vipu exam r1

Written Exam for Dr. Vipu

Norman S. Hoffman, PE 3651 Foremast. Dr. Galveston, TX 77554

Phone: (409)‐621‐6740 Email: [email protected]

April 1, 2010

April 1, 2010 Written Exam for Dr. Vipu Norman S. Hoffman, PE 3651 Foremast. Dr. Galveston, TX 77554 Phone: (409)‐621‐6740, Email: [email protected]

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Table of contents

Plain Concrete Constitutive Laws 3

Smeared Concrete Constitutive Laws 6

Nano‐particle Concrete 21

Figures and Tables

Figure 1, Smeared Constitutive Model for Concrete 7

Figure 2, Stress Strain relationship for Panels TEF‐3, ‐4, and ‐5 9

Table 1, Constitutive Equation Comparison for Plain Concrete and Fiber Concrete 11

Table 2, Softened Membrane Model (SMM) Smeared Constitutive Equation

Comparison for Concrete and Steel Fiber Concrete 17

References 26


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Summarize in complete and concise form the state of knowledge on the constitutive models

for concrete (stress‐strain and failure).

A survey of literature reveals that constitutive material models for plain concrete can be

categorized into three very broad groups based on loading situation, namely:

1) Unixial models

2) Biaxial models

3) Triaxial models

The basic uniaxial stress strain model for plain concrete consists of an ascending branch and a

descending branch. The peak of the curve occurs at a location called concrete compressive

strength while the corresponding strain is the peak compressive strain. There have been

numerous studies and approximations for modeling the stress strain curve of plain concrete

(Popovics, 1970). Important and straight forward approximations of the curve include those

ranging from the Hognestad, 1952 parabola (to which Hognestad attributes to Stussi, 1932), to

those of Desayi & Krishnan 1964 and Wang & Shah, 1978. (See Table 1). The basic approach

for researchers modeling the curve is to base the shape on key parameters that can be

obtained easily from physical tests of specimens, namely the failure criteria, fc’ and e0.

Interestingly, for all the research done on the stress strain curve for plain concrete, the basic

Stussi, 1932 model is still used today in flexural design.

Tension models of plain concrete depend on the testing procedure used (Belarbi, 1994) A

linear behavior is obtained up to the tensile capacity which then ends abruptly with a brittle

failure, thus providing no descending branch. Gopal & Shah, 1985 developed a testing

technique and corresponding analytic model for determining the descending branch of the

tensile curve.

Biaxial models for plain concrete include those developed Zaman 1993, and Gerstle, 1981.

Please refer to Table 1 for a summary and short description of these models.


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How will the constitutive models of concrete be affected if fibers are added?

The addition of fiber to the constitutive models for concrete will by necessity have the same

effect as on the physical or experimental test results. The addition of fibers to plain concrete

has an improving effect on the key material properties listed below, and thus affects the

constitutive model for fiber concrete (Rao 2009, Thomas 2007, Vandewalle 2002, Li 1992,

Traina 1991).

Unixial Compressive Strength, fc’

Uniaxial Peak strain at compressive strength, e0

Modulus of Elasticity, Ec

Uniaxial Tensile Strength, ft

Modulus of Rupture Ductility

Poisson’s Ratio,

The addition of fiber, particularly steel fiber, affects most significantly the tensile strength and

ductility of concrete. Thomas 2007 achieved 38.2% increase in tensile split cylinder strength

using just 1.5% fiber content by volume. Tests by Rao 2009 on standard Modulus of Rupture

test prisms (6” x 6” x 24”) show a marked post peak improvement in load carrying capacity of

fiber concrete as compared to plain concrete. Whereas the plain concrete specimens failed

completely and suddenly upon reaching peak load, the fiber concrete specimens sustained

significant load post‐peak (steel fiber content ranged from 0.5% to 1.5% by volume). Rao

attributes this to the bridging effect of the fibers across the tensile crack. Other researchers (Li

1992, Thomas 2007) show similar tensile results.

The affect of fiber on uniaxial cylinder compressive strength was modest, as compared to its

affect on the tensile strength. Thomas 2007 found an 8.3% increase in compressive capacity

using 1.5% fiber content by volume. He found that there is an increasing linear relation

between fiber content and compressive strength, up to 1.5% fiber, which was the limit of his

investigation.


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Although there is only a modest increase in compressive strength, there is a substantial

increase in compressive strain, e0. Thomas attributes the gain in strain to the confinement

effect of the fibers within the concrete matrix. Again, the relation is linear with increasing fiber

content and the improvement over plain concrete is up to 29% for 1.5% fiber by volume.

Thomas 2007 also found only a slight gain in Ec (8%) with increasing fiber content, up to 1.5%

by volume. There was a lesser gain in poisons ratio.

Key investigation into the biaxial behavior of fiber reinforced concrete was performed by

Kupfer 1969, Traina 1991 and Yin & Hsu 1989. These researchers established the basic failure

envelope for fiber concrete, with respect to plain concrete. Both show that there are

differences between the two types of concrete. The primary difference under biaxial loading is

the increase in compressive strength over plain concrete, for all stress ratios. Furthermore,

when compared to the uniaxial strength, the biaxial strength increases by as much as 85% with

1.5% fiber volume over plain concrete (Traina 1991).

Important Constitutive models for fiber concrete include biaxial models developed by Tan 1993

and Hu 2003. The Tan 1994 model considers only compression‐compression while Hu 2003

considers both compression and tension. Hu presents a single smooth biaxial failure curve.

These models utilize the Traina and Yin experimental data.

To what problems can you apply these constitutive models.

Fiber concrete has been used for many years to reduce cracking of floor slabs. It is anticipated

though that fiber concrete can be used in beams and girders to reduce or minimize shear

stirrups. The analytic models can be used in computer simulation models (finite element) to

predict behavior of plain and steel fiber concrete structures without the necessity of

performing actual testing. (Although it is good practice to perform some physical testing to

calibrate the finite element model).

Use Tables to Summarize equations. (See Table 1)


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Summarize in complete and concise form the state of knowledge on the Smeared constitutive

models for concrete.

The most current and complete model for prestressed and reinforced concrete is the Softened

Membrane Model or SMM (Hsu, 2009). This is a 2‐D model which applies to membrane

elements and broadly encompasses all shear members such as walls, girder webs, particularly

deep beams.

The individual SMM constitutive equations are summarized in Table2 and illustrated in Fig. 1 for

concrete. The equations cover the following:

Concrete in Tension (pre‐cracking and post cracking branches)

Concrete in Compression (ascending and descending branches)

Equilibrium Equations

Compatibility Equations

Post‐Cracking Hsu/Zhu (Poisson) Ratios

Uniaxial – Biaxial Transformation Equations

Embedded Mild Steel

Embedded Prestressing Tendon

Constitutive Models for Concrete, Discussion:

Constitutive models for concrete are being investigated by two general groups of concrete

researchers. There are those models that have been developed by materials researchers and

there are those models developed by researchers attempting to predict the behavior of whole

structural assemblies, including reinforcing. The latter group of models is generally referred to

as smeared models. The models overlap and indeed the materials models form the basis of the

structures models. It must be clearly understood that the distinctive difference between the

two sets of research is the presence of reinforcing steel such as deformed mild steel rebar or

stressing tendon. Concrete with reinforcing steel behaves differently than concrete without

reinforcing steel.

The research at the University of Houston with respect to concrete Dr. Hsu’s constitutive

models (such as the SMM) has focused on structural assemblies of concrete and reinforcing


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steel. These assemblies are tested to determine the constitutive properties on what is called a

smeared or average basis. Smeared model properties by definition span multiple cracks in the

concrete. The smeared constitutive model is a macro or full scale model which is used to model

whole structural behavior, particularly shear behavior of reinforced concrete continuums such

as walls beam webs, and other membrane structures. Smeared constitutive models are

designed and calibrated to full‐scale structures.

The materials research models for concrete focus on the mirco level of concrete. They

generally consider concrete on the single crack level, and may even model the cracks

themselves.

The overlap of model groups occurs at the concrete‐rebar interface. Constitutive bond

researchers model the bond between concrete and reinforcing. These models form a bridge

between the materials models and the smeared model research.

Fig. 1 Smeared Constitutive Model for Concrete (Hoffman, 2009)


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To what problems can we apply these constitutive models for Concrete with Fibers?

Fiber Concrete has been used for many years to reduce cracking of floor slabs. It is anticipated

though that concrete containing steel fibers (SFC) can be used in beams and girders to reduce

or minimize shear stirrups. Phase 2 of this research is underway at U of H (Rao, 2009) to test

full scale prestressed SFC bridge girders without shear steel. The models can also be used to

predict post peak behavior of any shear dominant structure, such as low‐rise shear walls, deep

beams, etc.

The distinctive feature of the SMM is that it can predict post peak shear behavior (descending

branch behavior) because Poisson effects (Hsu‐Zhu ratios) are included in the elasticity function

that bridges between stress and strain. Post‐peak behavior of concrete structures is important

in seismic design and research where prediction of deflection and strength of damaged

structures is critical.

The solution to the constitutive, equilibrium, and compatibility equations requires an iterative

algorithm in order to generate the complete monotonic shear stress‐strain curve. The

constitutive models are ultimately be used in finite element software, such as Open Sees, to

model structural behavior concrete structures.

By testing individual macro‐elements of prestressed SFC in the Universal Element Testing

machine, one can likewise determine the constitutive laws within the SMM framework that

govern prestress SFC behavior. These laws can then be put into Open Sees to model the

behavior of complete prestressed SFC structures.

The effect of steel fibers on the Constitutive Models:

Please refer to Table 2 for a comparison and remarks regarding the effect of steel fiber on the

SMM.

The softened membrane model consists of the constitutive equations listed in the summary

table below. These equations are bound together and satisfy the necessary conditions of

compatibility and equilibrium.

The smeared constitutive laws for concrete comprise those for concrete, for mild steel, and for


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steel prestressing strand. The laws are depicted graphically in the figures below. The effects of

fiber are given with respect to the Fiber Factor which is fiber volume times fiber aspect ratio:

FF=Vf*(lf/Df)

The two constitutive laws that will be influenced the most are the post cracking Tensile Stress

curve and the embedded mild steel and embedded tendon curves. The most significant though

is the steel fiber conctete in tension curve. Regular concrete undergoes a simple smooth decay

of strength post‐cracking. Steel fiber concrete on the other hand exhibits initially higher

tension stiffening characteristics just past cracking due to the fibers “bridging” the crack with a

tensile force. However, as cracking increases, the steel fibers begin to pullout or otherwise fail.

The point where this begins to happen is roughly at the yield strain of the steel. As the fibers

fail the bridging force decreases until finally a dominant crack opens where most of the bridging

fibers failed. The bridging force is transferred to the tendon.

The influence of steel fiber on the tensile behavior of concrete is illustrated in the figure above.

The results of prestresed steel fiber concrete panels (dashed lines) are plotted concurrently

Fig. 5.2 11 relationships of panels TEF-3, 4, and 5

Cracking of the panels

Tensile Strain, 1

Tens

ile S

tres

s σ

1 , M

Pa

(ks

i)

Fig. 2


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with the results for prestressed regular concrete (solid lines). The steel fibers increase the

stiffness post‐cracking and greatly improve toughness.

How the effects of fiber are being determined:

The purpose of the current testing is to determine the effect of fiber content, specifically Fiber

Factor, on the constitutive laws of prestressed SFC. To determine this, five prestressed SFC

panels were tested under the sequential loading. The data in the two loading stages were

recorded separately. In the first stage of tensile loading, all five panels were used to obtain the

tensile constitutive laws of SFC and prestressing tendons. In the second stage compressive

loading, the softening coefficients of prestressed SFC were determined as related to the

perpendicular tensile strains, and fiber factor. In the first series of panels TEF‐1, 2, and 3, the

prestress level and fiber aspect ratio was held constant while the fiber percentage was

increased. The tensile strain targets ranged from 0.005 to 0.016. Wang (2006) showed that the

softening coefficient, , decreases with increased tensile strain, 1 , in prestressed concrete.

The SFC results were normalized with respect to tensile strain level to show the effect of

softening with respect to fiber factor.

The tensile target strains for the sequential load panels (TEF) were not designed to examine

ultimate tensile properties of the SFC. Fortunately, panel TEF‐5 was tested (inadvertently) until

tensile failure of the strand, so there is data available for this range of tensile strain for SFC.

For proportional loading, it is anticipated that the simultaneous application of compressive

force in conjunction with tensile forces will increase the pull‐out capacity of the steel fibers,

thus enhancing their performance. Yin (1989) has shown that this is the case for biaxial

compression‐compression loading and Hu (2003) prepared an analytic model for compression‐

tension loading. This factor will show up in f1(x) as a function of the compressive force or

strain.


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Table1 Constitutive Equation Comparison for Plain Concrete and Fiber Concrete

Constitutive Property

Equation for Plain Concrete Equation for Fiber Concrete Remarks

Plain Concrete, or Fiber in Uniaxial Tension

Ascending Branch, Gopal and Shah, 1985

Descending Branch, Gopal and Shah, 1985

1.01

See Table 2, Eq. 17 & 18 for constitutive equation for Fiber Concrete in tension

Eq. 1

Eq. 2


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Plain Concrete or fiber Concrete in Uniaxial Compression

Hogenstad, 1952 and Stussi, 1932

k1 ranges from 0.7 to 0.9 k2 ranges from 0.35 to 0.45 k3 ranges from 0.85 to 1.0 eu ranges from 0.003 to 0.005

See Table 2, Eq. 21, 22 for constitutive equation for Fiber Concrete in Compression.

The Hognestad/Stussi Curve is what is used today for concrete design. The curve is defined by key parameters k1, k2, and k3. It is more a method for applying an assumed stress strain distribution or gradient to a concrete bending member, as opposed to being a true model relating stress to strain.


Desayi, 1964 (equation for the full curve)

Desayi equation is for the full ascending and descending stress strain curve. The descending branch inflection point is not well captured. Eq. 3


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Wang 1978 (equation for the full curve)

The Wang equation is for the full ascending and descending stress strain curve. This equation does a better job at capturing the descending branch inflection point than the Desayi equation.

Plain Concrete or fiber Concrete in Biaxial Compression

Gerstle, 1981

Hu 2003

where,

And kappa values are material constants determined from a experimental test curve for any specific fiber concrete …

Hu utilizes a single smooth biaxial failure curve. Requires test data points. Model is solved iterativly.

Eq. 4

Eq. 5

Eq. 6


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Plain Concrete or fiber Concrete in Biaxial Compression

Chen, 1994

Tan, 1994 Biaxial stress‐strain curve is given by

where,

,

,

, and

, finally,

where nu1, nu0, l, r’ , Vf and tau are

properties of the fibers with the concrete matrix.

Tan equations do not require test points, unlike Hu.

Eq. 7 Eq. 8

Eq. 9


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Tan 1994 further defines the failure envelope by:

where,

and,

Eq. 10


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Plain Concrete or fiber Concrete in Triaxial Compression

Warkne 1975

where:

Eq. 11


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Table 2 Softened Membrane Model (SMM) Smeared Constitutive Equation Comparison for Concrete and Steel Fiber Concrete


SMM Equation for Reinforced Concrete (RC), including prestressed Proposed SMM Equation for Steel Fiber Concrete (SFC), including prestressed Remarks

Concrete, or SFC in Tension

For SFC, there is a tension stiffening effect that occurs during Stage T2. After the reinforcing steel yields, there begins a softening region when the fibers bridging the cracks begin to pull out. This effect was not modeled by Mansour (2005) for sequential loading. Korb (2006) attempted to model this behavior from the results of his proportional load panel tests, but the cause was not realized and the equation used is awkward. Thus stage T2 is modified and Stage T3 is added to account for fiber failure across the crack bridge. Stage T3 will tend to zero as strain increases. Equations proposed by Mansour (2005) account for increases ductility and stiffness of the SFC, but do not tend to zero with increasing strain. Functions f1(F) and F2(F) are functions of the Fiber Factor, F = (Lf/Df)Vf Proper understanding of SFC in tension is crucial to being able to properly model it and predict experimental curves with analytic functions.

‐1

‐0.8

‐0.6

‐0.4

‐0.2

0

0.2

0.4

‐0.005 0 0.005 0.01 0.015 0.02

Concrete Stress (MPa)

Concrete Strain (mm/mm)

TEF‐3, Concrete Stress vs Strain (ksi)

Experimental Results, 1.5% Steel Fibers

Existing Theory (No Fibers)

Stage UC

Stage T1

StageT2 Stage T3

Eq. 12

Eq. 13

Eq. 14

Eq. 16

Eq. 17

Eq. 18

Eq. 15


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Concrete or SFC in Compression

Softening for RC:

.

Softening for presteressed SFC:

9.01 pc WffffFf , where

FFf 2.01

SFC will effect the descending branch of the SFC in compression curve, stage C2. Function f2(F) will be used to adjust the shape of the parabolic curve to fit the experimental data. Additionally, the experimental data shows that increasing the fiber factor, increases the softening coefficient (increases the peak compressive concrete stress). The softening factor will thus influence the shape of both Stage C1 and C2 of the SFC in compression curve.

Eq. 20

Eq. 19

Eq. 22

Eq. 21


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Mild Steel

))(293.0( 5 FfByn

Steel Fibers have two effects on embedded steel. First it lowers the apparent yield stress and second, it stiffens the post yield behavior, which for the bare bar is typically perfectly plastic. The factors F5(F) and F6(F) are used to adjust the apparent yield of steel and post‐yield slope. These factors are obtained from the experimental results.

Embedded Prestressing Strand

The prestressing tendon embedded in SFC displays stiffer post yield modulus than concrete without steel fiber. As such, function f4(F) will be used to adjust the post yield stiffness and will be fitted from the experimental results of the sequential load tests. Because of the shape of the knee of the tendon stress strain curve, the yield stress is not a well defined point as it is with mild steel. As such the apparent yield stress for tendon embedded in SFC should not require modification to the 0.7fpu apparent yield value currently being used for tendon embedded in regular concrete.

Equilibrium

Same as RC

Eq. 23

Eq. 25

Eq. 24

Eq. 27

Eq. 28

Eq. 26

Eq. 30

Eq. 29

Eq. 31

Eq. 32

Eq. 35

Eq. 34

Eq. 33


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Compatibility

Same as RC

Poisson (Hsu‐Zhu)

The Hsu‐Zhu ratios are presumed to be the same as RC, however, literature review for tests on un‐reinforced SFC (“plain” SFC) indicates that SFC tends to be increase confinement and this could influence (reduce) the Poisson ratio of plain SFC. However, panel cracking (reinforced concrete and reinforced SFC) is usually so extensive at the target tensile strains (usually 1% or more) that that the effect may not be significant.

For SFC, this could have effect on peak and post‐peak shear behavior. If experimental peak shear strains are difficult to correlate with the analytic model, then the Hsu‐Zhu relation may need refinement.

Uniaxial‐biaxial transformation 2

2112

121

21121 11

1

,

22112

12112

212 1

1

1

,

2221

22

122

2 cossin22

sincos

,

2221

22

122

2 cossin22

cossin

t .

The uniaxial‐biaxial strain relationships are the same as for RC Note that the stress and strains in the compatibility and equilibrium equations are biaxial. The key to a relatively simple solution algorithm is this straight‐forward transformation between uniaxial and biaxial strain.

Eq. 37

Eq. 38

Eq. 36

Eq. 41

Eq. 40

Eq. 39

Eq. 45

Eq. 44

Eq. 43

Eq. 42


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Concrete Mixtures with Nanoparticles

What are the types of nanoparticles used in concrete?

Although there are many types of nanoparticles that can be added to concrete, research has

generally focuses on three types of particles (Chung, 2004, Hui 2007, Gao 2009):

a) Nanosilica (SiO2)

b) Nanotitaniates (TiO2)

c) Nanocarbon fibers

Nanosilicates are spherical particles of SiO2 with a diameter on the order of 10‐9 m. Nanosilcates

are popular because they are readily available as they are a byproduct of the extraction of

metallic silica and alumina from electric arc furnaces.

Nano carbon fibers are extremely short length of carbon fiber with diameters on the order of

10‐9 m and aspect ratios ranging from (L/d) of 50 – 200.

Properties of nanoparticle concrete that are being studied include:

a) Increased Compressive strength

b) Improved Tensile toughness, strength, and fatigue behavior

c) Reduced Creep

d) Electrical resistivity, particularly for nanaocarbon fibers.

What factors must be considered in designing concrete mixtures with nanoparticles?

Factors to be considered in designing mixes include:

a) Dispersion due to high specific surface electrical charge

b) Effect of dispersants on the concrete mix

c) Influence of flocking or aggregation of nanoparticles and formation of inclusions or

weak pockets

d) Increase in density affects strength and creep for nanosilcate.

e) Increase in water demand vs. strength and workability (see discussion item 3)


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Dispersion:

To minimize aggregation of nanoparticles in the cement matrix where inclusions or weakness

could form, nanoparticle concrete additives must be thoroughly dispersed thoroughly and

uniformly dispersed throughout the cement paste during mixing.

Dispersants and special multi‐step mixing techniques and sequences are required for mixing

nanoparticles and fibers in concrete due to their high specific surface energy. High speed pre‐

mixing of small batches of nanoparticles or fibers in special “blenders” helps with dispersion, as

does the use of chemical surfactants, such as liquid soap and superplasticizers. Sonication is

another method to disperse nanoparticles. Soncation is the use of sound or ultrasound energy

to agitate and mix the nanoparticles by breaking the inter‐particle attraction. The un‐clumped

particles are then free to disperse within the water and ultimately the concrete paste.

Jo (2007), for example, used a 5 step procedure for mixing a 50 cubic inch batch of nanosilicate

concrete:

1) Mix nanoparticles with water at high speed for 1 minute (120 rpm)

2) Cement was added and mixed at medium speed (80 rpm) for 30 seconds

3) Sand was then added, gradually, at medium speed

4) Superplasticizer was added and stirred at high speed for 30 seconds

5) The mixture was allowed to rest for 90 seconds, then mixed for an additional minute at

high speed.

Creep:

A more creep resistant concrete (long term) is desirable; however, current concrete design

codes would need to be extended for applications where creep resistant concrete is used. Long

term creep influences residual steel forces in prestressed and post tensioned concrete

structures significantly. Serviceability considerations would be improved. Particular attention,

for example, would need to be paid to ACI sect. 18.4. For example, section 18.4.2 limits on

stress for structures with relatively high sustained service loadings could be increased for

structures using low creep concrete.


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What are the mechanisms that will explain the modified behavior of nanoparticle concrete?

Mechanisms that explain nanoparticle behavior, particularly silicates, revolve around the

increased homogeneity of Ca(OH)2 crystal growth during hydration. Hydrate products

crystallize around a nanoparticle “kernel” during hydration. If the appropriate content of

nanoparticles with respect to hydrate products is incorporated in the mix, the hydrate crystal

growth will tend to be controlled and the resulting product will be more homogeneous. This

homogeneity makes the cement matrix more compact and thus improves density and reduces

porosity. This denser, more uniform matrix helps explain the improved flexural and fatigue

properties found by researchers (Li 2002).

The hardness of the hydrated Ca(OH)2 crystals has been studied by Ulm (2009) in order to

determine how creep behavior is influenced. The controlled growth crystals are denser and

thus exhibit better creep performance. Because of the more uniform crystal growth around the

nano‐kernel, pores typically filled with water are now filled with these nanocrystals. The more

densely packed crystals inhibit C‐S‐H movement over time, thus reducing the movement

associated with creep.

Nanosilcia increased the water demand of the cement past. When the mix is designed with the

appropriate amount of water and super plasticizer, this increased hydration or pozzolanic

reaction of the nanosilica extender is the mechanism that improves the compressive strength.

Designing a mix using nanosilcia requires careful attention to water content. Due to the specific

surface area of the SiO2 nanoparticles the pozolanic reaction is more effective and efficient

than with larger particles of the same material. Higher nano SiO2 content must be

accompanied by adjustments to the water content and super plasticizer (Jo 2007). The higher

hydration of the nanoparticles required additional water to avoid desiccation, however too

much water reduced strength. The right balance of water and superplasticizer is required when

designing nanosilcia mixes.

The mechanism by which carbon nanofibers operate to improve the flexural toughness and

tensile characteristics of concrete (Chen 1992) are very much different from the chemical

interaction behavior of nanosilicate particles. The individual fibers act to hold the mortar

matrix together longer by bridging microcracks. Nanocarbon fibers bond to some extent to the

cement, but improved chemical bonding can be achieved by treating the fibers with ozone gas.


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This increases the surface oxygen concentration on the nanocarbon fibers and increases

wetability and thus the degree of fiber dispersion (Chung, 2003).

What are the important factors that have to be considered for the constitutive models for

nanoconcrete materials?

Factors to consider when developing constitutive models of nanoconcrete include the

modification of the compressive stress strain curve. Tests at U of H on carbon fiber

nanoconcrete specimens (Gao, 2009), for example, indicate increased ductility in compression

when compared to regular high strength concretes. These tests were performed with carbon

nanofiber contents ranging from 1% to 2.5% by volume.

The smeared tensile stress strain curve also needs to be developed for nanoconcretes due to its

increased toughness over normal concrete (Zhou, 2010).

The modification to the existing set of constitutive equations for normal concrete (eg., the

Softened Membrane Model framework) would likely be straightforward. It would require

materials test to determine the constitutive laws, but the general structure of each constitutive

equation and solution method would generally be the same.

How will you develop a constitutive model for these concrete materials?

To develop constitutive models, uniaxial compression tests must be performed to determine

the nanoconcrete’s stress strain curve (Gao, 2009). Specifically, strength, peak and maximum

strains, ductility and in particular the shape of the descending branch of the stress strain curve

need to be determined. This would be the un‐softened characteristic stress strain curve of the

nanoconcrete. To determine how tensile strain influences the peak compressive performance,

the softening factors would need to be determined from tests in the Universal Element Testing

Machine in the TC Hsu Structural Lab (Zhou, 2010).

Since nanoparticles improve fatigue and flexural strength, additional testing needs to be done

to develop the tensile properties of the material. The post‐cracking smeared stress strain curve

needs to be determined. Since tensile ductility is enhanced, the shape and area under of the


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smeared tensile strain curve will differ from that of regular concrete (larger area under the

curve). In order to determine this smeared curve, one will need to test specimens reinforced

with rebar and loaded in uniaxial tension. Tensile load would be applied to rebar embedded in

and projecting from the ends of prismatic nanoconcrete specimens. Foil strain gages should be

applied at intervals along the length of the rebar to better determine the constitutive curve for

the embedded rebar.


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Constitutive Model References:

Belarbi, A. and Hsu, T. T. C. (1994), “Constitutive Laws of Concrete in Tension and Reinforcing Bars Stiffened by

Concrete,” Structural Journal of the American Concrete Institute, Vol. 91, No. 4, pp. 465‐474.

Belarbi, A. and Hsu, T. T. C. (1995), “Constitutive Laws of Softened Concrete in Biaxial Tension‐Compression,”

Structural Journal of the American Concrete Institute, Vol. 92, No. 5, pp. 562‐573.

Chen, W.F. Constitutive Equations for Engineering Materials, Vol. 1: Elasticity and Modelling, Elsevier Publications, 1994.

Collins, M. P., Vecchio, F. J., and Mehlhorn, G. (1985), “An International Competition to Predict the Response of

Reinforced Concrete Panels,” Canadian Journal of Civil Engineering, Ottawa, Vol. 12, No. 3, pp. 626‐644.

Desayi, P. and Krishnan, S., Equation for the stress‐strain curve of concrete, ACI J., Vol. 61(1964)345‐350. Dhonde, H.B., Mo, Y.L., and Hsu, T.C. (2006), “Fiber Reinforcement in Prestressed Concrete Beams”, Texas

Department of Transportation Report 0‐4819, March 2006.

Gerstle, K.H. Simple formulation of biaxial concrete behaviour, ACI Journal, 78(1981)62‐68.

Gopalaratnam, Shah, Softening Response of Plain Concrete in Direct Tension, ACI Journal, May‐June 1985.

Hoffman (2009), Constitutive Properties of Steel Fiber Concrete, Proceedings of the October 2009 ASCE Texas

Section Fall Meeting, Houston, TX.

Hognestad, Fundamental Concepts in Ultimate Load Design, Journal of the American concrete Institute, V 23, June

1952.

Hsu, T. T. C. (1993), Unified Theory of Reinforced Concrete, CRC Press, Inc., Boca Raton, FL, 336 pp.

Hsu, T.T.C. (2009), Unified Theory of Reinforced Concrete, 2nd. edition, CRC Press, Inc. Boca Raton, FL.


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Hsu, T. T. C. and Zhang, L. X. (1996), “Tension Stiffening in Reinforced Concrete Membrane Elements,” Structural

Journal of the American Concrete Institute, Vol. 93, No. 1, pp. 108‐115.

Hsu, T. T. C. and Zhu, R. R. H. (2002), “Softened Membrane Model for Reinforced Concrete Elements in Shear,”

Structural Journal of the American Concrete Institute, Vol. 99, No. 4, pp. 460‐469.

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