A fracture mechanics-based method for prediction of cracking of circular and elliptical concrete rings under restrained shrinkage

1. A fracture mechanics-based method for
prediction of cracking of circular and elliptical
concrete rings under restrained
Shrinkage
by
Wei Dong, Xiangming Zhou , Zhimin Wua
Tamonash Jana
001411202019

2. Introduction
Residual stress development during shrinkage of
concrete.
Occurance and effect of crack formation.
Cracking potential test methods(Plate,Bar,Ring)
Why Ring test is preferred?

3. Circular ring for Ring Test
Long period of time is needed before the first cracking
occurs in a restrained circular concrete than elliptical ring
due to its geometry.
Initial cracking may appear anywhere
along the circumference of a circular
ring specimen.
Restraining effect from the central
steel ring to the surrounding concrete ring is uniform.
Geometry function depends on Ro/Ri only.

4. Elliptical ring for Ring Test
Higher degree of restraint can be provided by an elliptical
geometry than a circular one.
Comparetively short period of time is
needed before the first cracking than
the circuler one.
Crack initiates close to the vertices
on the major axis of an elliptical concrete ring.
Geometry function depends on their inner major and
minor radius-to-outer major and minor radius ratio, i.e.
R1/(R1 + d) and R2/(R2 + d).

5. Specimen Setup
The mix proportions for the concrete is1:1.5:1.5:0.5
(cement:sand:coarse aggregate:water) by weight.
Following specimen are prepared.
1. 100 mm-diameter and 200 mm-length cylinders(for
measuring mechanical properties of concrete)
2. 75 mm in square and 280 mm in length prisms(for free
shrinkage test)
3. Notched beams with the dimensions of (100×100×500
mm3 (for fracture test).
4. Series of circular and elliptical ring specimens.
Specimens were covered by a layer of plastic sheet and
cured in the normal laboratory environment for 24 h.

6. Subsequently, all specimens were de-moulded and moved
into an environment chamber with 23°C and 50% relative
humidity.
Mechanical properties of concrete, including elastic
modulus E, splitting tensile strength ft and uniaxial
compressive strength fc, were measured from the cylindrical
specimens at 1, 3, 7, 14and 28 days.
Three specimens tested for each mechanical property at
each age.

7. Material properties
It is found that the average 28-day compressive and
splitting tensile strength of the concrete are 27.21 and 2.96
MPa, respectively.
Age-dependent Equations of mechanical properties, in this
case are determined...
1) Elasticity Modulus in GPa
E(t)=0.0002t3 + 0.0134t2 + 0:3693t + 12.715 [t≤28]
2) Splitting tensile strength, ft, in Mpa
fc(t)=1.82t0.13 [t≤28]

8. 3) Critical Stress Intensity Factor in MPa mm1/2
KIC(t) = 3.92 ln(t) + 12:6
4) Critical Crack Tip Opening Displacement in mm
CTODC = 0.029t2 + 1.62t + 3.96
t is the age (unit: day) of concrete..

9. Free shrinkage tests
Free shrinkage of concrete was measured on concrete
prisms.
Prisms subjected to drying in 23°C and 50% relative
humidity.
Longitudinal length change was monitored by a dial gauge,
which was then converted into shrinkage strain.
Four different exposure conditions, i.e. representing four
different A/V ratios, were investigated on concrete prisms:
(1) all surfaces sealed, (2) all surface exposed, (3) two side
surfaces sealed and (4) three side surfaces sealed.
Double-layer aluminum tape was used to seal the surfaces.

10. Shrinkage strain of concrete obtained from free
shrinkage test

11. Restrained shrinkage ring tests
Thickness, Major and Minor radius of the elliptical ring is
taken 37.5, 150 and 75 mm.
Inner radius of the circuler one is taken 150.
Four strain gauges were attached, each at one equidistant
mid-height, on the inner cylindrical surface of the central
restraining steel ring.
The top and bottom surfaces of the concrete ring
specimens were sealed using two layers of aluminum tape.
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.

12. Cracking of concrete is indicated by the sudden drop in the
measured strain, recommended by ASTM.
Two concrete ring specimens were tested per geometry.
Cracking ages of the circular rings are 14 and 15 days,
respectively, for elliptical ones are both 10 days.

13. Numerical modelling restrained
shrinkage
Finite element analyses were carried out using ANSYS code
to simulate stress development and calculate stress
intensity factor.
Unlike circuler rings uniform internal pressure theory is
not applicable to elliptical ones.
A derived fictitious temperature field is applied to concrete
to simulate the shrinkage effect.
The elastic modulus and Poisson’s ratio of steel both
remain constant as 210 GPa and 0.3.
Poisson’s ratio of concrete is set constant as 0.2.

14. Derived fictitious temperature drop with respect
to A/V ratio for a concrete element

16. Crack driving energy rate curve
(G-curve)
Shrinkage effect is simulated through applying the
temperature drop on concrete in numerical analysis.
Internal stress is developed in concrete, which is uniformly
distributed in a circular ring but non-uniformly distributed in
an elliptical ring.
For circular ring pressure is
Distributed uniformly along its
inner circumference, and
the value is 0.59 MPa

17. Pressure on the elliptical
one distributes non-uniformly,
with the maxim
-um and minimum values
being 2.14 MPa and 0.28
Mpa.
The stress intensity factor (KI) for the circular and elliptical
rings are determined based on the classic fracture mechanics
theory

18. The fictitious temperature drops are applied on the ring in
combined thermal and fracture analysis to simulate the
mechanical effect of shrinkage of concrete.
Stress intensity factor KI in MPa mm1/2 of the circular ring can be
formulated from numerical simulation as
While that of the elliptical ring is
T (in °C)=fictitious temperature drop,
αc =linear thermal expansion coefficient of concrete
Energy supplied for crack propagation, i.e. G can be derived by

19. G-curves of circular and elliptical rings at
various ages

20. Resistance curve (R-Curve)
Based on the work of Ouyang and Shah.. The R-curve is
formulated as
α = ac /ao ac = critical crack length, ao =initial crack
length,
α,β=R curve coefficients, ψ= Resistance parameter

21. The values of α and β can be determined by
and
 In this study the R-curves for the circular and elliptical
rings were approximately taken as that of an infinite large
plate with a Side Edge Notch pre-crack.

22. Creep
The effect of creep on strain in a ring specimen is not taken
into account when using the fictitious temperature drop to
simulate the shrinkage effect.
But actually cracking of a concrete ring specimen is
affected by not only shrinkage but also creep.
the total strain is the sum of elastic strain and creep strain
and can be expressed as
σc (t0)=stress in concrete at the time of loading t0
= modulus of elasticity of concrete at loading time,
J(t,t0)=creep function, φ(t,t0)=Creep Coefficient

23. Effect of creep on nominal stress

24. Effect of creep on nominal stress

25. Cracking age
The restraining effect provided by the steel ring can be
regardedas the externally applied load on concrete.
The abrupt strain drop indicates that the
applied load has reached its peak. Corresponding crack
length a is the critical crack length ac.
 At the critical state corresponding to a = ac, the G and R
curves intersect and have the same slope.
i.e. mathematically
G = R = (KIC)2/E

26. Determination of cracking age for concrete
rings

27. Determination of cracking age for concrete
rings

28. Conclusion
A fictitious temperature field to simulate the shrinkage of
concrete to investigate the cracking behavior of concrete are
appropriate and reliable.
The maximum circumferential tensile stress in an elliptical
ring is about 3.6 times of that in a circular ring.
For a range of value of the critical crack length
[ >or< 24 mm] cracking tendency of circuler and
elliptical ring is different.

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