Restraint Calculation in Concrete Culvert First Casting

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Restraint Calculation in Concrete Culvert

First Casting

Majid Al-Gburi

ISSN 1402-1536 ISBN 978-91-7439-XXX-X (tryckt) ISBN 978-91-7439-882-3 (pdf) Luleå University of Technology 2014

Department of Civil and Environmental and Natural Resources Engineering Division of Division of Structural Engineering

Department of Civil and Environmental and Natural resources Engineering Division of Structural Engineering

Technical Report 2014

Restraint Calculation in Concrete Culvert

First Casting

Majid A. Al-Gburi

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First Casting

Majid A. Al-Gburi

Luleå University of Technology

Department of Civil and Environmental and Natural resources Engineering Division of Structural Engineering

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TABLE OF CONTENTS

1 Introduction 3

2 Derivative the Equation to Estimate Restraint at Early Age Concrete 4

3 Describe the collection data 5

3.1 Network Data Preparation 6

4 Study of Importance Geometry Factors Walls Models

5 Discuss The Results 18

5.1 Effect of Wall Thickness 18

5.2 Effect of Wall Height 21

5.3 Effect of Slab Thickness 21

5.4 Effect of Slab Width 21

6 Discuss The Results Of Restraint In The Roof 28

6.1 Effect of Roof Thickness 40

6.2 Effect of Wall High 40

6.3 Effect of Wall Thickness 40

6.4 Effect of Roof Width 41

6.5 Effect of Slab Thickness 41

7 ANN Model Development For Wall Restraint Prediction 52

7.1 The Design Formula For Wall 52

8 ANN Model Development For Roof Restraint Prediction 60

8.1 The Design Formula For Roof 60

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1. Introduction

Restraint volume changes associated with heat of hydration and shrinkage is one of the mine sources of through cracks at early. Such cracks may cause too early start of corrosion of rebars or penetration of harmful liquids or gases into the concrete body. These situations could result in significantly increased maintenance cost. It is important prevent or control these cracks.

If unrestraint, the concrete in structures expands and contract during early age heating and subsequent cooling process without stresses are induced. In practice, the concrete is nearly always restraint to some degree. Both externally by adjoining structures or internally by deferent temperatures and moisture in the component of structure itself, Emborg (1994). ACI (2002) proposed equation to estimate the degree of restraint for the new concrete casting an old supports. Bamforth, and et al (2010), proposed for typical ordinary structures such as wall-on-slab, and culvert. The values of restraint are typically in the range from 0.3 to 0.7, and it is tempting to assume an average value of 0.5. In JSCE (2010), suggested equations and diagrams to estimate the external restraint in slabs and walls. It discussed these studies the restraint in the member’s young casting resulting from adjacent buildings. It refers these studies in general to the importance influence of dimensional geometry on restraint value. Some of these studies gave a general restraint percent, and reality tells that every structure had a different restraint. It is dealing with some geometrical parameter influence on the restraint and omitted other parameters. In the present study, focus on all the geometric factors affecting on the restraint in the walls and roof’s new casting each one alone. Study the percentage effect of each parameter; study the effect of each parameter in raising or lowering value of the restraint.

Method calculating restraint is classified in two types: the first, software accurate, expensive and require expertise to use. The other type is too simplified loses the accuracy. The present study is addressed to combine accuracy and simplicity by use the relations and tables are programmed easily as an excel sheet, quick application is easy to use for calculate the restraint in the walls and roofs with a high-accuracy reaches up to 99%.

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2. Derivative the Equation to Estimate Restraint at Early Age Concrete

A typical structure of ANN consists of a number of processing elements (PE), or nodes, that are usually arranged in layers: an input layer, an output layer and one or more hidden layers as shown in Figure 1.

Each PE in a specific layer is fully or partially joined to many other PE via weighted

connections. The input from each PE in the previous layer (xi) is multiplied by an adjustable

connection weight (wji). At each PE, the weighted input signals are summed and a threshold

value or bias (

ɵ

j) is added. This combined input (Ij) is then passed through a nonlinear transfer

function e.g. sigmoid transfer function to produce the output of the PE (yj). The output of one PE

provides the input to the PE in the next layer. This process is summarized in Eqs. 1 and 2, which illustrated in Figure1.

𝐼𝑗 = ∑ 𝑤𝑗𝑖𝑥𝑖 + 𝜃𝑗 Summation (1)

𝑦𝑗 = 𝑓(𝐼𝑗) Transfer (2)

The propagation of information in ANN starts at the input layer where the network is presented with a historical set of input data and the corresponding (desired) outputs. The actual output of the network is compared with the desired output, and an error is calculated. Using this error and utilizing a learning rule, the network adjusts its weights until it can find a set of weights that will produce the input/output mapping that has the smallest possible error. This process is called “learning” or “training”. It should be noted that a network with one hidden layer can approximate any continuous function provided that sufficient connection weights are used. Once the training

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TW BF BR BW L TF

has to be validated using an independent validation set, see Yousif (2007).

3. Describe the collection data

324 cases have been used in calculation of the restraint in the walls, and 972 cases have been used in calculation of the restraint in the roofs, as shown in Table 1. Each of these calculations has been analyzed elastic three-dimensional finite-elements (3D FE) using the Abaqus software by use the following parameters; the modulus of elasticity of the old concrete was used

30.109 N/m2and for the fresh concrete 27.9.109 N/m2. The poison’s ratio v is 0.2 and thermal

dilation coefficient of new concrete α is 1.10-5o

C.

Results were taken at mid length of the structure, when calculate the average restraint through the walls, while select the bottom restraint value for the roofs. The result was treated with ANN tools; see Al-Gburi and et al, (2012b). 90% of results were used for training of ANN, while the remaining 10% used in the test.

Table 1. List of parameters and there value used in Finite Element Method calculation of variation in culverts structures.

Parameter

value wall roof

Roof Width = slab width BF= BR 10, 15, 20 3 3 Slab thickness TF 0.5, 1, 1.5, 2 4 4 Wall thickness TW 0.3, 0.8, 1.2 3 3 Wall high BW 3, 6, 9 3 3 Roof thickness TR 0.5, 1, 1.5 3 Length L 8, 16, 24 3 3 Total calculations 324 972

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3.1 Network Data Preparation

Preprocessing of data by scaling was carried out to improve the training of neural network, to avoid the slow rate of learning near end points, specifically of the output range due to the property of the sigmoid function which is asymptotic to values 0 and 1. The input and output data were scaled between the interval 0.1 and 0.9, the linear scaling equation:

𝑦 = �0.8_∆� 𝑋 + ( 0.9 − 0.8 𝑥𝑚𝑎𝑥

∆ ) (3)

where used in this study for a variable limited to minimum (Xmin) and maximum (Xmax) Values

given in Table 1 with:

∆ = 𝑋𝑚𝑎𝑥− 𝑋𝑚𝑖𝑛 (4)

It should be noted that any new input data should be scaled before being presented to the network and the corresponding predicted values should be un-scaled before use.

4. Study the Importance of the Geometry Factors of the Walls

Generally is the restraint value at the bottom of the wall at biggest values; gradually starts less as we go towards a free edge. The method of the partitioning weights, proposed by Garson (1991) and adopted by Goh (1995), was used within this study. In order to determine the relative importance of the various input parameters. The Figures, (2-33) are shown the influence of each factor on the restraint of the walls at early age.

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1 2 3 4 5 0 5 10 15 20 25 30 % Im po rt an ce Input Factors Thickness of Foundation

Width of Wall Thickness of Wall Length Width of Foundation 8.8235 25.6684 26.8737 15.4399 23.1945

Figure2. Relative Importance of Input Parameter for ANN Model at 0.1 Wall Heights

0.4 0.5 0.6 0.7 0.8 0.9 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 Targets T O u tput s Y , Li near F it : Y = (1) T + (0. 0 015) Outputs vs. Targets, R=0.97993 Data Points

Best Linear Fit Y = T 0.4 0.5 0.6 0.7 0.8 0.9 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 Targets T O u tput s Y , Li near F it : Y = (1) T + (-0. 0039) Outputs vs. Targets, R=0.98412 Data Points

Best Linear Fit Y = T

Figure3 Training F.E. with ANN model at 0.0 Wall Heights

Figure 4. Testing F.E. with ANN model at 0.0 Wall Heights

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0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Targets T O ut put s Y , L ine ar F it : Y = (1) T + (0. 0022) Outputs vs. Targets, R=0.97702 Data Points

Best Linear Fit Y = T 0.2 0.3 0.4 0.5 0.6 0.7 5 3 5 4 5 5 5 6 5 7 Targets T Outputs vs. Targets, R=0.99224 Data Points Best Linear Fit Y = T

Figure 6. Training F.E. with ANN model at 0.1 Wall Heights

1 2 3 4 5 0 5 10 15 20 25 30 35 40 % Im port anc e 8.6936 15.8009 37.2133 24.1431 width foundation Thickns foundation length 14.1491 thickness wall width wall

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0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Targets T O ut put s Y , L ine a r F it : Y = (1) T + (0.00095) Outputs vs. Targets, R=0.98316 Data Points

Best Linear Fit Y = T 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (-0.0067) Outputs vs. Targets, R=0.995 Data Points Best Linear Fit Y = T 0 5 10 15 20 25 30 35 40 % Im por tanc e 23.9702 16.3393 35.8541 18.3559 5.4804 width foundation width wall thickness foundation thickness wall length

Figure 8. Relative Importance of Input Parameter for ANN Model at 0.2 Wall Heights

Figure 9. Training F.E. with ANN model at 0.2 Wall Heights

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1 2 3 4 5 0 10 20 30 40 50 60 % Im port anc e Input Factors 12.3106 15.5141 3.2932 thickness foundation width foundation length thickness wall width wall 51.9786 16.9034 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (0.00076) Outputs vs. Targets, R=0.98238 Data Points Best Linear Fit Y = T 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (0.0015) Outputs vs. Targets, R=0.98951 Data Points Best Linear Fit Y = T

Figure11.Relative Importance of Input Parameter for ANN Model at 0.3 Wall Heights

Figure12.Training F.E. with ANN model at 0.3 Wall Heights

Figure13. Testing F.E. with ANN model at 0.3 Wall Heights

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1 2 3 4 5 0 5 10 15 20 25 30 % Im port anc e Input Factors 19.8972 23.8042 26.3831 24.4918 5.4237 length thickness wall width wall thickness foundation width foundation -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (0.00024) Outputs vs. Targets, R=0.98826 Data Points Best Linear Fit Y = T -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (0.0028) Outputs vs. Targets, R=0.99655 Data Points Best Linear Fit Y = T

Figure 14. Relative Importance of Input Parameter for ANN Model at 0.4 Wall Heights

Figure15.Training F.E. with ANN model at 0.4 Wall Heights

Figure16.Testing F.E. with ANN model at 0.4 Wall Heights

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-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (-8.7e -005) Outputs vs. Targets, R=0.98769 Data Points Best Linear Fit Y = T -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (-0.0002) Outputs vs. Targets, R=0.99565 Data Points Best Linear Fit Y = T 0 5 10 15 20 25 30 % Im po rt an ce 12.5644 26.8228 18.8254 28.0653 13.7220 thickness foundation width foundation width wall length thickness wall

Figure17.Relative Importance of Input Parameter for ANN Model at 0.5 Wall Heights

Figure18. Training F.E. with ANN model at 0.5 Wall Heights

Figure19.Testing F.E. with ANN model at 0.5 Wall Heights

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0 5 10 15 20 25 30 35 % Im port anc e 30.3630 29.1644 18.6031 8.1114 13.7580 width foundation thickness foundation width wall thickness wall length -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (-1.3e -005) Outputs vs. Targets, R=0.98267 Data Points Best Linear Fit Y = T -0.1 0 0.1 0.2 0.3 0.4 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (0.0018) Outputs vs. Targets, R=0.99467 Data Points Best Linear Fit Y = T

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-0.1 0 0.1 0.2 0.3 0.4 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Targets T O ut put s Y , L ine a r F it : Y = (1) T + (8.9e -005) Outputs vs. Targets, R=0.97281 Data Points

Best Linear Fit Y = T -0.1 0 0.1 0.2 0.3 0.4 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (-0.00029) Outputs vs. Targets, R=0.99019 Data Points Best Linear Fit Y = T length 0 5 10 15 20 25 30 35 % Im port anc e 27.0919 32.3795 2.8409 17.7239 19.9638 width foundation thickness wall thickness foundation width wall

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-0.2 0 0.2 0.4 0.6 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (1.8e -005) Outputs vs. Targets, R=0.91084 Data Points Best Linear Fit Y = T -0.2 -0.1 0 0.1 0.2 0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (0.0016) Outputs vs. Targets, R=0.97477 Data Points Best Linear Fit Y = T 0 5 10 15 20 25 30 % Im p o rt an ce 25.2876 25.3453 25.2403 18.0859 6.0408 width foundation thickness foundation thickness wall length width wall

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-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Targets T O ut put s Y , L ine ar F it : Y = (1.1)T + (0.0087) Outputs vs. Targets, R=0.97635 Data Points Best Linear Fit Y = T -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (0.00014) Outputs vs. Targets, R=0.87782 Data Points Best Linear Fit Y = T 0 5 10 15 20 25 30 % Im port anc e 23.2663 _22.4664 24.9487 20.6988 8.6198 length thickness foundation width foundation width wall thickness wall

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0 5 10 15 20 25 30 35 40 45 % Im port anc e thickness wall length thickness foundation width

foundation widthwall

5.3194 17.4039 23.4101 14.1979 39.6687 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (0.00046) Outputs vs. Targets, R=0.89036 Data Points Best Linear Fit Y = T -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (0.0028) Outputs vs. Targets, R=0.98055 Data Points Best Linear Fit Y = T

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5. Discuss the results

5.1 Effect of the wall thickness

Increase the wall thickness has combined with reduces of the restraint as shown in Figures (34-44). This behavior can be explained as follows, the bigger size of the young concrete means increasing the possibility of counteraction of outer constraints (old concrete, i.e. the slab in this case); therefore, a decrease restriction interpreted with amplified wall thickness. This behavior agrees with results in ACI (2002), Sang-Chel (2000) and Weiss et al (2000). On the other hand, notes increased the length of the structures increases the contact area between an old and new concrete. Therefore, the restraint is increased from old concrete where it is more along with structures leading to higher restraint value. So a fewer exterior restraints are caused by a smaller length. The same performance was shown in Kheder (1997) and Kianousha and et al, (2008).

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8 10 12 14 16 18 20 22 24 0.65 0.7 0.75 0.8 0.85 R es tr ain t Length of Wall TW = 0.3 TW = 0.525 TW = 0.75 TW =0.975 TW = 1.2 BF=15 TF= 1 BW = 6 8 10 12 14 16 18 20 22 24 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 res tr a in t length of wall TW = 0.3 TW = 0.525 TW = 0.75 TW = 0.975 TW = 1.2 BF = 15 TF = 1 BW = 6 8 10 12 14 16 18 20 22 24 0.2 0.25 0.3 0.35 0.4 0.45 0.5 res tr a in t length of wall TW = 0.3 TW = 0.525 TW =0.75 TW = 0.975 TW = 1.2 BF = 15 TF = 1 BW = 6 8 10 12 14 16 18 20 22 24 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 res tr a in t length of wall TW = 0.3 TW = 0.525 TW = 0.75 TW = 0.975 TW = 1.2 BF = 15 TF =1 BW = 1.2 8 10 12 14 16 18 20 22 24 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 res tr a in t length of wall TW = 0.3 TW =0.525 TW =0.75 TW = 0.975 TW = 1.2 BF = 15 TF = 1 BW =6

Figure34. Variation of restraint with wall thickness

and length as predict by ANN at 0.0 wall height

Figure37.Variation of restraint with wall thickness

8 10 12 14 16 18 20 22 24 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 r es tr a in t length of wall TW = 0.3 TW = 0.525 TW = 0.75 TW = 0.975 TW = 1.2 BF = 15 TF = 1 BW = 6

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8 10 12 14 16 18 20 22 24 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 res tr ain t length of wall TW = 0.3 TW = 0.525 TW = 0.75 TW = 0.975 TW = 1.2 BF =15 TF = 1 BW = 6 8 10 12 14 16 18 20 22 24 -0.07 -0.065 -0.06 -0.055 -0.05 -0.045 -0.04 -0.035 -0.03 re s tra int length of wall TW = 0.3 TW = 0.525 TW = 0.75 TW = 0.975 TW = 1.2 BF = 15 TF = 1 BW = 6 8 10 12 14 16 18 20 22 24 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 res tr a in t length of wall TW = 0.3 TW = 0.525 TW = 0.75 TW = 0.975 TW = 1.2 BF = 15 TF = 1 BW = 6 8 10 12 14 16 18 20 22 24 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 res tr a in t length of wall TW = 0.3 TW = 0.525 TW = 0.75 TW = 0.975 TW = 1.2 BF = 15 TF = 1 BW =6 8 10 12 14 16 18 20 22 24 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 res tr a in t length of wall TW = 0.3 TW = 0.525 TW = 0.75 TW = 0.975 TW = 1.2 BF =15 TF = 1 BW = 6

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5.2 Effect of wall height

Increases the wall height raises the wall inertia that is it reduces the effect of the external restraint (the slab in this case) in the wall. As shown in Figures (44-55), the restraint value decreases with increasing the wall height. It highlights the influence of wall height is evident whenever moved away from base to a distance 0.6 of wall height as shown in Table 2. On the other hand, observed increases the restraint with a rises (Length/Height) ratio, this behavior is compatible with results in Emborg (1989), Kheder et al,

(1994), and Nilsson (2003).

5.3 Effect of slab thickness

Increasing the slab thickness means growth stiffness, raises the value of the external restraint in the wall, as shown in Figures (56-66). The effect of increasing the length of the wall is also increasing the value of restraint. This result is well-matched with Kheder et al, (1994), ACI (2002), and Sang-Chel (2000).

5.4 Effect of the slab width

In the midst of these effects, the biggest loser of competing it the influence in the restraint value is the effect of slab width. This parameter did not have a significant role in effects on the restraint calculation in the walls, as shown in Figures (67-77). Where it is with the increasing of the slab width the contact area between the wall and the slab itself remains unchanged. Thus, it has the smallest effect on the restraint value.

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8 10 12 14 16 18 20 22 24 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 R es tr ain t Length of Wall BW = 3 BW = 4.5 BW = 6 BW = 7.5 BW = 9 BF= 15 TF =1 TW =0.8 8 10 12 14 16 18 20 22 24 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 res tr a in t length of wall BW = 3 BW = 4.5 BW = 6 BW = 7.5 BW = 9 BF = 15 TF = 1 TW = 0.8 8 10 12 14 16 18 20 22 24 0.2 0.25 0.3 0.35 0.4 0.45 0.5 r es tr a in t r atio length of wall BW = 3 BW = 4.5 BW =6 BW =7.5 BW =9 BF =15 TF = 1 BW = 6 8 10 12 14 16 18 20 22 24 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 res tr a in t length of wall BW = 3 BW = 4.5 BW = 6 BW = 7.5 BW = 9 BF = 15 TF = 1 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 res tr a in t length of wall BW = 3 BW = 4.5 BW = 6 BW =7.5 BW = 9 BF = 15 TF = 1 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.05 0 0.05 0.1 0.15 0.2 res tr a in t length of wall BW = 3 BW =4.5 BW = 6 BW = 7.5 BW = 9 BF =15 TF = 1.0 TW = 0.8

Figure45. Variation of restraint with wall width

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8 10 12 14 16 18 20 22 24 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 res tr a in t length of wall BW = 3 BW = 4.5 BW = 6 BW = 7.5 BW = 9 BF = 15 TF = 1 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.075 -0.07 -0.065 -0.06 -0.055 -0.05 -0.045 -0.04 -0.035 -0.03 -0.025 res tr ain t length of wall BW = 3 BW = 4.5 BW = 6 BW = 7.5 BW = 9 BF = 15 TF = 1 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 res tr a in t length of wall BW = 3 BW = 4.5 BW = 6 BW = 7.5 BW = 9 BF = 15 TF = 0.1 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 res tr a in t length of wall BW = 3 BW = 4.5 BW = 6 BW = 7.5 BW = 9 BF= 15 TF = 1 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 res tr a in t length of wall BW = 3 BW = 4.5 BW = 6 BW = 7.5 BW = 9 BF = 15 TF = 1 TW = 0.8

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8 10 12 14 16 18 20 22 24 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 R e s tr a in t Length of Wall TF = 0.5 TF = 0.75 TF = 1.25 TF = 1.625 TF = 2.0 BF = 15 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 res tr a in t length of wall TF = 0.5 TF = 0.875 TF = 1.25 TF = 1.625 TF = 2.0 BF=15 BW = 6 TW = 0,8 8 10 12 14 16 18 20 22 24 0.25 0.3 0.35 0.4 0.45 0.5 res tr ain t length of wall TF = 0.5 TF = 0.75 TF = 1.25 TF =1.625 TF = 2.0 BF = 15 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.2 0.25 0.3 0.35 0.4 0.45 0.5 res tr a in t length of wall TF = 0.5 TF = 0.75 TF = 1.25 TF = 1.625 TF = 2.0 BF = 15 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.05 0.1 0.15 0.2 0.25 0.3 res tr a in t length of wall TF = 0.5 TF = 0.75 TF = 1.25 TF = 1.625 TF = 2.0 BF = 15 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 res tr a in t length of wall TF = 0.5 TF = 0.75 TF =1.25 TF = 1.625 TF = 2.0 TF = 15 BW = 6 TW = 0.8

Figure56. Variation of restraint thick slab and

length as predict by ANN at 0.0 wall height

Figure57. Variation of restraint with thick slab and

Figure58.Variation of restraint with thick slab and

Figure60. Variation of restraint with thick slab and

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8 10 12 14 16 18 20 22 24 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 res tr a in t length of wall TF = 0.5 TF = 0.75 TF = 1.25 TF = 1.625 TF = 2.0 BF = 15 BW = 6 TW =0.8 8 10 12 14 16 18 20 22 24 -0.07 -0.065 -0.06 -0.055 -0.05 -0.045 -0.04 -0.035 -0.03 -0.025 -0.02 re s tr a in t length of wall TF = 0.5 TF = 0.75 TF = 1.25 TF = 1.625 TF = 2.0 BF =15 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 res tr a in t length of wall TF = 0.5 TF = 0.75 TF = 1.25 TF = 1.625 TF = 2.0 BF = 15 BW 0 6 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 res tr a in t length of wall TF = 0.5 TF = 0.75 TF = 1.25 TF = 1.625 TF = 2.0 BF = 15 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 res tr a in t length of wall TF = 0.5 TF =0.75 TF = 1.25 TF = 1.625 TF = 2.0 BF = 15 BW = 6 TW = 0.8

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8 10 12 14 16 18 20 22 24 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 R es tr ain t Length of Wall BF = 11 BF =13 BF = 15 BF = 17 BF = 20 TF=1 BW=6 TW=0.8 8 10 12 14 16 18 20 22 24 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62 res tr ain t length of wall BF =11 BF = 13 BF =15 BF =17 BF=20 TF =1 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.25 0.3 0.35 0.4 0.45 0.5 res tr a in t length of wall BF = 11 BF =13 BF =15 BF =17 BF = 20 TF = 1 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.2 0.25 0.3 0.35 0.4 res tr a in t length of wall BF =11 BF = 13 BF = 15 BF = 17 BF = 17 TF =1 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 res tr a in t length of wall BF = 11 BF = 13 BF =15 BF = 17 BF = 20 TF = 1 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 res tr a in t length of wall BF = 11 BF= 13 BF = 15 BF = 17 BF =20 TF = 1 BW = 6 TW=0.8

Figure67.Variation of restraint slab width and

Figure68Variation of restraint with slab width and

Figure69. Variation of restraint with slab width and length as predict by ANN at 0.2 wall height

Figure70.Variation of restraint with slab width and

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0.0 8 10 12 14 16 18 20 22 24 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 res tr a in t length of wall BF = 11 BF = 13 BF =15 BF = 17 BF =20 TF = 1 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.075 -0.07 -0.065 -0.06 -0.055 -0.05 -0.045 -0.04 -0.035 res tr a in t length of wall BF = 11 BF = 13 BF = 15 BF = 17 BF = 20 TF = 1 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.16 -0.15 -0.14 -0.13 -0.12 -0.11 -0.1 -0.09 -0.08 -0.07 -0.06 res tr a in t length of wall BF = 11 BF = 13 BF = 15 BF = 17 BF = 20 TF = 1 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 res tr a in t length of wall BF = 11 BF = 13 BF = 15 BF = 17 BF = 20 TF = 1 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 res tr a in t length of wall BF = 11 BF = 13 BF = 15 BF = 17 BF = 20 TF = 1 BW = 6 TW = 0.8

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6. Discuss the Results of Restraint in the Roof

It has been noted that the biggest restraint is not always at the edge of the roofs, but it is

sometimes at 0.1 BR distances from the edge of the roofs and start progressively decreases when

we reach to the middle of the roofs. As shown in Figures (78-110), the most prominent

influential dominance on the restraint of the roof is the roof thickness TR. In a close distance, the

influence of wall height BW comes and followed by the influence of the length of structure L.

After a distance, the effect of roof width BR and the slab width BF comes, leaving behind the

influence of the wall thickness TW. At the bottom of the list, the effect of slab thickness TF

comes. The following is the description effect of each factor separately.

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Targets T O ut put s Y , L ine ar F it : Y = (1) T + (0.00031) Outputs vs. Targets, R=0.99446 Data Points

Best Linear Fit Y = T 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (0.00017) Outputs vs. Targets, R=0.99339 Data Points Best Linear Fit Y = T

Figure78.Relative Importance of Input Parameter for ANN Model at 0.0 roof width

Figure79.Training F.E. with ANN model at 0.0 roof width

Figure80.Testing F.E. with ANN model at 0.0 roof width

0 5 10 15 20 25 30 35 % Im port anc e Thickness wall 32.9805 length 11.2600 12.0304 10.0697 3.1926 Thickness foundation width foundation width wall 30.4668 thickness roof

Roof Roof Slab Wall Wall Length Thickness Width Thickness Width Thickness

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Targets T O ut put s Y , L ine ar F it : Y = (1) T + (-0.00074) Outputs vs. Targets, R=0.9958 Data Points

Best Linear Fit Y = T 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Targets T O ut put s Y , L ine ar F it : Y = (1) T + (0.0011) Outputs vs. Targets, R=0.99414 Data Points

Figure83. Testing F.E. with ANN model at 0.1 roof width

0 5 10 15 20 25 30 35 % Im port anc e 6.0571 8.6152 30.9188 11.2781 30.1512 12.9796 thickness foundation thickness wall width wall length width roof thickness roof

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Targets T O ut put s Y , L ine ar F it : Y = (1)T + (-0.00021) Outputs vs. Targets, R=0.99466 Data Points Best Linear Fit Y = T 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Targets T O ut put s Y , L ine ar F it : Y = (1) T + (-0.0017) Outputs vs. Targets, R=0.99294 Data Points

Figure84. Relative Importance of Input Parameter for ANN Model at 0.2 roof width

Figure85. Training F.E. with ANN model at 0.2 roof width

0 5 10 15 20 25 30 % Im port anc e length thickness wall width wall thickness foundation width roof thickness roof 14.9489 11.2689 14.3546 22.9914 24.2176 12.2185

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Targets T O ut put s Y , L ine ar F it : Y = (1) T + (-6.2e -005) Outputs vs. Targets, R=0.99704 Data Points

Best Linear Fit Y = T 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Targets T O ut put s Y , L ine a r F it : Y = (0.99) T + (0.0019) Outputs vs. Targets, R=0.99573 Data Points

Figure89.Testing F.E. with ANN model at 0.3 roof width 0 5 10 15 20 25 % Im port anc e 23.2330 17.2111 5.5762 12.3425 22.3757 19.2615 width wall thickness wall length thickness foundation width roof thickness roof

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-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Targets T O ut put s Y , L ine a r F it : Y = (1) T + (-0.00039) Outputs vs. Targets, R=0.99506 Data Points

Best Linear Fit Y = T 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Targets T O ut put s Y , L ine a r F it : Y = (1) T + (-3.4e -005) Outputs vs. Targets, R=0.99541 Data Points

0 5 10 15 20 25 30 35 40 % Im po rt an ce 34.1302 9.0986 9.4287 11.9224 21.2475 14.1725 width roof thickness

roof _foundationthickness

width wall

thickness wall

length

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-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Targets T O ut put s Y , L ine ar F it : Y = (1) T + (0.00034) Outputs vs. Targets, R=0.9953 Data Points

Best Linear Fit Y = T -0.1 0 0.1 0.2 0.3 0.4 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Targets T O ut put s Y , L ine a r F it : Y = (1) T + (-0.00084) Outputs vs. Targets, R=0.99581 Data Points

0 5 10 15 20 25 30 35 % Im po rt an ce 26.5239 8.8195 8.9577 9.3976 14.4910 thickness foundation width roof thickness roof thickness wall width wall length 31.8103

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-0.2 0 0.2 0.4 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Targets T O ut put s Y , L ine ar F it : Y = (1) T + (0.00064) Outputs vs. Targets, R=0.99028 Data Points

Best Linear Fit Y = T -0.1 0 0.1 0.2 0.3 0.4 -0.1 0 0.1 0.2 0.3 0.4 Targets T O ut put s Y , L ine ar F it : Y = (0.99) T + (0.0022) Outputs vs. Targets, R=0.99166 Data Points

0 5 10 15 20 25 30 % Im po rt an ce 25.7303 12.2022 19.4155 11.9954 14.2929 16.3638 thickness roof width roof width wall thickness wall thickness foundation length

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-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Targets T O ut put s Y , L ine ar F it : Y = (1) T + (0.00021) Outputs vs. Targets, R=0.99214 Data Points

Best Linear Fit Y = T -0.1 0 0.1 0.2 0.3 0.4 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Targets T O ut put s Y , L ine a r F it : Y = (0.98) T + (-0.00075) Outputs vs. Targets, R=0.98924 Data Points

Figure101.Testing F.E. with ANN model at 0.7 roof width Figure99. Relative Importance of Input Parameter for ANN Model at 0.7 roof width

0 5 10 15 20 25 30 % Im po rt an ce 21.4405 15.2801 11.3384 7.5777 15.5096 thickness roof width roof thickness foundation width wall thickness wall length 28.8537

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-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Targets T O ut put s Y , L ine a r F it : Y = (1) T + (0.00016) Outputs vs. Targets, R=0.99031 Data Points

Best Linear Fit Y = T -0.1 0 0.1 0.2 0.3 0.4 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Targets T O ut put s Y , L ine a r F it : Y = (0.98) T + (-0.0012) Outputs vs. Targets, R=0.98678 Data Points

0 5 10 15 20 25 30 35 % Im po rt an ce 29.3997 13.4791 7.6918 11.0996 16.8518 21.4780 width roof thickness foundation width wall thickness wall length thickness roof

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-0.2 0 0.2 0.4 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Targets T O ut put s Y , L ine ar F it : Y = (1) T + (3.9e -005) Outputs vs. Targets, R=0.9924 Data Points

Best Linear Fit Y = T -0.1 0 0.1 0.2 0.3 0.4 -0.1 0 0.1 0.2 0.3 0.4 Targets T O ut put s Y , L ine ar F it : Y = (1) T + (-0.0014) Outputs vs. Targets, R=0.99263 Data Points

0 5 10 15 20 25 30 % Im po rt an ce 16.9663 21.4708 10.3324 6.4304 16.8750 27.9250 thickness roof width roof thickness foundation thickness wall width wall length

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-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Targets T O ut put s Y , L ine ar F it : Y = (1) T + (-0.00011) Outputs vs. Targets, R=0.9945 Data Points

Best Linear Fit Y = T -0.1 0 0.1 0.2 0.3 0.4 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Targets T O ut put s Y , L ine a r F it : Y = (0.99) T + (4e -005) Outputs vs. Targets, R=0.99483 Data Points

0 5 10 15 20 25 30 35 % Im po rt an ce thickness roof thickness foundation width wall thickness wall length width roof 29.9562 18.9554 9.3174 5.2524 19.3072 17.2113

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6.1 Effect of roof thickness

It is expected that increasing the roof thickness raises the heat generated inside the roof. Still this influence has fewer effects comparing with the power of inertia that obtains to the roof. It will make it more able to counteract with the effects of an old casting (the slab and the wall in this case). Therefore, decrease the restraint with the increased the roof thickness is noticed as shown in Figures (111-121). This result compatible with experimental observation in deck casting; increasing the deck thickness reduces deck cracking (Brown et al, 2001, French et al, 1999; Krauss and Rogalla, 1996; Kochanski et al., 1990, Ramey et al. and 1997; Mayers, 1982).

6.2 Effect of the wall height

The wall height effects as external restraint on the roof, the increase of wall height gives the power of inertia to the outer restraint. This role did not seem clear in the short structures, but it becomes evident in long structures as shown in Figures (122-132). This result is compatible with field observation of Saadeghvaziri and Hadidi (2003).

6.3 Effect of wall thickness

Increasing the wall thickness means enlarge the contact area between an old and a new parts. In addition, it provides more area to move the influence of the wall restraint and tries to restrict the roof as shown in Figures (133-143). Ducret et al. (1997) have measured that reducing the ratio of cross sectional area of girder to deck (which can be related to flexibility) reduces risk of cracking. Brown et al (2001) mentioned that girder size and spacing also affects restraint, larger girders provide more restraint and therefore induce more cracking in the deck.

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6.4 Effect of roof width

Increasing the width of the roof means moves far from the contact area between the roof and the

wall which is caused to stay absent from the source of external influence. Therefore, it has been observed that increasing the roof width decreasing the restraint in the roof as shown in Figures (144-154).

6.5 Effect of slab thickness

The effect of the slab thickness might does not play a major role affecting on the restraint in the roofs, because moving away from the area of transmission of external influence to the roof (contact area). In general, increases the restraint with the increases the slab thickness, but this effect is not relevant here as shown in the Figures (155-165). .

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8 10 12 14 16 18 20 22 24 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 res tr a in t length of wall TR = 0.75 TR = 1.0 TR = 1.25 TR = 1.5 BF = 15 TF= 1.5 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 res tr ain t length of wall TR = 0.5 TR = 0.75 TR = 1.0 TR = 1.25 TR = 1.5 BR = 15 TF = 1.5 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 re s tr a in t length of wall TR = 0.5 TR = 0.75 TR = 1.0 TR = 1.25 TR = 1.5 BR = 15 TF = 1.5 BW= 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.05 0.1 0.15 0.2 0.25 0.3 re s tr a in t length of wall TR= 0.5 TR = 0.75 TR = 1.0 TR = 1.25 TR = 1.5 BR = 15 TF = 1.5 BW = 6 TW =0.8 8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 re s tr a in t length of wall TR = 0.5 TR = 0.75 TR = 1.0 TR = 1.25 TR = 1.5 BR = 15 TF = 1.5 BW = 6 TW = 0.8

Figure111.Variation of restraint with roof thickness

and length as predict by ANN at 0.0 roof width

8 10 12 14 16 18 20 22 24 0.25 0.3 0.35 0.4 0.45 0.5 0.55 re s tr a in t length of wall TR = 0.75 TR = 1.0 TR = 1.25 TR= 1.5 BF = 15 TF = 1.5 BW = 6 TW = 0.8

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8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 r e s tr a in t length of wall TR = 0.5 TR = 0.75 TR = 1.0 TR = 1.25 TR = 1.5 BR = 15 TF = 1.5 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 r e s tr a in t length of wall TR = 0.5 TR = 0.75 TR 0 1.0 TR = 1.25 TR = 1.5 BR = 15 TF = 1.5 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 0.25 r e s tr a in t length of wall TR = 0.5 TR = 0.75 TR = 1.0 TR = 1.25 TR = 1.5 BR = 15 TF = 1.5 BW = 6.0 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.05 0 0.05 0.1 0.15 0.2 r e s tr a in t length of wall TR = 0.5 TR= 0.75 TR= 1.0 TR = 1.25 TR = 1.5 BR = 15 TF = 15 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 re s tr a in t length of wall TR = 0.5 TR = 0.75 TR = 1.0 TR = 1.25 TR = 1.5 BR = 15 TF = 1.5 BW = 6 TW = 0.8

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8 10 12 14 16 18 20 22 24 0.2 0.25 0.3 0.35 0.4 res tr ain t length of wall BW = 3 BW = 4.5 BW = 6.0 BW = 7.5 BW = 9.0 TR = 1 BF = 15 TF = 1.5 TW = 0.8 8 10 12 14 16 18 20 22 24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 re s tr a in t length of wall BW = 4.5 BW = 6.0 BW = 7.5 BW = 9.0 BF = 15 TF=1.5 TW = 0.8 8 10 12 14 16 18 20 22 24 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 re s tr a in t length of wall BW = 3 BW = 4.5 BW = 6 BW = 7.5 BW = 9 TR = 1 BR = 15 TF = 1.5 TW = 0.8 8 10 12 14 16 18 20 22 24 0.16 0.18 0.2 0.22 0.24 0.26 re s tr a in t length of wall BW =3 BW = 4.5 BW = 6 BW = 7.5 BW = 9 TR = 1 BR = 15 TF = 1.5 TW= 0.8 8 10 12 14 16 18 20 22 24 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 re s tr a in t length of wall BW =3 BW = 4.5 BW = 6 BW 07.5 BW = 9 TR = 1 BR= 15 TF = 1.5 TW= 0.8 8 10 12 14 16 18 20 22 24 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 re s tr a in t length of wall BW = 3 BW = 4.5 BW = 6 BW = 7.5 BW =9 TR = 1 BR = 15 TF = 1.5 TW = 0.8

Figure122Variation of restraint with wall height

Figure123Variation of restraint with wall

height and length as predict by ANN at 0.1 roof

Figure125Variation of restraint wall height width

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8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 r e s tr a in t length of wall BW = 3 BW = 4.5 BW = 6 BW = 7.5 BW= 9 TR= 1 BR= 15 TF= 1.5 TW= 0.8 8 10 12 14 16 18 20 22 24 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 re s tr a in t length of wall BW = 3 BW =4.5 BW = 6 BW = 7.5 BW = 9 TR = 1 BR = 15 TF = 1.5 TW = 0.8 8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 0.25 r es tr ai nt length of wall BW = 3 BW= 4.5 BW = 6 BW = 7.5 BW = 9 TR = 1 BR= 15 TF= 1.5 TW= 0.8 8 10 12 14 16 18 20 22 24 -0.05 0 0.05 0.1 0.15 0.2 r e s tr a in t length of wall BW = 3 BW = 4.5 BW = 6 BW = 7.5 BW = 9 TR = 1 BR = 15 TF = 1.5 TW= 0.8 8 10 12 14 16 18 20 22 24 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 re s tr a in t length of wall BW = 3 BW = 4.5 BW = 6 BW = 7.5 BW = 9 TR = 1 BR = 15 TF = 1.5 TW = 0.8

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8 10 12 14 16 18 20 22 24 0.2 0.25 0.3 0.35 0.4 0.45 0.5 res tr ain t length of wall TW = 0.5 TW = 0.7 TW = 0.9 TW = 1.2 TR = 1 BF = 15 TF= 1.5 BW = 6 8 10 12 14 16 18 20 22 24 0.2 0.25 0.3 0.35 0.4 0.45 0.5 re s tr a in t length of wall TW = 0.3 TW = 0.5 TW = 0.7 TW = 0.9 TW = 1.2 TR = 1 BF = 15 TF = 1.5 BW = 6 8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 re s tra int length of wall TW = 0.3 TW = 0.5 TW = 0.7 TW = 0.9 TW = 1.2 TR = 1 BR = 15 TF = 1.5 BW = 6 8 10 12 14 16 18 20 22 24 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 re s tr a in t length of wall TW = 0.3 TW = 0.5 TW= 0.7 TW =0.9 TW = 1.2 TR = 1 BR = 15 TF = 1.5 BW= 6 8 10 12 14 16 18 20 22 24 0.05 0.1 0.15 0.2 0.25 0.3 re s tr a in t length of wall TW = 0.3 TW = 0.5 TW = 0.7 TW= 0.9 TW = 1.2 TR = 1 BR = 15 TF = 1.5 BW = 6 8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 0.25 re s tr a in t length of wall TW = 0.3 TW= 0.5 TW = 0.7 TW = 0.9 TW = 1.2 TR = 1 BR =15 TF = 1.5 BW= 6

Figure133Variation of restraint with wall thickness

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8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 re s tr a in t length of wall TW = 0.3 TW= 0.5 TW = 0.7 TW = 0.9 TW= 1.2 TR = 1 BR= 15 TF= 1.5 BW = 6 8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 r e s tr a in t length of wall TW = 0.3 TW= 0.5 TW = 0.7 TW = 0.9 TW = 1.2 TR= 1 BR = 15 TF = 1.5 BW = 6 8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 0.25 r e s tr a in t length of wall TW = 0.3 TW = 0.5 TW = 0.7 TW = 0.9 TW = 1.2 TR = 1 BR = 15 TF = 1.5 BW = 6 8 10 12 14 16 18 20 22 24 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 re s tr a in t length of wall TW = 0.3 TW = 0.5 TW = 0.7 TW = 0.9 TW = 1.2 TR = 1 BR = 15 TF = 1.5 BW = 6 8 10 12 14 16 18 20 22 24 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 re s tr a in t length of wall TW = 0.3 TW = 0.5 TW = 0.75 TW = 0.9 TW = 1.2 TR = 1 BR = 15 TF= 1.5 BW = 6

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8 10 12 14 16 18 20 22 24 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 res tr a in t length of wall BR = 12.5 BR = 15 BR = 17.5 BR = 20 TR = 1 TF = 1.5 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 res tr ain t length of wall BR = 12.5 BR = 15 BR = 17.5 BR = 20 TR = 1 TF= 1.5 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 re s tr a in t length of wall BR = 10 BR = 12.5 BR = 15 BR = 17.5 BR = 20 TR = 1 TF = 1.5 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.05 0.1 0.15 0.2 0.25 0.3 re s tr a in t length of wall BR = 10 BR = 12.5 BR = 15 BR =17.5 BR= 20 TR= 1 TF = 1.5 BW= 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 re s tr a in t length of wall BR= 10 BR = 12.5 BR= 15 BR=17.5 BR= 20 TR = 1 TF= 1.5 BW= 6 TW= 0.8 8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 re s tr a in t length of wall BR = 10 BR = 12.5 BR = 15 BR = 17.5 BR = 20 TR= 1 TF = 1.5 BW = 6 TW = 0.8

Figure144Variation of restraint with roof width

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8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 0.25 0.3 r e s tr a in t length of wall BR = 12.5 BR = 15 BR = 17.5 BR = 20 TR = 1 TF = 1.5 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.05 0 0.05 0.1 0.15 0.2 0.25 r e s tr a in t length of wall BR = 10 BR = 12.5 BR= 15 BR =17.5 BR = 20 TR = 1 TF = 1.5 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 0.25 r e s tr a in t length of wall BR = 10 BR = 12.5 BR = 15 BR = 17.5 BR = 20 TR = 1 TF = 1.5 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 r e s tr a in t length of wall BR = 10 BR = 12.5 BR = 15 BR = 17.5 BR = 20 TR = 1 TF = 1.5 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 r e s tr a in t length of wall BR= 10 BR = 12.5 BR = 15 BR = 17.5 BR = 20 TR = 1 TF = 1.5 BW = 6 TW = 0.8

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8 10 12 14 16 18 20 22 24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 res tr a in t length of wall TF = 0.875 TF = 1.25 TF = 1.625 TF = 2.0 BF = 15 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.3 0.32 0.34 0.36 0.38 re s tr a in t length of wall TF= 0.5 TF = 0.875 TF = 1.25 TF = 1.625 TF = 2.0 8 10 12 14 16 18 20 22 24 0.22 0.23 0.24 0.25 0.26 0.27 0.28 re s tr a in t length of wall TF = 0.5 TF = 0.875 TF = 1.25 TF = 1.625 TF = 2.0 TR = 1 BR = 15 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 re s tr a in t length of wall TF = 0.5 TF = 0.875 TF = 1.25 TF = 1.625 TF = 2.0 TR = 1 BR = 15 BW = 6 TW = 0.8

Figure155Variation of restraint with found. Thick.

Figure157Variation of restraint with slab thick. and

length as predict by ANN at 0.2 roof width

Figure158Variation of restraint with slab

thick& length as predict by ANN at 0.3 roof width

8 10 12 14 16 18 20 22 24 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 re s tr a in t length of wall TF = 0.5 TF = 0.875 TF =1.25 TF =1.625 TF =2.0 TR= 1 BR =15 BW = 6 TW= 0.8 8 10 12 14 16 18 20 22 24 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 re s tr a in t length of wall TF =0.5 TF = 0.875 TF= 1.25 TF = 1.625 TF = 2.0 TR = 1 BR = 15 BW = 6 TW = 0.8

Figure159Variation of restraint with found. thick.

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8 10 12 14 16 18 20 22 24 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 r e s tr a in t length of wall TF =0.5 TF = 0.875 TF = 1.25 TF = 1.625 TF = 2.0 TR = 1 BR = 15 BW =6 TW = 0.8 8 10 12 14 16 18 20 22 24 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 r e s tr a in t length of wall TF = 0.5 TF = 0.875 TF = 1.25 TF = 1.625 TF=2.0 TR = 1 BR = 15 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 0.25 r e s tr a in t length of wall TF = 0.5 TF = 0.875 TF= 1.25 TF = 1.625 TF = 2.0 TR = 1 BR = 15 BW = 6 TW = 0.8 8 10 12 14 16 18 20 22 24 -0.05 0 0.05 0.1 0.15 0.2 r e s tr a in t length of wall TF = 0.5 TF = 0.875 TF = 1.25 TF = 1.625 TF = 2.0 TR = 1 BR = 15 BW= 6 TW = 0.8 8 10 12 14 16 18 20 22 24 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 re s tra in t length of wall TF = 0.5 TF = 0.75 TF = 1.25 TF = 1.625 TF = 2.0 TR = 1 BR = 15 BW = 6 TW = 0.8

Figure164Variation of restraint with found. thick.

Figure165Variation of restraint with slab.

Thick. and length as predict by ANN at 1.0 roof

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10 11 12

13

The ANN model used to derive the design formula for the restraint wall calculations and use multi-layer perceptions (MLP) that is trained with the back-propagation training algorithm for

feed forward ANNs. The model has five inputs representing the width of slab (BR), the thickness

of slab (TF), the thickness of the wall (TW), the height of the wall (BW) and the lengths of the

structure (L). All the parameters and their values are listed in Table 1.

The structure of the optimal ANN model is shown in Figure (166). The connection weights and threshold levels are summarized in Table 2-12.

7.1 The Design Formula for the Wall

The small number of connection weights of the neural network enables the ANN model to be translated into a relatively simple formula in which the predicted restraint can be expressed as follows: γ_𝑅= 1 1+𝑒 − ⎝ ⎜ ⎜ ⎛𝜃13+�𝑤13:6 1 1+𝑒−(𝑥1)�+�𝑤13:7 1 1+𝑒−(𝑥2)�+�𝑤13:8 1 1+𝑒−(𝑥3)�+�𝑤13:9 1 1+𝑒−(𝑥4)�+ �𝑤13:10 1 1+𝑒−(𝑥5)�+�𝑤13:11 1 1+𝑒−(𝑥6)�+� 𝑤13:12 1 1+𝑒−(𝑥7)� ⎠ ⎟ ⎟ ⎞ (5)

Figure (166) Structure of the optimal ANN model for the wall

3 2 1 6 5 4 7 9 8 BR TW TF BW L Restraint

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x1= θ6+w B6:1 R+w T6:2 F +w T6:3 W +w B6:4 W +w L6:5 (6) x2= θ7 +w B7:1 R+w T7:2 F +w T7:3 W +w B7:4 W +w L7:5 (7) x3= θ8+w B8:1 R+w T8:2 F +w T8:3 W +w B8:4 W +w L8:5 (8) x4= θ9+w B9:1 R+w T9:2 F +w T9:3 W +w B9:4 W +w L9:5 (9) x5= θ10+w10:1BR+w10:2TF +w10:3TW +w10:4BW +w10:5L (10) x6= θ11+w11:1BR+w11:2TF +w11:3TW +w11:4BW +w11:5L (11) x7= θ12+w12:1BR+w12:2TF +w12:3TW +w12:4BW +w12:5L (12)

It should be noted that before using Equations (5), all input variables need to be scaled between 0.1 and 0.9 using Eq. (5) and the data ranges in Table 1. It should also be noted that predicted restraint obtained from Eq. (5) is scaled between 0.1 and 0.9 and in order to obtain the actual value, this restraint has to be re-scaled.

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Table 2 .

Weights and threshold levels for the ANN model at 0.0 H of wall high.

Hidden layer

nodes wji (weight from node Hidden layer i in the input layer _{to node j in the hidden layer )} threshold (θj)Hidden layer

i=1 i=2 i=3 i=4 i=5

j=6 _1.196604043 _-2.710429 _4.0807997 _{2.3205384 -5.5005248} _-0.09263 j=7 _-1.159629301 _{16.843524 -1.8235015 -4.1688952} _-1.095523 _2.104549 j=8 _-0.605311879 _{3.6071244 -7.8725882 -1.9064524} _0.9766331 _0.410788 j=9 _23.28879002 _{15.954220 -13.489495} _{0.0440172 -29.929089} _26.90328 j=10 _{0.332934866 -0.4047448} _{0.2067261 -2.5616315} _1.5330059 _-3.62888 j=11 _{2.480026578 -39.107658 -15.520689} _17.536450 _9.9221829 _-10.6551 j=12 _{-1.453592191 -2.1011843} _7.919931 _{0.636391 -6.0713031} _-2.286 j=13 _1.75163238 _{2.3969273 -9.5042529} _-0.882554 _5.6912695 _3.250705 Output layer

nodes wji (weight from node i in the hidden layer to node j in the output layer )_i=6 _i=7 _i=8 _i=9 threshold (Output layer θj)

j=14 _{-1.08046 1.60290587 -1.100723 0.81477985} 2.167988

i=10 i=11 i=12 _i=13

8.481029937 0.56304 -3.44043 -3.0771

Table 3 .

Weights and threshold levels for the ANN model at 0.1 H of wall high.

Hidden layer

nodes wji (weight from node Hidden layer i in the input layer to node j in the hidden layer ) threshold (θj)Hidden layer

i=1 i=2 i=3 i=4 i=5

j=6 _0.27950487 _0.0756950 _{1.2699631 -1.7817459 -0.5550618} _-0.62897 j=7 _{-0.247027045 -1.3032092 -0.2390725} _7.3913537 _1.7243826 _-10.0047 j=8 _{-0.716541153 -9.6606539} _1.8950992 _1.9920320 _4.2273022 _-3.63998 j=9 _0.395758689 _0.7866268 _0.9958922 _{4.9979926 -0.0525373} _-14.35 j=10 _-0.331279365 _{0.0537805 -2.2000709} _1.0060188 _3.0667343 _1.08751 j=11 _1.317851585 _{11.388767 -1.5486832 -1.6122216 -6.2210557} _3.700381 j=12 _{-0.578333465 -2.5401480 -1.0605415 -2.3728923 -0.0565377} _11.81053 Output layer

nodes wji (weight from node i in the hidden layer to node j in the output _{layer )} Output _layer

threshold

(θj)

i=6 i=7 i=8 i=9

j=13 _5.246844233 _{5.431759 -5.621436 2.40803975}

-2.79486

i=10 i=11 i=12

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Table 4.

Weights and threshold levels for the ANN model at 0.2 H of wall high.

Hidden layer

nodes wji (weight from node Hidden layer i in the input layer to node j in the hidden layer ) threshold (θHidden layer j)

i=1 i=2 i=3 i=4 i=5

j=6 _0.391790855 _0.3816751 _{0.6224678 -6.4928588} _0.1221245 _-11.434 j=7 _0.019952133 _{11.891015 -4.2254759 -4.1367035 -1.4577617} _4.847053 j=8 _{-0.84638404 -0.2965388} _0.0711544 _4.4321223 _0.7001051 _1.314577 j=9 _{-1.990180366 -2.6094724 -3.3706394 -1.4760052 -0.0559774} _-8.57675 j=10 _0.072980109 _0.3277183 _{6.6257771 -0.2986818 -9.0191054} _-2.76859 j=11 _-1.759690203 _{0.0864652 -0.3331401} _6.2836101 _2.3977018 _0.006674 j=12 _-0.091389215 _0.0854995 _{0.0590207 -0.2838246} _1.3291375 _1.157792 Output layer

j=13 _2.31224768 _{0.777605 -6.619474 1.71248334}

1.597973 i=10 i=11 i=12

-1.721428456 2.10436999 3.023217

Table 5.

Weights and threshold levels for the ANN model at 0.3 H of wall high.

Hidden layer

nodes wji (weight from node Hidden layer i in the input layer to node j in the hidden layer ) threshold (θHidden layer j)

i=1 i=2 i=3 i=4 i=5

j=6 _0.411238992 _0.3899691 _{0.6097453 -6.4922599} _0.1504665 _-11.4348 j=7 _0.597418448 _3.8632831 _0.3658912 _8.4691217 _0.8938177 _-5.35167 j=8 _0.631972925 _4.1490121 _0.7382095 _11.265575 _1.2177835 _-6.78746 j=9 _{-0.486634828 -3.0284970 -7.5812416} _0.7726993 _0.9558808 _-2.68351 j=10 _{-0.03368563 -1.3631111} _0.3607524 _{9.0633259 -0.4280160} _-9.86609 j=11 _0.216736635 _0.7995455 _{3.8100486 -0.9792275 -6.8107268} _-1.59383 j=12 _{-0.181082307 -0.4682637} _{0.4740155 -2.7330037} _0.2461733 _6.669836 Output layer

j=13 _{2.311915414 4.94931342 -4.754895 4.39740247}

-0.32809 i=10 i=11 i=12