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Assumption of geometrical, material, and technological data

The wall of 4 m (13.12 ft) height was analyzed for the length of 10, 15, or 20 m (32.81, 49.21, or 65.62 ft) and two thicknesses: 40 and 70 cm (15.75 and 27.56 in.). The length-height ratios of these walls were, respectively, 2.5, 3.75, and 5. Six examples of combinations of these cases were considered. The analyzed walls were supported on a 4 m (13.12 ft) wide and 70 cm (27.56 in.) deep strip foun-dation of the same length. The wall and the founfoun-dation were assumed to be reinforced with a near-surface reinforcing net of ø16 bars (0.63 in.). The wall was reinforced at both surfaces with horizontal spacing of 20 cm (7.87 in.) and vertical spacing of 15 cm (5.91 in.). The foundation was reinforced with 20 x 20 cm (7.87 x 7.87 in.) spacing at the top and bottom surface. Due to a double symmetry of the wall, the model for finite element analysis was created for 1/4 of the walls. A uniform mesh was prepared and densified

at the free edges of the wall and within the contact surface between the wall and the foundation. A final geometry of the wall with a mesh of finite elements for one exemplary wall is presented in Fig. 4.

It was assumed that the analyzed wall was made of the following concrete mixture: cement CEM I 42.5R 375 kg/m³ (23.41 lb/ft³), water 170 L/m³ (10.61 lb/ft³), and aggregate (granite) 1868 kg/m³ (116.60 lb/ft³). Thermal and moisture coefficients necessary for calculations were set in Table 1.

The development of mechanical properties in time was assumed according to CEB-FIP MC90.¹⁸ The final values for 28-day concrete were assumed as follows: compres-sive strength fcm = 35 MPa (5.08 ksi), tensile strength fctm = 3.0 MPa (0.44 ksi), and modulus of elasticity Ecm = 32.0 GPa (4.64 Mpsi). It was also assumed that the foundation was erected earlier and had hardened, so the material properties were taken as for 28-day concrete, with the same final values as the wall. Environmental and technological conditions were taken as: ambient temperature 20°C (68°F), initial temperature of fresh concrete mixture 20°C (68°F), wooden formwork of 1.8 cm (0.71 in) plywood on the side surfaces, and foil protection of the top surface. It was also assumed that formwork was removed 28 days after concrete casting.

Thermal and shrinkage stresses

First, the temperature and moisture development in time was determined. Figure 5 presents a juxtaposition of temperature and moisture content development diagrams for two areas in the walls (Fig. 4) with different dimensions.

Although the character of both temperature and moisture content are independent of the dimensions of the wall, their magnitudes depend directly on these dimensions. Only the Fig. 4—Dimensions of analyzed walls with finite element

mesh.

Table 1—Thermal and moisture coefficients

Thermal fields

Coefficient of thermal conductivity λ, W/mK (Btu/s·ft°F) 2.52 (4.04 × 10^–4)

Specific heat c_b, kJ/kgK (Btu/lb°F) 0.95 (0.227)

Density of concrete ρ, kg/m³ (lb/ft³) 2400 (149.80)

Coefficient of thermal diffusion αTT, m²/s (ft²/s) 11.1 × 10^–7 (1.19 × 10^–5) Coefficient representing influence of

moisture concentration on heat transfer αTW, m²K/s (ft²·°F/s) 9.375 × 10^–5 (1.81 × 10^–3)

Thermal transfer coefficient αp, W/m²K (Btu/ft²s°F)

6.00 (29.41 × 10^–5) surface without protection, without considering wind

3.58 (17.56 × 10^–5) surface with plywood 5.80 (28.45 × 10^–5) surface with foil 0.81 (3.97 × 10^–5) bottom surface: soil Heat of hydration^* According to equationQ T t( ), ⁼Q e∞ ^ate,

− ^{−0 5} Q∞ = kJ/kg (218.90 Btu/lb); a = 513.62t_e^–0.17 Moisture fields

Coefficient of water-cement

proportionality K, m³/J (ft³/Btu) 0.3 × 10^–9 (1.12 × 10^–5)

Coefficient of moisture diffusion αWW, m²/s (ft²/s) 0.6 × 10^–9 (6.46 × 10^–9) Thermal coefficient of moisture diffusion αWT, m²/sK (ft²/s°F) 2 × 10^–11 (1.20 × 10^–10)

Moisture transfer coefficient βp, m/s (ft/s)

2.78 × 10^–8 (91.21 × 10^–9) surface without protection 0.18 × 10^–8 (5.90 × 10^–9) surface with plywood

0.10 × 10^–8 (3.28 × 10^–9) surface with foil 0.12 × 10^–8 (3.93 × 10^–9) bottom surface: soil

*Approximation made on basis of experimental results of heat of hydration.

thickness, not the length, of the wall is influenced by the temperature and moisture content development. The greatest temperatures are reached in the thicker walls, while the moisture removal rates are almost the same; only slightly greater loss of moisture was observed in thinner walls.

For the known thermal-moisture fields and strains, the stress state can be determined. The following cases were analyzed, and the results of stress distribution at the height of the wall in its internal midspan cross section were presented:

1. Thermal stresses with assumed uniform distribution of temperature in the wall (development of temperature in time was only considered) (Fig. 6(a) and (c));

2. Shrinkage stresses with assumed uniform moisture content distribution in the wall (moisture content change in time was only considered) (Fig. 6(b) and (d));

3. Thermal stresses with assumed real distribution of temperature in the wall (development of temperature in time was also considered) (Fig. 7(a), (c), and (e));

4. Shrinkage stresses with assumed real distribution of moisture content in the wall (moisture content change in time was also considered) (Fig. 7(b), (d), and (f)); and

5. Coupled thermal and shrinkage stresses with assumed real distribution of both temperature and moisture content (Fig. 8(a)).

The diagrams of nonuniform temperature distribution at the height of the wall (Fig. 7(a)) are presented at the moment the maximum hardening temperature is reached; in case of uniform distribution of temperature, the values of tempera-ture from the interior of the wall are applied the entire wall (Fig. 6(a)). Similarly, the moisture distribution is presented at the moment the maximum hardening temperature is reached.

Figure 6 presents distribution of thermal (Fig. 6(c)) and shrinkage (Fig. 6(d)) stresses in the midspan cross section of the interior of the wall under the assumption of uniform temperature and moisture content distribution in the wall.

Such an assumption is very common in the analysis of medium-thick externally restrained structures, especially when analytic methods of thermal-shrinkage stresses deter-mination are used, but also in numerical analyses in which a model is reduced to a two-dimensional problem. It is believed that such an approach provides a good approxi-mation because both the temperature and moisture content differences within the body of the wall are relatively small.

The resultant stress distribution at the height of the section is approximately linear with the maximum values of stresses at the joint between the wall and the foundation. On such a simple example, it can be observed that both the length and the thickness of the wall influence the resulting stresses.

The thickness of the wall determines the maximum value of stress; it should be noted that greater values of thermal stresses occur in thicker walls, which results from higher exerted temperatures; shrinkage stresses are greater in thinner walls, which is caused by a higher rate of water removal. The length of the wall (linear restraint) determines the distribution of stress at the height of the wall. For the analyzed walls characterized by length-height ratio (L/H) ≥ 2.5, tensile stresses occur at the whole height of the wall; in high walls—that is, the walls with L/H < 2.5—compression of top areas may be expected.

Nevertheless, heat and moisture are transported within the element and to the surrounding environment in the process of concrete curing. Therefore, the values of temperatures Fig. 5—Temperature distribution: (a) in interior and (b) on surface of wall; and moisture content development: (c) in interior and (d) on surface of wall.

and moisture content vary in different zones of the wall, as do the resulting stresses. Figure 7 shows thermal (Fig. 7(c)) and shrinkage (Fig. 7(d)) stress distribution at the height of the wall in its interior taking into account the real (nonuni-form) temperature and moisture content distribution. It is also noted that temperature and moisture content difference at the thickness of the wall leads to stress diversification in the internal and near-surface areas (Fig. 7(e) and (f)). It can be observed that the length of the wall influences the occur-ring stresses as much as its thickness in such a way that the length determines the character of stress distribution, while the thickness determines the maximum values of stresses.

It should be emphasized that considering real distribution of temperature and moisture the maximum stresses, and consequently the highest cracking risk, is observed at some distance above the joint, which complies with observations in References 4, 9, and 10.

The observation diagrams in Fig. 8 were prepared to present coupled thermal-shrinkage stress distribution in

the wall. The character of total thermal-shrinkage stresses results mainly from the character of their thermal compo-nent (Fig. 7(b) versus Fig. 8(a)); shrinkage stresses only add to the final value. Thus, the aforementioned conclusions remain valid. The location of the maximum stresses varied;

generally, it was the closest to the construction joint for the thinnest and the shortest walls (0.4 m = 1.31 ft), and was elevated as the thickness and the length of the wall increased (even up to 1.2 m [3.94 ft]). Thus, the results comply with observations in Reference 4, while the observations from References 9 and 10 seem more accurate for high walls (walls with a low L/H ratio). In all analyzed cases, because the formwork was detained for the whole process, greater total stresses were observed in the interior of the wall, which explains the occurrence of first cracks in the interior of the wall.⁵

It should be noted that Fig. 6 presents stress distribution at the height of the wall in the midspan cross section after 18.3 days under the assumption of uniform distribution Fig. 6—Case of uniform distribution of: (a) temperature; (b) moisture; (c) thermal; and (d) shrinkage stress distribution at height of wall in midspan cross section after 18.3 days.

of temperature in the wall, while in Fig. 7 and 8, the real, nonuniform distribution of temperature is taken into account.

It accounts for the visible differences in the obtained stress distribution, especially in thermal stress near the joint.

In this case, the self-induced stress arises in the wall due

to the nonuniform distribution of temperature in the wall.

The importance of the self-induced stresses is discussed in a following section. It is interesting that the same differ-ences in the stress distribution can be obtained with the use Fig. 7—Stress distribution at height of wall in midspan cross section after 18.3 days under assumption of real (nonuniform) distribution of: (a) temperature and (b) moisture content in wall; (c) thermal and (d) shrinkage stresses in the interior of the wall; and (e) thermal and (f) shrinkage stress in interior and on surface of 20 m (65.6 ft) long wall.

of analytical methods when the nonuniform distribution of temperature at the height of the wall is taken.¹⁹

Self-induced versus restraint stresses

In the next stage, the computations were made to evaluate the share of self-induced stresses in total stresses exerted

in the wall. Total stresses exerted in the wall supported on a stiff foundation due to the thermal-shrinkage effects are a sum of self-induced and restraint stresses. To assess the contribution of self-induced stresses in the analyzed walls, the impact of restraint in a form of foundation was mini-mized by reduction of the foundation’s stiffness to EF = 100 MPa (14.5 ksi) (flexible foundation was assumed).

Diagrams in Fig. 9 present development of self-induced (Fig. 9(a)) and total (Fig. 9(b)) stresses in time for the loca-tion in which the maximum value of stress was observed for the 20 m (65.6 ft) long walls of 70 and 40 cm (2.3 and 1.3 ft) thickness, assuming that both walls were detained in the formwork. Stress development was presented for one length of the wall for better visibility, as the values are similar.

The resulting self-induced stresses reach relatively low values compared with the total stresses. Moreover, their character is different and is closer to the behavior of typical massive concrete structures. In the first phase, the interior of the wall is subjected to compression, while surface layers are tensioned; in the second phase, stress body inversion is observed. This behavior is especially visible in thicker walls;

temperature and moisture content differences at the thick-ness of the thinner walls are smaller, so the resulting stresses are of lower value. It is worth noting that the generation of self-induced stresses is the cause of total stress difference in different zones of the wall.

Figure 10 presents a comparison of the diagrams of self-induced (Fig. 10(a)) and total stress (Fig. 10 (b)) distri-bution in the midspan cross section of the wall in both the heating (after 1.2 days) and cooling phase (after 18.3 days).

The character of the observed stresses is similar in each wall, so the diagrams are presented on the example of one wall (L_20,d_0.7) only. Massive concrete-like behavior can be observed in unrestrained walls, while the signs of total stresses are the same at the whole height of the wall.

Fig. 8—Coupled thermal-shrinkage stress distribution at height of wall in midspan cross section after 18.3 days under assump-tion of real (nonuniform) temperature and moisture content distribuassump-tion in wall: (a) total stress in interior of wall; and (b) stresses in interior and on surface of 20 m (65.6 ft) long wall.

Fig. 9—Thermal-shrinkage stress development in time: (a) self-induced stresses; and (b) total stresses in 20 m (65.6 ft) long wall.

Influence of time of formwork removal on stress distribution

Finally, an analysis was performed that investigated the influence of early formwork removal (3 and 7 days after concrete casting). Diagrams in Fig. 11 show stress devel-opment in time, while Fig. 12 shows stress distribution after 18.3 days in the midspan cross section for the walls from which the formwork was removed after 3 and 7 days. Because of the similar character, diagrams were also presented for one exemplary wall (L_20,d_0.7). If the wall is kept in the formwork long enough for the concrete to cool completely, the heat concentration in the interior of the wall leads to higher stress and possible first crack development in the internal parts of the wall (Fig. 9(b) and 10(b)). When form-work is removed in early phases of concrete curing, greater stresses are observed on the surface of the wall as a result of rapid cooling of the wall surface, which may lead to first crack formation in the near-surface areas (Fig. 11 and 12). It should be noted that in both cases, cracks may extend to the entire wall thickness. It is important to note that formwork removal accelerates moisture loss near the surface, which is why a significant increase of tensile stresses is observed in the vicinity of the top surface if it is no longer protected.

CONCLUSIONS

The problem with high temperatures arising during the hardening of concrete has been known since the 1930s, when dams were first built in the United States. Much effort has been focused on the creation of efficient methods for mitigation of the negative effects of concrete curing in massive structures; this problem is well known in concrete elements with considerable thickness. Nevertheless, forma-tion of cracks is also observed in medium-thick concrete elements, such as RC walls cast against an old set founda-tion. Such walls can also be sensitive to early-age cracking of thermal and shrinkage origins. Control of thermal and shrinkage cracking in early-age concrete is of great impor-tance to ensure a desired service life and function of struc-tures. It is a complicated problem due to the complex nature Fig. 10—(a) Self-induced and (b) total stress distribution at height of 20 m (65.6 ft) long, 70 cm (2.3 ft) thick wall in midspan cross section in heating (ph_I) and cooling (ph_II) phase.

Fig. 11—Stress distribution in midspan cross section of 20 m (65.6 ft) long, 70 cm (2.3 ft) thick wall after 18.3 days from which formwork was removed after 3 and 7 days.

Fig. 12—Stress distribution in midspan cross section of 20 m (65.6 ft) long, 70 cm (2.3 ft) thick wall after 18.3 days from which formwork was removed after 3 and 7 days.

of interacting phenomena and a large number of contributing factors. Important factors include the dimensions of struc-tural elements and the length-height ratio, which directly affect the level of a wall restraint in a foundation.

The contribution of self-induced and restrained stresses to total stresses induced in the wall with different dimen-sions was investigated. The results obtained with the use of the original numerical model were discussed. Very limited analysis of the stress distribution in the walls could be found in the literature concerning early-age concrete. This paper is an attempt to fill the vacancy in this field, in particular by providing information about development of stresses and the importance of self-induced stresses in externally restrained structures. Knowledge about the distribution of stresses is necessary in many practical cases, for example, in the eval-uation of crack risk of early-age concrete structures. The results of the analysis can be summarized as follows:

1. Thermal stresses play a predominant role in the total thermal-shrinkage stress development;

2. The total thermal-shrinkage stresses arising in an RC wall result mainly from restraint stresses generated by limited possibility of the wall deformation; the share of self-induced stresses increases with the increasing thickness of the wall, but even in relatively thin elements, it has an influence on total stress distribution in the wall; and

3. Three-dimensional numerical analysis results explain the following phenomena observed in externally restrained elements:

• The greatest thermal-shrinkage stress does not occur at the interface between the wall and the restraint but at some level above the restraint joint. That fact results from nonuniform distribution of temperature and mois-ture within the element which concentrate in its central parts. In this case, the self-induced stress is also consid-ered. The first crack can consistently be observed at some level above the restraint joint; and

• When the wall is detained in formwork until it cools down, stresses develop towards the interior of the wall;

thus, internal cracking may initially develop. When formwork is removed in early phases of concrete curing, an increased cooling rate leads to greater stresses in the surface zones, and first cracks can develop on the surface of the wall.

AUTHOR BIOS

Barbara Klemczak is an Associate Professor in the Department of Civil Engineering at the Silesian University of Technology, Gliwice, Poland.

She received her PhD and DSc from the Silesian University of Technology in the field of numerical modeling of early-age massive concrete. Her research interests include nonlinear analysis of reinforced concrete struc-tures, particularly numerical modeling of thermal and shrinkage effects in concrete structures at early ages.

Agnieszka Knoppik-Wróbel is a PhD Student in the Department of Civil Engineering at the Silesian University of Technology. Her research interests include cracking risk in early-age externally restrained concrete structures.

ACKNOWLEDGMENTS

This paper was done as a part of a research Project N N506 043440 enti-tled, “Numerical Prediction of Cracking Risk and Methods of Its Reduction in Massive Concrete Structures,” funded by the Polish National Science Centre. The co-author of the paper, Agnieszka Knoppik-Wróbel, is a scholar under the Project SWIFT, co-financed by the European Union under the European Social Fund.

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