i a l n o r m a l c o n c r e t e c o m p r e s s i v e s t r e s s i n n m(ri2 25t' p r i n t " t h e s h e a r s t r e s s l e v e l i s i n d i c a t e d by t e " : p r i n t 255 p r I n t d , f c , s 2o0 f o r i = l t o 5 j readt 2o5 g l = a l + a 2 * s + a 3 * l o g < s ) 270 g 2 = b l + b 2 * s + b 3 * l o g C s ) 275 g 3 = c 1 + c 2 * s + c 3 « I o g < s ) 285 wee I> = <gl+g2*t + g 3 * l o g c t ) ) « V 290 n e x t : r e s t o r e 315 f o r i = l t o 5 : r e a d t 320 g 4 = d l + d 2 « s + d 3 « l o g < s ) 325 y 5 = e 1 + e 2 * s + e 3 « l o g < s ) 330 y 6 = f 1 + f 2 « s + f 3 * 1 o g < s ) 340 se ( I > = '-y4 + g5«t+ g6*l og( t > > *v 345 next 350 al=-l.4991 355 a4= .7468 365 Dl= 1.9444 370 b4= - . 9 5 4 4 3S0 cl= - . 7 7 1 4 335 c4= .3348 395 j = l : t(j >=.10

400 res tore : t=l ogi; t (j ) ) 4 0 5 f or i = 1t o5:r e ad t e 410 dl= a l + a 2 » t + a 3 » ( t » t ) 415 d2= a 4 t a 5 « t + a 6 » < t * t ; 4 2 5 d4= b l + b 2 * t + b 3 * < t » t ) 430 05= b 4 + b 5 * t + b 6 * ( t » t > 440 d7= c 1 tc2*t + c3»i; t«t) 4 4 5 d8= c 4 + c 5 * t + c 6 * < t * t ) 4 5 5 e l = d l + d 2 * s 460 e2='d4 + d 5 « s 4 6 5 e 3 = d 7 * d 8 * s 470 w c ( i , J ) = e l + e 2 » t e + e 3 « l o g < t e ) 475 next 480 I f j = 8 then 490 4 8 5 j=j»l : t<J :> = 10*t< j-1 ) :goto4G0 490 al= - . 4 1 2 8 495 a4= .250 6 50 5 bl= .60 6 3 510 b4= - . 3 3 5 9 520 cl= -.20 41 5 2 5 c4= .1269 5 3 5 j = l :t< J > = 0 . 10 540 r e s t o r e : t = l o g < t < j ) ) 545 for 1 = 1to5:readte 550 dl= a l + a 2 * t + a 3 » < t * t ) 555 d2= a4+a5*t + a6*'; t*t) 5 c 5 d4= b l + b 2 » t + b 3 * ( t * t ) 570 d5= b 4 + b 5 * t + b 6 * ( t * t ) 580 d7= c l + c 2 * t + c 3 * < t * t ) 5 é 5 d8= c4 + c5«t+c6«<: t«t) 595 é i = d l + d 2 * 5 600 e 2 = d 4 + d 5 » s 605 e 3 = d 7 + d 3 « 5 610 scCi , J ) = e l + e 2 * t e + e 3 » l o g C t e ) a 15 next 620 i f j = S then 630 6 2 5 j = j + l : t < j ) = 1 0 » t ( j - l ) : g o t o 5 4 0 630 o p e n 4 , 4 : o p e n 2 , 4 , 1 :open3,4,2 a 2= a 5= b2= b5= c2= c5= .627900 - . 3 6 3 4 - . 7 0 6 3 .4121 .3466 - . 2 0 6 6 :a3= : a w = :b3= :b6= :c3= : c6= - . 1 6 1 3 .0861 . 1886 - . 0 9 9 2 - . 0 3 3 7 .0475

6o!ü 6 3 5 640 6 4 5 650 6 5 5 66(1 6 6 5 670 4>75 680 6 8 5 o90 o 9 5 70 0 705 710 7 1 5 720 7 2 5 730 7 3 5 740 7 4 5 750 7 5 5 76i 76^' o p e n 4 , 4 ; o p e n 2 , 4 , 1 : o p e n 3 , 4 , 2 pr i n t**4, "f unc t i o n s f 5 - f 8 " p r i n t # 4 , " c r e e p c o e f f i c i e n t s wc/wel p r i n t » 4 , " f c = "fc p r I n t « 4 , " s o = "s print*»4," d= "d tu=l .64?*(s" , 4 2 7 ) * C f c * .321> p r i n t » 4 , " t u = "tu p r i n t tt 4 , r e s t o r e f o r i = l t ü 5 : r e a d t e p r i n t » 4 , " p r i n t t t 4 , " t / t u w e 1 p r i n t » 2 , t e , w e < i ) , s e < i ) p r I n t « 4 , " f o r j = l t o 3 f w < j ) = w c c i , j ) / w e ( i ) f s ^ J ) = s c < i , j ) / s e ( i ) f w ( j ) = i n t ( 1 0 0 0 * f w < J ) ) / 1 0 0 0 f s < j ) = i n t ( 1 0 0 0 * f s < j ) ) / l O 0 0 p r i n t » 2 , t < j ) , f w < j ) , f s ( J > n e x t J i n e x t i c l o s e 4 : c l o s e 2 : c l o s e 3 1 i s t 2 3 5 a n d i e l e n d e n d d a t a s a v e . 5 , " óiO ; , . 7 , . 8 , . 9 . l p " , 8

Stevin-reports published by the division of concrete structures:

SR - 1 Leeuwis, M. "Krulp en krimponderzoek op ongewapend beton. Collec-taneum onderzoeken 1958-1970" (2 delen). Out of print.

(5-71-3)

SR - 2 Froon, M. "Hoogwaardig beton" (1972). Out of print. (5-72-1)

SR - 3 Walraven, J.C. "De meewerkende breedte van voorgespannen T-balken" (1973). Out of print.

(5-73-1)

SR - 4 Nelissen, L.J.M. "Het gedrag van ongewapende en gewapende beton-blokken onder geconcentreerde belasting" (197 3). Out of print. (5-73-7)

SR - 5 Nelissen, L.J.M. "Stress-strain relationship of light weight con-crete and some practical consequences" (1973). Out of print.

(5-73-8)

SR - 6 Bruggeling, A.S.G. "De constructieve beïnvloeding van de tijdsaf-hankelijke doorbuiging van betonbalken" (1974).

(5-74-2)

S R - 7 Stroband, J. , Tack, P.J. "KolomvoeCverbinding met geïnjecteerde stekeinden (1974).

(5-74-3)

S R - 8 Christiaanse, A.R., Vrande, L.W.J.W. van der, Rooden, R.J.W.M, van "Het gedrag van stalen voetplaatverbindingen" (2 delen) (1974). Out of print.

(5-74-4)

SR - 9 Uijl, J.A. den, BednSr, J. "Onderzoek naar het verankeringsgedrag van gebundelde staven" (1974).

(5-74-5)

S R - 10 Nelissen, L.J.M. "Twee-assig onderzoek van grlndbeton" (1970). Out of print.

S R - 11 Meuzelaar, L.C., Smit, D.R., Brakel, J. , Zwart, J.J. "Ponts a haubans en béton précontraint" (1974).

(5-74-6)

S R - 12 Bruggeling, A.S.G., Boer, L.J. den "Eigenschaften von stahlfaser-bewehrtem Klesbeton" (1974).

(11-71-10)

S R - 13 Boer, L.J. den "Fibre reinforced concrete" (1973). Out of print. Conference on properties and applications of fibre reinforced con-crete and other reinforced building materials. Out of print.

SR - 14 Uijl, J.A. den "Met bamboe gewapend beton onder herhaalde belas-ting" (1976). Out of print.

(5-76-1)

S R - 15 Dijk, H.A. van, Nelissen L.J.M., Stekelenburg, P.J. van "Het gedrag van kolom-balkverbindingen in gewapend beton" (1976).

(5-76-2)

SR - 16 Brunekreef, S.H. "Gedeeltelijk voorgespannen beton; Op buiging be-last" (1977).

(5-76-8)

S R - 17 Betononderzoek 1971-1975 (met samenvatting in het Engels) (1976). Out of print.

S R - 18 Bruggeling, A.S.G. "Time-dependent deflection on partially pres-tressed concrete beams" (1977).