Analysis of temperature stresses in concrete breakwater elements: Hollow cubes and Tetrapods

Report 25.5-94-2

Analysis of Temperature Stresses

in Concrete Breakwater Elements:

Hollow cubes and Tetrapods

February 1994 l.B. N o o r u - M o h a m e d

mm

i i i i i i i

Delft University of Technology Faculty of Civil Engineering

Report 25.5-94-2 February 1994

Analysis of Temperature Stresses in Concrete

Breakwater Elements: Hollow cubes and Tetrapods

M.B. Nooru-Mohamed

Mailing address:

Stevin Laboratory

Faculty of Civil Engineering Delft University of Technology Stevinweg 1

2628 CN Delft The Netherlands

Acknowledgements

I should thank Dr. Jan van Mier for availing the opportunity to work on this project, for the discussions, and reading the draft version of this report. Also my appreciations are extended to: Adri Vervuurt for fruitful discussions, Henk Spiewakowski for the drawings and Wim Jansze for introducing FEMGEN/FEMVIEW.

Contents

Abstract v

1. Introduction 1

2. Numerical tool 2 2.1 Material properties 2

2.2 Element geometry and schematization 3

3. Parameter variation 6

4. Solid versus hollow cubes: a comparison 9

5. Temperature histories of hollow cubes 11 5.1 Influence of initial concrete mix-temperature 11

5.2 Influence of cement content 14 5.3 Influence of hole diameter 16 5.4 Influence of specimen size 18 5.5 Influence of isolation 18

6. Temperature histories of tetrapods 24 6.1 Influence of cement content 24

6.2 Influence of size 24

7. Stress analysis of hollow cubes 28 7.1 Influence of hole diameter 30

7.2 Influence of size 35 7.3 Influence of cement content 38

8. Stress analysis of tetrapods 40 8.1 Influence of size 43

8.2 Influence of cement content 47

9. Conclusions 51

References 53

Abstract

In this report, the results of a numerical parameter study on temperature stresses caused by hydration of cement in concrete breakwater elements are shown. Two different geometries were analysed namely hollow cubes and tetrapods.

The problem encountered in solid cube breakwaters is the undesirable temperature build up between the core and the outside surface. The main reason for the temperature build up, is the thermal heat, which is generated due to the heat of hydration of cement from the time of casting. This thermal gradient induces eigen-stresses and causes the early age cracking. The choice of a hollow cube is a way to reduce the temperature gradient and thus the eigen-stresses.

A parameter study is undertaken in order to arrive at an optimum dimension based on the thermal gradient and the eigen-stresses. The variables are the cube dimension (i.e., height H = 2, 3 and 4 m with the inner diameter to the height ratio d,/H = 0.4), the size of the hole (d[ = 0.9, 1.2 and 1.5 m for the 3 m hollow cube only), the cement content (225 and 325 kg/m3), the initial concrete mix temperature (5, 15 and 25 °C) and the degree of isolation (with and without isolation). For the analyses of the tetrapods, two geometries were chosen. The total height H of the tetrapod was 3.11 and 4.15 m with corresponding diameters D = 1.5 and 2 m.

The favourable conditions for the reduction in the temperature gradient and the tensile stress peak for the hollow cubes were: the low mix temperature, the larger hole diameter, the low cement content and the smaller hollow cube. From the range of hollow cube specimens analysed, it appears the 2 m hollow cube is the best choice based on the temperature gradient as well as the tensile stress peak generated.

1. Introduction

As part of the European cooperative project on Rubble Mound Breakwater Failure Modes (Contract MAS2-CT92-0042), the results of a numerical parameter study on temperature stresses caused by hydration of cement in concrete breakwater elements are presented. Two different block geometries were studied, viz. hollow cubes and tetrapods.

The problem encountered in mass concrete, such as the solid cubes used in breakwaters (of dimension up to 4 meters), is the undesirable temperature build up between the core and the outside surface. The main reason for the temperature build up, is the thermal heat, which is generated due to the heat of hydration of cement from the inception of casting. Often the surface is exposed to the ambient temperature, provided there is no insulation. This thermal gradient induces eigen-stresses in the concrete element and causes the early age cracking. The choice of the hollow cube element is a way to reduce the temperature gradient and thus the eigen-stresses. A parameter study is undertaken in order to arrive at an optimum dimension based on the thermal gradient and the eigen-stresses. The variables are the cube dimension, the size of the hole, the cement content, the initial concrete mix temperature and the degree of isolation.

In this study, three different hollow cube geometries with a height H = 2, 3 and 4 meters were analysed. The inner diameter to the height ratio was constant (d,/H = 0.4). To investigate the influence of hole size on temperature and stress developments, three different diameters were studied for the 3 m hollow cube.

For the tetrapods, the two geometries adopted by Horden(1987) were chosen for the present analysis. The total height H of the tetrapods was 3.11 and 4.15 m with corresponding diameter D = 1.5 and 2.0 m.

2. Numerical Tool

The numerical analyses were carried out with the finite element package DIANA, and for the pre- and post processing FEMGEN/FEMVIEW was used. In the DIANA computing environment the two modules: (a) potential flow analysis, and (b) flow-stress analysis, are well suited for the present problem.

A typical analysis consists of two parts. First, the potential flow analysis was performed, in which a linear transient analysis was carried out. Second, the flow-stress analysis was executed where the temperature values at each time step(corresponding to each node), resulting from the previous flow analysis were input as the temperature load.

2.1 Material properties

For the D I A N A analysis a set of material properties are required as a priori. The heat analysis is based on an Arrhenius type curve. The following material properties for normal concrete are assumed:

REACTI : Degree of hydration r, which is equal to the ratio of the momentary heat production to that of the total heat production and where 0< r <1.

PRDKAR : Degree of hydration dependent heat production q(r): the maximum value of the diagram is scaled to 1.0. r = QCO/Q,^ , q = q^fCr) = ae b m ,f ( r )

CONDUC : Conductivity X = 2.6* 10 3 kW/(m)°K

CONREA : Conductivity as a linear function of the degree of hydration r. for r = 0 X = 4.0*10'3 and

for r = 1 X = 2.6*10"3 CAPACI : Heat Capacity = 2650 kJ/(m3)°K

CAP ART : Capacity diagram with the entries beginning with 2950 and ending with 2650: here, the number of entries must correspond to that of the REACT!.

CONVEC : Convection = 25 W/(m2)°K = 0.025 kW/(m2)°K, equal to the convection coefficient of air.

Convection = 6.8* 10"4 kW/(m2)°K when a 5 mm polyurethane isolation is present.

MAXPRD : Total heat production for the cement contents of 225 and 325 kg/m3 are qm a x = 73528 and 106207 kJ/m3 respectively.

The constant of Arrhenius c = 5995. The Young's modulus and the Poisson ratio were assumed as 17.5 GPa and 0.2, without accounting for the change in those values due to hydration of concrete. The coefficient of thermal expansion a = 11* 10"6 / °C.

2.2 Element Geometry and Schematization

hollow cubes

For the numerical analyses, three hollow cube geometries with a height H = 2, 3 and 4 m were adopted. In these geometries, a constant inner diameter to the height ratio d / H = 0.4

was used (Figure 1). d1-o.4H»i.6m.

d , - 0 . 3 H = 0 . 9 d , - 0 . 4 H - 1 . 2 d , - 0 . 5 H - 1 . 5

Figure 2. Three hole diameters for the 3 m geometry

To investigate the influence of hole diameters (dj) on the temperature and stress development, three different diameters were chosen for the 3 m hollow cube only, such that the ratio d,/H = 0.3, 0.4 and 0.5 (Figure 2).

Considering the symmetry of the hollow cube, it was decided to analyse one eighth of the geometry. The schematized one eighth of the 3 m hollow cube is shown in Figure 3. Even though, there are a variety of elements to choose from the DIANA software package, the restriction is, that only the CHX60 (20 noded brick) element must be used for a 3D temperature analysis. This is due to the fact that the element must be suitable for the potential flow as well as for the flow-stress analyses.

A problem was encountered during the initial stage of the study namely:

Problem : When the CHX60 element is used for the potential flow analyses, the temperature output produced bv DIANA of the middle plane nodes (always nodes 9.10.11 and 12) is twice as compared to the other nodes, (see the Appendix).

This problem was not fully described in the D I A N A manual.

tetrapod

The dimensions of the tetrapods are shown in Figure 4, in which the two heights H are 3.11 and 4.15 m are shown with the corresponding diameter D = 1.5 and 2 m. The schematized finite element mesh is shown in Figure 5. Considering the axi-symmetry, only one eighth of the tetrapod is analyzed using CQ16A (8 noded quadrilateral) and CT12A (6 noded triangular)

solid elements. For the convecting surface two noded B2AHT axi-symmetric potential flow elements are used.

Figure 3. Finite element mesh of an eighth of the hollow cube (H = 4 m, d,/H = 0.4)

H = 4.15m D = 2.0m

0.627 H = 3.11m

D = 1.5m

Figure 4. Two tetrapod geometries with heights H =3.11 m, 4.15 m and corresponding D values are 1.5 m and 2.0 m respectively.

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3.0 Parameter variation

hollow cube

In this study, the parameters investigated are: (a) the initial concrete mix-temperature, (b) the cement content, (c) the hole diameter and (d) the variation in hollow cube size. The ambient temperature was assumed 15 °C in all cases.

(a) the initial concrete mix-temperature: the three initial concrete mix-temperatures chosen were 5, 15 and 25 °C

(b) the cement content: the two cement contents studied are 225 and 325 kg/m3 with a maximum heat production of 73528 and 106207 kJ/m3 respectively.

The variation in (a) and (b) were only carried out for the 3 m hollow cube with the ratio d,/H = 0.4, see Table 1.

(c) the hole diameter: For the 3 m hollow cube, three different hole diameters 0.9, 1.2 and 1.5 m (d,/H = 0.3, 0.4 and 0.5) were adopted, see Table 2. The initial concrete mix tempera-ture was 25 °C and the surrounding temperatempera-ture was as mentioned before. The cement content was 225 kg/m3 for all the hole sizes. In addition, for the middle hole (d,=1.2 m) an analysis was performed with 325 kg/m3 cement content.

(d) the variation in hollow cube size: Three hollow cube geometries with the height H = 2, 3 and 4 m were analysed, see Table 2. In all these cases the ratio d,/H = 0.4. The temperatures of the concrete mix and the surroundings were 25 °C and 15 °C respectively. The cement content was 225 kg/m3. In addition, for H=3 m, a cement content of 325 kg/m3 was studied as well.

tetrapod

With regard to the tetrapods, the height H is 3.11 and 4.15 m with diameters D=1.5 and 2.0 m, respectively (see Table 3). The mix- and surrounding temperatures were 25 and 15 °C. The same two cement contents as discussed above were analysed for each of the geometries.

Table 1-3 shows an overview of the total parameter study (for the hollow cube and tetrapod geometries).

Table 1: Overview of temperature analyses for H=3 (d,/H = 0.4) with three different initial temperatures and two cement contents.

H (m) (d,=.4H) Cement Content (kg/m3) Concrete Mix Tem-perature Surrounding Temperature Isolation 5 15 no 225 _yes 15 15 no yes 25 15 no 3.0 1.2 yes 5 15 no 325 _yes 15 15 no yes 25 15 no yes

Table 2: The temperature/stress analyses for the hollow cube breakwater elements. H ( m ) 2.0 3.0 4.0 Mass (ton) d,=0.4H 16.78 56.63 134.2 M m ) 0.4H 0.3H 0.4H 0.5H 0.4H Cement Con-tent (kg/m3) 225 225 225 325 225 225 Concrete Mix Temperature 25 25 25 25 25 25 Surrounding Temperature 15 15 15 15 15 15

Table 3: The Temperature/Stress analysis for the Tetrapods.

H ( m ) 3.11 4.15 D ( m ) 1.5 2.0 Mass (ton) 20.25 48.0 Cement Con-tent (kg/m3) 225 325 225 325 Concrete Mix Temperature 25 25 25 25 Surrounding Temperature 15 15 15 15

4.0 Solid versus hollow cubes

Before the analyses were done, qualitatively the difference between a solid and a hollow cube was studied using the following geometries. The heights of the solid and the hollow cubes were 2.7 m (V=19.7 m3) and 3.0 m (V=21.7 m3). For the hollow cube, a 1.5 m hole was assumed. Considering the symmetry of the solid and hollow cubes, only an eighth of the geometry is analysed, see Figure 6a and 7a. With regard to the eighth of the solid cube, there are three convecting surfaces as identified in Figure 6a, the two outer sides and the bottom surface. For the hollow cube, there are four convecting surfaces, three sides as of the solid cube and in addition the curved surface of the hole.

In Figure 6b, the simulated temperature-time history for the 2.7 m solid cube is shown. In this simulation, the temperatures of the initial concrete mix and the surroundings are assumed to be 25 °C and 15 °C respectively. The cement content was 225 kg/m3.

When the solid cube is not isolated, the exterior surfaces are allowed to convect. As can be seen in Figure 6a, the corner node (182) cools down from 25 °C (the initial mix temperature) to 15 °C (the surrounding temperature) almost immediately. The interior of the solid cube becomes hot due to the heat of hydration of cement. As can be seen, NODE 1 in the centre of the cube, reached the temperature peak of 48.5 °C after 37 hours. Also the temperature gradient i.e., the maximum temperature difference between the in and outside of the cube at this moment is 33.1 °C. A dominant temperature gradient is seen in the solid cubes.

The simulated temperature history of the hollow cube is shown in Figure 7b. The conditions are similar as for the solid cube. The exterior NODE 109 (see Figure 7c) in the hollow cube behaves similar to the solid cube. However, the interior NODE 27 is not. The interior node reached the peak after 15 hours and the post peak temperature curve decreased more rapidly than its counterpart. In addition, the rate of decrease of the temperature gradient is faster for the hollow cube than the solid cube. A comparison is made in Table 4. The choice of a

hollow cube is in a way intended to reduce this temperature gradient. When compared to the temperature history of the solid cube the purpose is to a certain extent appears to be achieved. Hence, the additional cylindrical convecting surface area and the smaller dimension of the concrete section would have caused the lower temperature gradients in the hollow cube.

Table 4: Maximum temperature difference AT (the difference in temperature between the core and the exterior) at 15, 37, 50, 100 and 150 hours for solid and hollow cubes. AT, = T , " " - T ,m l n C O Solid cube H = 2.7m Hollow cube H = 3m, d,=1.5m A T1 5 28.5 25.1 A T3 7 33.1 18.8 A T5 0 32.2 13.9 A TI 0 0 21.6 3.8 A T1 5 0 12.4 1.1 H-=2.7m (a) 60.0 50.0 10.0 0.0 0.0

GO

Temperature versus Time

An eighth of a solid cube, H > 2.7 m, no Isolation

O — O N O O E 182 (X.1.35, Y«0, Z=1.35) —# NODE 134 (X*0, YeO, 2=0) • — D N O D E 1 (X»0, Y.1.35,Zrf>) O — O N O O E 25 (Xrfj.876. Y.1.35, Z-0.675) - • NOOE 485 (X>0.675, Y.0.675, Z-0) <J ONOOE 908 (X-0.675, Y.0.676, Z-0.875) ' - A N O D E 7 (X.1.35, Y-1.35.2>0)

concrete mix temperature - 25 'C surrounding temperature - 15 *C cement content« 225 kg/m1

convection - 0.025 kW/(m') Y

50.0 100.0 150.0

Time In hours

Figure 6. Nodal locations for the 2.7 m solid cube(a), the simulated temperature-history for the 2.7 m solid cube(b).

d1=0.5H=1.5m

Tin» In hours

Figure 7. Nodal locations for the eighth of the 3 m hollow cube (d,/H = 0.5) (a), simulated temperature histories for hollow cube in 7a (b).

5.0 Temperature histories of hollow cubes

As shown in Table 1, the 3 m hollow cube with constant d,/H = 0.4 was analyzed with varying initial concrete mix-temperatures (5, 15 and 25 °C) and two cement contents (225 and 325 kg/m3).

5.1 Influence of initial concrete mix-temperature

In Figure 8a, b, and c, the simulated temperature-time histories for the 3 m hollow cube with 225 kg/m3 cement content are shown. The relevant locations of the nodes are identified in Figure 9a and b. As can be seen in Figure 8a-c, the 5 °C initial concrete temperature produced a low temperature gradient, and pushed the peak in the T-t curve ahead of the other initial temperature simulations. That is, the analysis with the 5 °C initial mix-temperature reached the peak temperature at 47 hours, the 25 °C analysis already after at 17 hours. In Table 5, the

maximum and minimum temperatures, the time needed to reach the maximum temperature, and the temperature gradient for the two types of cement are shown. Therefore, by lowering the concrete mix to a low temperature, such as 5 degrees, it is possible to reduce the temperature build up between the core and the exterior of the hollow cube; in the present case A T ^ was 10.0 °C, where A T ^ = T ^ - T ^ . 60.0 50.0 40.0 30.0 20.0 10.0

Temperature versus Time

An eighth of a hollow cube, H = 3 m, ó\~ 0.4H no Isolation O — O NODE 79 (X=t.5, Y=0, Z.1.5) • — • N O D E 49 (X.0424. Y=0, Zrf).424) •—DNOOE 834 (Xrf).B54, Y.1.2. Z=0.854) 0 O NODE 1025 (X«0.839. Y.1.5, Z.0.639) <3—<]NOOE 956 (X=0, Y.1.5, Z^>.78) 4—•«NODE 926 (X=0.96, Y-1.2. Z-0) Ö—ANODE 1027 (Xrf>.424, Y.1.5, Z^).424) NOPE 1151 (X.1,5, Y.1.5, Z-0) concrete mix temperature . 5 °C surrounding temperature • 15 *C cement content - 225 kg/m* convection - 0.025 kW/(m') *K (a) 50.0 100.0 150.0 Time In hours 60.0 50.0 40.0 30.0 20.0 10.0

Temperature versus Time

An eighth of a hollow cube, H • 3 m, d1= 0.4H no Isolation O—O NODE 79 (X.1.5, Y-0, Z.1.5)

NODE 49 (X-0.424. Y-O, Z4).424) jO—DNODE 834 (X-0.854, Y-1.2. Z-0.854) 0—O NODE 1025 (X.0.639. Y.1.5, Z-0.839) <—ONOOE 956 (X.0, Y . I A Z.0.78) 4 *NODE 926 (X.0.96, Y-1.2, Z-0) |A—ANODE 1027 (X.0.424, Y.1.5. Z-0.424)

NODE 1151 (X.1.5, Y.1.5, Zrt) concrete mix temperature . 15 *C surrounding temperature • 15 °C cement content. 225 kg/m* convection . 0.025 kW/(m') V 100.0 150.0 Time In hours

CO

Figure 8. Simulated temperature-time histories for the initial concrete mix-temperatures 5 (a) and 15 °C (b) of H = 3.0 m, d,=1.2 m where the cement content = 225 kg/m3.

60.0

50.0

40.0

30.0

Temperature versus Time

An eighth ol a hollow cube, H • 3 m, d l . 0 . 4 H no isolation

20.0 10.0 P — O N O O E 79 (X.1.5, Y.O, 2.1.5) NODE 49 (X.0.424, Y . 0 . ZiO.424) • — O N O O E 834 (X-0.654. Y.1.2,2s0.854) O — O N O O E 1025 PWI.B39, Y.1.5, 2=0.639) < ANODE 956 (X.0, Y.1.5. Z-0.78) NODE 926 (X.0.96, Y.1.2, 2.0) | A — A N O D E 1027 (X.0.424, Y.1.6, Z-0.424) NODE 1161 (X.1.5. Y.1.5, Z-0)

concrete mix temperature • 2 5 ' C surrounding temperature . 1 5 * C cement content - 2 2 5 kg/m1

convection - 0 . 0 2 5 kW/(m') "K

50.0 100.0 150.0 (c)

Time In nouns

Figure 8c. Simulated temperature-time histories for the initial concrete mix-temperature 25 °C of H = 3.0 m, d,=1.2 m where the cement content = 225 kg/m3.

H - 3 m d ^ O . A H - l ^ m

00

CO

Figure 9. Locations of the nodes for one eighth of the 3 m hollow cube (d,/H =0.4) (a), and the top view (b) corresponding to Figure 8.

5.2 Influence of cement content

In Figure 10a, b, and c, the simulated temperamre-time plots for the same geometry discussed above but with a higher cement content are shown. A higher cement content releases more heat to the system and raises the temperature in the hollow cube. In the simulation, the choice of a higher cement content, i.e from 225 to 325 kg/m3, raised the temperature gradient by 7 °C. For the 5 °C initial temperature, A T ^ was 17 °C (see Table 5). Notably, the initial mix temperature with 25 °C, produced the temperature gradient of 35 °C. A comparison between the two cement contents is shown in Table 5.

60.0 50.0 j 40.0 .5 t 30.0 h 20.0 10.0 0.0 0.0

Temperature versus Time

An eighth of a hollow cube, H • 3 m, d1> 0.4H no Isolation

O — O N O O E 79 (X.1 £ , Y . 0 , 2.1.5) • — • N O D E 49 (X«0.424, Y-0.2*0.424) • — O N O O E 834 (X=0.854, Y.1.2,2*0.854) O—O NODE 1025 (X=0 639, Y.1.5,2V0.639) < — O N O O E 956 (X«0, Y.1.5.2-0.78)

4 ^NODE 026 (X =0.96, Y.1.2, ZcO)

A — A N O D E 1027 (X*>.424, Y.1.5, 2=0.424) ' NODE 1151 (X.1.5. Y.1.5. 2*0)

concrete mix temperature • 5 °C surrounding temperature . 15 *C cement content - 325 kg/m' convection - 0.025 kW/(m') *K 50.0 100.0 (a) 150.0 Time in hours 60.0 50.0 40.0 30.0 20.0 10.0 0.0 0.0

Temperature versus Time

An eighth of a hollow cube, H . 3 m, d l . 0.4H no Isolation

50.0

| 0 — O NODE 79 (X-1.5, Y . 0 , 2=1.5) • — • N O D E 49 (X.0.424, Y . 0 . Zo0.424) • • NODE 834 (XcO.654, Y.1.2, Z.O.B54) | 0 — O NODE 1025 (X.0.639. Y.1.5.2*1.639) < — O N O O E 956 (X.0, Y.1.5,2*1.78) 4 — 4 NODE 926 (X.0.96. Y.1.2, 2 * ) ) i£ ANODE 1027 (X.0.424, Y.1.5. 2=0.424)

NODE 1151 (X.1.5, Y.1.5, 2=0)

concrete mix temperature • 15 *C surrounding temperature . 15 *C cement content • 325 kg/m* convection • 0.025 kW/(m") °K (b) 100.0 150.0 Time In hours

Figure 10. Simulated temperature-time histories for the initial concrete mix-temperatures 5 (a) and 15 °C (b) of H = 3.0 m, d,= 1.2 m and the cement content = 325 kg/m3.

60.0 50.0 40.0 £ I c £ 2 a I E 30.0 20.0 10.0 0.0 0.0

Temperature versus Time

An eighth ol a hollow cube, H = 3 m, d1 = 0.4H no Isolation

O—ONOOE 78 (X.1.5, Y*l, 2.1.5) • • NOOE 49 (X.0.424, Y.0, 2*0.424) •—ONOOE 834 pw>.854, Y.1.2, 2*1.854) O—ONOOE 1025 (X.0.839, Y.1.6, Z-0.639) <—ONODE 956 (X*>, Y.1.5.2*0.78) NODE 926 (X.0.96, Yol.2, 2 * 1 ) \£s—ANODE 1027 (X.0.424, Y.1.5, 2*0.424) lil—ANODE 1151 (X.1.5. Y.1.5.2*1)

concrete mix temperature . 25 *C surrounding temperature • 15 °C cement content - 325 kg/m* convection - 0.025 kW(m') °K 50.0 100.0 Time in hours 150.0

Figure 10c. Simulated temperature-time histories for the initial concrete mix-temperature 2 5 °C of H = 3.0 m, d,= 1.2 m and the cement content = 325 kg/m3.

Table 5: Temperature difference AT ( = Tm a x - T ^ ) between the external and internal nodes of the 3.0 m hollow cube geometry, with respect to three concrete mix-temperatures. H = 3.0 m, d,=1.2 m, Surrounding temperature = 15 °C Concrete Mix-temperature (°C) 5 15 2 5 Cement content (kg/m3) 2 2 5 3 2 5 2 2 5 3 2 5 2 2 5 3 2 5 Maximum tem-perature T , ^ (°C) 2 5 . 3 3 2 . 5 3 2 . 3 3 9 . 6 4 3 . 7 5 1 . 5 Time(t) to reach T , ^ (hours) 4 7 5 0 2 9 3 3 17 2 0 Tjnin at t (°C) 1 5 . 3 1 5 . 4 1 5 . 6 1 5 . 8 1 6 . 2 1 6 . 6 AT = T - T max max mm (°C) 1 0 . 0 17.1 1 6 . 8 2 3 . 8 2 7 . 5 3 5 . 0

53 Influence of hole diameter

Figure 11a, b and c shows the temperature history simulation for the 3 m hollow cube with three different hole diameters di = 0.9, 1.2 and 1.5 m (i.e, d,/H = 0.3, 0.4 and 0.5). The corresponding locations of the nodes are shown in Figure 12, 9 and 7a. In all of these simulations, the initial concrete mix-temperature, the surrounding temperature and the cement contents were kept constant and were 25 °C , 15 °C and 225 kg/m3, respectively.

The maximum temperature for the smallest and the largest holes was 44.8 °C, and 41.4 °C. The difference between the simulated peak temperatures for the two holes seems not so significant. However, at different time steps, the temperature gradient for the specimen with the smallest diameter is the largest. It is due to the smaller convecting area. Furthermore, the maximum temperature gradient AT was 28.8 °C and 25.1 °C for the above two cases. Table 6 shows the overview of the values in these analyses.

Table 6: Temperature difference AT (= T , ^ - T^,,) between the external and internal nodes of the 3.0 m hollow cube geometry, for three different hole diameters.

H = 3.0 m, Surrounding & Mix-temperatures = 15, 25 °C, Cement content = 225 kg/m3 Hole diameter d, (m) 0.9 1.2 1.5 Maximum Tem-perature (°C) 44.8 43.7 41.4 Time t to reach T ^ (hours) 19 17 15 at t (°C) 16.0 16.2 16.4 AT = T - T max max mm ( ° Q 28.8 27.5 25.1

60.0

50.0

40.0

An eighth of a hollow cube, H = 3 m, d1 = 0.3H - 0.9 m, no Isolation

30.0 20.0 10.0 0.0 O—ONOOE 109 (X.1.5, Y.O, Z.1.5) •—«)NOOE 104 (X.0.32, Y.0, Z>0.32) • DNODE 338 (X.0.79, Y.1.2, Z-0.79) O ONOOE 26 (X-0.55, Y.1.5, Zrf>.55) O—ONODE 644 (X*), Y.I.5, Z.0.66) -<NOOE 222 (X.0.87, Y.1.2, Z=0) |A—ANODE 25 (X.0.32, Y.1.5, Z=0.32) - A NOPE 6 (X.1.5, Y.1.6, Z=K))

concrete mot temperature . 25 "C surrounding temperature . 15 °C cement content > 225 kg/m convection . 0.025 kW/(m') 'K 0.0 60.0 50.0 100.0 Time in hours

An eighth of a hollow cube, H . 3 m, d U 0 . 4 H . 1.2 m, no Isolation

(a) 150.0 60.0 50.0 40.0 30.0 50.0 100.0 Time In hours

An eighth of a hollow cube, H • 3 m, d1 = 0.5H • 1.5 m, no Isolation

(b) 150.0 20.0 10.0 h 0.0 O—ONODE 109 (X.1.5, Y.0, Z.1.5) •—ANODE 105 (X.0.53, Y.0, Z*>53) D—ONOOE 351 (X.1.01, Y.1.2. Z.1.01) O—ONOOE 27 (X.0.77, Y.1.5, Zrf>.77) <—ONODE 646 (X.0, Y.1.5. Z*>.94) * 41 NODE 235 (X.1.12, Y.1.2, Z*)) A—ANODE 26 (X.0.53. Y.1.5, Z.0.53) NODE 6 (X.1.5, Y.1.5, Z*0)

concrete mix temperature . 25'C surrounding temperature . 15 °C cement content . 225 kg/m convection . 0.025 kW/(m') *K (c) 50.0 100.0 150.0 Time in hours

V;5 0,45 . , y 104 - 0.32 0 0.32 109 - . 1.5 0 1.5 222 - 0.87 1.2 0 338 - 0.79 1.2 0.79 644 - 0 1.5 0.66

Figure 12. Locations of the nodes for the eighth of the 3 m hollow cube where d! = 0.9 m.

5.4 Influence of specimen size

In Figure 13a, b, and c the simulated temperature-time plots are shown for three hollow cube geometries namely 2, 3 and 4 m. For the nodal locations see Figure 14, 9, and 15. As can be seen in Figure 13a-c, the larger the size of the element, the higher the temperature produced. Moreover, the largest hollow cube geometry (4 m) maintained the thermal gradient for a longer period than the 2 m hollow cube. In addition, the maximum thermal gradient A T , ^ was 31, 27.5 and 21.5 °C for the 4, 3 and the 2 m hollow cube geometries. In Table 7, the simulated maximum and the minimum temperatures as well as the maximum temperature gradient etc. are shown. It can be noticed that the temperature peaks for the 2 m hollow cube occurred at 12 hours, whereas for the 4 m geometry it was at 23 hours. This is also pronounced in the stress analysis of the same geometry. As can be seen in Figure 28b, 24b, and 29b, the highest Gyy reached was at 10 hours for the 2 m hollow cube and at 20 hours for the other.

5.5 Influence of isolation

The temperature gradients can also be reduced by applying isolation to the concrete surfaces of the breakwater elements after casting. In order to simulate the effect of isolation, the four

convecting surfaces are assigned a very low convection. In the present case, it was 0.00068 kW/(m2)°K, equal to the convection when a 5 mm polyurethane isolation material is present against the surfaces. As shown in Table 1, the 3 m hollow cube element is used in these simulations with two cement contents (225 and 325 kg/m3), three initial concrete mix-temperatures (5, 15 and 25 °C) and 15 °C ambient temperature.

60.0 50.0 I 40.0 S « 8 30.0 20.0 10.0

Temperature versus Time

An eighth of a hollow cube, H = 2m,d1= 0.4H - 0.8 m, no isolation |0—ONODE 90 (X=1, Y=0, Z=1) NODE 86 <X=0.20, Y=0, Z=0.2B) O • NODE 300 (X«0.64, Y=0.6, Zrf>.64) |0 ONOOE 22 (X-0.46. Y.t, Z-0.46) < ONODE 632 (Xrfl, Y . l , Z-0.55) NODE 197 (Xrf.7, Y.0.6. Z«0) A—ANODE 21 (X-0.28, Y«1, Z-0.28) NODE5(X=1,Y«1,Z-0) concrete mix temperature - 25'C surrounding temperature - 15 eC cement content • 225 kg/m1 convection - 0.025 kW/(m") *K 50.0 100.0 Time In hours 150.0 (a) 60.0

Temperature versus Time

An eighth of a hollow cube, H > 3 m, d U 0.4H » \ 2 m, no Isolation

50.0 100.0

Time In hours

150.0

00

60.0 50.0 40.0 30.0 20.0 10.0 0.0

Temperature versus Time

An eighth ot a hollow cube, H • 4 m, d1= 0.4H « 1.6 m, no Isolation O ONODE 155 (X.2, Yrf), 2=2) NODE 150 (X=0.56, Y=0. Z=0.56) •—ONODE 630 (X=1.14, Y*1.0 Z=1.14) O—ONODE 36 (X*).65, Y=2.0. Z=0.B5) O—<NODE 1385 fX=0. Y.2.0, Z=1.04) NODE 392 (Xcl.52, Y=1.0 Z«0) A — A NODE 37 (X.0.56, Y-2.0, Z=0.56)

NODE 6 Ofa2,0, Y=2,0, Z=0)

surrounding temperature • 15 "C cement content• 225 kg/m1

convection • 0.025 kW/(m') *K

0.0 50.0 100.0 150.0

Figure 13c. Simulated temperature-time histories for specimen size: 4 m.

df-O.+H-O.Bm H-2m NODE X y z 5 - 1.0 1.0 0 21 - 0.28 1.0 0.28 22 - 0.46 1.0 0.46 86 - 0.28 0 0.28 90 - 1.0 0 1.0 197 - 0.70 0.6 0 300 - 0.64 0.6 0.64 532 - 0 1.0 0.55 H-2m d,-0.4H-0.Bm

Figure 14. Locations of the nodes for the eighth of the 2 m hollow cube, d,/H = 0.4.

d f - 0 . 4 H - 1 . 6 m H - 4 m 1385 NODE X y z 6 = 2.0 2.0 0 37 - 0.56 2.0 0.56 38 - 0.85 2.0 0.85 150 - 0.56 0 0.56 155 - 2.0 0 2.0 392 - 1.52 1.0 0 630 - 1.14 1.0 1.14 1385 - 0 2.0 1.04 H=4m d.-0.4H-1.6m

Table 7: Temperature difference AT (=1^ - T ^ ) between the external and the internal nodes of the three hollow cubes (2, 3 and 4 m), all having the constant ratio d / H = 0.4

dx/H = 0.4, Surrounding & Mix-temperatures = 15, 25 °C,

Cement content = 225 kg/m3

Hollow cube size 2.0 m 3.0 m 4.0 m

Maximum Tem-perature T ^ ( ° Q 38.1 43.7 46.7 Time t to reach T , ^ (hours) 12 17 23 T ^ at t (°C) 16.6 16.2 15.7 AT = T - T . WMX max min 21.5 27.5 31.0

Figure 16a-c shows the simulated temperature histories of the 3 m hollow cube for 5, 15 and 25 °C initial concrete mix-temperatures with 225 kg/m3 cement content. Similarly, Figure 17a-c shows the temperature histories for the same initial temperature, but a higher 17a-cement 17a-content (325 kg/m3). The relevant nodal locations are shown in Fig. 9. As can be seen in Figures 16 and 17, the isolation prevents the external nodes from cooling at a faster rate. Evidently, for the 5 °C initial mix-temperature, the difference between the core and the surface temperatures up till almost 50 hours of casting is minimal(see Figure 16a and 17a). Furthermore, irrespective of the cement content, the temperature gradient is the lowest for a lower initial concrete mix-temperature (this was the case when there were no isolation too). For the analyses with isolation, the temperature gradient obtained was the minimum, and for the no isolation it was the maximum. Hence, the combination of high initial mix temperature as well as cement content led to a higher thermal gradient when compared to the case where a low initial mix temperature and or low cement content was chosen. Note that when the isolation would be removed after two- or three-days, still cooling of the outer layers of the breakwater element would occur. However, at that time the tensile strength has developed sufficiently to avoid cracking.

to K ' 3 °cra' 5"° Is?. to 3 toe . • & 3 o "3 2 a> 3 r> c» ?; Ö c «-!• l-l O ? O p. S 3 cr & 2 . . <T> j 0 i cr i-> o

1

È

Temperatura In degrees Celsius Temperature in degrees Celsius O b cn O b cn o b

ran

Temperature In degrees Celsius O b 3 a

s

mm

Jfifi!

{!

I

60.0

50.0

An eighth of a hollow cube, H = 3 m, (11= 0.4H full isolation

40.0 30.0 20.0 10.0 0.0 O ONODE 79 (X.1.5. Y-O. Z-1.5) -•NODE 49 (X=0.424, Y-O, Z-0.424) •—GNOOE 634 (X=0.854, Y-1.2, 2=0.854) O ONOOE 1025 (X-0.639, Y=1.5, Z=0.639) O ONODE 956 (X=0, Y=1.5, Z=0.7B) —«I NODE 926 (X=0.96, Y . 1.2, Z=0) A — A NODE 1027 (X=0.424, Y=1.5. Z-0.424) -ANODE 1151 (X=1.5,Y=1J, Z=0)

concrete mix temperature • 5 *C surrounding temperature • 15 °C cement content - 325 kg/m1 convection - e^-IO^kW/lm^'K (a) o.o 60.0 50.0 100.0 Time in hours

An eighth of a hollow cube, H - 3 m, d l - 0.4H full Isolation

150.0 0.0 50.0 100.0 150.0 (b) 60.0 50.0 40.0 Time In hours

An eighth ol a hollow cube, H - 3 m, d l - 0.4H lull Isolation

e i 30.0 20.0 10.0 0.0 |0—ONODE 79 (X=1.5, Y=0, Z-1.5) NODE 49 (X=0.424, Y=0, Z=0.424) IB E1NODE 834 (X=0.654, Y=1.2, Z=0.B54) |0—ONODE 1025 (X-0.639, Y=1.5, Z-0.639) O—ONODE 956 (X=0, Y=1.5. Z=0.78) 4 4 NODE 926 (X-0.96, Y-1.2. Z-0) A—ANODE 1027 (X=0.424, Y-1.5, Z=0.424) NODE 1151 (X-1.5, Y-1.5, Z=0) concrete mix temperature - 25 °C

surrounding temperature • 15 °C cement content - 325 kg/m1

convection . 6.8,10"'kW/(m')''K

(c)

0.0 50.0 100.0 150.0

6.0 Temperature histories of Tetrapod

The tetrapod geometries chosen for the simulations are shown in Table 3. For the analyses two different cement contents (225 and 325 kg/m3), and ambient and mix-temperatures of 15 °C and 25 °C respectively were assumed.

6.1 Influence of cement content

Figure 18 a and b shows the temperature simulations for the cement contents of 225 and 325 kg/m3 of the 3.11 m tetrapod. The corresponding nodes are shown in Figure 19a. The simulated temperature-time histories of the tetrapod also show the usual trend: in which the core (node 655) reached the highest temperature and the extreme edge (node 221) immediately dropped from 25 °C, thereby almost reaching the ambient temperature (15 °C). As shown before the simulation with the high cement content produced the highest temperature gradient A T ^ = 39.1 °C for the 3.11 m tetrapod and 43 °C for the 4.15 m tetrapod. The choice of a higher cement content increased the temperature gradient by 10 degrees, see Table 8.

6.2 Influence of size

The influence of the size of the tetrapod on the simulated temperature histories can be seen by comparing Figure 18a and 20a, corresponding to the 3.11 and 4.15 m tetrapods with a 225 kg/m3 cement content. (Also see Figure 18b and 20b, for the 325 kg/m3 cement content). Figure 19 shows the locations of the nodes for the tetrapods. The choice of a large tetrapod, increased the gradient of the temperature peak by 2 degrees irrespective of the cement content, see Table 8. However, a shift in the occurrence of the simulated temperature peak, in this case by 10 hours, is seen for the 4.15 m tetrapod. Moreover, the simulated rate of decrease of the temperature gradient for the 4.15 m tetrapod was lower as compared to the 3.11 m tetrapod. In other words, the 4.15 m tetrapod maintained a high thermal gradient for a longer time span as compared the 3.11 m tetrapod.

60.0

Temperature versus Time An eighth of a tetrapod, H = 3.11m, D=1.5m, no isolation

(a)

50.0 100.0 150.0 Time In hours

Figure 18. Simulated temperature-time histories of the tetrapod H = 3.11 m and D =1.5 m: cement content 225 kg / m3 (a) and 325 kg/m3 (b).

0.627 (a) 0.470 (axial) (radial) Z (tangential) to (b)

Figure 19. Locations of nodes for the tetrapod: H = 3.11 m, D = 1.5 m (a) and H = 4.15 m, D = 2.0 m (b).

Table 8: The maximum temperature difference A T ^ (= T ^ - T ^ J between the external and internal nodes of two tetrapods having the height H (3.11 and 4.15 m with D = 1.5 and 2.0 m).

Surrounding and Mix-temperature = 15, 25 °C Tetrapod size H (m) D ( m ) 3.11 1.5 4.15 2.0 Cement content kg/m3 225 325 225 325 Maximum tempera-ture T , ^ (°C) 47.4 56.5 49.1 59.9 Time t to reach T , ^ (hours) 21 23 29 32 Tm l n at t (°C) 16.8 17.4 16.7 16.9 AT = T - T max D U X A nun 30.5 39.1 32.3 43.0

60.0

Temperature versus Time

An eighth ol a tetrapod, H •= 4.15m, D=2.0m, no Isolation

50.0 40.0 8" 30.0 20.0 NODE 1 (X-0, Y-0.656) |0 ONODE 11 (X-0, Y.1.614) \&—ANODE 21 (X.0, Y-2.672) P—DNODE 1006 (X-0, Y-O) < ONOOE 106 (X-O.499, Y-0.5S6)

NODE 116 (X-0.406, Y-1.614) * *NOOE 316 (X.1.0, Y-0.556) O ONOOE 336 (X-0.627, Y.2.672) 10.0 0.0 0.0

concrete mix temperature • 25 °C Gurrounding temperature • 15 °C cement content • 225 kg/m* convection - 0.025 kW/(m*) *K (a) 50.0 100.0 150.0 Time In hours

Figure 20. Simulated temperature-time histories of the tetrapod H = 4.15 m and D = 2.0 m: cement content 225 kg/m (a) and 325 kg/m (b).

7.0 Stress analysis of hollow cubes

As shown in Table 2, three hollow cube geometries with heights 2, 3 and 4 m were analyzed. For the 2 and 4 m hollow cubes, a constant ratio of d,/H = 0.4 was used, whereas for the 3 m hollow cube, three d / H ratios (viz. 0.3, 0.4 and 0.5) were adopted.

The chosen time steps for the stress analyses were 1, 5, 10, 20, 25, 30, 50, 100 and 150. The following analyses were carried out at each of the time steps. First, the potential flow analysis was performed with as output the nodal temperatures. For example, for the 3 m hollow cube (d,/H = 0.4) 1152 nodal temperatures corresponding to 264 CHX60 solid elements were calculated. Second, the temperature output resulting from the first step was corrected for the middle plane nodes of the CHX60 elements (which showed twice as high temperature values as compared to the other nodes as discussed in the Appendix). Third, the corrected temperature 'load' was used as input for the stress analysis.

The support conditions assumed for the hollow cube are shown in Figure 21. Every node on the planes X Z (bottom), Y Z and X Y is fixed along the directions Y, X and Z respectively. These support conditions were assigned with the assumption that these plane sections should remain plane at any moment during the loading history.

For the six cases analyzed, stresses are plotted in terms of a ^ , Cy-, and c-^ along the radial direction. For example, the top view of the eighth of the 2 m hollow cube is shown in Figure 22. The elements a, b, c and d, adjoining the radial line at 0 = 45° are chosen (0 = 0 along the axis Z, and the counter clock wise direction is positive). This geometry is schematized into five layers, and the actual location of the elements in the Y direction is between Y = 0.4 and 0.6 m, see Figure 22b. Out of the 27 integration points only three are chosen for each element. The integration points for elements a, b, c and d fall along the radial line such that 6 is between 39-41°. For the rest of the geometries the top view of the elements is the same except for the Y coordinates of the elements.

In all the stress analyses, the Young's modulus and the Poisson's ratio were assigned constant values of 17.5 GPa and 0.2. The change in the above values due to progressive hydration were not considered.

c t t -(c fe-lt: 0.6m i i l _ L l I - - l _ L I I r • r L . _ 1 _ I. Side view

Figure 21. Support conditions assigned for the stress analyses of hollow cubes.

(a) (b)

H=2m

d,=0.4H--0.8m

7.1 Influence of hole diameter

The chosen diameters for the 3 m hollow cube are 0.9, 1.2 and 1.5 m respectively. The three stress values, G^, Gyy and corresponding to three diameters are shown in Figures 23, 24 and 25.

These simulated stress histories confirm problems often encountered in mass concrete. The occurrence of a temperature gradient between the interior and the exterior of a non-insulated concrete mass is shown in the previous section on temperature histories. As shown before, the interior became hot whilst the surface lost heat to the environment. The interior is restrained from f u l l thermal expansion, and as a consequence a compressive stress is induced. Because no external reaction forces may occur, these compressive stresses must be balanced by tensile stresses in other parts of the breakwater element. The above phenomenon is simulated in Figure 23. As can be seen in Figure 23a, b, and c, the stress distribution is such that, tensile stress prevails near both edges, that is, at the surface of the hole and the far face of the cube, whereas compressive stress dominates in the interior. A gradual increase in the tensile stress occurred from the time of casting up-till an age of 20 hours and there after a decrease is seen. Out of the three stress components, Gyy reached the highest tensile stress of 3.5 MPa at 20 hours of casting. Also, the compressive stress in the interior followed similar suit. The above observation is for the heating (or hydration) phase.

As the concrete starts to cool and contract, the tensile stress near the edges is relieved. In Figures 23, the decrease in tensile stress takes place after 20 hours. Similarly compressive stress in the interior also decreases. Since the interior wants to contract more than the exterior, the strain in the former is restrained and a tensile stress is induced, with the balancing stress in the exterior (Neville & Brooks(1987)). At 150 hours after casting the simulated stresses Gxx, O Y Y , and 0 - 3 reach the zero axis in the entire cross-section both the exterior and the interior. This means compression in the exterior and tension in the interior is induced as to balance the opposite trend that prevailed at an age of around 20 hours. However, it would not be completely balanced due to the nonlinear effect of the progressive increase in the Young's modulus. Thus, the tensile stress in the interior is induced due to internal restraint caused by

cooling down of the hollow cube. As a consequence, crack would appear in the interior i f the net effect of the compression and the tension exceeds the tensile strength of the material.

o » ( N / m m ' ) H - 3 m , d,=0.3H - 0.9m a„ (N/rnm- ) H - 3m, d , - 0 . 3 H - 0.9m

0 ) (b)

°u ( N / m m1) H - 3 m , d , - 0 . 3 H - 0.9m

element cross section ot Y - 0 . 5 6 6 m from the bottom

1.5 H

Radial distance r in meters

Figure 23. Simulated a^-r (a), G^-r (b), G-z-r (c) plots of the 3 m hollow cube, d! = 0.9 m, cement content = 225 kg/m3.

o » ( N / m m1) H - 3 m , d , - 0 . 4 H = 1.2m 3.5 -\ 2.5 1.5 H 0.5 - 0 . 5 -1.5 - 2 . 5

element cross section at Y - U . 6 3 4 m from the bottom

dMone» r «tan I 3 9 -0.0 25 hrs 30 hrs 50 hrs 150 hrs 1.5 2.0 2.5 3.0 Radiol distance r in meters

O \ Y ( N / m m1) H •= 3 m , d , - 0 . 4 H - 1.2m 3.5 H 2.5 1.5 0.5 - 0 . 5 -1.5 - 2 . 5

element cross section at Y — U . b J 4 m from the bottom

dbtanc- r aton- f - 39-41 **g-MS 0.0 30 hrs 50 hrs 150 hrs 1.5 Radial distance ~2Ü 3.0 • in meters (a) (b) "a ( N / m m1) H - 3 m , d , - 0 . 4 H - 1.2m 3.5 2.5 1.5 0.5 - 0 . 5 -1.5 - 2 . 5 element c r o s s section at Y - U . 6 J 4 m from the bottom

d-tww* r «fene I - 39-41 fegnMi o 1 hr a 5 hrs 4 10 hrs 0 20 hrs 30 hrs 50 hrs 150 hrs 0.0 0.5 U ) K 5 2X> ï!s 3.0

Radial distance r in meters

Figure 24. Simulated c^-r (a), a-^-r (b), a^-r (c) plots of the 3 m hollow cube, d, = 1.2 m, cement content = 225 kg/m3.

o-w ( N / m m ' ) H = 3m, d,=0.5H = 1.5m 3.5 2.5 1.5 0.5 - 0 . 5 -\ - 1 . 5 - 2 . 5

element cross section ot Y - U . s b b m from the bottom

dttonc. r «Ion- f - 40-42 4 q n « o 1 hr o 5 hrs 4 10 hrs 0 20 hrs 25 hrs 30 hrs 50 hrs 150 hrs O.Ö 0.5 1.0 1.5 2.0 2.5 3.0

Radiol distance r in meters

3.5 2.5 1.5 0.5 - 0 . 5 -1.5 H - 2 . 5 art ( N / m m1) H 3 m , d , 0 . 5 H

-element cross section at Y=U.566m from the bottom

««•tont. r «tong ( - 40-42 A - u 1.5m o 1 hr • 5 hrs 4 10 hrs 0 20 hrs 0.0 25 hrs 30 hrs 50 hrs 150 hrs 0.5 1.0 1.5 2.0 2.5 3.0

Radial distance r in meters

00

0>)

ozz ( N / m m1) H 3m, d , 0 . 5 H

-element cross section at Y=U.5ö6m 1.5m 3.5 2.5 1.5 H 0.5 H - 0 . 5 H - 1 . 5 (C) - 2 . 5

from the bottom

d-UMK4 r «lens I - 40-42 d t - M 0.0 o 1 hr s 5 hrs 4 10 hrs 0 20 hrs 1 0.5 25 hrs 30 hrs 50 hrs 150 hrs 1.0 _{1.5 2.0} 1 1 Radial distance 2.5 3.0 r in meters

Figure 25. Simulated o ^ - r (a), c ^ - r (b), oH- r (c) plots of the 3 m hollow cube, d, = 1.5 m, cement content = 225 kg/m3.

Figures 24 and 25 also show the stress distributions for the other two hole diameters. In every respect the trend is the same as in Figures 23. From the Oyy - r plots in Figure 23b, 24b and 25b, the tensile stress peak values (at 20 hours) were obtained for cubes with three different hole diameters, and are plotted in Figure 26. Figure 26 shows the influence of the hole diameter on the tensile stress peak. As the hole diameter increases, the tensile stress peak decreases. When the hole diameter is increased, the concrete mass is decreased and less heat is produced due to the heat of hydration, see Fig. 11 and Table 6. Moreover, hollow cube with the largest hole diameter has a relatively large surface area, and thus convect faster than the hollow cube with small hole diameters. Consequently a low temperature gradient is produced. For example the temperature gradients at t=50 hours for the hole diameters 1.5,1.2 and 0.9 are 13.9, 17.4 and 20.5 °C/hours (Fig. 11). Hence, the lowest stress is found for the hollow cube with the largest hole diameter.

For a better illustration of the stress state in the hollow cube, the maximum principal stress contours were plotted in Figures 27 at 10 hours of casting. The examples are all for a 3 m hollow cube with a hole diameter d, = 1.2 m (i.e., d,/H = 0.4).

4.0 rjYY ( N / m m2) H = 3 m 3.0 H 2.0 hollowcube with o c e m e n t c o n t e n t = 2 2 5 k g / r r \ • c e m e n t c o n t e n t = 3 2 5 k g / m 1.0 1 r 0.0 0.5 1.0 1.5 2.0

Hole diameter in meters

M O D E L : C32251 L C 2 : Linear load nr 2 PRINC STRESS PMAX

R E S U L T S W E R E C A L C U L A T E D

M A X ---.309R7 M I N = -.U1E7

7.2 Influence of size

The simulated stresses (Oxx, dyy, for the three hollow cubes of 2, 3 and 4 m are shown in Figure 28a-c, 24a-c and 29a-c. In the heating phase tensile stress occurred near the edges, i.e., both at the hole surface and near the exterior face of the cube, whereas compressive stress dominated in the interior. From the cY Y - r plots of the three geometries, it can be seen that the stress peaks occurred at 10 hours for the 2 m hollow cube and at 20 hours for the 3 and 4 m hollow cubes. This trend was also seen in temperature histories where an early temperature peak was observed for the small geometry, see Figure 13a.

o » ( N / m m * ) H - 2 m , d , - 0 . 4 H element cross section at Y=U.bm

from the bottom .

•long I - 39-41 -teg-Ms 0.8m 1.5 H 0.5 H - 0 . 5 H - 1 . 5 H 0.0 0.5 1.0 O 1 hr * • 5 hrs & 10 hrs • V 20 hrs • 1 1.5 i 2.0 25 hrs 30 hrs 50 hrs 150 hrs 2.5 3.0 Radial distance r in meters

(a) ay, ( N / m m ' ) H = 2 m , d,=0.4H = 0.8m 3.5 2.5 1.5 0.5 - 0 . 5 -1.5 - 2 . 5 0.0

element cross section at Y = 0 . 6 m from the bottom

dvtonc* r «tong f - 39-41 tftgrMt O 1 hr 25 hrs O s hrs 30 hrs 10 hrs a 50 hrs 20 hrs _• 150 hrs 1 0.5 I 1.0 1 1.5 2.0 Radial distance 2.5 3.0 in meters (b) on ( N / m m ' ) H - 2 m , d , - 0 . 4 H - 0.8m 3.5 H 2.5 H 1.5 0.5 - 0 . 5 H -1.5 H (c) - 2 . 5 0.0

element cross section at Y - 0 . 6 m from the bottom

Q 1 hr 25 hrs • 5 hrs 30 hrs A 10 hrs • 50 hrs 0 20 hrs * 150 hrs I 0.5 I 1.0 1 1.5 2.0 2.5 3.0 Radial distance r in meters

3.5 2.5 1.5 0.5 - 0 . 5 -1.5 - 2 . 5 ( N / m m2) H 4 m . d , 0 . 4 H

-element cross section at Y = U . 9 / 2 m from the bottom

•bbjnc* r along I - 41-42 dograaa 1.6m try, ( N / m m1) H - 4 m , d,=0.4H - 1.6m 0.0 O 1 hr • 5 hrs 10 hrs 0 20 hrs 1 0.5 2.0 2.5 3.0 Radial distance r in meters

3.5 H 2.5 1.5 H 0.5 - 0 . 5 H - 1 . 5 - 2 . 5 0.0

element cross section at Y = 0 . 9 7 2 m from the bottom

dtatwKO r along I - 41-42 dagroai

1.5 2.0 Radial distance 2.5 3.0 in meters (a) _(b) r /n ( N / m m1) H - 4m, d , - 0 . 4 H - 1.6m 3.5 -\ 2.5 1.5 H 0.5 - 0 . 5 -1.5 H (c) - 2 . 5 0.0

element cross section at Y - 0 . 9 7 2 m from the bottom

dlitonca r along • — 41-42 <

1.5 2.0 2.5 3.0 Radiol distance r in meters

CYY ( N / m m2) , a f t e r 20 hours 4.0 -i 0 3.0 -\ 2.0 -\ i.o H 0.0 O • d d d d 0.3H, cc = 2 2 5 k g / m3 0.4H, cc = 2 2 5 k g / m3 0.5H, cc = 2 2 5 k g / m3 0.4H, cc = 3 2 5 k g / m o.o 1.0 2.0 3.0 4.0 5.0 H ( m )

Figure 30. Simulated Cyy peak (at 20 hours) versus hollow cube size.

For the three geometries, the stress peaks ( 0 ^ obtained from Figure 28b, 24b and 29b, at the 20th hour, were plotted with respect to the specimen size H, see Figure 30. In this analysis the d,/H ratio was constant. Hence, the largest specimen consists of more concrete mass, and in turn produces more heat during the hydration process of cement. As a consequence, the temperature gradient between the interior and the exterior of the non-insulated 4 m hollow cube is higher compared to the 2 m hollow cube, see Figures 13. Therefore, the stress peak of 4 m hollow cube should be the highest. This is shown in Figure 30, where an increase in the specimen size led to higher tensile stress peak oYY. In the same

figure, the stress peaks corresponding to the 3 m hollow cube with three different hole diameters are plotted. As observed before, the hollow cube with the smallest and the largest diameters produced the highest and the lowest stress peaks.

73 Influence of cement content

Figures 24 and 31 show the simulated stresses for the two cement contents 225 and 325 kg/m3, for a 3 m hollow cube with a hole diameter d,= 1.2 m (d,/H = 0.4). The choice of a higher cement content leads to a high heat evolution and subsequent rise in temperature. Therefore, the rate of heat evolution and the total heat produced are higher for the above case

and result in high tensile stress. This is shown in Figures 24 and 31, where the simulation with the highest cement content produced the highest stress peak. Also in Figure 30, the tensile stress peak (at 20 hours) for the simulation with 325 kg/m3 cement content is plotted. Noticeably, the 3 m hollow cube with high cement content produced the same stress peak as of the 4 m hollow cube with low cement content.

<r„ ( N / m m ' ) H - 3 m , d , - 0 . 4 H - 1.2m 3.5

2.5

0.5 -\

element cross section at Y=0.B66n from the bottom

dstonco r along f - 39-41 dograoi comont contant - 323 kg/m ffyv ( N / m m ' ) H - 3 m , d , - 0 . 4 H - 1.2m 25 hrs 30 hrs 50 hrs 150 hrs 1.5 2.0 2.5 3.0 Radiol distance r in meters

3.5 H 2.5 H

0.5

1.5 2.0 2.5 3.0 Radial distance r in meters

00

element cross section at Y - 0 . 8 6 6 m from the bottom

•Manco r along f - 39-41 dograoa oomont contant - 323 kg/m 25 hrs 30 hrs 50 hrs 150 hrs 1.5 2.0 2^> 3.0 Radiol distance r in meters

Figure 31. Simulated c ^ - r (a), oY Y- r (b), oa- r (c) plots of the 3 m hollow cube, d, = 1.2 m and the cement content = 325 kg/m3.

4.0

CTYY ( N / m m2) , after 20 hours

- 2 . 0 -1.0 H 0.0 2.0 4 3.0 H 1.0 H - 4 . 0 - 3 . 0 H o H = 2 m , d , = 0 . 8 m , c c - 2 2 5 k g / m3 • H = 3 m . d , - 0.9m, c c « 2 2 5 k g / m3 4 H - 3 m . d , - 1.2m, c c - 2 2 5 k g / m3 * H - 3 m , d , = 1.2m, c c - 3 2 5 k g / m3 a H - 3 m , d , = 1.5m, c c - 225 k g / m . H - 4 m , d , - 1.6m, c c - 2 2 5 k g / m3 0.0 1.0 2.0 3.0

Radial distance r in meters

Figure 32. Simulated a ^ - r plot at 20 hours, for all three hollow cubes sizes and two cement contents.

Figure 32 shows the combined Cyy distribution at 20 hours, for all the six hollow cube simulations. The maximum principal stresses from the above six simulations (at 20 hours) are shown in Table 9 as well. The locations and the directions of the three principal stress components r j „ o2, and c3 at 20 hours are shown in Table 9 as well. From the components of a2, the 2 m hollow cube produced the lowest principal tensile stress and the 4 m hollow cube the highest for the same cement content. However, the simulation with 325 kg/m3 cement content for the 3 m hollow cube produced the highest a2. Figure 33a and b shows the simulated principal stress contours for the 3 and 4 m hollow cubes with 325 and 225 kg/m3 cement content.

As discussed before, stress analyses were carried out for two tetrapod geometries. The ambient and the concrete mix temperatures were 15 and 25 °C and the two cement contents were 225 and 325 kg/m3.

Y Z J X (a) • .300E7 • .243E7 .186E7 .129E7 .714E6 .I43E6 -.429E6 -.100E7 (b)

Table 9: Locations and directions of the simulated principal stresses at 20 hours for the three hollow cube geometries; the cement content is 225 kg/m3 and mix and surrounding temperatures are 25 and 15 °C.

H d, X Y Z direction"

° 2 direction"

direc-m m m m m MPa MPa MPa tion"

2' 0.8 .245 .423 .347 0.99 .79, 0.-.61 2.06 0, 1, .01 0.12 .61.-.01..79 3 0.9 .279 .566 .394 1.64 .80, 0, -.59 3.47 -.01, 1.-.01 0.24 .59..01..8 3 1.2 .362 .634 .514 1.73 .8, 0.-.6 3.19 0, 1,0 0.15 .6, 0, .8 3* 1.2 .362 .634 .514 2.22 .8, 0,-.6 4.05 0, 1,0 0.20 .6, 0, .8 3 1.5 .474 .566 .624 1.60 .77..08.-.63 2.54 -.07, 1..04 0.15 .63..01..77 4 1.6 .557 .778 .633 2.24 .78, 0.-.63 3.81 .01,1,0 0.23 .63.-.01..78

* principal stress results are obtained after 10 hours

* simulated results correspond to 325 kg/m3 cement content

* X, Y, Z directions in DIANA is defined as X : 1.0 0.0 0.0

Y : 0.0 1.0 0.0 Z : 0.0 0.0 1.0

A user defined axis that makes 45° with the X axis is: 0.5 0.5 0.0

Considering the axi-symmetry of the tetrapod, only an eighth of the element is analyzed. In the analysis the axes X , Y and Z correspond to the radial, the axial and the tangential direc-tions. The time intervals for the analyses were 1, 5, 10, 20, 25, 30, 50, 100, 150 and 300 hours. The procedure was similar to that in the stress analyses of the hollow cubes. First, the flow stress analysis is performed and the output is the nodal temperatures. Subsequently, the nodal temperatures are input for the stress flow analysis. The elements that are available for

the potential flow and the stress analysis, CQ16A and CT12A, gave no undesirable temperature rise in certain nodes as was observed in the CHX60 elements used in the hollow cube analysis.

8.1 Influence of size

The stress values in the quadrilateral elements along line AB (3.11 m tetrapod) and CD (4.15 m tetrapod) (see Fig. 19) are representative for the stress distribution in the tetrapod. In Figure 34a-c and 35a-c, the simulated radial (a„), axial (o^) and tangential ( a ^ plots are shown for the 3.11 m and 4.15 m tetrapods respectively. The cement content used was 225 kg/m3. The X axis represents the radial direction and X = 0 corresponds to the centre of the tetrapod, whereas X = 0.75 and 1.0 correspond to the edge of the two tetrapods respectively. A l l three stress components showed an increase in compression in the interior during the heating phase (0 to 30 hours), whereas at the surface, only the axial and the tangential components showed an increase in tension during the heating phase(0 to 30 hours), see Figures 34 and 35. In the cooling phase, that is after 30 hours, the compression in the core as well as the tension along the surface were relieved. For example, for the 3.11 m tetrapod, the axial compressive peak was -4.2 MPa in the interior, and the axial tensile peak was 5.3 MPa at the surface at 20 hours (see Table 10). A t 300 hours, the stresses in the interior and at the surface were -1.02 and 1.48 MPa respectively. The same arguments advocated in the hollow cube analyses are also valid here.

The axial(ayY) and the tangential(az z) peak tensile stresses were 5.35 and 2.75 MPa at 20 hours for the 3.11 m tetrapod and 6.0 and 3.25 MPa for the 4.1 m tetrapod. It appears that the difference in size has less influence on the axial tensile peak stress values, and differ by about 1 MPa only. Figure 36a and b show the principal stress contours of the two tetrapods at 25 hours for the cement content of 225 kg/m3. In Figure 37a and b, the locations of the maximum and the minimum principal stresses for the two tetrapods are shown. As can be seen in Figures 36 and 37, the tensile peak stress occurs at the surface of the leg of the tetrapod which has the largest diameter. Thus, the minimum stress point is located in the core of the tetrapod.

(c) 8.0 6.0 4.0 2.0 0.0 °zz ( N / m m1) Tetrapod H - 3.1 l m , d i - 1 . 5 m - 2 . 0 H - 4 . 0 H - 6 . 0

X - radial distance along line AB cement content = 2 2 5 k g / m 0.0 0.3 1 hr • 5 hrs 10 hrs 0 20 hrs i 0.5 0.8 30 hrs 50 hrs 100 hrs 1.0 X (m)

Figure 34. Simulated Gxx-X (a), Cyy-X (b), o ^ - X (c) plots of the 3.11 m tetrapod (D=1.5 m) and the cement content = 225 kg/m3.

Figure 35. Simulated an- X (a), o^-X (b), G^-X (c) plots of the 4.15 m tetrapod (D=2.0 m) and the cement content = 225 kg/m3.

MAX--.530E7 MAX ••- .623E7

Figure 36. Simulated principal stress contours (at 25 hours) of the 3.11 m (a) and 4.15 m (b) tetrapod, and the cement content = 225 kg/m3.

MAX

(a) (b)

Figure 37. The locations of the maximum and the minimum principal stresses of the 3.11 m (a) and 4.15 m (b) tetrapods.

8.2 Influence of cement content

The two tetrapod geometries were also analyzed with 325 kg/m3 cement content. In Figure 38a-c and 39a-c, the radial, the axial, and the tangential stresses for the 3.11 and 4.15 m tetrapods are shown. To see the influence of the cement content, compare Figures 34 and 38 for the 3.11 m tetrapod, and Figures 36 and 40 for the 4.15 m tetrapod. Hence, the difference in the axial tensile stress peak in both cases were 1.2 and 1.4 MPa (see Table 10). In Figure 40a and b, the principal stress contours (at 25 hours), for the simulation with the 325 kg/m3 cement content are shown.

Note that the tensile stress values obtained from the tetrapod analyses are higher compared to the hollow cube results.

Table 10. Simulated axial tensile stresses (a^) at 20 hours for the two tetrapod geometries. Tetrapod dime-nsion H = 3.11 m D = 1.5 m H = 4.15 m D = 2.0 m Cement Con-tent (kg/m3) 225 325 225 325 aY Y MPa (ten-sion) 5.3 6.5 6.1 7.5 cY y MPa (com-pression) -4.2 -5.1 -4.3 -5.2

oxx ( N / m m1) Tetropod H - 3 . 1 1 m , d , - 1 . 5 m cy, ( N / m m1) Tetrapod H - 3.11m, d , - 1 . 5 m

Figure 38. Simulated a ^ - X (a), Oyy-X (b), G^-X (c) plots of the 3.11 m tetrapod (D=1.5 m) and the cement content = 325 kg/m3.

Figure 39. Simulated a ^ - X (a), a ^ - X (b), a ^ - X (c) plots of the 4.15 m tetrapod (D=2.0 m) and the cement content = 325 kg/m3.

Figure 40. Simulated principal stress contours (at 25 hours) of the 3.11 m (a) and 4.15 m (b) tetrapod, the cement content = 325 kg/m3.