A fracture mechanics based method for prediction of

“A FRACTURE MECHANICS-BASED METHOD
FOR PREDICTION OF CRACKING
OF CIRCULAR AND ELLIPTICAL CONCRETE
RINGS UNDER RESTRAINED
SHRINKAGE”
Published by
1. Mr.Wei Dong
2. Mr.Xiangming Zhou
3. Mr.Zhimin Wu
Presented by
Sajith Babu George
2MTMD -1567204
Christ University

 Introduction
 Experimental program
 Material properties
 Free shrinkage tests
 Restrained ring tests
 Numerical modelling
 Modelling of restrained shrinkage
 Crack driving energy rate curve (G-curve)
 Resistance curve (R-Curve)
 Cracking age
 Conclusions
 References

INTRODUCTION
 A new experimental method, utilizing elliptical ring specimens, is developed for assessing the
likelihood of cracking and cracking age of concrete subject to restrained shrinkage
 To investigate the mechanism of this new ring test, a fracture mechanics-based numerical
approach is proposed to predict crack initiation in restrained concrete rings by using the R-curve
method.
 When volume change of concrete from autogenous, drying or thermal shrinkage is restrained,
residual stress will be developed and crack may occur once the residual tensile stress exceeds the
tensile strength of concrete
 Cracking in concrete can reduce load carrying capacity and accelerate deterioration, which
shortens the service life of concrete structures and increases maintenance costs.
 The circular ring test has been widely used for assessing cracking tendency of concrete and other
cement-based materials due to its simplicity and versatility.It has subsequently become a
standard test method for assessing cracking potential of concrete and other cement-based
materials recommended by American Association of State Highway and Transport Officials
(AASHTO)

 Elliptical ring test as a better tool than the circular ring test for estimating the cracking
tendency of concrete or other cement-based materials.

EXPERIMENTAL PROGRAM
 The mix proportions for the concrete used for this study was 1:1.5:1.5:0.5 (cement:sand:coarse
aggregate:water) by weight with the maximum aggregate size of 10 mm.
 100 mm-diameter and 200 mm-length cylinders for measuring mechanical properties of concrete
 75 mm in square and 280 mm in length prisms for free shrinkage test
 notched beams with the dimensions of 100 100 500 mm3 for fracture test and a series of
circular and elliptical ring specimens for restrained shrinkage test
 Then the concrete specimens were covered by a layer of plastic sheet and cured in the normal
laboratory environment for 24 h. Subsequently, all specimens were de-moulded and moved into
an environment chamber with 23 C and 50% relative humidity (RH) for continuous curing/drying.

MATERIAL PROPERTIES
 It has been found that the average 28-day compressive and splitting tensile strength of the
concrete are 27.21 and2.96 MPa, respectively
 In this study, fracture properties, including the critical stress intensity factor KIC and the critical
crack tip opening displacement CTODC, of concrete were derived based on the two-parameter
fracture model (TPFM)

FREE SHRINKAGE TESTS
 Free shrinkage of concrete was measured on concrete prisms with the dimensions of 280 mm in
length and 75 mm square in cross section conforming to ISO 1920-8, subject to drying in the
same environment condition as for curing concrete cylinders and ring specimens
 Considering that concrete shrinkage depends on the A/V ratio of a concrete element, four
different exposure conditions, i.e. representing four different A/V ratios, were investigated on
concrete prisms in free shrinkage test.
 (1) all surfaces sealed, (2) all surface exposed, (3) two side surfaces sealed and (4) three side
surfaces sealed, representing A/V ratio of 0, 0.0605, 0.0267, and 0.0133 mm1, respectively
 In experiment, double-layer aluminium tape was used to seal the surfaces which were not
intended for drying

A fracture mechanics based method for prediction of

RESTRAINED RING TESTS
 The restrained circular ring test has been widely used to assess cracking tendency of
concrete and other cement-based materials.
 In restrained ring test, four strain gauges were attached, on the inner cylindrical surface of
the central restraining steel ring and they were connected to a data acquisition system
which is able to automatically record the circumferential strain of the inner surface of the
restraining steel ring continuously.
 The strain gauges were then connected to the data acquisition system, and the
instrumented ring specimens were finally moved into an environmental chamber for
continuous drying under the temperature 23 C and RH 50% till the first crack occurred.
 It was found that the cracking ages of the circular rings are 14 and 15 days, respectively,
while those of the elliptical ones are both 10 days

NUMERICAL MODELLING
 In this study, finite element analyses were carried out using ANSYS code to simulate stress
development and calculate stress intensity factor in concrete ring specimens under
restrained shrinkage
Numerical process
Thermal Structural Fracture analysis
2-D 8- Node thermal
elements(plane77)
2-D 8-Node elements
(plane183)

MODELLING OF RESTRAINED SHRINKAGE
 In order to take into account the non-uniform stress distribution in elliptical concrete rings, a
numerical approach was proposed to simulate the effect of geometry factor on cracking in
concrete ring specimens subject to drying shrinkage.
 In this approach, a derived fictitious temperature field is applied to concrete to simulate the
shrinkage effect so that a combined thermal and structural analysis can be adopted to analyze
cracking in a concrete ring specimen caused by restrained shrinkage
 With the implementation of the Fictitious temperature field, shrinkage of concrete caused by the
temperature field is restrained by the inner steel core, resulting in compressive stress developed
in the steel core and tensile stress in the concrete ring.
 For a given concrete ring with certain exposure condition (i.e. certain A/V ratio), the relationship
between fictitious temperature drop and concrete age can be derived by linear interpolation
from the relationship between A/V ratio and concrete age obtained in the given graph.

CRACK DRIVING ENERGY RATE CURVE (G-CURVE)
 When shrinkage of concrete is restrained by the inner steel ring, internal stress is developed in
concrete, which is uniformly distributed in a circular ring but non-uniformly distributed in an
elliptical ring

 The elliptical geometry dramatically influences the distribution of restraining pressure provided by the central
steel ring when concrete is shrinking
 The pressure enforced on the circular concrete ring distributes uniformly along its inner circumference and the
value is 0.59 MPa; while the pressure on the elliptical one distributes non-uniformly, with the maximum and
minimum values being 2.14 MPa and 0.28 MPa, and occurring on the vertices of the major and minor axes,
respectively.
 In this study, finite element method was used to calculate the stress intensity factor (KI) for the circular and
elliptical rings. Based on the classic fracture mechanics theory, KI can be defined as following
 The given equations are stress intensity factor for circular and elliptical rings, respectively.

In which T (in C) is the fictitious temperature drop acting on a concrete ring representing the effect of drying
shrinkage, E is the elastic modulus of concrete (in GPa), and ac is the linear thermal expansion coefficient of concrete
(in 1/C).
The given figure illustrates the finite element meshes of concrete rings with an initial crack length a = 12 mm, in which
Fig. 7(a) and (b) present the overall mesh in a circular and an elliptical ring, respectively, and Fig. 7(c) shows the refined
mesh at crack tip.

 Based on classical fracture mechanics theory, the energy supplied for crack propagation, i.e. G-curve, can be
derived based on KI and modulus of elasticity of material E as following:
The figure illustrates the derived
G-curves of both the circular and
the elliptical rings at the ages
of 5, 10, 15, 20 days, respectively.

RESISTANCE CURVE (R-CURVE)
• The R-curve is formulated as
• It should be noted that the R-curve derived from Eqs. (9)–(13) is based on the geometry of an infinitely large plate.
• R-curve is defined as an envelope of G-curves with different specimen sizes but the same initial crack length

CRACKING AGE
 In the restrained shrinkage ring test, cracking age of a concrete ring is determined by the abrupt drop observed
from the measured steel strain
 By comparing the critical fictitious temperature drop obtained from fracture analysis with the age-dependent
one derived from free shrinkage test, the cracking age can be determined.
 Alternatively, cracking age of a restrained concrete ring can be determined by maximizing nominal stress r and
comparing it with the maximum allowable tensile stress
 According to the condition of G = R, the relationship between crack length a and allowable nominal stress r can
be determined

 Collecting rmax for each day, the relationship between allowable nominal tensile stress and concrete age can
be established, which is presented below.
 The cracking age of the circular ring is approximately 16 days and that of the elliptical one is approximately
12 days

 These values are compared with their counterparts from experiment (see Table). It can be seen that
they agree reasonably well with their experimental counterparts, indicating that the fracture
analysis model proposed in this study for predicting crack initiation in concrete ring specimens
subject to restrained shrinkage is reliable.
 Based on the cracking age derived from Fig. 12, the G- and R-curves for circular and elliptical rings
are illustrated in Fig. 13(a) and (b), respectively. At the cracking ages, G- and R-curves intersect and
also have the same slope. The crack length corresponding to the point of intersection is the critical
crack length afc for a ring specimen

CONCLUSIONS
 Cracking ages from numerical analyses agreed well with experimental results for circular and
elliptical ring specimens.
 It indicates that using a fictitious temperature field to simulate the shrinkage of concrete and
introducing resistance curve to investigate the cracking behaviour of concrete in restrained
shrinkage test are appropriate and reliable.
 Based on experimental and numerical results, it can be seen that the elliptical ring with R1/R2 = 2
cracks earlier than the circular ring, which can shorten the cracking period in restrained shrinkage
ring test.
 The numerical results indicate that restraining pressure caused by shrinkage enforced on a
circular concrete ring distributes uniformly along its circumference, while that on an elliptical one
distributes non-uniformly.
 For the normal strength concrete investigated in this research, when afc< 24 mm, the driving
energy in an elliptical ring is greater than that in a circular ring indicating that elliptical geometry
can provide higher degree of restraint. In contrast, when afc > 24 mm, things are different. The
driving energy in an elliptical ring becomes less than that in a circular ring, indicating that an
elliptical ring needs a longer period to crack.

REFERENCES
[1] Parilee AM, Buil M, Serrano JJ. Effect of fiber addition on the autogenous shrinkage of silica fume concrete. ACI Mater J
1989;86(2):139–44.
[2] Kovler K. Testing system for determining the mechanical behavior of early age concrete under restrained and free uniaxial shrinkage.
RILEM Mater
Struct 1994;27(6):324–30.
[3] Kraai PP. A proposed test to determine the cracking potential due to drying shrinkage of concrete. Concr Construct 1985;30(9):775–8.
[4] Shales CA, Hover KC. Influence of mix-proportion and construction operations on plastic shrinkage cracking in thin slabs. ACI Mater J
1988;85(6):495–504.
[5] Carlson RC, Reading TJ. Model of studying shrinkage cracking in concrete building wall. ACI Struct J 1998;85(4):395–404.
[6] Grzybowski M, Shah SP. Shrinkage cracking of fiber reinforced concrete. ACI Mater J 1990;87(2):138–48.
[7] Weiss WJ. Prediction of early-age shrinkage cracking in concrete. Ph.D Dissertation, Northwestern University, Evanston, Illinois; 1999.
[8] Weiss WJ, Yang W, Shah SP. Influence of specimen size/geometry on shrinkage cracking of rings. ASCE J Engng Mech 2000;126(1):93–
101.
[9] Weiss WJ, Shah SP. Restrained shrinkage cracking: the role of shrinkage reducing admixtures and specimen geometry. RILEM Mater
Struct
2002;35(3):85–91.
[10] Bentur A, Kovler K. Evaluation of early age cracking characteristics in cementitious systems. RILEM Mater Struct 2003;36(3):183–90.
[11] Moon JH, Pease B, Weiss J. Quantifying the influence of specimen geometry on the results of the restrained ring test. J ASTM Int
2006;3(8):1–14.

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