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UNIVERSITY OF TRANSPORT AND COMMUNICATIONS NGUYỄN XUÂN LAM APPLICATION OF HOMOGENIZATION THEORY TO ANALYZE THE STAGE OF TEMPERATURE DISTRIBUTION AND STRESS DUE TO THE HEAT OF HYDRATION

UNIVERSITY OF TRANSPORT AND COMMUNICATIONS

NGUYỄN XUÂN LAM

APPLICATION OF HOMOGENIZATION THEORY TO ANALYZE THE STAGE OF TEMPERATURE DISTRIBUTION AND STRESS DUE TO THE HEAT OF HYDRATION OF CEMENT IN REINFORCED CONCRETE WORKS

Field: Transport Construction Engineering

Mã số : 9.58.02.05

Major: Bridge and Tunnel Construction Engineering

SUMMARY OF DOCTORAL THESIS

HÀ NỘI – 2022

Communications

Người hường dẫn khoa học:

1 PGS.TS Nguyễn Ngọc Long

Academic Supervisors:

1 Assoc Prof.Dr Nguyễn Ngọc Long

2 Assoc Prof.Dr Nguyễn Duy Tiến

At……,…….,…….2022

The thesis can be found at:

1 University of Transport and Communications Library

2 National Library

INTRODUCTION

I URG ENCY OF TH E SUB JECT

Concrete is a widely used building material around the world because it has many features to meet the requirements of different types of structures, high formability, good structural properties and durability However, in the construction process, the formation of heat at early-age appears in reinforced concrete structures due to the influence of heat of hydration This is one of the important issues to study because the heat distribution has a direct influence

on the stress-strain state of reinforced concrete structures at the construction stage

In details, tensile stresses due to a combination of temperature differences, heat of hydration and ambient conditions, natural strains, and boundary conditions, often exert significant intrinsic effects on concrete structures Whenever such stress reaches the tensile strength of concrete, cracking will occur, which in turn can reduce the serviceability and durability of structures The handling, repair and overcoming of these cracks are costly in terms of money and cause difficulties in construction, as well as in maintenance and exploitation of works

Thermal cracking in early-age concrete structures occurs frequently, such as large concrete blocks such as foundations, dams and bridge structures Thermal cracking of these structures is closely related to the binder content, the ambient temperature during construction, the temperature of fresh concrete and the geometrical characteristics of the structures The formation of a heat source in concrete structures depends on many factors, of which the important ones are concrete mix and construction technology

In Vietnam, according to the standard TCVN 9341:2012 “Large concrete – Construction and acceptance” [1], to prevent the formation of cracks in concrete structures, we must ensure two factors: The temperature difference

∆T between points or areas in the concrete block does not exceed 20oC: ∆T <

20oC; The temperature difference modulus MT between points in the concrete block is not more than 50oC/m; MT< 50oC/m Currently, bridge works often use high-strength concrete (from 25MPa to 40MPa), so it needs to be reconsidered due to a number of factors as follows: High-strength concrete often contains large cement content (can be more than 400kg/m3) resulting in a much higher heat of hydration of cement than roller compacted concrete and hydraulic concrete Especially, the pier concrete structure uses reinforcement

at the edge near concrete surface, so they change the coefficient of thermal conductivity and tensile strength on the surface of concrete

Research on temperature sources and stress fields of concrete structures (temperature distribution and deformation) is of interest to many scholars However, the limitations of these studies are that researched structures are

concrete blockes without reinforcement and the test mix is not suitable for the lower part of the structure bridge works (grades C30 and C35)

Therefore, the topic: "Application of homogenization theory to analyze the

state of temperature distribution and stress due to the heat of hydration of cement in reinforced concrete works" was chosen to study The research

proposes a theoretical, verified computational model through field measurements to analyze and evaluate the behavior due to the heat of hydration of cement in the reinforced concrete structure

II RESEARCH OBJECTIVES

The first objective is: to determine equivalent thermal conductivity coefficient, influence range and equivalent material characteristics of reinforced concrete shells with some typical reinforcement diameters

Next, the second objective is: to perform a chamber adiabatic test of some common concretes used for bridge construction to measure their adiabatic temperature curves

Finally, the third objective is: to use the tested temperature generated values of the room and equivalent thermal conductivity coefficient, equivalent material characteristic of the material reinforced concrete to develop a program analysing temperature distribution and stress due to heat of hydration of cement in reinforced concrete structures

III REASEARCH CONTENTS

The thesis contents include:

1 Literature review;

2 Determination of equivalent thermal conductivity coefficient and equivalent material characteristics of reinforced concrete layer by homogenization method;

3 Experimental study to determine the adiabatic temperature from the hydration process of cement for ordinary concrete used in bridge construction;

4 Application of homogenization theory to analyze the state of temperature and stress distribution due to heat of cement hydration in reinforced concrete structures at early-age

IV SCIENTIF IC AND P RACTICAL CO NTRIB UTION

 Firstly, develop a program to calculate the thermal characteristics of reinforced concrete using homogenization theory (TCon1 program): equivalent thermal conductivity coefficient, specific heat, homogenization range of reinforced concrete materials of typical reinforced concrete shell structures of bridge piers

 Second, measure adiabatic curves for some concrete mixes used in the lower part

of bridge structures (concrete C30, C35) according to the adiabatic method in the laboratory and the semi-adiabatic method in the field

 Third, develop a program to calculate the distribution and change of temperature and stress over time due to the heat of cement hydration (Program TCon2) to compare with the actual measurement results in the field

LITERATURE REVIEW

1.1 Overview of non-mechanical crack formation in reinforced concrete

structures

1.1.1 Analysis of non-mechanical crack types

The types of cracks caused by temperature in the cement hydration process, the shrinkage, the creep of concrete structures and by the restraining deformation

in concrete blocks at early-age are the types of cracks that are not directly affected by mechanical impact Tensile stresses due to a combination of temperature differences, heat of hydration and ambient conditions, natural deformations and boundary conditions, often exert significant intrinsic effects

on concrete structures Whenever such stress reaches the tensile strength of concrete, cracking will occur, which in turn can reduce the serviceability and durability of structures Thermal cracking in early-age concrete structures occurs frequently, such as large concrete blocks including foundations, dams and bridge structures Thermal cracking ability of these structures is closely related to binder content, ambient temperature during construction and the temperature of fresh concrete, the geometrical characteristics of the structures

1.1.2 Concept of heat of hydration of cement in concrete

The heat of hydration is the heat released during the cement hydration process, which causes an increase in the temperature of concrete blocks during the first

72 hours The heat of cement hydration increases the uneven temperature in the concrete mass, creating a temperature Gradient and thermal expansion, which is one of the possible causes of cracking of reinforced concrete structures

The hydration of cement caused by the constituent minerals generates a certain amount of heat That heat can be monitored and measured with an isothermal device Under normal conditions, the heat generated during the hydration of cement is classified into 5 stages

Figure 1 1 Hydration heat release of Portland cement

1.1.3 Regulations on non-structural crack control for bridge contructions in Vietnam

According to TCVN 11823:2017: “Design standard for motorway bridges” [2]

to control the temperature due to the heat of hydration of cement that forms non-structural cracks: For concrete used for saltwater and coastal structures, the water/cement ratio should not exceed 0.45; The total amount of Portland cement and other cement-containing materials must not exceed 475 kg/m3 of concrete, except for high-performance concrete, the amount of Portland cement and other cements shall not exceed 593 kg/m3

According to the standard TCVN 9341:2012 [1], using ordinary Portland cement, the amount of heat of hydration after 7 days is not more than 70cal/g; low heat Cement, with heat of hydration after 7 days not exceeding 60 Cal/g; pozzolan and Portland cement, or Portland- slag cement should be used for construction projects in coastal areas that are exposed to acidic water

1.2 Methods of analyzing the formation of heat of hydration of cement in reinforced concrete structures at early-age around the world and in Vietnam

1.2.1 Methods around the world

One of the complete approaches to estimating the size of a texture mass is hydrothermal diffusion characterization proposed by Ulm and Coussy [55], which considers both the dimensional size of structures and its thermal conductivity characteristics Another approach is related to the geometry of structures [12]

Besides, De Schutter and Taerwe [26] studied the American Concrete Institute (ACI) concepts of block size and proposed to use the equivalent thickness as a measure of the size of structures, where M is the mass of the structure and γa is the form factor with respect to heat flow

A thermo-mechanical analysis using the finite element method to evaluate the structural safety based on the FIB Model Code 2010 [24] by nonlinear analysis [23] to evaluate the safety for the wind power pylon foundation using high-strength concrete and reinforced or non-reinforced concrete tunnels with the formation of heat of hydration of cement at early-age

1.2.2 Methods in Vietnam

A study on the influence of bulk concrete structure size on the formation of temperature field and cracks due to cement hydration of [3] examined the effect of concrete block size on the temperature field at early age However, it was a pure concrete block without reinforcement inside the structure Another study on the degree of hydration and strength development in high-strength concrete [7] based on the degree of hydration determined from the adiabatic temperature experiment However, these studies still have limitations as the structure is simply a concrete block without reinforcement and the test concrete mix is not suitable for the lower part of bridge works

The above studies can use the finite element method or experimental methods,

or actual data investigation methods, but none of them mentioned the method

of homogenizing reinforced concrete materials of the structural shell.

1.3 Some solutions to prevent and limit non-mechanical crack in concrete and reinforced concrete structures of abutments at the construction stage

A number of solutions have been applied in practice, including: methods of cooling down aggregates, using less heat-emitting cement, curing concrete, controlling concrete temperature during construction and using mineral

additives

1.4 Conclusion for Chapter 1

This chapter gives the overview of non-mechanical crack formation in reinforced concrete structures and the causes of cracks in large-sized constructions Next, the author reviews methods to evaluate the formation of cracks, analyze and handle these types of cracks The analysis and evaluation can be done by field experiment method and simulation method through structural analysis software (FEM/FEA)

DETERMINATION OF EQUIVALENT THERMAL CONDUCTIVITY COEFFICIENT AND MATERIAL CHARACTERISTICS OF REINFORCED CONCRETE LAYERS BY HOMOGENIZATION

2.1.2 Multi-level concept

Multi-level model is a research direction in which different models at different levels (quantum mechanics, molecular dynamics mechanics, continuum mechanics ) are used simultaneously to describe the behavior of physical systems

2.1.3 The concept of homogenization

The multi-level material homogenization method considers the material at the level obeys the laws of continuous environmental mechanics At the macro level (structure level) the building is considered as a continuum characterized

by theoretical elementary volume, the

volume element is infinitely small of

considered material system More

specifically, if we denote L and l are the

dimensions of the building and the volume

element respectively, then l << L needs

to be guaranteed in order to be able to use

differential calculations describing the

continuous environment Next, the volume

element is expected to be large enough to

have a representative characteristic of all the properties of the constituent material, hence the more complete name is representative elementary volume, named abbreviated as REV as shown in Figure 2.1

2.1.4 Material homogenization according to heat problem

Consider a domain  of representative elementary

volume (REV), in which  is the outer boundary of

 The domain  contains two component

materials: concrete (material 1) and reinforcement

(material 2) The homogenization method is followed

by a rule of temperature behavior expressed by

Fourier's rule:

q( ( ))T x  K( ) T( )x x (2.1)

The thermal conductivity coefficient K(x) at each

position x in the concrete block is determined by

the formula :

In which, K(1 ) and K(2 ) are the thermal conductivity coefficients of steel and

concrete respectively χ(x) is a position function, having the value 1 if x is in the reinforcement region, and 0 if x is in the concrete The heat flow q(T(x))

must satisfy the heat balance equation

2.1.5 Material homogenization with strain boundary conditions to determine equivalent material properties of reinforced concrete structures

Theoretical basis of homogenization method with deformation boundary conditions

The object of study is a linear elastic continuous material medium whose

mechanical properties vary with spatial position x, characterized by the

quaternary elastic stiffness tensor ( )x or elasticity tensor ( )x with 1

Figure 2 1

Representative elementary volume REV containing two equivelent component materials

(REV), which must be large enough relative to the microstructures to represent the properties of the constituent material and at the same time be small enough relative to the size of the microstructure object for the determination of meaningful macroscopic properties Representative elementary volume is composed of n component material phases (referred to as phases) Each component phase has its own geometrical characteristics, occupies space and has uniform mechanical properties characterized by elastic stiffness tensor or elastic softness tensor with i= 1, 2, ,n connecting to each other through the interface which is considered perfect (characterized by the condition of continuity of displacement vectors and force vectors)

Hình 2 2 Representative elementary volume REV of r einforced concrete material (circle is

reinforcement, rest is concr ete): (a) Representative elementary volu me REV; (b) triangle mesh

for REV

Applying the homogenization method in the finite element method

The overall behavior of material is expected to be linearly elastic, characterized by the overall elastic modulus eff

With i is the mean value in phase i of the strain concentration tensor

i is the index running from 1 to n phases

The matrix of elastic stiffness tensor after homogenization according to strain boundary conditions has the form:

ff

1 1

( ) ( ) e e e( ) ( ) e e

e

e e d

2.2.1 Differential equation of heat transfer

The heat transfer problem models that take into account the heat release in the cement hydration process are based on the well-known heat transfer

(2.40)

differential equation given in [32, 43, 46]:

In which: q- Heat generated per unit volume (kJ/m3)

T - increase in temperature over time t (hour)

- Volumetric mass of the material (kg/m3)

C- Specific heat capacity of the material (kJ /kg.K)

kx, ky- heat diffusion coefficient in each direction x,y (m2/s) 2.2.2 Calculation parameters of heat source

The formula for determining the unit heat source q and the law of adiabatic temperature rise in concrete In 1986, this formula was recognized by the American Society of Civil Engineers - ASCE [16, 38]

2.2.3 Heat transfer formula in finite element method

Applying Galekin's criterion in the time domain: T(t)= Ti(t)Ni +Tj(t)Nj for each element, with Ni=1-t/∆t and Nj=t/∆t

2.3.1 Determine equivalent thermal conductivity coefficient

The thermal conductivity coefficient K(x) at each position x in the reinforced concrete block is determined by formula (2.2) In which, K(1) =50W/m.K and K(2) =1,6W/m.K are the coefficients of thermal conductivity of steel and concrete respectively χ(x) is a position function, which has a value of 1 if x is

in the reinforcement region, and 0 if x is in the concrete

The thermal conductivity coefficient of the whole object is characterized as follows:

2.3.2 Determine the thickness of the concrete layer

The thickness of the reinforced concrete layer after homogenization is determined by the distance from the outer edge of the structure to the boundary

of the area where the temperature field is the same at all vertical points of the simulated structure

2.3.3 Xác định nhiệt dung riêng của lớp BTCT

Assuming the volume of the reinforced concrete layer after homogenization is

, the volume and specific heat capacity of steel and concrete materials are respectively (1),C(1 ) and (2),C(2) The specific heat capacity of the reinforced concrete layer is determined by the following formula:

Figure 2 3 Schematic diagram of the stress and

temperature field analysis process in bulk concrete

Figure 2 4 Program interface of Tcon1

The implementation process consists of steps described by the block diagram

as Figure 2.8 in the program which is written in Matlab language, named

Tcon1 The program interface is shown in Figure 2.9

After calculating, we can determine the temperature field and equivalent thermal conductivity coefficient of reinforced concrete block as Keff according

to the results presented in the table below:

Table 2 1 Equivalent thermal conductivity coefficient (W/mK) of reinforced concrete for some

typical reinforcement diameters

Table 2 2 The thickness of the reinforced concrete layer (mm) after homogenization for some

typical reinforcement diameters

2.4.2 Determination of equivalent material characteristics of reinforced concrete structures that change over time by homogenization method

Using homogenization theory with deformation boundary conditions mentioned in section 2.1.5 to determine equivalent material characteristics of reinforced concrete structures that change over time during the formation of strength of concrete (from the time of pouring concrete until the age of 28 days)

After homogenization, we can determine the tables of values of Ex, Ey, Poisson coefficient yx of reinforced concrete at the age of 28 days from Table 2.3 to Table 2.5 respectively and specific heat capacity of reinforced concrete structure for some typical diameter of reinforcement in Table 2.6

Table 2 3 Uniform elastic modulus Ex (MPa) for

some typical reinforcement diameters

Table 2 4 Uniform elastic modulus Ey (MPa) for some typical reinforcement diameters

Table 2 3 Uniform Poisson coefficient υx y for

some typical reinforcement diameters

Table 2 4 Specific heat capacity C(J/kg.K) of reinforced concrete structures for some typical

2.5.1 Mathematical modeling