Fatigue Strength of Concrete

COE Report No. 24

FATIGUE OF CONCRETE SUBJECTED TO BIAXIAL

LOADING IN THE COMPRESSION REGION

By

Technical Report of Research

Supported by the

Under Cooperative Agreement

DOT 05-C-AT-UIUC

FEDERAL AVIATION ADMINISTRATION

DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING

UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

ADVANCED TRANSPORTATION RESEARCH AND

ENGINEERING LABORATORY (ATREL)

January 2004

S. P. SHAH Bin MU

Final report (April 1, 02 – March 31, 03) 2

ACKNOWLEDGEMENT/DISCLAIMER

The final report is prepared from a study conducted in the Center of Excellence for

Airport Pavement Research. Funding for the Center of Excellence is provided by the

Federal Aviation Administration under Research Grant Number 03-128/DOT-95-C-

001/A18. The Center of Excellence is maintained at the University of Illinois at Urbana-

Champaign and is in partnership with Northwestern University and the Federal Aviation

Administration. Dr. Patricia Watts is the FAA-COE Program Director and Dr. Satish

Agrawal is Manager of the FAA Airport Technology R&D Branch. The authors also

acknowledge the support from the NSF Center for ACBM, Northwestern University,

during the course of this investigation.

The contents of this report reflect the views of the authors who are responsible for the

facts and accuracy of the data presented within. The contents do not necessarily reflect

the official views and policies of the Federal Aviation Administration. This report does

not constitute a standard, specification or regulation.


Table of Contents:

1. Project Summary……..…………………………………………….……………………………....4

2. Research Background……....………………………………….…………………………….……6

3. Experimental Details for Concrete Cylinders under

Static and Fatigue Compression……………………………...…………………………….....11

4. Failure Mechanisms of Concrete Cylinders under

Static and Fatigue Compression………………………………………………………………17

5. Instrumentation and Test Control for Concrete Hollow Cylinders under

Combined Compression and Torsion…………………………………….…...………….....20

6. Experimental Results for Concrete Hollow Cylinders under

Combined Compression and Torsion…………………………….…………...………….....27

7. Model for Fatigue Failure of Concrete in c-C-T Region………….……………..……..36

8. Conclusions……………………………………………………………………...………………….50

References

Appendix I EXPERIMENTAL DATA – Compressive test

Appendix II EXPERIMENTAL DATA – Combined compressive & torsional test

Appendix III Publications from This Project


1. PROJECT SUMMARY

This is the final report for the project: Fatigue of Concrete Subjected to Biaxial Loading

in the Compression Region, sponsored by the FAA Center of Excellence for Airport

Technology (FAA-COE). The major objective of this study is the material property

characterization in the biaxial c-C-T region, where the principal compressive stress is

larger in magnitude than the principal tensile stress. The aim of this project is to

understand the response of concrete subjected to high amplitude, low cycle biaxial

fatigue loading in the c-C-T stress space and verify/extend the previously proposed

analytical model in/to the whole Compression-Tension (C-T) space. In the project, a great

deal of attention has been placed on the interaction between characterization of the

material response under static loading and establishment of the failure mechanism under

fatigue loading. This research is important in view of the fact that 3D tensile-compressive

stress states occur in airport rigid pavements under traffic loads. The work of this project

can be summarized as below:

• Static and fatigue tests were conducted on concrete cylinders and hollow cylinders

subjected to pure compression, and torsion superimposed on compression,

respectively. Closed-loop testing methods to avoid the test instability immediately

after the peak load were systematically studied. Static test results were compared with

fatigue test results with respect to compliance, crack length, or inelastic displacement,

as applicable.

• The failure mechanism for pure compressive tests was explained by the Band

Damage Zone model (Jansen and Shah 1997). In the combined compressive and

torsional tests, the failure mechanism was modeled using fracture mechanics.

• The previously proposed methods, theories and models in the t-C-T region were

verified with the new experimental results in the c-C-T region. The previous research

conclusions in the t-C-T region were found to be applicable to the whole biaxial C-T

region.


The details will be illustrated in the following chapters. Chapter 2 gives a review of

the research background. In Chapter 3, the detailed experimental procedures for the static

and fatigue compressive tests are introduced. The failure mechanism of the specimens

under the static and fatigue compressive loading is discussed in Chapter 4. Experimental

instrumentation and the controlling method of concrete hollow cylinders subjected to

static combined compression and torsion are presented in Chapter 5. Chapter 6 presents

detailed experimental results for both static and fatigue compressive and torsional tests.

Chapter 7 discusses the two typical failure modes in the c-C-T region. The previously

proposed models in the t-C-T region are also verified with the new results in this chapter.

Chapter 8 is the final chapter containing a summary and conclusions.


2. RESEARCH BACKGROUND

Fatigue behavior of concrete recently has received considerable attention from

investigators. The present state of the art in designing structures against fatigue distress is

largely empirical, gained by many years of experience. A more rational design procedure

requires a clear understanding of the behavior of material subjected to fatigue thus

permitting design for newer applications.

FF TT

Pavemen

Subgrade

Figure 2.1 A concrete pavement subjected to triaxial and a biaxial stress state at the top and bottom surfaces

Airport pavements are subjected to repeated high stress amplitude loads due to

passing aircraft. In addition, introduction of newer heavier aircraft subjects these

pavements to increased magnitude of fatigue stresses. Using Westergaard theory a

pavement structure subjected to point load shows the state of stress to be triaxial at the

top and bottom surfaces (Fig. 2.1). When resolved in terms of principal stresses, the

stresses are predominantly in the compression-tension (C-T) region of the triaxial stress

space, where the principal stresses have opposite signs. It is vital to understand behavior

of concrete under multi-axial stresses, such as those present in airport pavements, to be

able to assess reduction in stiffness due to the repeated loading of the structure. A

thorough understanding of the fatigue response of concrete subjected to such loading

would also enable determination of the remaining service life of the pavement.

Characterization of the material response to such loading will enable determination of the

response of the pavement to increasing loading in the future. The information could be


used in a mechanistic design of future airport pavements and for assessing the need for

rehabilitation and/or replacement of old airport pavements. Considering the complexity

of the 3-D fatigue loading case, the investigation began with the 2-D fatigue loading case.

One of the tensile stresses was eliminated for simplicity (Fig. 2.1). Then the concrete

pavement is subjected to biaxial fatigue loading in the C-T region.

Considerable work has been done to understand the response of concrete

subjected to uniaxial stress (monotonically increasing and fatigue loads). Tensile,

compressive and tensile-compressive loading schemes considering constant amplitude,

variable amplitude cyclic loading and random loading have been studied. Fatigue curves,

which predict the number of cycles to failure for a given loading scheme, have been

formulated. The failure is understood to be caused by the propagation of cracks in

tension. Cracks are formed from pre-existing flaws and these cracks propagate due to

cyclic or continued monotonic loading, resulting in increased deformation and

redistribution of stresses. After the crack is initiated, the propagation is studied using

fracture mechanics. Fracture mechanics has been applied to gain insight into the failure

mechanism of concrete under tensile loading and successfully applied to predict the

response of structures subjected to monotonic and fatigue loading. Relatively few studies

have been conducted to obtain the response of concrete subjected to multiaxial loading.

The difficult and tedious experiments required for fatigue research under multiaxial

loading are the reason for the scarcity of experimental results under such loading. Most of

the experimental results in the literature pertain to biaxial and triaxial compressive

loading (both monotonic and fatigue). The first systematic investigation into the

performance of concrete subjected to biaxial stresses was conducted by Kupfer et al.

(1969, 1973). In their investigation concrete plates were subjected to in-plane loading for

different biaxial stress combinations spanning the entire biaxial stress space. The results

from this investigation continue to be used today for developing biaxial constitutive

relationships of concrete. The mode of failure as seen in the crack patterns is different in

the three regions of biaxial stress space (C-C, C-T, and T-T). In the C-C region,

microcracking parallel to the free surface is the main damage mechanism. The behavior

of concrete in the C-T region is similar to that in the C-C region as long as the tensile

stress is less than 1/15th of the compressive stress. For larger tensile stresses concrete fails


by a single crack forming perpendicular to the maximum principal tensile strain. Similar

behavior is also observed for specimens in the T-T region. Several continuum mechanics

based theories have been used to model the material response under such loading

conditions (Kupfer and Gerstle 1973; Gerstle 1981). These constitutive formulations are

valid for monotonic loading up to ultimate stress (peak stress). No data are available in

the literature for the behavior of concrete subjected to biaxial loading after the peak

stress. Controlled experiments are needed to obtain the post-peak part of the response in

the C-T region. Complete understanding of fatigue response of concrete in the C-T region

of the biaxial stress space is still lacking. A thorough understanding of the material

response in the C-T space under a monotonically increasing load is essential to establish

the mechanism of failure of concrete. The evolution of damage under fatigue can then be

studied and a fatigue damage law can be established.

|||| 21 σσ >

0 0 12 >= σσ

|||| 21 σσ =0 0 21 <= σσ |||| 21 σσ <FF TT

viv

F

ii

TT

iii

F

i

FF TT

σ2

σ1

t-C-T c-C-T

Figure 2.2 Biaxial C-T region (Subramaniam 1999)

The biaxial C-T region is bounded by uniaxial compression at one end and by

uniaxial tension at the other end (Fig. 2.2). The C-T region in the biaxial stress space

represents a transition from a cracking type failure to a crushing type failure as the ratio

of the tensile principal stress to the compressive principal stress is varied from infinite to

zero. There is also a significant change in the strength of the material associated with a

change in the mode of failure in this region; the strength of concrete subjected to uniaxial


tensile stresses is approximately a tenth of the uniaxial compressive strength. Hence it is

of interest to investigate the behavior of concrete in the C-T region and identify the

failure mechanisms at different stress ratios.

In a previous FAA-COE sponsored project (Fatigue of Concrete Subjected to

Biaxial Loading in the Tension Region, 1999) an investigation to characterize the quasi-

static and low-cycle fatigue response of concrete subjected to biaxial stresses in the

tensile-compression-tension (t-C-T) region, where the principal tensile stress is larger in

magnitude than the principal compressive stress, was performed at the Center for

Advanced Cement Based Materials (ACBM) of the Northwestern University. The

damage imparted to the material was examined using mechanical measurements and an

independent nondestructive evaluation (NDE) technique based on vibration

measurements. The failure of concrete in the t-C-T region was shown to be a local

phenomenon under quasi-static and fatigue loading, wherein the specimen fails owing to

a single crack. The crack propagation was studied using the principles of fracture

mechanics. It was shown that the crack propagation resulting from the t-C-T loading

could be predicted using mode I fracture parameters. It was observed that crack growth in

constant amplitude fatigue loading is a two-phase process: a deceleration phase followed

by an acceleration stage. The quasi-static load envelope was shown to predict the crack

length at fatigue failure. A fracture-based fatigue failure criterion was proposed, wherein

the fatigue failure can be predicted using the critical mode I stress intensity factor. A

material model for the damage evolution during fatigue loading of concrete in terms of

crack propagation was proposed. The crack growth acceleration stage was shown to

follow the Paris law. The model parameters obtained from uniaxial fatigue tests are

shown to be sufficient for predicting the considered biaxial fatigue response. Details of

the experimental work and the analytical model for the material response are available in

the references (Subramaniam et al. 1998; Subramaniam et al. 1999, Subramaniam et al.

2000, and Subramaniam et al. 2002).

The proposed analytical model can be implemented in mechanistic or numerical

pavement response prediction models. However, for proper implementation of the

numerical procedures, a material model that predicts the material response under any

biaxial stress state in the Compression-Tension (C-T) region is required. Hence, it is of


interest to extend the model to the compression-Tension-Compression (c-T-C) stress state

(where the magnitude of the principal compressive stress is larger than that of the

principal tensile stress) within the Compression-Tension (C-T) stress space. In this

research project, the existing test setup was utilized to generate biaxial stresses in the c-

C-T region, by applying torsion with superposed axial compression to hollow concrete

cylinders. By varying the magnitudes of the torsional and axial forces the entire tension

compression stress state can be obtained. The crack formation and propagation under

controlled conditions for both static and fatigue loading were studied. The research

started with the uniaxial compressive test (point - iv in Fig. 2.2). Then the behavior of

concrete subjected to combined compressive and torsional test (point-v) in the c-C-T

region was studied.


3. EXPERIMENTAL DETAILS FOR CONCRETE CYLINDERS

UNDER STATIC AND FATIGUE COMPRESSION

Concrete cylinders (4-in diameter and 8-in length) were used in the static and fatigue

compressive test. The mixture proportions by weight of the constituents were, cement:

water: coarse aggregate: fine aggregate = 1.0:0.5:2.0:2.0.

In the static test, the samples were unloaded and reloaded in the post-peak period

at 90%, 80% and 70% of the peak strength, respectively to obtain the compliance of the

specimen in the post peak period. A closed-loop testing procedure was used with a

circumferential extensometer mounted around the cylinders until the first unloading

started at 90% of the peak strength in the post-peak period. The circumferential

displacement was used as the feedback signal in the MTS. At the first unloading process

at 90% of the peak strength, the control mode was shifted to displacement for the rest of

the testing region.

In the fatigue test, three different load ranges (90% - 5%, 80% - 5% and 76% -

5%) were employed in a sinusoidal waveform at a frequency of 2 Hz. The test was in

load control. To prevent the data file from becoming too large, a continuous data

acquisition process was designed such that data were acquired every seven load cycles, or

every time the stiffness of the specimen changed by a certain threshold value. Within

each load cycle data were acquired in a time increments of 0.05 s.

For each loading case at least three samples were loaded in a Material Testing

System (MTS). The typical load-displacement relationships for both the static and fatigue

tests are shown in Fig. 3.1. From Fig. 3.1(a), there is a progressive decrease in the

structural stiffness due to crack initiation and propagation as evidenced by the decrease in

the slope of the loading and unloading curves in the post peak. In the fatigue result (Fig.

3.1(b)), it can also be observed that there is a steady decrease in the axial stiffness of the

specimen with repeated loading as seen by the decreasing slope of the load-unloading

curves.


0

20

40

60

80

100

0.00 0.02 0.04 0.06 0.08 0.10

Axial Displacement (in)

Load

(kip

)

Ci

Cr

0

20

40

60

80

100

0 0.02 0.04 0.06 0.08 0.1

Axial Displacement (in)

Load

(kip

)

(a) (b)

Figure 3.1 Load-displacement response of a cylinder in compressive test (a) Static test (b) Fatigue test

The axial normalized stiffness/compliance represents the secant

stiffness/compliance calculated between the minimum and maximum load levels divided

by the initial stiffness/compliance (Kr/Ki or Cr/Ci). The damage evolution in concrete

during compressive fatigue, in term of the measured stiffness/compliance, follows a

three-stage trend (Fig. 3.2). The normalized cycles in Fig. 3.2 are obtained from the

current cycle divided by the total number of fatigue cycles (N/Nf). The curve is S-shaped.

A remarkable reduction in the stiffness during the first few cycles (region I) is followed

by a region of gradual, almost linear change (region II). This is followed by a sharp

decrease in stiffness prior to failure (region III). Fig. 3.3 shows a plot of the slope of the

decrease in compressive stiffness, dK/dN, in stage II of fatigue response versus the

fatigue life, Nf, for all the specimens (where K, N and Nf are stiffness, cycle and fatigue

life, respectively). The response of all the fatigue specimens tested at three different

loading range follows a linear trend. The best-fit relationship between the slope of stage

II and the fatigue life is found to be:


log(Nf)=-0.9444*log(dK/dN)+2.5134 (3.1)

0.6

0.8

1

1.2

0 0.3 0.6 0.9 1.2

Normalized cycle

Nor

mal

ized

stif

fnes

ssample-1sample-2sample-3sample-4

III

III

Figure 3.2 Normalized stiffness vs. normalized cycles (76%-5%)

0

1

2

3

4

5

-2 -1 0 1

Log(dK/dN)

Log(

Nf)

90%-5% load80%-5% load76%-5% load

Log(Nf)=-0.9444Log(dK/dN)+2.5134

Figure 3.3 Relationship between the decrease rate of stiffness in stage II of the fatigue response and fatigue life


It is interesting to observe that the relationship between the slope of stage II and

fatigue life is independent of the load range (Fig. 3.3). This suggests that the fatigue life

can be determined or predicted from Eq. (3.1) for any load range if the slope in stage II of

the fatigue response is known. In practice, stage I may represent the first few months of

use and stage II may represent a long period of time before the airport pavement needs to

be repaired. Since stage I is quite short, and the slope of stage II shows a nearly linear

trend, engineers are able to predict the fatigue life of the airport pavement by conducting

the fatigue test over just a few cycles. Whenever stage II and the stiffness change rate in

stage II are determined, Eq. (3.1) can be employed.

60

70

80

90

100

1 1.5 2 2

Normalized compliance

% p

ost-p

eak

load

static

fatigue: 90%-5%

fatigue: 80%-5%

fatigue: 76%-5%

0

20

40

60

80

100

0 0.02 0.04 0.06 0.08 0.1

Displacement (in)

Load

(kip

)

staticfatigue: 90%-5%fatigue: 80%-5%fatigue: 76%-5%

.5

(a) (b)

Figure 3.4 A comparison of static and fatigue response (a) Load – displacement (b) Load - compliance

A comparison of the static response and fatigue response is shown in Fig. 3.4,

both in terms of axial displacement (Fig. 3.4(a)) and axial compliance (Fig. 3.4(b)). In

Fig. 3.4(a) the curves represent the static load-displacement and each data point

represents a different specimen tested in fatigue. It can be seen that the axial

displacement at fatigue failure, which is defined as the maximum displacement at the


higher fatigue loading level, is not comparable to post-peak displacement obtained from

the static response; the axial displacement at fatigue failure are considerably smaller than

the axial displacement at the corresponding load in the static post-peak. Therefore, a

failure criterion based on static deflections may not be suitable for plain concrete

subjected to uniaxial compressive fatigue. Similar observations have been reported for

the concrete airport pavement in the biaxial t-C-T region. However, from Fig. 3.4(b) it

can be seen that the compliance at the fatigue failure compares favorably with the

compliance at the corresponding load in the static post peak period response. This

suggests that the compliance at fatigue failure can be obtained from the static response.

Hence the static response acts like an envelope to the fatigue response when framed in

terms of compliance.

Figure 3.5 Compliance change rate versus compliance

-8

-6

-4

-2

00.9 1 1.1 1.2 1.3 1.4 1.5

Normalized compliance, Cr/Ci

Log(

∆C

/ ∆N

)

sample-1 sample-2sample-3 sample-4sample-5 sample-6sample-7 sample-8sample-9 sample-10

deceleration acceleration

inflection point

The rate of compliance change ( NC ∆∆ / ) as a function of compliance for all of

the specimens is shown in Fig. 3.5. The rates of compliance follow a two-stage process; a

deceleration stage and an acceleration stage up to failure. There is a distinct inflection

point marking a critical compliance where the rate of compliance change crosses over


from deceleration to acceleration. This critical compliance in the fatigue test corresponds

to the compliance at the peak load in the static test, as can be seen from Fig. 3.6.

0.8

0.9

1

1.1

1.2

70 75 80 85 90 95 100

Max. load in fatigue as % of peak

Nor

mal

ized

com

plia

nce

at inflection in fatigue test

at the peak in static test

Figure 3.6 Comparison of the compliance at fatigue inflection point and that at static peak load

All of the results at point-iv in the c-C-T region (Fig. 2.2) are similar to those in

the t-C-T region. This means that the theory, method and conclusions, except for the

terms related to the length of crack, obtained from the previous project can be easily

extended to the point-iv. As to the crack information, there is no simple way to

characterize such information in the compression test, because the cracks are not

distinctly localized as that in tension. However, using the concept of inelastic

displacement to replace the term of crack length, the previously proposed theoretical

model can still be applied to the compression test. The detailed modeling work will be

presented in Chapter 4 & 7.


4. FAILURE MECHANISMS OF CONCRETE CYLINDERS UNDER STATIC

AND FATIGUE COMPRESSION

In compression, the damage zone has a nonuniform distribution as shown in Fig. 4.1a.

From a previous study (Jansen and Shah 1997), this damage zone can be modeled as a

band (Fig. 4.2b). Using the band model, two distinct areas within the concrete sample can

be defined, bulk concrete and damage zone. Localization initiates at the peak stress

(Torrenti et. al. 1993) or just prior to the peak stress (Shah and Sankar 1987). In either

case, the shape of the stress-strain curve up to the peak can be considered approximately

the same in the bulk concrete and the eventual damage zone. During prepeak the same

amount of energy is dissipated in the bulk concrete and the damage zone due to

microcracking (Fig. 4.1c). During postpeak, the bulk concrete unloads. Additional energy

is dissipated in the damage zone. If one assumes that the total strain in the damage zone,

zε , is composed of two parts: one same as the bulk concrete, bε and the additional

inelastic strain in the localized damage zone, then one can write

lbz /δεε += (4.1)

where l is the length of the damege zone. The inelastic post-peak displacement δ is

obtained by substracting the prepeak displacement response of the concrete specimen

from the postpeak displacement. Displacement δ is an approximate measure of the

inelastic, localized deformations occuring during strain softening. It is assumed to be

independent of the length of the damage zone, as shown in Fig. 4.1c. The assumption that

δ is independent of l implies that the postpeak inelastic behavior of the concrete

specimen in compression is the same no matter the length of the specimen. This

assumption has been verified by the previous study (Jansen and Shah 1997) by pressing

concrete cylinders with different lengths.


σ

bε

σ

l/δ

zε

Postpeak Energy

Prepeak Energy

FF

lL Band-damage zone

(a) (b) (c)

Figure 4.1 Band model for localization behavior in compression: (a) Distributed damage; (b) Band-damage distribution; (c) Bulk concrete behavior and damage zone behavior

This band damage zone model is employed to explore the fatigue failure

mechanism characteristic of concrete airport pavements. Fig. 4.2 gives the comparison of

these inelastic postpeak displacements at different loading levels from the static test and

the fatigue test. In a fatigue test, the inelastic post-peak displacement is calculated as the

difference between the failure displacement and the displacement at the compliance

inflection point. A good agreement is found between static and fatigue tests. This

suggests that the static postpeak displacements, which approximately describe the

localization displacements in static compression, govern the fatigue failure mechanism.

The fatigue failure under compression is also a localized phenomenon. The plot of the

load as a function of the inelastic postpeak displacement obtained from static tests acts as

the envelope curve for fatigue test.


0

20

40

60

80

100

0 0.01 0.02 0.03 0.04 0.05 0.06

Inelastic post-peak displacement, δ (in)

Load

(kip

)staticfatigue: 90%-5%fatigue: 80%-5%fatigue: 76%-5%

Figure 4.2 A comparison of static and fatigue response: Load – post-peak inelastic displacement in damage zone


5. INSTRUMENTATION AND TEST CONTROL FOR CONCRETE HOLLOW

CYLINDERS UNDER COMBINED COMPRESSION AND TORSION

5.1 Specimens, loading and instrumentation

Concrete hollow cylinders subjected to combined torsion and compression were used to

represent the loading condition of point-v (Fig. 2.2). Concrete hollow cylinders were 4-in

in outer diameter, 2.25-in in inner diameter, and 8-in in length. The mixture proportion is

the same as that used in the compressive test, i.e. cement: water: coarse aggregate: fine

aggregate = 1.0:0.5:2.0:2.0 (by weight). The ends of the hollow cylinders were reinforced

with 0.5 in x 0.5 in steel wire mesh cages that were embedded into the concrete to ensure

that failure happens in the gage section away from the ends. The mesh extends a length of

2.5 in from both ends and is continuous in the circumferential direction (Fig. 5.1a). The

mesh cage has a diameter of approximately 3.6 in. The average age of specimens used for

the test was over 6 months to ensure mature concrete.

θ

σ

τ

TP

(a) (b)

Figure 5.1 Hollow cylinder specimens (a) Geometry (b) Stress state under compression and torsion


During the test, the ratio of compression to torsion was kept constant so as to keep

the ratio of the two principal stresses constant at 1.5 throughout the loading process (i.e.

proportional loading was used). For the hollow cylinder shown in Fig. 5.1b, the shear

stress at the mid-thickness of the cylinder due to applied torsion T, can be calculated as:

JTr /=τ (5.1)

where J is the section modulus equal to and R2/)( 440 πiRR − 0 and Ri are the inner and

outer radii, respectively, of the hollow cylinder. The radius at the mid-thickness, r, is

equal to (R0+Ri)/2. The normal stress resulting from the applied axial force, P, can be

calculated as:

)( 22

0 iRRP−

=π

σ (5.2)

The ratio of shear to axial stress in a stress element at the mid-thickness of the cross-

section is given as:

)()(

220

0

i

i

RRPRRT

++

=στ (5.3)

The principal stresses can be calculated as:

22

2,1 22τσσσ +

±−= (5.4)

Hence, the ratio of the principal tensile and compressive stresses can be calculated as:

c=

+

+−

−

+

=

)121(

121

2

2

2

1

στ

στ

σσ (5.5)

Therefore,

)ˆ1(

ˆc

c−

=στ (5.6)

where c . Substituting Eq. (5.6) into Eq. (5.3) yields: c−=ˆ

)()(

)ˆ1(ˆ

0

220

i

i

RRRR

cc

PT

++

−= (5.7)


For the case of c equal to –0.667 (where the magnitude of principal compressive stress is

50% larger than that of the principal tensile stress), the ratio of axial force to torque is

2.88 ft-1.

Instrumentation was specially designed to measure rotational and axial

deformation of the gage section of the specimen. Two steel rings were mounted at the

ends of the 2.5 in long gage section. The relative movements of the two rings were

monitored using linear variable differential transducers (LVDTs) mounted on one of the

rings. Four equally spaced LVDTs along the circumference were used to measure the

relative axial displacement. The relative rotation of the two rings was measured by two

LVDTs mounted on one ring in the horizontal direction (tangential to the ring). The

measured change in length is converted to the change of angle between the two rings

(Fig. 5.2). The axial LVDTs have a range of ±0.02 in and rotational LVDTs have a range

of ±0.01 in.

(a) (b)

Figure 5.2 Experimental setup (a) LVDTs (b) Connection to MTS

The procedure to mount the LVDTs and specimens is described below.

• Mount the two rings on the hollow cylinder. Ensure that the two rings are parallel

and level. Then mount the LVDT fixtures.


• Cover the rings and fixtures with plastic sheet before applying epoxy, so the

epoxy will not flow on the rings and fixtures.

• Attach the two steel caps to the MTS fixtures. The specimen will be connected

between these two steel caps by epoxy.

• Apply epoxy cement to the specimen and the two steel caps. Slowly lower the

actuator of the MTS in load control and apply an initial compression of

approximately 0.2 kip. Shift to torque control and set the torque to zero. Leave it

in load and torque control for 24 hours.

• After 24 hours, remove the plastic sheet, and reduce the applied axial load to zero.

Mount and zero the LVDTs.

5.2 Test control

The complete load-deformation response of unnotched concrete hollow cylinders

subjected to compression and torsion is difficult to obtain because of test instabilities.

The test instability is a result of snapback in the post-peak part of the torque-twist or

force-displacement response of these specimens. The three reasons for test instability are

summarized by Rokugo et al. (1986): a) the load is increased at a constant rate, or b) the

stiffness of the testing machine is lower than the specimen stiffness, even through the

displacement rate is kept constant, or c) the elastic energy in the specimen is larger than

the total fracture energy of the specimen. Reason a) is distinctly present in a test, while

reasons b) and c) can be either distinctly or both present in a test.

The sudden failure of specimen in cases a) and b) can be controlled by testing

concrete in a stiff testing machine at a constant displacement rate. In case c) the failure of

the specimen itself is unstable and the use of a stiff testing machine at a constant

displacement rate is not sufficient to obtain the complete load-displacement response.

The load-displacement response in case c) exhibits snapback in the descending branch of

the load-displacement wherein both force and displacement simultaneously decrease. The

snapback phenomenon is a result of localization of damage and simultaneous elastic

recovery of the unchanged portion of the specimen, and is dependent on the specimen

dimension. The total fracture energy required to break the specimen remains constant.

The elastic energy stored in the undamaged portion, however, increases with increasing


specimen length. Therefore the phenomenon of the simultaneous decrease of load and

displacement becomes more pronounced as the gage length is increased as shown in Fig.

5.3 (Jansen 1996).

Failure zone L 1 f +

L1-Lf L-Lf

P

P

P

Lf L1 L P

δe

=

δ

δf

P

(a)

Figure 5.3 Test instabidisplacement response (Elastic response (P ~ eδ )displacement response of

In order to limit tes

exhibit snapback behavior

throughout the destructio

sensitive parameter as the

it can be seen that as the

proportionally without af

parameter which monitors

failure zone would be an

Nishimatsu (1985) propose

, δ

a

n

f

, δ

L

lity)

an sp

t in

, a

o

ee

ga

fec

th

ap

d

L

(b)

due to snapback in the Specimen with a failure zond response of failure zone (P ecimen with different gage len

stability and therefore have sta

measurable test parameter, wh

f the specimen is needed. In

dback signal can eliminate the t

ge length decreases the elastic

ting the response of the fail

e response of a very small gag

propriate feedback signal to co

a control method in which a lin

(c)

post-peak part of force-e of finite length, Lf, (b) ~ fδ ), (c) Combined load-gths, ( fe δδδ += ).

ble control in specimens that

ich increases monotonically

this way, using a failure

est instability. From Fig. 5.3

displacement, , reduces

ure zone. Therefore a test

e length, which includes the

ntrol snapback. Okubo and

ear combination of load and

eδ


displacement is used as the feedback signal in a closed loop servo controlled test

machine. This feedback signal was used to obtain the complete load-displacement

response of rock specimens that exhibited snapback when subjected to unaxial

compression. The feedback signal is defined as

0

KFSignalFeedback αδ −= (5.8)

where δ is axial displacement of the specimen, and K0 is the initial tangent stiffness of

the entire specimen. F is axial force, and α is a coefficient, which must be a positive

number less than 1.0. Such a signal has been also successfully put into use to obtain the

complete load-displacement response of high strength concrete specimens subjected to

compression and exhibiting snapback in the descending portion of the load-displacement

response (Jansen and Shah 1993). For a specimen of length L, where the damage

localizes in a region of finite length L)1( α− and the rest of the material Lα behaves

elastically, the calculated value of this feedback signal by Eq. (5.8) corresponds to the

displacement of the failure zone. The symbol α physically represents the fraction of the

specimen that unloads elastically. Under tension, the failure in the concrete localizes

along a plane perpendicular to the applied force and the value of α may be chosen close

to one. High strength concrete specimens subjected to uniaxial compression have a

failure zone with finite dimension and a value of α equal to 0.6 was found to produce the

best results in 8-in high specimens (Jansen and Shah, 1993).

In this study of concrete hollow cylinders subjected to combined compression and

torsion, a feedback signal which is a linear combination of gage rotation and torque was

used to obtain the complete torque-gage rotation curves. It has the form

0

KTSignalFeedback αθ −= (5.9)

where K0 is the initial tangent modulus of the torque-gage rotation curve and is

approximately 25,000 ft-lb/deg. The applied torque and the rotation of the gage section

are T and θ , respectively.

Theoretically, Eq. (5.9) can eliminate the elastic snapback at the peak point of a

torque-rotation curve as illustrated in Fig. 5.3. However, the selection of the parameter,

α , is critical in the experiment. If α is too small, elastic snapback may not be


completely eliminated (Fig. 5.4a) and if α is too large, the feedback signal described in

Eq. (5.9) may not be monotonic (Fig. 5.4b). Since the initial tangent modulus, K0, can

only be estimated from the previous test results, it might also have influence on the value

of α . In the current static test, a value of approximate 0.8 for α is found to give a stable

control (Fig. 5.4c).

=

(1-α)L (1-α)L(1-α)L

αL αLαL

L L L

P

P

−

P

=

δ'

P

=δe

P−

δ

P

δ'

P

−

δ

P

δe

P

δ

δe

δ'

(a) (b) (c)

Figure 5.4 Test instability of force-displacement response

(a) too small α , (b) too large α , (c) suitable α

In the test, two combined signals are defined as input signals (calculation signal

type) in the MTS configuration file. The first is given by Eq. (5.9) and a rate of 2.0x10-5

deg/sec was used. The second was used to achieve a constant ratio of the principal


compressive stress to the principal tensile stress. The second input signal given by Eq.

(5.7) was defined on the axial control channel and was kept constant (close to zero)

throughout the loading process.


6. EXPERIMENTAL RESULTS FOR CONCRETE HOLLOW CYLINDERS

UNDER COMBINED COMPRESSION AND TORSION

6.1 Static combined compression and torsion

A typical torque vs. gage rotation curve for tests under combined compression and

torsion is shown in Fig. 6.1. The torque and gage rotation values have been normalized

with respect to peak torque and the gage rotation corresponding to the peak torque,

respectively. During the test, the samples were unloaded and reloaded in the post-peak

period at approximately 90%, 80% and 70% of the peak strength, respectively, to obtain

the compliance of the specimen in the post peak period. Closed-loop testing was used

with a combined signal given by Eq. (5.9) until the first unloading started at 90% of the

peak strength in the post-peak period. At the first unloading point at 90% of the peak

strength, the control mode was shifted to the rotation gage for the rest of the testing

process. The percentage increase in rotational compliance is calculated with respect to the

initial rotational compliance (the most linear part of the pre-peak torque-gage rotation

curve). The rotational compliance corresponding to each unloading-reloading loop

represents secant compliance calculated between the top and the bottom intersection

points of the unloading-reloading curves. The increase in rotational compliance in the

post-peak part can be attributed to the propagation of a crack in the gage region.

Fig. 6.2 shows the extension of the four axial LVDTs with respect to normalized

test time. The crack growth can be traced by the variation of the axial LVDTs. It can be

seen that LVDT 2 & 3 increased while LVDT 1 & 4 decreased after the peak load

(normalized time = 1.0). The difference of the extensions of the four axial LVDTs after

the peak load apparently indicated a misalignment, because no significant variation was

noticed before the peak load. This misalignment was produced by the initiation and

propagation of a single inclined crack as observed in the experiment. This crack formed

between LVDT 2 & 3. The crack was inclined at an angle of approximately 51o with

respect to the horizontal (Fig. 6.3).


0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

Normalized gage rotation

Nor

mal

ized

torq

ue

Figure 6.1 Torque vs. gage rotation for tests in combined signal control

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0 1 2 3 4 5 6 7 8

Normalized time

Exte

nsio

n of

axi

al L

VDTs

(in) LVDT-1

LVDT-2LVDT-4LVDT-3 2

41

3

Figure 6.2 Extension of the four axial LVDTs


Figure 6.3 An inclined crack in the hollow cylinder specimen

0

0.0002

0.0004

0.0006

0.0008

1 1.5 2 2

Normalized rotational compliance

Ave

rage

axi

al d

ispl

acem

ent (

in)

.5

Figure 6.4 Increase in rotational compliance versus the average axial deformation

The average axial displacement (from the four axial LVDTs) of the specimen

versus the increase in the rotational compliance is plotted in Fig. 6.4. The rotational


compliance is normalized to the initial rotational compliance. It can be seen that the

average axial deformation is monotonically related to the increase in the rotational

compliance of the specimen. This increase in the rotational compliance of the specimen

can be attributed to the propagation of a crack in the gage region.

6.2 Fatigue combined compression and torsion

Fatigue tests were conducted at three different torque ranges. Torque was applied

between two fixed torque levels in a sinusoidal waveform at a frequency of 2 Hz. The

tests were performed in a torque control. The lower limit of the torque was kept as 5% of

the average static peak torque value. Three different upper limits were used

corresponding to 90%, 85% and 80% of the average static peak torque.

0

200

400

600

800

0 0.01 0.02 0.03 0.04 0.05

Gage rotation (deg)

Torq

ue (f

t-lb)

90%-5%

Figure 6.5 A typical fatigue response of a specimen

A typical fatigue response of a specimen is shown in Fig. 6.5. There is a steady

increase/decrease in the rotational compliance/stiffness as seen by the decrease slope of

the loading-unloading curves. The change in rotational compliance/stiffness during

fatigue loading for the same specimen is shown in Fig. 6.6. The number of cycles has

been normalized with respect to the number of cycles to failure (Nf) for the specimen.


The rotational compliance/stiffness represents the secant rotational compliance/stiffness

of the specimen calculated between the minimum and maximum torque levels. The

damage evolution during the fatigue combined compressive and torsional loading, in

terms of the measured rotational stiffness, can be divided in to three stages. There is a

measurable drop in the stiffness in stage I, which lasts for a few cycles. Subsequently in

stage II, there is a gradual, almost linear change suggesting that the damage accrues at a

constant rate in this stage. Stage III is marked by a large and rapid reduction in stiffness

immediately followed by failure.

0.6

0.8

1

1.2

0 0.3 0.6 0.9 1.2

Normalized cycles, N/Nf

Nor

mal

ized

stif

fnes

s

90%-5%

I II III

Figure 6.6 Decrease in rotational stiffness during the fatigue test

Fig. 6.7 shows a plot of the slope of the decrease in rotational stiffness, dK/dN, in

stage II of fatigue response versus the fatigue life, Nf, for all the specimens (where K, N

and Nf are stiffness, cycle and fatigue life, respectively). The response of all the fatigue

specimens tested at three different loading range follows a linear trend. The relationship

between the slope of stage II and the fatigue life was obtained by linear regression:

log(Nf)=-0.8213*log(dK/dN)+2.7572 (6.1)

Similar to the previous research in the t-C-T region and compression tests of c-C-

T region, the relationship between the slope of stage II and fatigue life is independent of


the load range. This suggests that the fatigue life can be determined or predicted from Eq.

(6.1) for any torque-range if the slope in stage II of the fatigue response is known.

1

2

3

4

5

6

-4 -2 0 2

Log(dK/dN)

Log(

Nf)

90%-5% load85%-5% load80%-5% load

Log(Nf)=-0.8213Log(dK/dN)+2.7572

Figure 6.7 Correlation between the decrease rate in rotational stiffness in stage II of the fatigue response and fatigue life

The axial displacements measured by the four axial LVDTs mounted on the same

specimen during the fatigue test are shown in Fig.6.8. The trends in axial displacement

correspond well with the observed change in rotation stiffness. The axial displacements

also show a three-stage change, I, II & III, similar to rotation stiffness. This suggests a

possible connection between the two phenomena. LVDT-3 shows an increase in axial

displacement while LVDT-1 shows a decrease in the measured axial displacement. In the

experiment, it was observed that an inclined crack initiated and propagated between

LVDT-2, 3 & 4 (Fig.6.8). From the static response, it has been established that the

different extensions of axial LVDTs are explained by crack propagation in the gage

section. Thus, formation and propagation of a crack in the gage section appears to be the

mechanism of failure for the fatigue loading.


The extensions of axial LVDTs suggest that the damage localized into a single

crack in the first few cycles of loading. This corresponds to a measurable reduction in

stiffness in the first few cycles (stage I). The crack propagates at a steady rate in the

subsequent cycles. This corresponds to gradual reduction in the rotational stiffness in

stage II. Failure occurs when the crack grows to a size that cannot safely support the

applied load.

-0.0001

0

0.0001

0.0002

0.0003

0.0004

0 0.3 0.6 0.9 1.2

Normalized cycles (N/Nf)

Exte

nsio

n of

axi

al L

VDTs

(in) LVDT-1

LVDT-2LVDT-3LVDT-4

24 3

I II III

Figure 6.8 Extensions of axial LVDTs during the fatigue test

6.3 Comparison between static and fatigue results

As known, for static test, the specimen can still sustain loading after the peak load. The

static failure displacement at a certain load level should be in the post-peak part of the

load displacement curve. A comparison of gage rotation at fatigue failure and that at the

corresponding load level in the post-peak part of the static response is shown in Fig. 6.9.

Each data point in the graph represents a different specimen loaded in fatigue. It can be

seen that the rotation at fatigue failure, which is defined as the maximum rotation at the

higher fatigue loading level, is not comparable to rotation obtained from the static post-

peak response; the rotation at fatigue failure are considerably smaller than the rotation at

the corresponding load in the static post-peak period. Therefore, a failure criterion based


on static rotation may not be suitable for plain concrete subjected to biaxial fatigue

loading in the c-C-T region. Similar observations have been reported for the concrete

airport pavement in the biaxial t-C-T region (Subramaniam 1999).

0

0.3

0.6

0.9

1.2

0 0.03 0.06 0.09 0.12

Gage rotation (deg)

Nor

mal

ized

torq

ue

Static90%-5%85%-5%80%-5%

Figure 6.9 A comparison of static and fatigue response: Torque - rotation

1

1.2

1.4

1.6

75 80 85 90 95% post-peak torque

Nor

mal

ized

rota

tiona

l com

plia

nce

FatigueStatic

Figure 6.10 A comparison of static and fatigue response: Torque – compliance


A comparison of rotational compliance at ultimate failure in fatigue and at

different points in the post peak part of the static response is shown in Fig. 6.10. The

normalized rotational compliance in fatigue was calculated with respect to the rotational

compliance in the first cycle. The figure indicates that for rotation compliance, the static

response is comparable to the fatigue response. This suggests that static response acts like

the failure envelope for fatigue behavior when change in compliance is considered. Since

change in compliance is associated with crack propagation, the fatigue response indicates

that the crack length at failure may be obtained from the static response.

-4

-3

-2

-10.8 1 1.2 1.4

Normalized rotational compliance

Log(

∆C

/ ∆N

)

90%-5%


Inflection point

Figure 6.11 Rotational compliance change rate versus rotational compliance

The rates of compliance change as a function of compliance for a typical

specimen as shown in Fig. 6.11. The rates of compliance follow a two-stage process; a

deceleration stage followed by an acceleration stage up to the failure. There is a distinct

inflection point marking a critical compliance where the rate of compliance changes from

deceleration to acceleration. This critical compliance corresponds to the compliance at

the peak static loading.


7. MODEL FOR FATIGUE FAILURE OF CONCRETE IN c-C-T REGION

7.1 Combined compressive and torsional test

Experiments conducted on concrete under combined compression and torsion discussed

in Chapter 6 showed that for a point in the biaxial c-C-T region (point-v), where the

principal compressive stress is 50% larger than the principal tensile stress, both the static

and fatigue failures are due to initiation and propagation of a single inclined crack.

Experimental evidence suggests that the increase in rotational compliance during the

compressive and torsional test is a result of this crack growth. This failure is very similar

to those observed in the t-C-T region when loading is pure torsion. Hence, it is important

to obtain information about the crack growth due to combined compressive and torsional

loading. The problem can then be studied by fracture mechanics, and a fracture-based

crack growth criterion can be established for such loading. The previously proposed

models and methods in the t-C-T region can then be extended and verified.

τ

τ σ

σ

τ τ θ

τi τ τoRi=2.25 in

Ro=4 in

(a) (b)

Figure 7.1 Stress state at the inner, middle and outer radius of the cylindrical wall

Direct measurements of crack length are impossible in the concrete hollow

cylinders. The relationship between the crack length and rotational compliance of the

specimen can only be established by the finite element method (FEM) using the

principles of linear elastic fracture mechanics (LEFM). Detailed FEM simulation for

concrete hollow cylinders under pure torsion was carried out by Subramaniam (1999). He

fully discussed the crack front profiles, stress intensity factors (KI, KII, KIII), and crack


propagation criterion, and found that the crack growth in concrete hollow cylinders under

pure torsion was due to the principal tensile stress and was governed by the Mode I stress

intensity factor (SIF), KIC.

KI=KIC (7.1)

Using FEM, he presented the relationship between the Mode I SIF, KI, crack

length, a, and applied torque as:

KI = (2.6296x10-6(a) + 7.2064x10-6) * Torque (7.2)

where KI is in N/mm3/2, Torque is in N-mm, and a is in mm.

Torque

CiCu

KIC

Rotation

Figure 7.2 Failure analysis using Mode I SIF in the static

combined compressive & torsional test

Eq. (7.2) was obtained from the pure torsional case, i.e. (Fig.

5.1b). For this loading case, the crack surface was observed to be planar along the

thickness direction. However, under the combined compressive and torsional loading

case, the crack surface is no longer a plane as observed in the case of pure torsional

loading, because of the influence of the compressive stress (

o45 and 0 == θσ

0≠σ ). The crack surface

formed a helically curved surface in space. However, the angles of inclination of the

helical crack along the thickness direction (θ in Fig. 7.1b) calculated from the elastic

theory were found to be quite close among each other: 50.76o: 52.91o: 49.53o (1: 1.04:

0.98) for the case of c=-2/3. Even for c=-1/4, the ratio of the calculated angles of

inclination of the crack is not very large: 63.43o: 68.08o: 60.17o (1: 1.07: 0.95). So it can


be assumed that the crack surface is approximately a plane in the thickness direction. The

inclined crack in the specimen was shown in Fig. 6.3 at an angle of approximately 51o

with respect to the horizontal (for c=-2/3).

Since the failure of hollow specimens under the combined compression and

torsion is quite similar to those under the pure torsion (both due to the crack propagation

by the principal tensile stress), and the ratio of the inclined angles along the hollow

cylinder thickness direction is not large, it is reasonable to extend the relationship of Eq.

(7.2) to the combined compressive and torsional loading case. However, due to a change

of direction and magnitude of the principal tensile stress, the applied torque, which

generates the principal tensile stress, should be modified by multiplying a factor, c (Eq.

5.5 & 5.6). For example, at pure torsion, σ1=τ, and at combined compression & torsion, if

c=-2/3, σ1= c * τ = 0.817τ. Then Eq. (7.2) can be modified as

KI = (2.63x10-6(a) + 7.21x10-6) * c * Torque (7.2a)

where KI is in N/mm3/2, Torque is in N-mm, and a is in mm.

1

1.2

1.4

1.6

0 0.5

Crack length (in)

Nor

mal

ized

com

plia

nce

ExperimentFitting curve

y=3.35x2-1.80x+1

1

0

0.2

0.4

0.6

0.8

1 1.2 1.4 1.6

Normalized compliance

Cra

ck le

ngth

(in)

ExperimentFitting curve

y=sqrt(-4.02x2+11.32x-7.30)

(a) (b)

Figure 7.3 Normalized torsional compliance vs. crack length in static test


In the previous research on the t-C-T region, evidence was presented to show that

both static and fatigue failure are due to Mode I crack propagation and governed by the

Mode I SIF, KIC. KIC = 38 N/mm3/2. Under combined compression and torsion, the failure

observed was very similar to failures in the t-C-T region (due to the crack propagation).

Thus, it is assumed that under static combined compression and torsion the post-peak

torque vs. rotation curve was also governed by the Mode I SIF, KIC (Fig. 7.2). The crack

length at any point in post-peak part of the response can be calculated from the increase

in unloading/reloading compliance. From Fig. 6.1 and 7.2 and Eq. (7.2), the relationship

between crack length and compliance can be established (Fig. 7.3).

Torsional compliance = 3.35*(a)2 - 1.80* (a) + 1 (7.3a)

a = SQRT(-4.02* (torsional compliance)2 + 11.32*(torsional compliance) - 7.30)

(7.3b)

where a is in inches.

(a) (b)

0

0.2

0.4

0.6

0.8

1

0 0.3 0.6 0.9 1.2

Normalized cycles, N/Nf

Cra

ck le

ngth

(in)

90%-5%

Inflection point

-4

-3

-2

-10.2 0.4 0.6 0.8

Crack length (in)

Log(

∆a/

∆N

)

90%-5%


ainflection

Figure 7.4 Crack growth during fatigue compressive and torsional test

(a) crack length vs. cycles (b) rate of crack growth


In Fig. 6.11, it was shown that under compression and torsion the compliances are

comparable between static and fatigue tests. Therefore, Eq. (7.3) and Eq. (7.2) are also

applicable for fatigue loading. This implies that the fatigue failure is also governed by the

Mode I SIF. Thus, fracture mechanics can be employed to analyze the failure mechanism

of concrete under fatigue compression and torsion.

The fatigue crack length vs. number of cycles is shown in Fig. 7.4a, and the rate

of fatigue crack growth (∆a/∆N) for the same specimen is shown in Fig. 7.4b. It can be

seen that similar to the change rate of torsional compliance, the crack growth rate also

follows a two-stage process: a deceleration stage followed by an acceleration stage up to

failure. There is an inflection point (minimum) corresponding to a critical crack length,

acritical, where the rate of crack growth changes from deceleration to acceleration. This

critical crack length is estimated to be 0.44 inch based on Fig. 7.4b. On the other hand,

from Eq. (7.2) it can be calculated that the crack length corresponding to the static peak

load, apeak, is approximately 0.48 inch. Then peakcritical aa ≈ . Taking crack length as the

indicator, the inflection point in fatigue test corresponds to the peak point in the static

test. The deceleration stage typically accounts for the first 45% the fatigue life of the

specimen.

0

0.2

0.4

0.6

0.8

1

75 80 85 90 95

Load range (%)

Cra

ck le

ngth

(in)

FatigueStatic

Figure 7.5 A comparison of static and fatigue response: Torque – crack length


In Fig. 6.9, it was shown that a deflection based failure criterion is not suitable for

fatigue behavior of concrete under combined compression and torsion. The static failure

deformation (gage rotation) at the post-peak period is not comparable with that of the

corresponding fatigue test. However, as shown in Fig. 6.10, under the torsional

compliance, the static post-peak behavior is favorably comparable to the corresponding

fatigue behavior. This suggests the possibility of using crack/fracture based failure

criterion for predicting fatigue failure using the results from the static test. A comparison

of crack length at fatigue failure to those at different unloading/reloading points in the

static post-peak period is shown in Fig. 7.5. It can be seen that the crack lengths at fatigue

failure compare favorably with the crack lengths at the corresponding load in the static

post-peak response of the specimen. This suggests that the crack length at fatigue failure

can be obtained from the static response. Hence the static response acts as a failure-

envelope to the fatigue response if framed in terms of crack lengths. Further more, Eq.

(7.1) (KI=KIC) represents the failure criterion for fatigue loading.

From the previous research in t-C-T region, it is proposed that the crack growth

rate at the deceleration stage is governed by the increasing resistance (R curve, Eq.7.4a),

and it is governed by the Mode I SIF (Paris law, Eq.7.4b) in the acceleration stage. The

concept of the R-curve is defined by Shah et. al. (1995). In the deceleration stage, the

crack growth rate can be expressed as:

1)( 01naaC

Na

−=∆∆ (7.4a)

or )()()( 011 aaLognCLogNaLog −+=

∆∆

In acceleration stage, the crack growth rate can be expressed as:

2)(2n

IKCNa

∆=∆∆ (7.4b)

or )()()( 22 IKLognCLogNaLog ∆+=

∆∆

Where a0 is the initial crack length (approximately 2 mm). Log(C1), n1, Log(C2) and n2

are constants. These parameters were calibrated from the previous study in the t-C-T

region (Table 7.1). The units of crack length, a, and SIF, KI, are mm and N/mm3/2,

respectively.


Table 7.1 Parameters in the models

Log(C1) n1 Log(C2) n2 t-C-T region -0.65~-2.05 -1.25~-1.7 -17.5~-42.5 9~24

c-C-T region (Tensile failure) 59.6 -68.8 -28.6 17.33

c-C-T region (Compressive failure) -10.25 -2.02 -6.46 21.33

A comparison of the experimental data from the combined compressive and

torsional tests and the model predictions from the deceleration and acceleration stages is

shown in Fig. 7.6. In the deceleration stage (Eq. 7.4a), the values of Log(C1) and n1 were

taken to be 59.6 and –68.8, respectively. The Paris law constants (Eq. 7.4b), Log(C2) and

n2, were taken to be –28.6 and 17.33, respectively (the average value from the previous

research, Table 7.1). It can be seen that there is a reasonable match between the

experimental data and model predictions. So the previously proposed model can be

successfully extended to the c-C-T region. However, the parameters in the deceleration

stage are different from those calibrated from the flexural response of concrete. The

fatigue crack rate growth at the acceleration stage can be predicted using the uniaxial

material parameters.

(a) (b) (c)

Figure 7.6 A comparison between analytical prediction and experiment:

loading equal to (a) 90%-5% (b) 85%-5% and (c) 80%-5%

-6

-4

-2

0

2

5 10 15 20 25

Crack length (mm)

Log(

∆a/

∆N

)

DecelerationAccelaration

-6

-4

-2

0

2

5 10 15 20 25

Crack length (mm)

Log(

∆a/

∆N

)

DecelerationAceleration

-6

-4

-2

0

2

5 10 15 20 25

Crack length (mm)

Log(

∆a/

∆N

)

DecelerationAcceleration


7.2 Pure compressive test

From the band-damage zone model (Fig. 4.1), it can be seen that the post-peak inelastic

displacements in the damage zone of the static test are comparable with those of the

fatigue test (Fig. 4.2). This suggests that the fatigue failure mechanism is the same as that

of the static failure mechanism. As mentioned by Shah and Sankar (1987), failure

localization initiates just prior to the peak stress. This phenomenon is clearly observed in

the experiment. For example, in the static compressive test, the pre-peak load-

displacement curve becomes nonlinear at approximately 70% of the peak load (Fig. 3.4a).

This means that the inelastic displacement of the specimen starts prior to the peak load.

However, since it is found that the total displacement for fatigue is not comparable with

the static test, it follows that the pre-peak inelastic displacements are also not comparable

between static and fatigue tests. This further implies that the deformation mechanisms are

different between the two tests in the pre-peak period. On the other hand, the pre-peak

compliances are comparable between the two tests. This provides a way to define the pre-

peak inelastic displacements for both static and fatigue test, based on compliance.

d

F

Cr

Cpeak

Ci

δ

Displacement

Compression

δ0

Figure 7.7 Definition of inelastic displacements

It is assumed this inelastic displacement starts at 70% of peak load, in both static

and fatigue tests. Prior to this load level, the specimen is elastic and the normalized

compliance (Cr/Ci) is unity. In the static test, the total inelastic displacement, δ*, consists

of two parts: pre-peak inelastic displacement, δ0, and post-peak inelastic displacement, δ:


(7.5) δδδ +=∗0

where δ is distributed with in the band damage zone only, and δ0 is distributed along the

whole length of the specimen including the band damage zone. The definition of the pre-

peak inelastic displacement, δ0, for the static test is illustrated in Fig. 7.7. It is equal to the

total displacement less the elastic displacement:

δ0 = d – F*Cr, (7.5a)

This pre-peak inelastic displacement, δ0, leads to the inelastic strain part in bε (Eq. 4.1).

For the fatigue test, it is assumed that the inelastic displacement also consists two

parts: pre-peak inelastic displacement, 0δ ′ , and post-peak inelastic displacement, δ, (same

as that of static test).

(7.5b) δδδ +′=∗0

where 0δ ′ is a pre-peak inelastic displacement. The relationship between the static pre-

peak inelastic displacement, δ0, and fatigue pre-peak inelastic displacement 0δ ′ is

established through the comparable pre-peak compliances of the two tests.

y = 0.22x3 - 0.92x2 + 1.26x - 0.57

0

0.004

0.008

0.012

0.016

0.02

0.024

1 1.2 1.4 1.6 1.8 2

Normalized compliance, Cr/Ci

Inel

astic

dis

plac

emen

t, δ

∗, (

in)

Experiment

Figure 7.8 Inelastic displacement vs. compliance in static compressive test

In pure compression, the cracks are distributed, unlike in the t-C-T region, where

individual cracks can be identified and measured, so it is not possible to verify the


previously proposed models in the t-C-T region, which are based on crack information.

To verify/extend the previous models for the pure compression point, this inelastic

displacement is used analogously to the crack length in the previous model.

δ

0

0.004

0.008

0.012

0.016

0.02

0 0.3 0.6 0.9 1.2

Normalized cycles (N/Nf)

Inel

astic

dis

plac

emen

t, ∗, (

in) 76%-5%

Inflection point

-7

-6

-5

-40 0.005 0.01 0.015 0.02

Inelastic displacement, δ ∗ , (in)

Log(

∆δ

∗/ ∆

N)

76%-5%Inflection point

(a) (b)

Figure 7.9 Increment of inelastic displacement during fatigue compressive test

(a) inelastic displacement vs. cycles (b) rate of inelastic displacement increment

Fig. 7.8 shows the relationship between the compressive compliance and inelastic

displacement in the static compressive test. The fatigue inelastic displacement vs. number

of cycles is shown in Fig. 7.9a, and the rate of fatigue inelastic displacement increment

(∆δ*/∆N) for the same specimen is shown in Fig. 7.9b. (In Fig. 7.9, nominal pre-peak

inelastic displacement is used). Similar to the previous results of crack length vs. number

of cycles in t-C-T region and point-v in c-C-T region (Fig. 7.4a), a S-shape curve is found

in Fig. 7.9a. The inelastic displacement change rate also follows a two-stage process: a

deceleration stage followed by an acceleration stage up to failure. There is an inflection

point (minimum) corresponding to a critical nominated inelastic displacement, where the

rate of inelastic displacement increment changes from deceleration to acceleration. This

critical inelastic displacement corresponds to the critical compliance in Figs. 3.5 and 3.6.


This means that when analyzed in terms of inelastic displacement, the inflection point in

the fatigue test corresponds to the peak point in the static test. The deceleration stage

typically accounts for the first 40% of the fatigue life of the specimen for pure

compression loading case.

In Fig. 4.2, it was shown that in terms of the post-peak inelastic displacement, δ ,

the static post-peak behavior is similar to the corresponding fatigue behavior. This

suggests the possibility of using an inelastic displacement-based failure criterion for

predicting fatigue failure using the results from the static test. From the previous research

in t-C-T region and current research at loading point-v (combined compression and

torsion) in the c-C-T region, it has been seen that the crack growth rate in the deceleration

stage is governed by the increasing resistance (R curve), and it is governed by the Mode I

SIF (Paris law) in the acceleration stage (Eqs. 7.4a and 7.4b). Here, it is reasonable to

assume that for pure compressive loading the change rate of inelastic displacement in the

deceleration stage is governed by the increasing compressive resistance, which is a

function of the increment of the pre-peak inelastic displacement, and it is governed by the

compliance in the acceleration stage (post-peak inelastic displacement). Based on this

assumption the previously proposed model, Eqs. 7.4a and 7.4b, can be modified. In the

deceleration stage, the inelastic displacement change rate can be expressed as:

1)(1nC

N∗

∗

=∆∆ δδ (7.6a)

or )()()( 11∗

∗

+=∆∆ δδ LognCLog

NLog

In acceleration stage, the crack growth rate can be expressed as:

2)(2n

i

r

CCC

N=

∆∆ ∗δ (7.6b)

or )()()( 22i

r

CCLognCLog

NLog +=

∆∆ ∗δ

Where δ* is the inelastic displacement in inches, and Cr and Ci are the reloading and

initial compliance, respectively.

A comparison of the experimental data from the pure compressive tests and the

model predictions from the deceleration and acceleration stages is shown in Fig. 7.10. In


the deceleration stage (Eq. 7.6a), the values of Log(C1) and n1 were taken to be –10.25

and –2.02, respectively. In the acceleration stage (Eq. 7.6b), Log(C2) and n2, were taken

to be –6.46 and 21.33, respectively (Table 7.1). A reasonable match between the

experimental data and model predictions is found. So by substituting crack length with

the inelastic displacement the previously proposed model can be extended to the pure

compressive loading point in the c-C-T region.


-8

-6

-4

-2

00 0.005 0.01 0.015 0.02

Inelastic displacement, δ ∗ , (in)

Log(

∆δ

∗/ ∆

N)


(a)

-8

-6

-4

-2

00 0.005 0.01 0.015 0.02

Inelastic displacement, δ *, (in)

Log(

∆δ

*/ ∆N

)


(b)

-8

-6

-4

-2

00 0.005 0.01 0.015 0.02

Inelastic displacement, δ *, (in)

Log(

∆δ

*/ ∆N

)


(c)

Figure 7.10 A comparison between analytical prediction and experiment in

compressive loading: (a) 90%-5% (b) 80%-5% and (c) 76%-5%


7.3 Transition point between tensile failure mode and compressive failure mode

As can be seen in the c-C-T region (Fig. 2.2), there are two typical failure modes caused

by two typical loading cases. One is the tensile failure mode similar to that observed in

the t-C-T region, which is dominated by crack initiation and propagation, as represented

by point-v (torsion & compression). The other is the compressive failure mode, which is

dominated by inelastic displacement, as represented by point-iv (pure compression). In

the case of the tensile failure mode, a crack based fracture mechanics criterion is

employed to explain the failure mechanism. In the case of compressive failure mode, an

inelastic displacement based criterion is employed.

0 0 21 <= σσc-C-T region t-C-T region

σ1

σ2

|||| 21 σσ >

iii |||| 21 σσ =ii

iv

v|||| 21 σσ <

Transition point

151

2

1 =σσ

Inelastic displacement dominated compressive failure Crack dominated tensile failure

i 0 0 12 >= σσ

Figure 7.10 Transition point from tensile failure to compressive failure

Kupfer et. al. (1969), Kupfer and Gerstle (1973), and Nelissen (1972)

systematically investigated the performance of concrete subjected to biaxial stresses. In

their study concrete plates were subjected to in-plane loading for different biaxial stress

combinations spanning the entire biaxial stress space. The mode of failure was found to

be different in the biaxial stress space. It has been found that in the c-C-T region, the

failure is similar to that in compression as long as the principal tensile stress is less that

151 of the principal compressive stress. For larger principal tensile stress concrete fails by

a single crack forming perpendicular to the maximum principal tensile stress. Their

conclusions support the current research results. Hence it is reasonable to postulate that

the transition point between tensile failure mode and compressive failure mode is


determined by the ratio of the principal tensile to compressive stresses (Fig. 7.10), as

expressed as

151

=−−

−−stressecompressivprincipal

stresstensileprincipal (7.7)

So, a larger principal tensile to compressive stress ratio (>151 ) leads to the tensile

failure mode where failure is governed by the crack based fracture mechanics, while a

smaller principal tensile to compressive stress ratio (<151 ) leads to the compressive

failure mode where failure is governed by the inelastic displacement.

Load Load

A

Fatigue

Static envelope governed by KI=KIC

δ0/δ0' δ

70% peak load

δcritical

C

B

A

Fatigue inelastic displacement growth

Inelastic displacement, δ*

Static envelope

Fatigue

acritical

C

B

Fatigue crack growth

Crack length

No. of cycles No. of cycles

(a) (b)

Figure 7.11 Schematic representation of (a) crack growth, and (b) inelastic

displacement growth in static and fatigue loading

In the tensile failure mode, the crack growth in the static and fatigue loading can

be schematically represented by Fig.11a, and in compressive failure mode, the inelastic

displacement growth in the static and fatigue loading can be schematically represented by

Fig.11b.


8. CONCLUSIONS

The objective of this work is to characterize the static and fatigue response of concrete

subjected to biaxial stresses in the c-C-T region, where the principal compressive stress is

larger in magnitude than the principal tensile stress. By extending the previous methods

and results to the biaxial compression region, a full description of fatigue in the C-T

region was obtained.

Previous research on high amplitude fatigue response of concrete subjected to

biaxial stresses in t-C-T region, where the principal tensile stress is larger in magnitude

than the principal compressive stress, suggested the following: (a) the change rate of

structural compliance and crack length in constant amplitude fatigue loading is a two-

phase process: a deceleration phase followed by an acceleration stage; (b) a static load

envelope was shown to predict the crack length or structural stiffness/compliance at

fatigue failure. The primary mode of failure in this biaxial stress region was shown to be

crack propagation. The fatigue crack growth models were developed using fracture-based

parameters.

An experimental investigation of material behavior in the biaxial c-C-T region

was conducted. Two typical loading cases were selected to represent the biaxial c-C-T

loading cases: compression and combined compression and torsion. The experimental

setup consisted of two test configurations: concrete cylinders subjected to pure

compression, and hollow concrete cylinders subjected to torsion with a superimposed

axial compressive force. The damage imparted to the material was measured by

mechanical means.

In the investigation, the static load response can be visualized as a failure

envelope curve, where each point in the post-peak region is an equilibrium point

representing the maximum load that can be supported for a given level of damage.

Therefore every point on the post-peak load envelope can be characterized by a given

damage level. Further, it can be implicitly assumed that the change in compliance of a

specimen is due to accruing damage in the specimen and the increase in compliance is


indicative of the increase in level of damage. In both tests, the favorable comparison

between the percentage increase/decrease in compliance, or crack length (if applicable),

or inelastic displacement (if applicable) in fatigue and static post-peak load indicates that

the damage level is comparable for the two loadings. Some specific conclusions of this

project are listed below:

• The previously proposed methods, theories and models in the t-C-T region are

applicable in the whole C-T region if a suitable length/displacement is selected to

describe the failure mode. For tensile failure mode, the crack length is used, and for

compressive failure mode, the inelastic displacement is used.

• A deflection based fatigue failure criterion is not suitable for concrete subjected to

biaxial C-T loading because the displacement at fatigue failure is not comparable to

the static post-peak displacement at the corresponding load.

• In the biaxial C-T region, the static response acts like an envelope to constant-

amplitude, low-cycle fatigue response when framed in terms of compliance, or

crack length, or inelastic displacement.

• The change rate of compliance, or crack length, or inelastic displacement is a two-

phase process in the C-T region: a deceleration stage followed by an acceleration

stage up to failure.

• The damage evolution in C-T region, in term of the measured stiffness, follows a

three-stage trend. Fatigue life can be predicted by the slope of the stiffness change

rate in stage II, and the relationship between the slope of stage II and fatigue life is

independent of the fatigue load range.


REFERENCES

Gerstle, K. H., (1981). “Simple Formulation of Biaxial Concrete Behavior,” ACI

Materials Journal, 78, 62-68.

Jansen, D. C. and Shah, S. P., (1997). “Effect of Length on Compressive Strain Softening

of Concrete,” Journal of Engineering Mechanics, ASCE, 123(1), 25-35.

Jansen, D. C., (1996). Postpeak Properties of High Strength Concrete Cylinders in

Compression and Reinforced Beams in Shear, Ph.D thesis, Northwestern

University.

Jansen, D. C. and Shah, S. P., (1993). “Stable Feedback Signals for Obtaining Full Stress

Strain Curves of High Strength Concrete”, Proceedings – Utilization of High

Strength Concrete, V.2, 20-24, Lillehammer, Norway.

Okubo, S. and Nishimatsu, Y. (1985). “Uniaxial Compression Testing Using a Linear

Combination of Stress and Strain as the Control Variable,” International Journal

of Rock Mech. Min. Sci. & Geomech. Abstr. 22(5), 323-330.

Rokugo, K., Ohno, S., and Koyanagi, W., (1986). “Automatic Measuring System of

Load-Displacement Curves Including Post Failure Region of Concrete

Specimens”, In Fracture Toughness and Fracture Energy of Concrete, Edited by

F. H. Wittman, Elsevier Applied Science.

Kupfer, H. B., Hilsdorf, H., and Rusch, (1969). “Behavior of Concrete Under Biaxial

Stresses,” ACI Materials Journal, 66(8), 656-666.

Kupfer, H. B., and Gerstle, K. H., (1973). “Behavior of Concrete under Biaxial Stresses,”

Journal of Engineering Mechanics, ASCE, 99(4), 853-866.

Nelissen, L. M. J., (1972). “Biaxial testing of normal concrete,” Heron (Delft), 18(1).

Shah, S. P., Swartz, S. E., and Ouyang, C., (1995). Fracture Mechanics of Concrete:

Applications of Fracture Mechanics to Concrete, Rock and Other Quasi-Brittle

Materials, Wiley, New York.

Shah, S. P. and Sankar, R., (1987). “Interanl cracking and strain-softening response of

concrete under uniaxial compression,” ACI Materials Journal, 84(3), 200-212.


Subramaniam, K. V., Popovics, J.S., and Shah, S.P., (2002). “Fatigue Fracture of

Concrete Subjected to Biaxial Stresses in the Tensile C–T Region,” Journal of

Engineering Mechanics, ASCE, 128(6), 668-676.

Subramaniam, K. V., O’Neil, E., Popovics, J.S., and Shah, S.P., (2000). “Flexural Fatigue

of Concrete: Experiments and Theoretical Model,” Journal of Engineering

Mechanics, ASCE, 126(9), 891-898.

Subramaniam, K. V., (1999). “Fatigue of Concrete Subjected to Biaxial Loading in the

Tension Region,” PhD dissertation, Northwestern University, Evanston, IL.

Subramaniam, K. V., Popovics, J.S., and Shah, S.P., (1999). “Fatigue Behavior of

Concrete subjected to Biaxial Stresses in the C-T Region,” ACI Materials Journal,

96(6), 663-669.

Subramaniam, K. V., Popovics, J.S., and Shah, S.P., (1998). “Testing Concrete in

Torsion: Instability Analysis and Experiments,” Journal of Engineering

Mechanics, ASCE, 124(11), 1258-1268.

Torrenti, J. M., Benaija, E. H. and Boulay, C., (1993). “Influence of boundary conditions

on strain softening in concrete compression test,” Journal of Engineering

Mechanics, ASCE, 119(12), 2369-2384.


APPENDIX I EXPERIMENTAL DATA – Compressive test

Table A-1 Test data at the peak loads in static compressive test

Specimen Peak load (kip) Axial displacement (in) Normalized compliance (Cr/Ci)

1 76.06 0.035 1.13

2 75.29 0.033 1.04

3 75.83 0.034 1.09

Average 75.73 0.034 1.05

Table A-2 Post-peak normalized compliance (Cr/Ci) and axial displacement in static compressive test

Specimen

Pre-peak

Ci/Ci (Disp,

in)

90% post-peak

load

Cr/Ci (Disp, in)

80% post-peak

load

Cr/Ci (Disp, in)

70% post-peak

load

Cr/Ci (Disp, in)

1 1.0 (0.026) 1.21 (0.038) 1.43 (0.040) 1.56 (0.041)

2 1.0 (0.026) 1.12 (0.038) 1.39 (0.040) 1.59 (0.043)

3 1.0 (0.027) 1.18 (0.040) 1.43 (0.043) 1.56 (0.045)

Average 1.0 (0.026) 1.16 (0.038) 1.41 (0.041) 1.56 (0.043)

Note: i) The axial displacement at the initial compliance stage was measured at the starting point of the nonlinear load-displacement curve in pre-peak load period.

ii) The axial displacement at x% post-peak load was measured at the starting unloading point in the load-displacement curve.


Table A-3 Results from fatigue tests for the load cycle of 90% ~ 5% of the average static compressive strength

At fatigue failure At inflection

Specimen

Cycles to

failure Axial disp.

(in)

Normalized

compliance

Normalized

compliance

1 3621 0.038 1.27 1.03

2 338 0.034 1.28 1.03

3 66 0.031 1.21 1.02

Average 1342 0.034 1.25 1.03

Table A-4 Results from fatigue tests for the load cycle of 80% ~ 5% of the average static compressive strength


Specimen

Cycles to

failure Axial disp.

(in)

Normalized

compliance

Normalized

compliance

1 1966 0.035 1.32 1.05

2 1390 0.032 1.24 1.06

3 3984 0.038 1.35 1.10

Average 2447 0.035 1.30 1.07

Table A-5 Results from fatigue tests for the load cycle of 76~ 5% of the average static compressive strength


Specimen

Cycles to

failure Axial disp.

(in)

Normalized

compliance

Normalized

compliance

1 15256 0.035 1.39 1.04

2 15906 0.035 1.72 1.04

3 18656 0.037 1.54 1.03

4 11174 0.033 1.43 1.04

Average 15248 0.035 1.52 1.04


APPENDIX II EXPERIMENTAL DATA – Combined compressive & torsional test

Table AA-1 Test data at the peak torque static combined compressive & torsional test

Specimen Peak torque (ft-lb) Gage rotation (deg)

1 838 0.0371

2 829 0.0411

3 835 0.0408

Average 834 0.0397

Table AA-2 Cycles for fatigue combined compressive & torsional test

Cycles to failure

Cycles to failure

Specimen

(90%-5%) (85%-5%)

Cycles to failure

(80%-5%)

1 1458 14162 90642

2 128 46074 -

3 274 3370 -

4 2134 - -

Average 999 21202 90642


APPENDIX III Publications from This Project

1. Bin Mu, Kolluru V. Subramaniam and S. P. Shah, (2003). “Failure Mechanism of

Concrete under Fatigue Compressive Load,” Journal of Materials in Civil Engineering, submitted (MT/2003/022651).

2. Bin Mu and S. P. Shah, (2003). “Failure Mechanism of Concrete Under Biaxial

Fatigue Load,” Fifth International Conference on Fracture Mechanics of Concrete and Concrete Structures, April 12 – 16, 2004, Vail Colorado, accepted.

3. Bin Mu and S. P. Shah, (2003). “Fatigue Behavior of Concrete Subjected to

Biaxial Loading in the Compression Region,” Materials and Structures, submitted.

Fatigue Strength of Concrete

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