Experimental Investigation on Time-Dependent Flexural Crack Growth

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Delft University of Technology

Report 25.5-13.01

Experimental Investigation on

1 May 2013

Time-Dependent Flexural Crack Growth

Experimental tests and Analytical Model

Ir. R. Sarkhosh

Ir. J.A. den Uijl

Dr. ir. C. R. Braam

Prof.dr.ir.

J.C.

Walraven

Mailing

address:

Delft University of Technology (TU-Delft)

Faculty of Civil Engineering and Geosciences

Concrete Structures Section

Stevin Laboratory II

Stevinweg

1 2628

CN

Delft

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Acknowledgment

The experiments of this research have been carried out in the Stevin laboratory of Delft University of Technology (TUDelft).

The assistance of Mr. Fred Schilperoort during the execution of all the tests is very much appreciated.

Furthermore, the authors are greatly indebted to the Ministry of Transport, Public Works and Water Management of the Netherlands (RWS) and the Netherlands Organisation for Applied Scientific Research (TNO) for their support.

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Preface

Crack initiation, crack growth and time-dependency of the crack in unreinforced and reinforced concrete have been an interesting topic for decades. Parallel to the research on the ‘shear capacity of concrete beams under sustained loading’, in which the shear capacity of large-scale beams with longitudinal reinforcement under high sustained loading is being investigated, this research is done to investigate the behaviour of crack in unreinforced concrete.

A series of short-term and long-term loading has been performed to acquire the short-term capacity of the beams (in case of short-term loading), displacements, and time of failure with different load ratios (in case of long-term loading).

In order to simplify the initiation of the crack and aggregate interlock effect, a notch is created in the middle of the concrete beam, and the beam was tested under 3-point bending test. In this manner, the crack initiates in front of the notch tip and grows along that. The crack opens in tension and the effect of aggregate interlock is the minimum.

The aim of this research is to investigate the time-dependent growth of a single crack in a plain concrete beam subjected to sustained loading. The results should be useful to propose a model for crack rate dependency and predict the crack growth in time under different load ratios.

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Summary

The experimental investigation concerns to the behaviour of plain concrete beams with a single notch under sustained loading. The notch is located at the bottom in the middle of the beam. In order to get insight into the short-term capacity of the beams, nine specimens were tested in short-term loading. The experimental program comprised four specimens in cast I, five specimens in cast II, six specimens in cast III, five specimens in cast IV and six specimens in cast V.

The test results comprise of measured material characteristics such as development of concrete compressive strength. The measured crack width and crack length are given for each specimen. The crack width (at notch tip), the notch opening (at mouth of the notch), the deflection in midspan and the applied force were measured and registered continuously.

In cast I, three specimens were tested in short-term loading and one specimen was tested in long-term loading under 90% of the average ultimate short-term capacity measured in the former tests. However, the latter specimen failed under a load equal to 88% of the average short-term capacity.

In cast II, three specimens were tested in short-term loading and two specimens were tested in long-term loading under 80% and 73% of the average short-term capacity. The former failed when it reached the desired load (80%) and the latter failed after 16 hours which gave the only time-dependent result in this test series. In cast III, two specimens were tested in short-term loading and two specimens were tested in long-term loading. The load ratios of tested specimens were 71% and 83% of the short-term capacity. The beam with the highest load failed after 3,5 hours and the other failed after 14 days.

In cast IV, one specimen was tested in short-term loading and four specimens were tested in long-term loading, one of which failed during load application. The load ratios were chosen to be 71,5 and 77%.

In cast V, all of the specimens were tested in long-term loading, with different load ratios between 65% and 73%. Later, the crack rate dependency of the tested specimens was analysed and a viscoplastic model by means of ‘Generalised Burger’s model’ was proposed in order to the creep data.

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1. Test method

1.1. Variables

The experimental program comprised only one variable, which is the load ratio (

P

/

P

u). The other parameters such as concrete class, mix design and specimen geometry were kept intact.

1.2. Experimental procedure

At concrete age of 28 days or older an initial notch was applied to the specimens by means of a saw cut. The notch is located at the bottom in the middle of the beam. In order to get insight into the short-term capacity of the beams, three specimens in each cast were tested in short-term loading. The experimental program comprised four specimens in cast I, five specimens in casts II and IV, and six specimens in cast III and V.

As the notch was assumed to be a pre-existing crack, the crack mouth opening displacement (CMOD) at the bottom of the notch and the crack tip opening displacement (CTOD) at top of the notch were measured on the surface of the beam on both sides of the crack plane.

Table 1: Overview of the tests

Cast Number of short-term

tests Number of long-term tests Depth [mm] Notch Width [mm] at test, Age of concrete t0 [days]

I 3 1 40 4 28 II 3 2 40 4 52 III 2 4 40 4 29 IV 1 4 40 4 29 V 0 6 40 4 28 IV 5 5 40 4 150

1.3. Concrete mix

Table 2: Mix proportions of the concrete per m3

Component * Wet weight [kg] Dry weight [kg]

CEM III/B 42,5 N 330 330 Sand 0,125-0,250 mm 74,72 74,72 Sand 0,250-0,500 mm 242,85 242,85 Sand 0,500-1 mm 242,85 242,85 Sand 1-2 mm 149,45 149,45 Sand 2-4 mm 93,4 93,4 Gravel 4-8 mm 373,61 373,61 Gravel 8-16 mm 691,18 691,18 Water 165 Super plasticizer 0,27 Total 2363

* Rounded sand and gravel

1.4. Curing condition

The concrete beams were cast in steel moulds. Synthetic moulds were used for the cubes. Immediately after casting all specimens were covered with plastic sheets and were kept in the lab temperature (20°C). After 3 days the specimens were demoulded and stored in the fog room (20°C, 99% RH). Next, at an age of 28 days, after making the notch on the beam, they were placed in the laboratory (20°C, 50% RH) and kept for a week (or more) prior to the test to be dried.

1.5. Specimens

The concrete specimens were similar to the type that Zhou 42 [2] used for long-term experiments. The dimensions of the beams were 1000×125×125 mm (Fig. 1). Using a diamond saw, a notch with 40 mm depth and 4 mm width was applied in the middle of the beam.

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Fig. 1: Geometry of the specimens (all dimensions are in mm)

1.6. Measurement

The measuring equipment consists of:

 Two horizontal LVDT’s in front and rear side of the beam at height of 40 mm from bottom, for measuring the notch tip opening displacement (CTOD), See Fig. 1.

 Two horizontal LVDT’s in front and rear side of the beam at height of 40 mm from bottom, for measuring the notch mouth opening displacement (CMOD).

 Tow vertical LVDT’s in front and rear side of the beam, installed at the middle bottom of the beam in order to measure the midspan deflection relative to the supports.

 Load-cell, which is installed over the loading plate.

1.7. Loading procedure

In short-term tests, displacement-controlled loading is applied with a loading rate of 0,005 mm/s until total failure of the specimen. In this way, a full load-crack opening curve is obtained. In long-term tests, however, both displacement-controlled and load-controlled procedure are used with the same loading rate.

125 loading plate 50 50 ₄₅₀ 450 50 125 Notch CTOD CMOD 50 50 4 40

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2. Standard tests on concrete

2.1. Compressive strength

Each casting batch comprises at least six 150×150×150 mm cubes tested at 28 days and the day of test:  Three cubes to determine the compressive strength at 28 days

 Three cubes to determine the compressive strength at the day of test (

t

0)

The results of compressive test are presented in Table 3.

Table 3: Compressive cube strength [MPa]

7 days 14 days 28 days 52 days 56 days 84 days 150 days

38,14 48,82 59,56 Cast I 39,98 49,24 57,04 39,22 48,48 57,68 48,49 48,52 Cast II 48,24 54,65 48,29 52,64 32,34 33,86 31,48 44,24 _54,31 54,18 50,82 55,43 Cast III 47,63 _44,43 54,04 _54,58 48,43 54,56 Cast IV 29,72 _29,73 42,97 46,21 53,35 45,34 50,14 50,38 43,37 49,45 51,57 Cast V 45,31 59,84 44,52 66,48 44,60 -

2.1. Development of compressive strength

Due to hydration of cement, the compressive strength of concrete increases as a function of age. To model the test results, it is necessary to have information on the compressive strength development of concrete. In addition, the development of compressive strength can be drawn by the Eurocode 2 ’s expression [1]:

f

cm

(t) = f

cm

(28) · exp[s(1 – (28 / t)

0,5

]

(1)

where

t

is the age of concrete in days, s is the coefficient based on the type of cement and

f

cm (28) is the mean compressive strength of the concrete at an age of 28 days. The test results show a very good agreement with Eq. (1).

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Fig. 2: Development of cube compressive strength

The boundaries of the 90% confidence interval of Eq. (1) were determined for the concrete compressive strengths:

Lower bound

= f

cm

(t) · (1 – 1,645 · COV)

(2)

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3. Three-point bending tests on specimens with a

notch

3.1. Test setup and equipment

The specimen geometry has been described in Fig. 1. Prior to the actual test, each specimen is notched in vertical direction in the middle. The notch is produced by using a saw-machine and has 4 mm width and 40 mm depth. Next, in order to dry out the specimens, they were moved to the laboratory and stored for the next few days before testing.

For three point bending test, the specimen was placed in a 10 kN capacity setup, as shown in Fig. 3. The estimated capacity of the beams are 2,75 kN. The moving part of setup was the bottom supports. The top part was fixed where the load-cell with an accuracy of 0,005 kN is installed. To compensate for pressure losses due to gradually increasing deformation of the specimen two oil-accumulators were added to the hydraulic system. The loading system including the speed of loading, was controlled by the computer.

Fig. 3: Three point bending test setup

3.2. Testing procedure

In short-term tests, the beams were loaded by the deflection-controlled method with a loading rate of 0,005 mm/sec. In long-term tests the beams were loaded by the load-controlled method with the same speed. During the test, the crack width, the crack length and the midspan deflection were measured by means of six LVDT’s. A pair of LVDT’s was used to measure the crack mouth opening displacement, CMOD (at the bottom of the notch) in front and back of the beam. The crack tip opening displacement (CTOD) was also registered by another pair of LVDT’s at the notch tip at the front and back of the beam. Two more LVDT’s were used to measure midspan deflection relative to the supports. In this way, complete load-CMOD and load-CTOD curves could be collected as well as load-deflection curve. Moreover, the crack length development was recorded by means of a digital camera. In each photo, shot by the camera, the load is displayed in a monitor next to the specimen. Therefore, a complete load-crack length curve was drawn for each specimen.

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In summary, the order of experiments is as follows: I. Preparation of specimens

The specimens were taken from the fog room (20°C, 99% RH) at an age of over 28 days, notched by the saw cut and stored in the laboratory (20±3°C, 50±10% RH).

II. Short-term tests

- The actual test started at

t

0 = 61 days for cast I,

t

0 = 52 days for cast II and

t

0 = 29 days for cast

III.

- Zero measurement of displacements was applied when the beam placed in the setup unloaded. - Six LVDT’s recorded the crack width and the midspan deflection on both sides of the beam (front

and back).

- The midspan deflection is measured relative to the supports by means of two steel bars (one in front side and another one in back) connected to a beam above the supports.

- Roller supports were provided to have free deformation of the beam on both sides.

- Centering the specimen in the test setup, the moving part of the setup pushed upward to let the beam tightened between the loading plate and the supports with a little force (approximately 0,1 kN).

- Application of the load was done by a computer-controlled hydraulic pump. The load was applied - Displacement-controlled method (DCM) with a rate of 0,005 mm/s is used for loading of the beams

in short-term.

- A digital camera with high resolution is used to record the appearance of the cracks during the test. in order to record the corresponding load in each photo-shoot, a monitor is used next to the specimen with force-value appearing on the screen.

- During the application of load, the crack tip opening displacement (CTOD), crack mouth opening displacement (CMOD) and the midspan deflection were measured on each load step (every 0,01 mm moving of the actuator)

III. Long-term tests

- The actual test started on the same day as short-term tests.

- All preparation of the specimens and the setup was as mentioned above.

- Load-controlled method (LCM) with a rate of 0,005 mm/s is used for loading of the beams in long-term.

- During the application of load, the displacements were measured on each load step (every 0,01 mm moving of the actuator)

- Immediately after application of the sustained loading, the displacements were recorded every 60 seconds with sensitivity to 0,0001 mm change in any LVDT.

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3.3. Handling of experimental data

During the test, each measuring scan consisted of  Date and time

 Displacement of the actuator  Magnitude of the load  CMOD in front and back  CTOD in front and back

 Midspan deflection in front and back All data are recorded in an EXCEL file.

3.4. Measured displacements

2.1.1. Introduction

The specimens were assigned an identifying code indicating the cast and beam number; C1B1 represents cast I, beam 1.

Mean values of the CMOD, CTOD and midspan deflection (in back and front surface) are presented in the following sections.

2.1.2. Observations during the tests

During short-term loading, the crack initiation and the length of the crack could be visually checked. So a digital camera is used to record the initiation of the crack. The crack initiates from the notch tip and continues along the notch. In some of the specimens, the crack initiates with an angle up to 30° and continues upward, yet remains under the loading plate.

Specimen C1B3 failed (under a load of 2,02 kN) before reaching the desired load level (2,08 kN) Specimen C2B1 unloaded before reaching the peak-load when the central hydraulic pump in the laboratory turned off accidentally. There was no crack initiated in the specimen but a permanent deflection of 0,01 mm was recorded. The beam reloaded again when the pump turned on. Specimen C2B3 was subjected to a high load level comparing to the other beams. It is noticed that in the front side of the beam, the crack was initiated from the right side of the notch, 10 mm below the notch tip, then with a sinusoidal path, continued upward in the left side.

In cast III, two beams (C3B1 and C3B2) were tested under short-term loading and the results were 3,19 kN and 2,86 kN.

With a coefficient of variation of 12,9%, a mean value of 2,74 kN for all the short-term tests (including C1B4 and C2B4) can be drawn.

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Table 4: Status of the tested specimens

Label Short-term Long-term Pmax [kN] Psus [kN] CTOD [µm] CMOD

[µm] t [sec] I C1B1 ✓ 2.56 - 34.1 60.0 - C1B2 ✓ 2.28 - 22.6 38.9 - C1B3 ✓ 2.11 - 22.8 40.3 - C1B4 ✓ 2.02 - 30.7 52.0 - C2B1 ✓ 2.64 - 18.6 35.8 - C2B2 ✓ 2.88 - 22.3 42.2 - II C2B3 ✓ 2.94 - 27.8 47.5 - C2B4 ✓ 2.25 - 25.2 45.8 - C2B5 ✓ - 2.05 33.8 60.5 12077 III C3B1 ✓ 3.19 - 17.9 38.2 - C3B2 ✓ 2.86 - 12.6 26.9 - C3B3 ✓ - 2.20 23.9 44.3 1527 C3B4 ✓ - 1.88 67.6 115.6 1218857 IV C4B1 ✓ 2.60 - 17.7 36.6 - C4B2 ✓ - 2.10 46.5 80.6 199416 C4B3 ✓ - 2.10 26.7 42.4 135 C4B4 ✓ 1.95 - 17.6 32.4 - C4B5 ✓ - 1.95 89.2 153.6 844432 V C5B1 ✓ - 1.90 49.0 91.0 484414 C5B2 ✓ - 2.00 36.8 64.1 11174 C5B3 ✓ - 1.85 103.0 182.7 6319914 C5B4 ✓ - 1.85 70.8 127.9 2236953 C5B5 ✓ 1.97 - 23.4 43.1 -

* The load, which was selected for testing the beam under sustained loading ✗ Cross marks represent the specimens, which failed during load application and before the desired load ✓ Tick marks represent the specimens, which are tested successfully under short-term or under long-term loading

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4. Presentation of the test results

4.1. Results of short-term loading

Mean values of the measured crack tip opening displacements (CTOD), crack mouth opening displacements (CMOD), and midspan deflection (δ) on the front and rear side of the beam are presented as functions of the applied load P. The displacement-controlled method was used in order to obtain the post-peak curves.

The measured displacements of all the specimens during the application of short-term load are presented in Fig. 5-8.

Fig. 5: P-CMOD, P-CTOD and P-δ curves of short-term tests, cast I

The average short-term capacities of the beams were obtained as 2,32 kN, 2,68 kN and 3,02 kN in cast I, II and III respectively. The corresponding coefficients of variation were 9,8% and 11,7% for cast I and II. In cast III, only two specimens were tested in short-term loading. However, the average short-term capacity of all ten beams (casts I-III), which failed in short-term loading (including the two beams that were supposed to be loaded in long-term but failed before the desired load was applied), was 2,74 with a coefficient of variation of 12,9%.

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Fig. 7: P-CMOD, P-CTOD and P-δ curves of short-term tests, cast III

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Fig. 9: Normal distribution function of the tested specimens

Table 5: Results of the short-term and long-term tests

Label Short-term Long-term

_P

_[kN] CTOD [µm] CMOD [µm]

_δ

[µm]

t

[s]

Cast I C1B1 ✓ - 2,56 34,1 60,0 - - C1B2 ✓ - 2,28 22,6 38,9 - - C1B3 ✓ - 2,11 22,8 40,3 - - C1B4 - ✗ 2,02 30,7 52,0 147,5 - C2B1 ✓ - 2,64 18,6 35,8 121,9 - C2B2 ✓ - 2,88 22,3 42,2 140,4 - Cast II C2B3 ✓ - 2,94 27,8 47,5 157,5 - C2B4 - ✗ 2,25 25,2 45,8 138,0 - C2B5 - ✓ - 33,8 60,5 167,5 12077 Cast III C3B1 ✓ - 3,19 17,9 38,2 136,2 - C3B2 ✓ - 2,86 12,6 26,9 101,4 - C3B3 - ✓ - 23,9 44,3 117,0 1527 C3B4 - ✓ - 67,6 115,6 325,9 1218857 C4B1 ✓ - 2,60 17,7 36,6 124,4 - C4B2 - ✓ - 46,5 80,6 232,6 199416 Cast IV C4B3 - ✓ - 26,7 42,4 92,8 145 C4B4 - ✗ 1,90 17,6 32,4 102,9 - C4B5 - ✓ - 89,2 153,6 399,0 844432 Cast V C5B1 - ✓ - 49,0 91,0 264,1 484414 C5B2 - ✓ - 36,8 64,1 154,6 11175 C5B3 - ✓ - 106,0 188,0 560,0 6320000 C5B4 - ✓ - 70,8 127,9 355,4 2236953 C5B5 - ✗ 1,97 23,4 43,1 128,4 - C5B6 - ✓ -

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4.2. Results of long-term loading

Long-term loading was performed directly after short-term loading. The load-controlled method was used in order to apply a constant load in time. All the plotted displacements of the beams are average values of observations on the front and rear side of the beams.

The average short-term capacity of cast I that was obtained from short-term tests on beams C1B1, C1B2 and C1B3 was 2,32 kN and the desired long-term load for specimen C1B4 was 2,08 kN, which was 90% of the average short-term capacity. Nevertheless, specimen C1B4 failed at 2,02 kN before reaching the desired load. Thus in the first cast, no long-term loading was made.

The average short-term capacity of cast II, obtained from short-term tests on beams C2B1, C2B2 and C2B3 was 2,68 kN and the desired long-term load for specimen C2B4 was 2,25 kN, which was 84% of the average. Nonetheless, specimen C2B4 failed just a few seconds after load application, seems that the short-term capacity of the beam was 2,25 kN. Thus, no term results observed for this specimen. For specimen C2B5, the long-term load was chosen to be 2,05 kN, which was 76% of the average. The beam resisted for 12000 seconds under sustained loading.

In cast III, two beams (C3B1 and C3B2) were tested under short-term loading and the results were 3,19 kN and 2,86 kN. Yet with a coefficient of variation of 12,9%, a mean value of 2,74 kN for all the short-term tests (including C1B4 and C2B4) can be drawn.

According to the compressive strength test results, and with attention to the fact that the same mix design was used for all the casts, it could be concluded that all the series had more or less the same concrete strength, but with a high coefficient of variation which is acceptable for plain concrete. Thus, only one beam in cast IV was tested in short term (C4B1) with a results of Pmax = 2,60 kN, which was close the mean value of previous tests

(2,74 kN) and all of the beams in cast V were tested under long-term loading.

The meaning of ‘Failure’ on the figures of this section is the point where the sustained loading drops.

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Fig. 12: P-CMOD, P-CTOD and P-δ curves of long-term tests, cast IV. The applied load on specimen C4B2 was not constant because a new setup was used to apply the load which had a little inconsistency on controlling the force after relaxation. Later a hard rubber was used to overcome this problem. Still in some cases like C4B3 the inconsistency of the load is visible. Specimen C4B5 experienced two times load drop due to a problem with the power of the setup.

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Fig. 13: P-CMOD, P-CTOD and P-δ curves of long-term tests, cast V

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4.2.1. Time dependent displacements

Similar to the results of the tests by Zhou [2], deformations (CTOD, CMOD or midspan deflection) increase with time in three phases. In the primary phase, the deformation rate decreases and become constant in the secondary phase. In the tertiary phase, the deformation rate increases rapidly until failure. The secondary phase dominates the entire failure lifetime, and the secondary deformation rate appears to have a good correlation with the failure life. This behaviour is in a very good agreement with the theoretical creep curves.

As shown in Fig. 14-17, in most of the cases, the duration of the primary phase and the tertiary phase are likely the same.

Fig. 14: Development of CMOD CTOD and δ in time, cast II

Fig. 15: Development of CMOD CTOD and δ in time, cast III

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Fig. 16: Development of CMOD CTOD and δ in time, cast IV. In specimen C4B5 the loading was dropped for a while (due to a problem in the setup), but went back on the desired level. That is the reason of flat curves of CMOD and CTOD between time 220000 and 500000 sec. Yet, the curve shows a linear growth in the secondary phase when the load went back on.

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Fig. 17: Development of CMOD CTOD and δ in time, cast V

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4.3. Crack path and aggregate crush

The crack path after the failure of some of the beams is presented in Fig. 18-24. In these figures, the crack path in front and rear side of the beam is presented as well as the surface of the crack on each side. The crushed aggregates are distinguished by their green colour.

Fig. 18: Failure of specimen C2B1, area of crushed aggregates = 1100 mm2_{, area of uncrushed aggregates =}

2800 mm2

Fig. 19: Failure of specimen C2B2, area of crushed aggregates = 650 mm2_{, area of uncrushed aggregates = 3400}

mm2

Notch

Front Rear

Left (A)

Right (B)

A

B

Front

B A

Front

Notch

Front Rear

Left (A)

Right (B)

A

B

Front

B A

Front

Notch

Front Rear

Left (A)

Right (B)

A

B

Front

B A

Front

Notch

Front Rear

Left (A)

Right (B)

A

B

Front

B A

Front

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2650 mm2

Fig. 23: Failure of specimen C3B1, area of crushed aggregates = 650mm2_{, area of uncrushed aggregates = 2000}

mm2

2100 mm2

4.4. Fracture Energy

Fracture energy GF can be determined from three-point bending tests on notched beams according to the RILEM

recommendation (1985). It can be evaluated using the following formula:

G

F

= (A

1

+ A

2

+ A

3

)/ [b · (h-a)] = (A

1

+ mgδ

0

)/ [b · (h-a)]

(4)

where

m

is mass of the beam, and

g

is gravity acceleration.

A

1 is the area under the load-deflection curve when

the deflection due to self-weight is neglected and

δ

0 is the deformation when the force has fallen to zero, as

shown in Fig. 25.

In Table 6, the calculated values of fracture energy are presented. Unfortunately, in cast I, the values of deformation δ was not accurate enough, since it was measured by the moving part of the setup and not by the LVDT on the beam. So the results of cast I is neglected in this table.

Notch

Front Rear

Left (A)

Right (B)

A

B

Front

B A

Front

Notch

Front Rear

Left (A)

A

B

Front

B A

Front

Notch

Front Rear

Left (A)

Right (B)

A

B

Front

B A

Front

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Fig. 25: RILEM recommendation for

G

F testing.

A

2

= A

3

= mgδ

0

/2

Table 6: Fracture energy

Label Pu [kN] A1 [Nmm] Δ0 [mm] GF (Eq. 3-3) [N/m] GF, MC2010 [N/m] GF, MC90 [N/m] C2B1 2.64 1031.0 1.380 137.1 145.5 87.8 C2B2 2.88 1419 1.665 182.0 C2B3 2.94 1578.5 1.865 202.8 C3B1 3.19 1260.5 1.477 161.6 C3B2 2.86 893.3 1.236 120.1 C4B1 2.60 1123.0 1.652 153.8

b

a

h

A

1

A

2

A

3

P

δ

0

δ

P

1

P

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5. Statistical analysis of measured crack width

5.1. Crack opening and resistance time before failure

In order to get insight into the effect of load of sustained loading, the applied load is given as a ratio to the short-term capacity. The load ratios in the following graphs are based on the average value of all short-short-term tests (10 beams) instead of the three beams in the same cast.

In Fig. 26-28, the load ratio is presented in vertical axis and the displacements are shown in the horizontal axis. The interesting part is that with a lower load ratio, the displacement at the failure is larger. E.g. specimens C3B3, C2B5 and C3B4 those were loaded under 83%, 75% and 71% of short-term capacity respectively, had a

maximum CTOD of 0,024 mm, 0,035 mm and 0,071 mm at the time of failure, see Fig. 27. A similar result was observed when looking to the CMOD and midspan deflection.

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In Fig. 29-Right, the magnitude of CTOD, CMOD and δ in different load ratios are presented in a graph vs. time. This curves verify that the longer the beam remains loaded, the failure occur in larger displacements. In the other words, under sustained loading, the concrete fails due to larger strains depending on the load ratio/time of loading. Further tests are required to derive a relationship.

Moreover, the load ratio vs. the corresponding time to the failure of the beam in sustained loading is shown in Fig. 29-Left. This graph show somehow a logaritmic relationship between the sustained-loading time and the load ratio.

Fig. 27: Relative P-CTOD and P-CMOD curves of long-term tests and comparison with two short-term tests

Fig. 28: Relative P- δ curve of long-term tests and comparison with two short-term tests

Fig. 29: Left: Sustained loading time in different load ratios. Right: Maximum displacements at failure increases in time.

The values of the points in the graphs of Fig. 29 are presented in Table 7. The graph in Fig. 29-Right shows a distrubuted values of CTOD, CMOD and

δ

in time; each triple corresponds to a specimen with a known load ratio.

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According to Zhou, the time of failure has a logarithmic relation to the load ratio [2]. If a logarithmic regression is drawn for the graph in Fig. 29-Left, the R-squared value of the regression line would be 0,81. That gives us the standard deviation of 9,5%. However, for the further modelling, it is required to neglect the scatter of the results, and use the regresion line for the relation of time and load ratio. later, the results of proposed model will be compared to the experimental results. The results of using the regression line for for the Fig. 29-Left is presented in Fig. 30, which brings all the points into the regression line. Using the values of the regression line, the load ratio and therefore the ultimate capacities are fixed in Table 7.

Table 7: Results of sustained loading

Nevertheless, the question remains if it possible to acquire the capacity of the beam and failure time by knowing the initial displacement and the rate of crack opening. To answer this question, the development of crack opening displacement in time (see Fig. 31) should be analyzed.

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5.2. Crack Rate Dependency

4.2.2. Elasto-visco-plastic model

The rate process of the breakage of bond in the fracture process zone (here, the notch tip), which causes the softening law for the crack opening to be rate-dependent can be modeled by a cohesive crack growth model in a viscoelastic material [4]. This research tries to develop an elastic-visco-plastic model with damage for the rate dependency of the crack strain. Typically, the viscoplastic constitutive equations are developed from a number of spring and dashpot elements arranged in series and parallel.

Fig. 32: Generalized Burger’s model.

Two commonly used models are the generalized Maxwell chain and generalized Burger’s model where in the former the same strain is shared across all the elements and the stress is additive and in generalized Burger’s model the strains are additive and the stress is the same for each element [5]. The generalized Burger’s model will be adopted here because it shares the same framework as classical visco-plasticity models and allows non-linearities based on stress to be accommodated more easily. It can be seen from Fig. 32 that the generalized Burger’s model comprises an elastic element in series with a number of viscoelastic (Kelvin-Voigt) elements and a viscoplastic element. The stress transmitted through each element and strains are additive such that:

ε(t) = ε

el

(t)+ ε

ve

(t) + ε

vp

(t)

(5)

0 (6)

where

σ

is stress and

E

0 is modulus of elasticity of the elastic element. The viscoelastic and

viscoplastic components can be calculated using the Hereditary Integral formulation [6]:

(7)

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Where

J

veand

J

vp are the viscoelastic and viscoplastic creep compliances and

t′

is a dummy integration variable which is in our case the age of the specimen when the first load applies. It can be shown that the first derivatives of the viscoelastic and viscoplastic creep compliances and the initial creep compliances for the generalized Burger’s model shown in Fig. 32 are given by:

(9)

τ

i

= η

i

/ E

i (10)

(11) Where

η

i and

E

i are viscosity and modulus of elasticity of the ith Voigt viscoelastic element,

η

The above-mentioned rheological model with three Voigt elements was used to evaluate the cracking strain rate according to the test results.

Fig. 33: Cracking strain vs sustained loading time.

If the crack opening strain after the beginning of long-term loading (the moment that the load application is complete and the load is constant in time) is plotted versus time, the cracking rate is obtained. It should be mentioned that at this point, the crack is already occurred. As shown in Fig. 33, three different zones can be distinguished in this graph; the primary phase, which can be

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fitted with an elastic and a viscoelastic model, the secondary phase, which can be fitted with a viscoplastic model and the tertiary phase that is the fracture phase. The latter is not fitted with any model since that is the irreversible zone and the fracture already occurred.

4.2.3. A model for the current tests

As shown in Fig. 14-17, the primary phases of the tested specimens are in a wide range between 25 sec and 250000 sec and the secondary phases last up to 2,4 million sec. In order to cover the entire the creep function, six Voigt elements with a wide range of

τ

i between 5 and 100000 sec are chosen.

τ

i is the ratio of viscosity (ηi) of the ith Voigt element to its modulus of elasticity (

E

i) and represents the time of contribution of the element into the creep function.

The cracking strains for different load ratios are plotted in Fig. 34 against time. These curves only show the primary and secondary phases of the cracking strain. In order to fit a viscoplastic model as mentioned before to each graph, six Voigt element with

τ

1

= η

1

/ E

1

= 5

sec,

τ

2

= η

2

/ E

2

6

/

E

6

= 100000

sec were chosen.

The load ratio has a direct effect on the modulus of elasticity of the elastic element (

E

0) as:

E

0

= σ(t) / ε

el

(t)

(15)

where,

σ

is the stress and

ε

el is elastic strain. In the other hand, the elastic strain is also a function of load ratio as the higher load ratio, results into a higher elastic strain. Therefore, the modulus of elasticity of the elastic element (

E

0) from Fig. 34 can be presented as a function of

load ratio as plotted in Fig. 35. The strains in the experimental results are obtained from CTOD divided by the measuring length which was 40 mm.

Moreover, the viscosity of the viscoplastic element (

η

0) that represents the rate of crack growth

(slope of the creep curve) in Fig. 34 is plotted against load ratio in Fig. 35, which shows a power relation to the load ratio.

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Fig. 34: Cracking strain vs. sustained loading time and the viscoplastic model.

Lastly, from each graph in Fig. 34 the properties of Voigt elements (

η

i and

E

i) were extracted and were plotted into curves against load ratio, see Fig. 37. All of the graphs in Fig. 37 show a power relation to the load ratio. This power relation can be described as:

E = A · λ

B (16)

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In order to find a relation for the two parameters

A

and

B

, they are plotted into a graph versus

τ

i, see Fig. 36. According to this graph, the values of parameters A and B for any assumed value of τ can be drawn from:

A = 13600 – ln (τ)

(17)

B = 8,75 – 0,53 ln(τ)

(18)

Subsequently, the modulus of elasticity and viscosity of each Voigt element with an assumed value of

τ

can be acquired from Eqs. (15) and (9) as a function of

λ

.

Fig. 35: Modulus of elasticity of the elastic element and viscosity of the viscoplastic element vs. load ratio.

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Fig. 37: Modulus of elasticity and viscosity of the elastoplastic elements vs load ratio. (x and axis represents the load ratio

λ

)

As a result, for a known load ratio, the presented model, can give the properties of Voigt elements as well as the properties of viscoplastic element and elastic element. This model will fit to any results of sustained loading test acquired from the same test. The result of the model is already presented in Fig. 34.

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6. Employment of the model for crack opening rate

6.1. Step by step method to employ the proposed model

The rheological model, which was presented in the previous section, should be explained by an example. In order to employ that model, a step-by-step method is drawn as following:

 Step 1: Choosing a load ratio is the first step. For example, a load ratio of

λ = 80%

=

800

sec and

τ

6

= η

6

/ E

6

= 8000

sec were chosen

 Step 3: Using the graphs in Fig. 36, in order to obtain parameters

A

and

B

. For the assumed

τ

i, the parameters would give the results of

A

1

= 12214, B

1

= –8,02, A

2

= 10892, B

2

= –7,31, A

3

= 9688,

B

3

= –6,68, A

4

= 8989, B

4

= –6,09, A

5

= 6915, B

= 11228 MPa.

 Step 5: Using Eq. (10) in order to obtain the viscosity of the elements:

η

1

= 292191 Ns/mm

2

, η

2

=

835734 Ns/mm

89828545 Ns/mm

2

_.

 Step 6: Using Fig. 35, in order to obtain

E

0 and

η

0. For this example,

E

0

= 2886 MPa

and

η

0

=

469220098 Ns/mm

2

_.

 Step 7: Using Eqs. (5-14) in order to plot the crack rate, as presented in Fig. 38.

Fig. 38: Crack rate for the load ratio of 80%

Such a curve for rate of the crack could be plotted in different load ratios. Fig. 39 show a semi-logarithmic plot of crack rate for several load ratios from 60% to 95%. A comparison should now be made to the experimental results, to see the accuracy of the model. As shown in Fig. 40, the results of cracking strain rate between model and experimental tests are very close.

6.2. Prediction of the failure

In the previous section, a model proposed to predict the crack opening rate in different load ratios. However, the question remains that, what is the time of failure? This can only be answered if a limit to the crack opening is made in time. For that purpose the strain of the crack, where the failure occurs should be defined. The P-CTOD curves in Fig. 27 show that the CTOD, at the point of failure depends on the load ratio. Thus a P-CTOD curve is plotted in Fig. 41 together with the

σ-ε

curve of the notch tip (the post-crack

σ-w

curve is also presented in the format of stress

σ

, versus relative strain

ε

). In these two graphs, point A is the beginning of the loading, where the strain is zero. Point C is the cracking point (

ε

= 0,00015 and

w

= 0,0 mm), thus the AC part of the

σ-ε

curve

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represents the elastic or elasto-plastic pre-crack behaviour and the slope of this part of the graph equals the modulus of elasticity. However, after cracking at point C, the load still increases with an elasto-plastic behaviour until point E. At point E, the load is the short-term capacity. The average crack width measured in short-term tests is 0,0164 mm (relative strain = + 0,00015 + 0,0164 / 40 = 0,00056). The softening curve based on the fracture energy calculated in Section 4.4, meets the horizontal axis at a crack width of 0,325 mm (relative strain = 0,00015 + 0,325 / 40 = 0,008278). At this point (point F), there is no resistant forces in crack at notch tip, but the specimen is not yet failed. The total failure occurs at point G, where the load becomes zero.

Fig. 39: Crack rate for load ratios between 60% and 95%

Fig. 40: Comparison of the cracking strain rate between model and experimental results

In Fig. 41, points B and D represent the strain/crack width at load ratios of 50% and 80%, respectively. At a high load ratio (e.g. point D) the beam is already cracked, so under a constant load, the crack width, follows path D-D′ and at point D′ the load drops and the failure begins. As it is obvious in this graph, the failure under long-term loading (e.g point D) occurs at a higher strain than short-term loading (point E). The lower the load ratio, the longer the path X-X’ is considered, and the crack opening rate is smaller.

For an uncracked beam, the crack opening rate is meaningless. However, another time-dependent function (i.e. creep) should be considered. Under a sustained loading near to point C, creep effect would increase the strain so that exceeds 0,00015. Afterwards, the behaviour is just like a cracked beam. In order to investigate the effect of creep, a numerical modelling is required. There is a limit for the opening of the uncracked region which is shown with a creep curve. Thus for the point B, the limit of widening would be B′.

net) and the presented graphs can be used. The other method is to use the ratio of initial strain to the cracking strain (

ε / ε

cr) in all the graphs Fig. 39 and Fig. 37 instead of the load ratio. The initial strain at the notch tip can be obtained by means of FE modelling.

Fig. 41: σ-ε curve at notch-tip and the initial strains with different load ratios.

6.3. Comparison with the results of Zhou

[2]

Zhou [2] performed similar experimental results, which are comparable to the current research. A comparison between the geometry and the test parameters are shown in Fig. 42, Table 8 and Table 9. Later, a comparison of the test results are presented.

Fig. 42: Geometry of the specimens

a

b

h

L

L′

B′

P/P

u

w [mm]

0,0164

– 0,006

1 0,325

C

B

A

D

E

F

0,65

0,8

0,5

0,6

G

D′

C′

Crack

0 Creep limit

σ

f

t

0,00828

C

B

A

D

E

F

Load decreases

Load increases

D′

C′

Creep limit

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Table 9: Comparison of the test parameters between Zhou’s research and current research

f

c,cube, 28 days

[MPa] [MPa]

f

t [GPa]

E

[Nm/m

G

F 2_]

Characteristic

_{length l}

ch[m]

Zhou 38 2,8 35 80 0,38

Current research 46,4 2,95* 33* 136,5 0,52

* Calculated according to the Eurocode 2 According to Zhou, for notched beams, a regression relation for stress-failure time curve can be drawn as:

t

cr

= 6 λ

–22 (20)

Current tests show an exponential relation to the load ratio:

t

cr

= 5E19 + e

–45λ (21)

However, it can also be presented as a power function as:

t

cr

= 1,75 λ

–37 (22)

In Fig. 43, a comparison of the time of failure between current research and Zhou’s research is presented together with the related equations (Eqs. 20 and 22). The difference between the slope of the tests are possibly due to differences in concrete class and (

a / h

) ratio. Another comparison of the initial strain vs. time is made in Fig. 44 for the current tests and Zhou’s tests.

Fig. 43: Comparison of the time of failure between Zhou and current tests.

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References

[1] Eurocode 2 commentary, European Concrete Platform ASBL, June 2008

[2] Zhou, F.P. (1992), Time-dependent crack growth and fracture in concrete. PhD thesis, Report TVBM-1011, Lund University of Technology, Sweden.

[3] Zhou, F.P., Hillerborg, A. (1992), Time-dependent fracture of concrete: testing and modelling. In: Ba zant, Z.P. (Ed.), Fracture Mechanics of Concrete Structures. Elsevier Applied Science, The Netherlands, pp. 906– 911.

[4] Bažant, Z.P. and Li, Y.N., 1997, Cohesive crack with rate-dependent opening and viscoelasticity: I. mathematical model and scaling. Int. J. of Fracture, 86 (3), pp.247–265.

[5] Collop, A.C, Scarpas, A., Kasbergen, C. and de Bondt, A., 2003, Development and FE implementation of a stress dependent Elasto-Visco-Plastic constitutive model with Damage for asphalt, 82nd TRB Annual meeting, Washington DC.

Experimental Investigation on Time-Dependent Flexural Crack Growth

Delft University of Technology

Report 25.5-13.01