How to Structurally Design a Concrete Slab Culvert? RC Slab Deck Design Using the FORTRAN-95 Program

TABLE OF CONTENTS

Contents

ACKNOWLEDGEMENTS

LIST OF FIGURES

LIST OF TABLES

LIST OF ABBREVIATIONS

ABSTRACT

1 Introduction
1.1 Motivations
1.2 Problem Statement
1.3 Objective
1.3.1 Specific Objective
1.4 Scopes and Limitations
1.4.1 Scopes
1.4.2 Limitations
1.5 Methodology for the Design & Program Development
1.5.1 AASHTO LRFD Method
1.5.2 Design Methodology
1.5.3 Writing the FORTRAN Code and Testing
1.6 Substructure of slab Culvert

2 Literature Review
2.1 Overview
2.2 Slab Culvert Structure Usage in Ethiopia
2.3 Concrete Slab Culvert
2.3.1 At Grade Slab Culvert Typical Drawing
2.3.2 At Fill Slab Culvert Typical Drawings
2.4 Reinforced Concrete Slab Culvert Deck
2.5 Construction Material Strength of Slab Culvert Deck
2.5.1 Concrete
2.5.2 Steel Reinforcement Bar
2.5.2.3 Fatigue Strength
2.6 Types of Culvert
2.7 Design Problems of Slab Culverts
2.8 Considerations for the Design of Concrete Slab Culvert Deck
2.8.1 Design Method
2.8.2 Design Philosophy
2.8.3 Loadings of Slab Culvert
2.8.4 Ultimate Limit States
2.8.5 Serviceability Limit States
2.8.6 Fatigue Limit States
2.9 Substructure Design of Slab Culvert
2.10 ERA Slab Culvert Design Practice
2.10.1 Concrete Slab Culvert Deck Design Practice
2.10.2 Substructure Design Practice

3 Analysis and Design
3.1 Definition of Slab Culvert Design
3.2 Dimensioning of Culverts
3.2.1 Span Determination of Slab Culvert [S]
3.2.2 Deck Width Determination of Slab Culvert [RW,TSW]
3.2.3 Edge Beam Dimensions
3.2.4 Minimum Deck Thickness
3.2.5 Fill Height of the Deck [H]
3.2.6 Post and Railing Dimensions
3.3 Live Load Interior and Edge Strip Width Computation (EI, EE)
3.3.1 Interior Strip Width
3.3.2 Edge Strip Width
3.4 Loadings and Combinations
3.4.1 Permanent load
3.4.2 Temporary Load
3.4.3 Analysis for Design Moment and Shear in the Interior Strip
3.4.4 Analysis for Design Moment and Shear in the Edge Strip
3.4.5 Absolute Maximum Design Moment and Shear force of the Slab Deck
3.4.6 Calculation of Bending Moment and Shear Force Resistance
3.5 Design of Slab Culvert
3.5.1 Thickness check for Limit State
3.5.2 Reinforcement Bar Design
3.6 Fatigue Limit State Check
3.6.1 Investigation of Main Reinforcement Bar Stress for Fatigue Limit State
3.6.2 Allowed Stress of Main Reinforcement Bar for Fatigue Limit State

4 Program Development
4.1 General
4.2 Programming Development Process
4.3 Algorithm for Slab Culvert Deck Design
4.4 Input and Output
4.4.1 Input Data
4.4.2 Output Data

5. Results and Discussions
5.1 Concrete Slab Culvert Deck Design Input
5.2 Concrete Slab Culvert Deck Design Output
5.2.1 At grade Slab Culvert Deck Output
5.2.2 At Fill Slab Culvert Deck Output
5.3 Discussions on the Design Results
5.3.1 At Grade Slab Culvert Output
5.3.2 At Fill Slab Culvert Output
5.3.3 Distribution Reinforcement Bar
5.3.4 Temperature & Shrinkage Reinforcement Bar

6 Conclusion and Recommendation
6.1 Conclusion
6.2 Recommendation

REFERENCES

Appendix 1 DevelopedProgram Script

Appendix 2 Input Data Example for At Fill and At Grade Slab

Appendix 3 ERA 2002 Design Manual for Slab Culvert Deck

Appendix 4 Ethiopia Roads Axle Load Data

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my advisor Dr. Ing. Bedilu Habte. He provided me with invaluable information and list of reference materials necessary for the successful completion of my work. He has guided me throughout the preparation of this thesis by giving me feedback on time and continuously encouraging me.

Ethiopian Roads Authority which granted me a full scholarship for this MSc program. It has been supporting me until the completion of the program. I would like to extend my great appreciation and gratitude to the authority for giving me this opportunity.

Finally, I would like to express my great thanks to my classmates and collogues for sharing reference books and important information.

LIST OF FIGURES

Figure 1.1 Equivalent per Meter Width Model for Analysis

Figure 1.2 Equivalent per Meter Width Loading Model for Analysis

Figure 2.1 Typical Longitudinal Section of At Grade Slab Culvert

Figure 2.2 Typical X-Section of At Grade Slab Culvert

Figure 2.3 Typical At Grade 3D View of Slab Culvert

Figure 2.4 Typical Longitudinal Section of At Fill Slab Culvert

Figure 2.5 Typical X-Section of At Fill Slab Culvert

Figure 2.6 Typical 3D View of At Fill Slab Culvert

Figure 2.7 Steel Stress Strain Curve

Figure 2.8 Characteristics of the Design Truck

Figure 2.9 Characteristics of Design Tandem

Fig re 2. 10 5 Year ERA Front A!le Load Reading With Respect To Design Code

Fig re 2. 11 5 Year ERA Front A!le Load Reading With Respect To Design Code

Figure 2.12 Vehicle Loading Configuration for Deflection Computation

Figure 2.13 Typical Masonry Abutment

Figure 2.14 Typical Masonry Wing Wall Drawing

Figure 2.15 Typical Masonry Pier Drawing

Figure 3.1 Typical X-Section of At Grade Slab Culvert Deck

Figure 3.2 Typical X-Section of At Fill Slab Culvert

Figure 3.3 Soil Load on the Slab Deck

Figure 3.4 Live Load Distribution with Fill Height

Figure 3.5 Design Truck Moving to the Right Loading Configuration

Figure 3.6 Design Truck Moving to the Left Loading Configuration

Figure 3.7 Design Tandem Moving to the Right Loading Configuration

Figure 3.8 Design Tandem Moving to the Left Loading Configuration

Figure 3.9 Internal Forces in a Cracked Slab without Stirrups

Figure 3.10 Equivalent Rectangular Stress Block

Figure3.11 Uniform Service Loading for Elastic Deflection Calculation

Figure 3.12 Design Truck Loading Configuration for Deflection Computation

Figure 3.13 Design Tandem Loading Configuration for Deflection Computation

Figure 3.14 FBD of Strain and Stress of the Deck X-Section

Figure 3.15 Loading Arrangement for Fatigue Limit States

LIST OF TABLES

Table 2.2 Values of Secant Modulus of Elasticity Ecm in GPas

Table 2.3 Multiple Presence Factors “MPF”

Table 2.4 Dynamic Load Allowance for Track Live Loads (IM)

Table 2. 5 Load Factor

Table 3.1 SlabCulvert Recommended Width According to ERA Design Manual Standard

Table 3.2 Multiple Presence Factors “MPF”

Table 3. 3 Multiplication Factor for Number of Lanes

Table 5. 1 Developed Program Thickness Output for At Grade Slab Culvert Deck

Table 5. 2 Developed Program Reinforcement Output for At Grade Slab Culvert

Table 5. 3 Developed Program Slab Thickness Output for At Fill Slab Culvert Deck

Table 5. 4 Developed Program Main Reinforcement Output for At Fill Slab Culvert Deck

Table 5. 5 Developed Program Dist. Reinf. Output for At Fill Slab Culvert Deck

Table 5. 6 Developed Program Shr. & Temp. Rein. Output for At Fill Slab Deck

LIST OF ABBREVIATIONS

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ABSTRACT

Concrete slab culvert is an important structure used to convey trucks and pedestrian along a road corridor or in one of a range of other situations. This structure is highly constructed in highway road projects in Ethiopia. In this study, a FORTRAN program is developed for the structural design of reinforced concrete slab culvert deck according to the provisions given in AASHTO LRFD Bridge 2005 Edition.

The developed program is expected to assist the structural designers and users to design the superstructure part of a reinforced concrete slab culvert deck efficiently with great accuracy.

Both at grade and at fill slab deck thicknesses are computed according to the specification specified in AASHTO LRFD Bridge 2005 Edition. The reinforcement bars are also designed based on the requirements specified in the code.

The program is developed in four steps. The first step is to define and analyze the problem; the second step is to develop an optimal solution and designing the program, the third step is coding the program and the final step is testing and documenting the program.

FORTRAN 95 is a programming language used in the fields of scientific, numerical, and engineering fields. In this thesis, this language has been used to develop the program for the structural design of reinforced concrete slab culvert deck.

The input data for at grade and at fill slab culverts are saved on a note pad in the external file folder which constitute the material properties, geometric features and proposed diameter of reinforcement bars of the slab culvert and its deck in the folder which contains FORTRAN 95 program. The output data is written on the note pad in the external folder based on the format assigned for each output in the folder which contains the design results of slab deck thickness and area, spacing and length of main, distribution and temperature reinforcement bars. Besides Edge beam design parallel to the traffic is executed and shown in the output result by the developed program.

1 Introduction

Slab culverts are important hydraulic structures used in the construction of highway roads. These structures are widely constructed in Ethiopia second to pipe culverts. Slab culverts are classified in to two based on the fill height (embankment height). These are namely at-grade and at fill slab culverts.

Slab culverts are used in Ethiopia to convey water across a road corridor or in one of a range of other situations. Slab culverts must be designed to convey this flow in an acceptable way, considering the hydraulic conditions and the required performance (level of flood immunity) of the road. Environmental and other requirements may also need to be incorporated depending on the specific circumstance.

As road transportation is the main and vital means to transport people and goods in Ethiopia, the failure of culverts and bridges have enormous impact on the economic development of the country, access to health institutions and other daily activities of the society.

This thesis focuses on the development of a FORTRAN 95 program for the structural design of the superstructure part of a concrete slab culvert.

1.1 Motivations

It is believed that standardizing the construction of slab culverts as well as using them widely benefits our country greatly. Some of the advantages of slab culverts are:

- Slab culvert structure is relatively simple structure to design and construct.
- The construction methodology is simple compared to other culverts like pipe culvert and box culvert.
- Slab culvert may not need protection like pipe culverts.
- Slab culverts do not use resources and skilled laborers during construction time like slab bridges, pipes and box culverts.
- Unlike pipes, it can be constructed for low fill or no fill roads.

1.2 Problem Statement

Generally it is assumed that the structural or civil engineer uses proper design manuals and specifications in designing a given structural part. This standard work practice is highly violated in our country due to many reasons. There are many design codes and specifications that are available for buildings and bridges which are based on our country’s conditions. They are sometimes not properly referenced probably due to lack of proper knowledge about their importance or lack of enough attention. Most designs are as a result not economically designed or they are not up to the expected loading conditions and serviceability of the structure during its life time.

Even though ERA design manual 2002 edition prepared a design manual in the table form for the design of concrete slab culvert deck, some design offices and consulting firms in the country design the culvert deck by referencing different design codes and using personal judgment. It is known that, the analysis and design of a concrete slab culvert deck is not a big challenge for engineers in our country. However, in most cases the final design of the structure varies from person to person. This variation is caused by the use of different design codes and manuals adopted as well as the designer’s experience and personal judgment. This lack of uniformity in the design outputs is a challenge for the safety and a durability of the structures constructed.

The substructure part of the slab culvert is almost all the time constructed by the drawing specification stated in the Ethiopian Roads Authority (ERA) design manual 2002 edition or Addis Ababa City Roads Authority (AACRA) design manual 2004 edition. There are detailed geometric specifications developed by ERA or ACCRA for inner and outer wing wall slopes, wing wall thickness, and wing wall deflection angle, abutment top thickness for superstructure placing, outer abutment slopes and inner abutment slopes. However the design of the superstructure part is sometimes performed by the various offices although there are general guidelines available in a table form similar to the substructure part of the slab culvert.

1.3 Objective

The general objective of the thesis is to help in achieving reasonable uniformity in the superstructure design of at-grade and at fill slab culvert deck in Ethiopia at the national level. It further expands the slab culvert design practice in Ethiopia by developing design standards. The study has been conducted based on the AASHTO LRFD Bridge design manual 2005 edition with the aim of establishing basic design techniques for economical design of at-grade or at fill slab culvert designs for all roads throughout the country. Additionally, this paper analysis the current ERA Bridge Design Manual 2002 and forwards improvement ideas.

It is the intention of this thesis to propose the standardization of the design of the super structure part of the slab culvert by performing proper study. The proposal includes an optimal solution to design slab culvert deck incorporated in a proper design and specification manual. It is believed that this contributes to higher quality designing of concrete slab culvert decks by further strengthening the existing provisions that ensure sufficient strength, safety and higher durability.

1.3.1 Specific Objective

The specific objective of this thesis is to develop a computer program for the analysis and design of the superstructure part of at grade and at fill concrete slab culvert based on AASHTO LRFD Bridge design manual 2005 edition, which is a nationally accepted standard. Besides, some references like Ethiopian Roads Authority Bridge Design Manual 2002 edition and AACRA Bridge Design Manual 2004 edition, specifications and Civil Engineering Books are used as a literature review.

Also the paper is expected to describe strength and drawbacks of the nationally used design manual which is namely ERA Bridge Design Manuals 2002 edition for the design of slab culvert deck.

The developed program is also expected to reduce slab culvert superstructure designing time, assure uniformity of design output in the national level and to avoid errors. The outputs of the developed program will be:

- Optimum total slab deck thickness D.
- Main reinforcement bar spacing for a given diameter size.
- Distribution Reinforcement spacing for a given diameter size.
- Temperature and Shrinkage reinforcement spacing for a given diameter size.
- The program also fixes edge beam dimensions and designs edge beam for main reinforcement bar.

1.4 Scopes and Limitations

The program development for the structural design of concrete slab culvert deck has scopes and limitations in the long run.

1.4.1 Scopes

The scope of this thesis is to help in achieving reasonable uniformity in the design of slab culvert deck in Ethiopia. Besides, it may be used for research purpose on the slab culvert and slab bridge designs in the research centers or universities available in the country.

1.4.2 Limitations

1) Main reinforcements are placed parallel to center line of roadway.
2) The bottom of the slab is assumed level.
3) Center to center of support is assumed perpendicular to supports.
4) The developed program is used only for the design of singly reinforced concrete slab culvert deck.
5) The program is used for monolithically casted concrete slab culvert deck.
6) The program is developed for simply supported single span culverts.
7) We can use the program for skewed angle culverts in a conservative way. Up to 25 degree skew angle, the skewness effect is negligible.
8) The additional amount of concrete thickness for cross fall is not considered for the flexural design, but the weight is included as dead load.

1.5 Methodology for the Design & Program Development

The research method is analytical research type rather than descriptive research method. The methods of research utilized in descriptive research are survey methods of all kinds, including comparative and correlation methods. In analytical research, on the other hand, the researcher has to use facts or information already available and analyze these to make a critical evaluation of the material.

Due to this, the thesis work focuses on the proper utilization of empirical and derived formulas and information available in AASHTO LRFD Bridge Design Manual 2005 edition, ERA Bridge Design Manual 2002 edition, Structural Engineering reference books and different journals from the Internet that are related to the thesis works for the analysis and design of slab culvert deck and development of the program.

1.5.1 AASHTO LRFD Method

Load and Resistance Factor Design (LRFD) method is a design methodology that makes use of load and resistance factors based on the known variability of applied loads and material properties. The LRFD Specifications are written in a probability-based limit state format requiring examination of some, or all, of the four limit states defined below for each design component of a bridge or a culvert.

- Strength limit state
- Serviceability limit states
- The Fatigue and Fracture Limit State
- The Extreme Event Limit State

The design incorporates all the limit state rules except extreme event limit state during developing the program for the structural design of slab culvert deck.

AASHTO LRFD method was used for the development of ERA bridge design manual 2002. Since this design methodology is highly used in Ethiopia, It is preferred to use this design manual for the design culvert deck and development of the program.

1.5.2 Design Methodology

An approximate method of analysis in which the deck is subdivided in to strips perpendicular to the support. The design of slab culvert is performed on a per meter width of slab deck. In general maximum design shear force and maximum design moment per meter strip width are used for the design of reinforcement bar and deck thickness.

An approximate method of analysis in which the deck is subdivided into strips perpendicular to the supporting components shall be considered acceptable for decks. Due to loading differences they are divided as interior strip and edge strip. These strips are designed as a simply supported reinforced concrete beam structure with single reinforcement bars.

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Figure 1.1 Equivalent per Meter Width Model for Analysis (Author’s own work, 2013)

During designing the program, it is assumed that both at grade and at fill slab culverts are simply supported structure. Therefore, In the case of simply supported slab culverts, only the bending moments and shear forces are considered for design purpose, the other internal forces

(Torsion and Axial forces) are insignificant.

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Figure 1.2 Equivalent per Meter Width Loading Model for Analysis (Author’s own work, 2013)

For uniform loads, the following general formulas are used for moment and shear force determination for the strip width.

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For concentrated axle loads the following formulas are used for moment and shear force determination for the interior and exterior strip width. During the calculation of the cumulative maximum shear force and bending moment for multiple moving axle loads, Influence lines computations are used.

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But in this case, the influence line method is used since two or more concentrated loads exist at given time due to design truck and design tandem axle loads.

The maximum moments and shear forces are calculated due to live loads and dead loads for interior and edge strips. The absolute maximum shear and bending moments of the interior and edge strips are used for the design of reinforced concrete slab deck thickness and reinforcement bar according to single reinforcement design method of AASHTO LRFD Bridge Design Manual 2005 Edition.

The computed values of slab deck thickness and reinforcement bars are checked for strength limit state, serviceability limit state and fatigue limit state. Therefore, the interior and edge strip of the slab culvert deck is designed for the absolute maximum moment and shear obtained from the analysis. The positive main reinforcement for the slab deck is calculated on the basis of formula developed in AASHTO Bridge Design Manual 2005. Distribution reinforcement is taken as a percentage of the main reinforcements required for positive moments as specified in the design code of AASHTO. Temperature and Shrinkage reinforcement bars are used based on the concrete cross sectional area and the yield strength of reinforcement bar used as stated in the design code.

1.5.3 Writing the FORTRAN Code and Testing

Computer is machine that performs tasks such as calculation or electronic communication. This is done under the control of a set of instructions called a program. The physical computer and its components are known as Hardware. The program that runs the computer is called software. The developed program use FORTRAN 95 software for the structural design of concrete slab culvert deck.

1.5.3.1 Programming Development Process

The steps in developing a program are similar to other problem solving procedures. Here there are four major steps that are followed during the design and development of the program.

- The Problem is Defined and Analyzed
- A solution is Planned and Designed
- A Program is Coded
- The program is finally Tested and Documented

The following general steps are used for developing the program:

- First, will insert material property of the concrete slab deck like steel tensile yield strength and concrete cylindrical compressive strength and material densities like concrete, bituminous layer and fill material.
- Second, inserting dimensions of the slab culvert like clear span, road way width, curb width, curb depth, support width, bituminous thickness, railing depth, railing width, post thickness, post width, post height and fill height.
- Third, inserting the used reinforcement bar dimensions for main, distribution and temperature reinforcement.
- During inserting the dimensions, different inputs for at grade and at fill slab culvert are prepared in separate files.
- Fourth, Calculation of slab thickness using do loop by iterating input and calculated variables. The do loop executes the depth determinations first for ULS & then for SLS successively and independently.
- Fifth, calculate the design moment and shear forces for interior and edge strips and taking the maximum load for both cases for main reinforcement design purpose.
- Six, Calculation of reinforcement bars and checking of the reinforcement bar used for serviceability limit state of crack width and Fatigue limit state.
- Finally, we will prepare and format all the outputs of the design. These outputs are slab thickness and spacing, length & area for main, distribution and temperature reinforcement bars.
- The program also recommends and designs longitudinal edge beam to resist higher stress at the edge due to stress concentration at the line of discontinuity. The design output for edge beam include beam width, beam height and main reinforcement bar at the bottom side of the beam.

1.6 Substructure of slab Culvert

Substructure units of slab culverts are constructed by masonry structure and it constitutes abutments. However, sometimes double spans of slab culverts may be constructed and in that case a pier between abutments is used. Both abutments and piers will be supported on spread footings (in the judgment of the engineer responsible for exploration), the minimum depth of the subsurface exploration shall extend below the anticipated bearing level.

If the foundation material of slab culvert is good then the concrete footing need not be recommended by the engineer except lean concrete (C5). However, sometimes the footing of the slab culvert is poor i.e the foundation material is black cotton soil with ground water, in that case the poor foundation material is replaced by select material or hard core material and then after the class C concrete footing is recommended.

2 Literature Review

2.1 Overview

This chapter briefly summarizes definition of culvert, Slab culvert deck conceptual information, loading issues, analysis and FORTRAN 95 program used for slab culvert deck design which are discussed in the academic literatures. Also this chapter includes a summary of culvert design practices in Ethiopia and procedures as determined based on ERA design manual 2002 edition. This information informs the evaluation of the developed program with respect to the practices and procedures and provides insight about loadings and analyses issues encountered during developing the program.

2.2 Slab Culvert Structure Usage in Ethiopia

Form the minor structure constructions; slab culverts are the highly constructed hydraulic structures second to pipe culverts in major road projects in Ethiopia. These structures like bridges are more sensitive structure compared to buildings and dams since they are highly exposed to repeated dynamic axle loads. [Condition Survey]

2.3 Concrete Slab Culvert

A slab culvert is a structure that is designed hydraulically to take advantage of submergence to increase hydraulic capacity. It is also a structure used to convey surface runoff through embankments. The traditional definition of slab culvert is based on the span length rather than function or structure type. The definition of slab culvert includes the opening measuring less than 20ft (6.1m) along the center line of the road. Structure over 20ft (6.1m) in span parallel to the road way is called bridge. [11]

Slab culverts are three major structural parts which are namely foundation footing (if any), masonry or concrete abutments and super structure reinforced concrete deck. The intention of the thesis focuses on program development for the structural design of the super structure part of the slab culvert based on mainly AASHTO LRFD Bridge Design Manual 2005 Edition.

2.3.1 At Grade Slab Culvert Typical Drawing

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Figure 2.1 Typical Longitudinal Section of At Grade Slab Culvert

(Adapted from Ethiopian Roads Authority, Standard Detail Drawings, 2002)

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Figure 2.2 Typical X-Section of At Grade Slab Culvert

(Adapted from Ethiopian Roads Authority, Standard Detail Drawings, 2002)

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Figure 2.3 Typical At Grade 3D View of Slab Culvert

(Adapted from Ethiopian Roads Authority, Standard Detail Drawings, 2002)

2.3.2 At Fill Slab Culvert Typical Drawings

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Figure 2.4 Typical Longitudinal Section of At Fill Slab Culvert

(Adapted from Ethiopian Roads Authority, Standard Detail Drawings, 2002)

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Figure 2.5 Typical X-Section of At Fill Slab Culvert

(Adapted from Ethiopian Roads Authority, Standard Detail Drawings, 2002)

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Figure 2.6 Typical 3D View of At Fill Slab Culvert

(Adapted from Ethiopian Roads Authority, Standard Detail Drawings, 2002)

2.4 Reinforced Concrete Slab Culvert Deck

The super structure part of slab culvert is a reinforced concrete deck which is mostly cast in place. Sometimes precast reinforced concrete deck is used during the construction of slab culvert in Ethiopia. Cast-in-place concrete structures are often constructed monolithically and continuously. They usually provide a relatively low maintenance cost and better earth quake resistance. Besides, precast culvert deck is not economical due to loading and unloading of precast concrete and transportation cost. [12].

2.5 Construction Material Strength of Slab Culvert Deck

The strength and other data of concrete and steel are determined based on standard tests. [6]

2.5.1 Concrete

Concrete is manmade structure which is made by proper mixing of Portland cement, sand and water.

2.5.1.1 Compressive Strength of Concrete

Generally, the term concrete strength is taken to refer to the uniaxial compressive strength as measured by a compressive test of a standard test cylinder, because this test is used to monitor the concrete strength for quality control or acceptance purpose.

The specified compressive strength Fc is measured by compression tests on 15cm by 30cm cylinder tested after 28 days of moist curing. This is the strength specified on the construction drawings and used in the calculations.

2.5.1.2 Tensile Strength of Concrete

The tensile strength of concrete falls between 8 and 15 percent of the compressive strength. The actual value is strongly affected by the type of test carried out to determine the tensile strength, the type of aggregate, the compressive strength of concrete and the presence of a compressive stress transverse of the tensile stress.

Although the tensile strength increases with an increase in the compressive strength, the ratio of the tensile strength to the compressive strength decreases as the compressive strength increases. Thus, the tensile strength is approximately proportional to the square root of the compressive strength. The mean split cylinder strength Fct from large number of tests of concrete from various localities has been found to be. [10]

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ACI Code Section 9.5.2.3 defines the modulus of rapture for use in calculating deflections as

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Here, FR and Fc are in psi.

Where λ = 1.0 for normal weight concrete.

2.5.1.3 Deformation Properties of Concrete

The values of material properties required for calculation of instantaneous and time dependent deformations of concrete depend not only upon the grades of concrete but also upon the properties of the aggregates and other parameters related to the mix design and the environment.

Any idealized stress - strain diagram which results in prediction of strength in substantial agreement with the results of compressive tests may be used. [7]

Table 2.1 Values of Secant Modulus of Elasticity Ecm in GPas

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(Ref. Table 2.5, Structural Use of Concrete, Ethiopian Building Code of Standard (EBCS-2), 1995.)

The vales Ec are based on the following equation: [4]

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AǤ Creep and Shrinkage of Concrete

Values of shrinkage and creep shall be used to determine the effects of shrinkage and creep on the loss of prestressing force in bridges other than segmentally constructed ones. These values in conjunction with the Moment of Inertia shall be used to determine the effects of shrinkage and creep on deflections. [7]

2.5.2 Steel Reinforcement Bar

Steel reinforcement bar is a structure which is made in the furnace by properly mixing mainly iron and carbon. This is finally molded to a required shape, size and length. Because concrete is week in tension, it is reinforced with steel bars or wires that resist the tensile stresses. The most common types of reinforcement for non prestressed members are hot rolled deformed bars and wire fabric.[10]

2.5.2.1 Strength of Steel

The yield stress Fy and the tensile strength Fsk are defined respectively as the characteristics value of the yield load, and the characteristic maximum load in direct axial tension, each divided by the nominal cross sectional area.

2.5.2.2 Stress – Strain Diagram

The behavior of steel reinforcement is usually characterized by the stress–strain curve under uniaxial tension loading. Typical stress–strain curves for steel Grade 300 and420 are shown in Figure below. The curves exhibit an initial linear elastic portion with a slope calculated as the modulus of elasticity of steel reinforcement Es= 200,000 MPa; yield plateau in which the strain increases (from εy to εh) with little or no increase in yield stress (fy); a strain-hardening range in which stress again increases with strain until the maximum stress (fu) at a strain (ɛu) is reached; and finally a range in which the stress drops off until fracture occurs at a breaking strain of εb. [12]

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Figure 2.7 Steel Stress Strain Curve

(From James K. Wight and James G. Mac Gregor, Reinforced Concrete Mechanics and Design. Fifth Edition.)

2.5.2.3 Fatigue Strength

Some reinforced concrete elements, such as bridge decks, are subjected to a large number of loading cycles. In such cases, the reinforcement may fail in fatigue. Fatigue failures of the reinforcement will occur only if one or both of the extreme stresses in the stress cycle is tensile.[10]

2.6 Types of Culvert

In general, Culvert structures can be classified by the method of construction or construction material used. The following are the major culvert types in the road sectors. [11]

1. Prefabricated concrete pipes of circular section, hereinafter called 'concrete pipe culverts'.
2. Prefabricated corrugated metal pipes and pipe arches, hereinafter called 'metal culverts'. These may be of Steel or Aluminum construction.
3. Box culverts may be cast in place or prefabricated. In-situ reinforced concrete box culverts constructed in accordance with culverts other than pipe culverts are hereinafter called portal culverts or rectangular culverts.
4. Slab culverts may be cast in place or prefabricated. In-situ reinforced concrete slab culverts constructed in accordance with culverts other than pipe culverts are hereinafter called portal culverts or rectangular culverts.
5. Masonry arch culverts
6. Timber culverts

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