# Lattice Modeling of Early-Age Behavior of Structural Concrete

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## Abstract

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## 1. Introduction

## 2. Modeling Framework

#### 2.1. Program Structure

#### 2.2. Domain Discretization

#### 2.3. Cementitious Materials Hydration

#### 2.4. Primary Analysis Modules

#### 2.4.1. Thermal Analysis

- Convection—Convective heat exchange across exposed surfaces depends on the difference between the solid surface temperature ${T}_{s}$ and that of the surrounding ambient medium ${T}_{a}$$${q}_{conv}={\mathrm{\Lambda}}_{T}({T}_{s}-{T}_{a})$$
- Solar radiation—The amount of solar radiation reaching the concrete surface depends on several factors including the structure’s location, surface orientation, altitude, atmospheric conditions, time of the day, and day of the year. Incoming heat due to solar radiation is$${q}_{sun}={\gamma}_{abs}{q}_{inc}$$
- Thermal radiation - Heat loss to the surroundings due to grey-body radiation is calculated using$${q}_{sky}=\sigma \u03f5({T}_{sK}^{4}-{T}_{sky}^{4})$$

- The heat capacity of the cement paste is estimated using an approach given by Bentz [35], in which heat capacity is a function of degree of reaction of the cement. The heat capacity of the concrete is then determined from the heat capacities of the cement paste and aggregates, according to the mass fractions of each using an ordinary rule of mixtures.
- Thermal conductivity of the concrete is estimated by taking the average of the Hashin-Shtrikman bounds for a two-phase composite formed of paste and aggregates [35].

#### 2.4.2. Hygral Analysis

#### 2.4.3. Structural Analysis

#### 2.5. Stiffness and Creep Representation

## 3. Validation Exercises

#### 3.1. Stiffness and Basic Creep Development

#### 3.2. Strength Development

#### 3.3. Autogenous and Drying Shrinkage Tests

#### 3.4. Analysis of Concrete Bridge Decks

#### 3.4.1. Model Definition

**Thermal analysis inputs:**The initial temperature of the simulated, cast concrete was set to its measured value at the time of casting (19.5 ${}^{\circ}$C). The initial temperature of the simulated, mature concrete was set to the value measured at mid-height within one of the girder stems at the time of casting (14.9 ${}^{\circ}$C).

**Hygral analysis inputs:**The ultimate value of relative humidity, associated with self-desiccation, was assumed to be ${h}_{su}$ = 0.9 with s = 3 governing the rate of self-desiccation according to Equation (16). The parameter values expressing hygral diffusivity, according to Equations (17) and (19), are: $\tilde{{D}_{0}}$ = 0.017 mm${}^{2}$/h; $\tilde{{D}_{1}}$ = 9 mm${}^{2}$/h, and n = 5.

**Mechanical analysis inputs:**The parameters for the solidification and microprestress modeling of concrete behavior are presented in Table 4. The creep parameter values were set according to the B4 model [85], based on the actual mixture composition. We have made no effort to distinguish between tensile creep, of interest herein, and compressive creep (even though tensile creep may be much larger than compressive creep for the same levels of applied stress [86]).

#### 3.4.2. Simulation Results

**Deck temperature histories:**The simulated temperature histories are compared with the field measurements in Figure 16, for the case of thermocouples TC2 and TC5 located within the mid-deck region and above the girder stem, respectively. The recorded ambient temperature history is also plotted in the figures. Comparisons with the entire set of thermocouple readings are presented in Figure 17. Several comments can be made.

- The influence of environmental factors is evident from the oscillatory behavior of the temperature history recorded by each thermocouple. After the first day, locations closer to the surface exhibit larger temperature swings, whereas deeper locations are less affected by environmental changes. This meets expectations.
- Peak temperatures occur at about 10 h after concrete casting (Figure 18). The lower temperatures over the supporting girders are due to conduction of heat toward the cooler substrate concrete. Conversely, the insulative properties of the plywood formwork give rise to higher temperatures between the supporting girders. These temperatures are significantly higher than the ambient temperature. The differences between the ambient and measured deck temperatures largely diminish over the first two days after casting, in contrast to mass concrete applications in which large temperature differences can exist for several weeks. Despite the discrete, irregular discretization of the domain, the iso-contours of temperature do not exhibit artifacts associated with mesh bias.
- After removal of the curing sheets, not only do diurnal variations in deck temperature increase, but also the temperature gradient through the deck thickness tends to increase. The implications of the larger thermal gradients are discussed later.

**Strength development:**Concrete cylinder specimens were cast on site and kept local to the bridge deck for 7 days, prior to laboratory storage and measurement of splitting tensile strength. Strength development was simulated using Equation (28), with ${\alpha}_{0}$ = 0.20, and the same lattice model adopted for the creep test simulations (Figure 6). Ambient temperature for the modeling exercise was constant and set equal to the average recorded temperature over the curing period. Figure 19a compares simulated strength development with the measured values, where the 28-day splitting tensile strength is used as a normalizing factor.

**Shrinkage development:**For measuring the drying shrinkage properties of the concrete mixture, prisms were cast on-site and transferred to the laboratory within 24 h, after which they were kept in a moist condition until exposure to a drying environment at t = 7 days. Testing was done according to ASTM C157, except for the 7 days of moist pre-conditioning according to Caltrans recommendations. The measured and simulated shrinkage strains are compared in Figure 19b. Initial expansion of the prisms was simulated by prescribing h = 0.97 as an initial condition prior to moist curing at 1 day. Whereas the sources of expansion may be due to other factors [91], this simulation of swelling demonstrates the expected workings of the model. As seen in the previous examples of autogenous and drying shrinkage, the shape of the simulated shrinkage curve does not conform to that of the experimental curve: the simulated rate of shrinkage over the first several days is lower. This could be remedied by increasing hygral diffusivity and/or the shrinkage coefficient, but the ultimate shrinkage strain would then be overestimated. Here, too, it is apparent that effects of microcracking need to be incorporated into the analyses.

## 4. Parametric Study

#### 4.1. Curing Protocol

#### 4.2. Structural Configuration

## 5. Conclusions

- This form of discrete model is capable of simulating the multi-field quantities associated with early-age concrete behavior, despite its discontinuous representation of the problem domain. There are no major disadvantages of this discrete approach with respect to continuum approaches, such as the finite element method. Advantages of this form of discrete model include its simplicity and adeptness at simulating the transition from diffuse damage to localized cracking.
- Stress-to-strength ratio is lacking as a practical measure of cracking potential. Sharp hygral or thermal gradients near exposed surfaces typically cause high stresses, which is indicative of cracking. However, the cracking may only be superficial. Knowledge of the stress conditions through the structural cross-section is also necessary for evaluating the severity and potential consequences of cracking. In this sense, numerical modeling nicely complements knowledge gained by laboratory testing and field observations.
- Structural configuration plays a key role in determining the magnitude and distribution of stresses caused by volume instabilities of the concrete material. Under largely restrained conditions, both thermal and hygral effects were found to be primary contributors to cracking potential, leading to crack propagation through the depth of the deck. Three-dimensional simulations are needed to assess the influence of longitudinal restraints, thermal flexing of the mature concrete girders, and the effects of reinforcing bars.
- Realistic simulations of the early-age behavior of structural concrete require a wealth of information regarding the material constituents, production/curing processes, and structural and environmental boundary conditions. As shown by the parametric studies conducted herein, cracking potential is sensitive to input quantities that are typically not well defined, especially in field applications. Ultimately, the assessment of cracking potential needs to be cast in a probabilistic framework that accounts for uncertainties in the various inputs to the modeling effort.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

Symbol | Definition |

${h}_{ij}$ | Euclidean distance between nodes i and j [m] |

t | time [s, h, d] |

${A}_{ij}$ | area of Voronoi facet common to nodes i and j [m${}^{2}$] |

Thermal analysis | |

$a/c$ | aggregate to cementitious materials ratio [kg/kg] |

c | cementitious materials content [kg/m${}^{3}$] |

${c}_{p}$ | specific heat capacity [J/(kg · K)] |

${q}_{conv}$, ${q}_{conv}^{\prime}$ | heat flux due to convection and its modified value [W/m${}^{2}$] |

${q}_{sun}$, ${q}_{sun}^{\prime}$ | heat flux due to solar radiation and its modified value [W/m${}^{2}$] |

${q}_{sky}$, ${q}_{sky}^{\prime}$ | heat flux due to radiation and its modified value [W/m${}^{2}$] |

${t}_{e}$ | concrete equivalent age [h] |

$w/c$ | water-to-cementitious materials ratio [kg/kg] |

D | thermal diffusivity [m${}^{2}$/s] |

${E}_{a}$ | apparent activation energy [J/mol] |

H | cumulative amount of heat produced by hydration [J/kg] |

${H}_{cem}$ | heat of hydration of cement [J/kg] |

${H}_{u}$ | total heat available for reaction [J/kg] |

Q | rate of heat production by cementitious materials hydration [J/(kg · s)] |

R | universal gas constant [J/(mol · K)] |

T | temperature [${}^{\circ}$C] |

${T}_{a}$, ${T}_{c}$, ${T}_{r}$, ${T}_{s}$ | ambient, concrete, reference, and surface temperatures [${}^{\circ}$C] |

${T}_{sky}$ | sky temperature [${}^{\circ}$C] |

α, ${\alpha}_{u}$ | degree of cementitious materials hydration and its ultimate value |

β, τ | hydration model parameters |

${\beta}_{T}$, ${\beta}_{Th}$ | coefficient of thermal expansion and its long-term value [1/${}^{\circ}$C] |

${\gamma}_{abs}$ | solar absorptivity of concrete |

ϵ | emissivity of concrete |

${\eta}_{1}$, ${\eta}_{2}$, ${\eta}_{3}$ | heat flux reduction factors |

λ | thermal conductivity [W/(m · K)] |

ρ | mass density of concrete [kg/m${}^{3}$] |

σ | Stefan-Boltzmann constant [W/(m${}^{2}$ · K${}^{4}$)] |

χ | magnifying factor for coefficient of thermal expansion at early-ages |

${\mathrm{\Lambda}}_{T}$ | coefficient of convective heat transfer [W/(m${}^{2}$ · K)] |

${\mathbf{M}}_{\mathbf{e}}$, $\mathbf{M}$ | elemental and system capacity matrices |

${\mathbf{K}}_{\mathbf{e}}$, $\mathbf{K}$ | elemental and system conductivity matrices |

Hygral analysis | |

h | relative humidity [-] |

${h}_{a}$ | ambient relative humidity [-] |

${h}_{s}$, ${h}_{su}$ | relative humidity associated with self-desiccation and its ultimate value [-] |

n | parameter relating hygral diffusivity and relative humidity |

${q}_{h}$, ${q}_{h}^{\prime}$ | moisture flux due to convection and its modified value [m/s] |

s | parameter relating humidity decrease and self-desiccation |

${D}_{0}$, $\tilde{{D}_{0}}$ | diffusivity values associated with fully dried material [m${}^{2}$/s] |

${D}_{1}$, $\tilde{{D}_{1}}$ | diffusivity values associated with saturated material [m${}^{2}$/s] |

${D}_{h}$ | hygral diffusivity [m${}^{2}$/s] |

${\beta}_{h}$ | hygral shrinkage coefficient |

${\mathrm{\Lambda}}_{h}$ | hygral convection coefficient [m/s] |

Mechanical analysis | |

b | element thickness [m] |

${f}_{c}$, ${f}_{cu}$ | compressive strength and its asymptotic limit [MPa] |

f | tensile strength [MPa] |

${k}_{n},{k}_{s},{k}_{t}$ | stiffness coefficients of the uniaxial springs [N/m] |

${k}_{\varphi x},{k}_{\varphi y},{k}_{\varphi z}$ | stiffness coefficients of the rotational springs [N · m] |

${n}_{\alpha}$ | parameter relating rate of solidification to degree of hydration |

${q}_{1}$, ${q}_{2}$, ${q}_{4}$ | instantaneous elastic, viscoelastic, and viscous strain parameters [1/MPa] |

${t}_{0}$ | age of loading [h] |

${t}_{e}$ | equivalent age [h] |

${t}_{r}$ | reduced time representing the thermal-hygral effects on creep [h] |

${t}_{is}$, ${t}_{fs}$ | times of initial and final concrete setting [h] |

w | crack opening [m] |

${w}_{c}$ | traction-free crack opening [m] |

${A}_{ij}^{P}$ | projected area of element facet [m${}^{2}$] |

E | elastic modulus of concrete [MPa] |

${E}_{v}$, ${E}_{S}$ | activation energies for creep and microprestress processes [J/mol] |

${F}_{n}$, ${F}_{s}$, ${F}_{t}$ | spring-set forces in the normal and two tangential directions [N] |

${I}_{11},{I}_{22}$, ${J}_{p}$ | principal planar and polar second moments of facet area [m${}^{4}$] |

$\mathbf{D}$ | material constitutive matrix |

${\mathbf{K}}_{\mathbf{e}}$, $\mathbf{K}$ | elemental and global stiffness matrices |

${\alpha}_{0i}$ , ${\alpha}_{0}$ | degrees of hydration at initial and final setting |

$\phi $ | coefficient governing concrete setting behavior |

γ | viscoelastic microstrain |

${\epsilon}^{h}$, ${\epsilon}^{T}$ | hygral and thermal strains |

${\epsilon}^{i}$, ${\epsilon}^{v}$, ${\epsilon}^{f}$ | instantaneous elastic, viscoelastic, and viscous flow strains |

ζ | coefficient governing material property development |

η | effective microscopic viscosity [MPa/s] |

κ | stress-to-strength ratio |

${\kappa}_{0}$ | microprestress model parameter [1/(MPa · d)] |

${\kappa}_{1}$ | microprestress model parameter [MPa/K] |

ν | Poisson’s ratio |

${\theta}_{R}$ | inclination of tensile force resultant relative to the element axis |

${\vartheta}_{1}$, ${\vartheta}_{2}$ | parameters defining the break-point in the tension softening relation |

σ | axial stress [MPa] |

${\sigma}_{R}$ | stress resultant [MPa] |

${\sigma}_{p}$, ${\sigma}_{fs}$ | penetration resistance and its value at final setting [MPa] |

$\psi ,{\psi}_{S}$ | reduced time coefficients for creep and microprestress processes |

ξ | factor relating normal and shear spring stiffnesses |

ω | degree of damage |

Φ | non-aging micro-compliance function [1/MPa] |

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**Figure 1.**Lattice model: (

**a**) domain discretization based on Delaunay and Voronoi tessellations; (

**b**) lattice element $i-j$.

**Figure 2.**Assumed relationship between coefficient of thermal expansion and degree of cementitious materials reaction.

**Figure 3.**Dependence of hygral diffusivity on relative humidity according to Equation (17) with $\psi (T)$ = 1.

**Figure 4.**(

**a**) Determination of tensile stress for planar analyses; and (

**b**) tension softening relation. Parameters ${\vartheta}_{1}$ and ${\vartheta}_{2}$ define the break point of the softening diagram in terms of tensile strength $f({\alpha}_{u})$ and traction-free crack opening ${w}_{c}$, respectively.

**Figure 6.**Discretization for creep test simulations: (

**a**) lattice representation; and (

**b**) volume rendering of cylindrical specimens.

**Figure 8.**Basic creep curves produced from a single element and from a fully discretized cylindrical specimen.

**Figure 9.**Strength development as a function of degree of hydration: dependence on setting parameter, ${\alpha}_{0}$.

**Figure 10.**Discretization for shrinkage test simulations: (

**a**) dimensions (in mm); (

**b**) lattice representation; and (

**c**) volume rendering of a symmetric portion of the prism specimens.

**Figure 12.**Lattice representation of symmetric portion of Markham Ravine bridge deck. The locations of thermocouples TC1, TC2, TC3 (midspan) and TC4, TC5, TC6 (over girder stem) are indicated. Dimensions are in meters.

**Figure 15.**Determination of degrees of hydration at initial and final sets using recorded penetration resistance data [78].

**Figure 16.**Temperature variation within the Markham Ravine Bridge deck: (

**a**) mid-deck at location TC2; (

**b**) solar radiation profiles selected for each day; and (

**c**) above girder stem at location TC5.

**Figure 19.**Property development in on-site cast and simulated specimens: (

**a**) splitting tensile strength; and (

**b**) shrinkage strain.

**Figure 20.**Simulated stresses at mid-deck location when the curing media is absent: (

**a**) hygral stress component (* daily rainfall, in mm, recorded at the Carmichael 0.9 S meteorological station); (

**b**) thermal stress component; and (

**c**) total stress.

**Figure 21.**Simulated stresses at mid-deck location when the curing media is present during $0.25<t<9$ days: (

**a**) hygral stress component; (

**b**) thermal stress component; and (

**c**) total stress.

**Figure 22.**Spatial maps of cracking potential at t = 11.42 days (8 pm on day 11) for the case where the curing sheets were removed at t = 9 days: (

**a**) thermal stress component; and (

**b**) total stress.

**Figure 23.**Simulated stresses at mid-deck location based on 28-day strain limit of 450 $\mathsf{\mu}$m/m: (

**a**) the curing sheet is absent; (

**b**) curing sheet is present; and (

**c**) curing sheet and curing compound used.

**Figure 25.**Simulated stresses at mid-deck location in the presence of constraint: (

**a**) thermal stress component; (

**b**) hygral stress component (due to self-desiccation); (

**c**) hygral stress component (due to external drying); (

**d**) total stress.

**Figure 26.**Simulated deck cracking: (

**a**) without restraint; (

**b**) with restraint associated with an internal diaphragm; and (

**c**) crack opening histogram for the restrained case (at 12 pm on Day 16).

Primary Category | Subcategory |
---|---|

Materials composition/proportioning | cementitious materials blend |

admixtures | |

water content | |

aggregate | |

fiber reinforcement | |

Environmental boundary conditions | heat exchange: conduction, convection, radiation |

moisture exchange | |

contaminant exposure | |

Processing and curing | concrete temperature at placement |

time of placement | |

methods of consolidation | |

curing | |

Structural boundary conditions | girder spacing and restraint conditions |

deck depth | |

reinforcing steel | |

anticipated loading |

${\mathit{\beta}}_{\mathit{T}}/{\mathit{\beta}}_{\mathit{Th}}$ | Hydration Degree Interval |
---|---|

χ | $\phantom{\rule{17.07182pt}{0ex}}\alpha \le {\alpha}_{0}$ |

$\chi -(\chi -1)(\alpha /{\alpha}_{0}-1)/(\phi -1)$ | ${\alpha}_{0}<\alpha <\phi {\alpha}_{0}$ |

1 | $\phantom{\rule{22.76228pt}{0ex}}\alpha \ge \phi {\alpha}_{0}$ |

Parameter | $\mathit{w}/\mathit{c}$ = 0.25 | $\mathit{w}/\mathit{c}$ = 0.45 |
---|---|---|

$\tilde{{D}_{0}}$ | 0.017 mm${}^{2}$/h | 0.017 mm${}^{2}$/h |

$\tilde{{D}_{1}}$ | 9 mm${}^{2}$/h | 9 mm${}^{2}$/h |

n | 6 | 5 |

${h}_{su}$ | 0.76 | 0.88 |

s | 1.4 | 3.0 |

${\mathrm{\Lambda}}_{h}$ | 0.25 mm/h | 0.25 mm/h |

${\beta}_{h}$ | 0.0025 | 0.0021 |

Parameter | Value * |
---|---|

${q}_{1}$ | 29.0 × 10${}^{-6}$/MPa |

${q}_{2}$ | 69.9 × 10${}^{-6}$/MPa |

${q}_{4}$ | 5.50 × 10${}^{-6}$/MPa |

${n}_{\alpha}$ | 2.2 |

${\kappa}_{0}$ | 0.01/(MPa · d) |

${\kappa}_{1}$ | 5 MPa/K |

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## Share and Cite

**MDPI and ACS Style**

Pan, Y.; Prado, A.; Porras, R.; Hafez, O.M.; Bolander, J.E. Lattice Modeling of Early-Age Behavior of Structural Concrete. *Materials* **2017**, *10*, 231.
https://doi.org/10.3390/ma10030231

**AMA Style**

Pan Y, Prado A, Porras R, Hafez OM, Bolander JE. Lattice Modeling of Early-Age Behavior of Structural Concrete. *Materials*. 2017; 10(3):231.
https://doi.org/10.3390/ma10030231

**Chicago/Turabian Style**

Pan, Yaming, Armando Prado, Rocío Porras, Omar M. Hafez, and John E. Bolander. 2017. "Lattice Modeling of Early-Age Behavior of Structural Concrete" *Materials* 10, no. 3: 231.
https://doi.org/10.3390/ma10030231