nazli azimikor_composite deck design report

CIVL 510 Design of Steel-Concrete Composite Decks Nazli Azimikor

Nazli Azimikor_Composite Deck Design Report.docx 4/28/2010 PAGE 1 OF 23

Design of Steel-Concrete Composite Decks

For: Dr. Stiemer CIVL 510 University of British Columbia By: Nazli Azimikor 41055021 Date: April 24, 2010

Abstract The efficiency and structural performance of steel-concrete decks is highly improved in composite action. This is because the high tensile resistance of steel complements the compression strength of concrete in bending. Composite steel-concrete floor decks are typically constructed by connecting steel girders with concrete slabs by means of shear connectors such as Nelson studs. As such, they may be analyzed and designed as composite T-beams based on principals of structural mechanics. The objective of this project is to produce a tool for the rapid evaluation of applied loads and analysis of member bending moment and shear capacity based on user-specified loads, member sizes and material strengths. For this purpose, a spreadsheet and a complementary Visual Basic software tool have been developed. The spreadsheet accepts information about the applied loads, the dimensions of the floor, member sizes and material strengths and strength factors from the user. Given the above information, the spreadsheet then calculates cross-sectional properties such as the location of neutral axis, the section's transformed moment of inertia and section modulus, as well as its bending moment and shear capacity in a step-by-step procedure according to established codes and standards. It also determines the number of shear connectors required to ensure composite action is achieved. As such, the spreadsheet developed for this project can be used as an efficient design tool that is easy to follow and document. The complementary Visual Basic software was developed based on the same procedures used in the spreadsheet and outputs the same cross-sectional properties. Therefore, it is a great analysis tool that allows the rapid evaluation of cross-sectional properties for cases in which a number of iterations are required.



Table of Contents Design of Steel-Concrete Composite Decks ................................... 1

Abstract ....................................................................................... 1Table of Contents ............................................................................ 2Table of Figures .............................................................................. 21.0 Introduction ............................................................................... 32.0 Background ............................................................................... 33.0 Statement of the Problem and the Solution Approach .............. 44.0 Microsoft Excel Spreadsheet .................................................... 5

4.1 Calculation of Specified and Factored Loads ....................... 64.2 Analysis and Design Procedure ............................................ 6

5.0 Visual Basic Analysis Tool ..................................................... 125.1 Limitations on the Use of the Visual Basic Analysis Tool . 135.2 Program Set-Up .................................................................. 13

6.0 Conclusion .............................................................................. 157.0 Bibliography ........................................................................... 16

Appendix A: Visual Basic Code ............................................... 17

Table of Figures Figure 1: Composite T-Beam cross section . Error! Bookmark not defined.Figure 2: Stress distribution along member's width and equivalent width. .............................................................................................. 7Figure 3: The possible locations of neutral ..................................... 8



1.0 Introduction Design of composite steel and concrete structures has become an essential component of engineering due to the widely popular use of the two materials in construction. Applications of design with composite sections range from buildings, to bridges, to foundations, and to special structures.

The high tensile resistance of steel and the compression strength of concrete complement each other in construction and their combination makes for highly efficient design. Therefore, steel-concrete composite sections can be advantageous in that they allow for use of shallower steel beams in construction, consequently reducing the steel weight. The highly efficient cross section also means stiffer floors and/or decks for the same depth and therefore increased span. To ensure composite action between concrete and steel, shear connectors such as Nelson studs are required. Therefore some of the disadvantages of composite section design can be the extra cost of shear connector and their presence as a tripping hazard during construction. Also, during service, the vibration of the floor/deck may sometimes be an issue due to the shallow depth of the sections. Finally, design of composite sections requires more engineering time and effort. Therefore, it is worthwhile to develop tools to help with rapid analysis and design of steel-concrete sections and to assess the usefulness of their application for projects. The objective of this project is to develop simple and easy to use tools to allow rapid engineering calculation and documentation. As such, a spreadsheet has been developed that takes user input information with regards to loads and dimensions of the section and performs step-by-step analysis to aid with design. For the purposes of very quick

analysis, complementary visual basic software has also been developed that calculates important section properties upon the click of a button. This report provides a general overview of the theory behind composite design and construction. The approach taken to develop the above design tools is discussed. Thereafter, an overview of the step-by-step design procedure outlined in the spreadsheet is provided, followed by a detailed description of the methods used in implementing the complementary visual basic software. The directions for use and the limitations of each tool are also discussed. It is our assumption that the engineer using the spreadsheet and the complementary software presented in this project has an adequate grasp of the fundamentals that govern how composite structures work. As such, the procedures outlined in the subsequent sections are derived mainly by using codes of practice or by the direct application of prescribed equations as quick design procedures for composite members.

2.0 Background Composite structural members are made by joining a steel component to a concrete component. For the purposes of this project, the composite steel-concrete section for a deck is analyzed. Such a section consists of a steel member, such as a wide flange steel beam, connected to a concrete component, such as a floor slab. The connections between the materials are created by the use of shear connectors such as Nelson studs, as is shown in figure 1.



Figure 1: Composite T-Beam cross section (Bradford, 1999) When a bending member, such as a floor deck, is subjected to a positive bending moment, the top face of the member undergoes compression stress, while the bottom is subjected to tension. Composite sections are a great way to combine concrete and steel in constructing highly efficient bending members. High resistance against positive bending moments is achieved in composite sections because the high strength of concrete in compression complements that of steel in tension, resulting in resistance to the internal stresses. The complementary relationship between steel and concrete is most effective when there is an efficient connection at the interface of the two materials. This connection allows for transfer of shear forces at the interface between the materials, hence preventing their vertical separation. In the absence of such a connection, the forces developed in one component would not be transferred to the other component and the section would behave as two separate members. Note that although calculations may be completed based

on the assumption of 100 percent shear connection, this may not be possible due to various factors such as fatigue, installation, spacing, etc.

3.0 Statement of the Problem and the Solution Approach Composite section design can be a time consuming process with many steps involved in calculating shear transfer between the materials and full and partial moment resistance for a given section. Therefore, a well-laid-out spreadsheet or a computer program would be an asset to any engineer who completes composite section design on a regular basis. Although limit state design requires checks for both ultimate limit state and serviceability limit state, the focus of this project has been on ultimate limit states. The project presented here provides the following two complementary tools for rapid analysis and design of composite sections: 1. An Excel spreadsheet outlining the step by step analysis

procedure to aid with design according to codes and standards 2. A visual basic software implemented within Microsoft Excel

that simply outputs important section properties

Both the Excel worksheet and the Visual Basic software accept user input for material properties and sectional dimensions. The Excel worksheet also accepts specified loads and calculates factored applied bending moment and shear.



The main outputs of the spreadsheet and the software herein presented are the following: Effective width of concrete slab, b1 Moment resistance of composite section, Mr, for full and partial

shear transfer Sum of factored resistances of all shear connections, Qr, for 100

percent connections Transformed moment of inertia, It Transformed section modulus, St

In addition to the above outputs, the Excel spreadsheet performs checks to ensure conditions during construction are satisfactory and determines the number of shear connectors required. In the sections that follow, the functions of each tool and their methods of development are discussed in more detail.

4.0 Microsoft Excel Spreadsheet To ease with the ease of analysis and design of steel-concrete composite deck sections, and to provide a tool that allows easy and accurate documentation, a spreadsheet has been developed for this project that takes performs analysis tasks in a stepwise manner. Within the spreadsheet, the definition of all parameters and the symbols used to represent them are given. At each computational step, the equations used are clearly displayed, and where provisional clauses in codes and standards are used, they are referenced. The Excel spreadsheet developed for this project consists of four worksheets labeled as follows:

1. CISC Sections 2. Loads 3. User Input 4. Analysis

Worksheet "CISC Sections" is simply a database of steel sections that provides section properties for specified steel wide-flange members. The user need not utilize this worksheet. Worksheet "Loads" accepts user input loads, dimensions and limit states load factors and determines specified and factored applied bending moments and shear on the member. Worksheet "User Input" accepts material properties, factors, and sectional dimensions for the concrete deck, steel beam and shear connectors to be used in analysis and design. The worksheet is set up as to allow the user to specify whether the steel beam is a wide flange member or a built-up section from a drop-down menu. If a wide-flange section is used, then the worksheet will use the information from the "CISC Sections" worksheet to automatically display section properties of the specified section based on its designation. Another drop-down menu is incorporated to allow the user to specify whether the section is stiffened or not. For the concrete slab, the user is also able to specify whether the concrete slab was solid, poured on steel deck parallel to ribs or perpendicular to ribs. Worksheet "Analysis" uses the information in the above three worksheets and outlines the analysis and design procedures for the composite section in a stepwise manner. Major outputs of this worksheet are highlighted for clarity. The following section explains in detail each analysis step as laid out in this worksheet.



4.1 Calculation of Specified and Factored Loads The construction of composite sections consists of the following three major stages: 1. Stage 1: Steel beams/girders are installed 2. Stage 2: Decking and/or formwork is laid above the steel beams and wet concrete is poured 3. Stage 3: Concrete has hardened and acts together with steal During the first stage, the steel member must have enough capacity to withstand its own weight. At the second stage, the steel member and the concrete slab are still in non-composite action since concrete has not yet hardened. As such, the steel member must hold up its own weight as well as the live loads during construction due to the placement of decking/formwork and pouring of the concrete slab. Finally, once concrete has hardened at the third stage, steel member and the concrete slab must resist all specified loads in composite action. During stages 1 and 2, the steel beam is treated like a temporary structure. Work Safe B.C. requires that all temporary structures have capacity to resist a minimum of 2 kPa construction live load. This construction live load may require the selection of deeper steel section for construction purposes even though a shallower member would suffice once composite action is achieved. This may be a source of inefficiency in design where shallower members are crucial. The factored applied moment and shear are calculated according to the provisions of National Building Code (NBCC) 2005 and using the live and dead load factors as appropriate. These factored applied moment and shear are used in the analysis section to assess the strength of the section in composite action. It must be noted

that in calculating the live load, the area reduction factor Rf is utilized. This factor which is influenced by the tributary area of the composite deck section, is introduced in NBCC 2005 clause 4.1.5.9.(3) to account for the unlikelihood of the event that the entire specified live load is applied at the same location on the roof/floor deck. Section 17.11 of the Canadian Standard Association (CSA) S16-01 requires that the stresses in the tension flange of the steel section due to the loads applied before the concrete strength reaches 0.75f'c plus the stresses at the same location due to the remaining specified loads considered to act on the composite section shall not exceed Fy. Therefore, moments due to specified loads (unfactored) during phases 1, 2 and 3 of construction as previously discussed are also calculated in this section. These unfactored applied moments are then used in combination with the transformed section modulus to determine stresses during each construction phases. It is then ensured that the sum of these stresses does not exceed the yield strength of steel.

4.2 Analysis and Design Procedure In this section, each step of analysis and design as outlined in the spreadsheet are discussed in detail. The equations and CSA references employed at each step are also provided. Step 1: Check to ensure steel beam/girder has adequate capacity under construction loads As was mentioned before, before concrete is poured or hardened, the steel beams installed act as temporary structural members. The



moment resistance of the steel section is determined based on its yield strength and its plastic section modulus as follows:

(1) If the steel section's moment capacity as determined above is lower than the factored bending moment due to construction loads as previously discussed, then the user must specify a larger steel beam/girder. Next, section properties in composite action must be determined. Calculating the section's strength and its properties requires finding the location of its neutral axis. This is because at the neutral axis, the section does not experience any strains and therefore, this point is an important reference point in determining the magnitude and direction of internal forces and subsequently the section's bending moment capacity. The sections moment of inertia and section modulus are also dependent on the location of the neutral axis. The neutral axis can be located by satisfying equilibrium conditions. The location of the neutral axis may vary depending on the value of the compressive strength of concrete in relation to the tensile strength of steel. Assuming a rigid-plastic approach, the unfactored axial strength of steel can be determined by multiplying the area of steel in tension, As, by its yield strength, Fy. Similarly, the compressive strength of concrete is equal to the area of the concrete in compression, Ac, multiplied by the compressive resistance of concrete, fc. However, before the area of concrete in compression can be calculated, the effective width of the concrete slab must first be determined. The concept of effective width is useful in design of composite steel-concrete structures since the stress is non-

uniformly distributed along the width of the element as shown in Figure 2.

Figure 2. Stress distribution along the element width and equivalent width As can be seen from Figure 2, the actual width of concrete subjected to stress approaches infinity. However, for practical purposes, an equivalent width may be defined over which stresses can be assumed to be uniformly distributed. According to CSA standard S16-01, section 17.4.1, the effective width of the concrete slab in compression in a composite steel-concrete T-beam is taken to be the minimum of one quarter of the length of the concrete



member and the span of the member. This effective width is then used in the subsequent calculations. As mentioned before, the location of the neutral axis can be determined by establishing the location at which equilibrium of forces is achieved. For this the concept of shear flow is used. Steps 2 through 4: Determine the shear transfer Assuming 100 percent shear connection, the compressive resistance of concrete, Cr, and tensile resistance of steel, Tr, are calculated based on cross sectional dimensions and material properties. Tr and Cr are determined using the following two equations as provided in CSA S16-01 sections 17.9.5 and 17.9.6 respectively: C f A (2) T f A (3) When using equation (2), the effective width of concrete previously determined is used in calculating the area of concrete. According to CSA S16-01, section 17.9.5, the shear transfer is taken as the minimum of the two values of Tr and Cr. This logic allows for the satisfaction of static equilibrium of the cross section, which requires that compressive and tensile forces be equal and assumes that the tensile resistance of concrete is negligible. Therefore, in step 4, the maximum possible shear flow is taken as the minimum of Tr and Cr. Having determined the shear flow, it can be determined whether the neutral axis lies within the concrete slab or the steel section. This is done in step 5.

Step 5: Determine whether the neutral axis is in concrete or in steel section The location of the neutral axis can be determined through comparing the values for Tr and Cr. Figure 3 illustrates the possible locations of the neutral axis in a composite steel-concrete section.

Figure 3: The possible locations of neutral axis (Bradford, 1999) As shown in Figure 3a, if the compressive strength of the concrete component is greater than the tensile resistance of the steel section, then the shear transferred between the two materials is as much resistance as the steel can offer and therefore the neutral axis will be in concrete. This is because either there must be more tensile force available across the cross section of the concrete slab or less



compression. Since the tensile resistance of concrete is assumed to be zero, the only available direction for the neutral axis to move, from the steel-concrete interface, to establish static equilibrium is upward into the concrete. Similarly, as shown in Figure 3b, if the compressive strength of concrete is smaller than the tensile resistance of the steel section, then the shear transferred across the interface will be as much resistance as concrete can offer. Therefore, the neutral axis in this case will be in the steel section. Taking the above concepts into account, Step 5 on the spreadsheet identifies whether the neutral axis is in concrete or in steel. Once this is identified, it becomes easier to find the exact location of the neutral axis. Step 6: Factored horizontal shear force Before moving on to finding the exact location of the neutral axis and subsequently the section's moment resistance, having identified whether the neutral axis is in steel or in concrete, the amount of horizontal shear force can easily be determined for full or partial shear transfer as follows: Q

CT % shear transfer

if N. A. in steelif N. A. in concrete (4)

Step 7 through 10: Find the exact location of neutral axis and corresponding moment resistance Whether the neutral axis was found to be in concrete or in steel, solving the equation of equilibrium for Tr = Cr provides the exact location of the neutral axis in either case.

If the neutral axis was found to be in concrete, the location of the neutral axis relative to the top fibre of the section is determined in Step 7. If the neutral axis was determined to be in steel, the exact form of the equation of equilibrium and the moment resistance of the section will depend on whether the neutral axis is in the steel flange or the steel web. Therefore, if the neutral axis was found to be in steel, Steps 8a through 8c help determine whether it is in steel flange or steel web. Steps 9a through 9c, determine the exact location of the neutral axis, the resulting internal compressive and tensile forces at the cross-section of the composite member and their corresponding moment arms. Given the above information the moment resistance of the section for all three possible locations of the neutral axis; in concrete, in steel flange and in steel web can be found respectively. In step 10, based on the actual location of the neutral axis determined in steps 5 and/or 8c, the appropriate moment resistance of the composite cross-section is chosen and displayed. Once the section bending moment resistance is determined, it is time to check for total stresses. But before stresses can be found, the section's transformed moment of inertia and elastic section modulus must be determined. For this, the location of the centroid of the section in the vertical direction must first be identified. Steps 11 through 13: Determine the location of the composite section's centroid in the vertical direction The following general equation is used in order to find the composite section's centroid: y A A (5)



In order to determine the area of the concrete slab in compression, first the transformed width of the concrete slab is determined using the modular ratio. The modular ratio is the ratio of the elastic modulus of steel to that of concrete. Depending on whether the neutral axis was determined to be in concrete or in steel, the location of centroid is solved using equation (5) with the appropriate depth of concrete. If the neutral axis is in steel, solving equation (5) is simple since the entire cross section of concrete and steel provide compression resistance. However, if the neutral axis is in concrete, only the portion of the concrete slab in compression provides resistance. This is because for simplicity concrete tensile resistance is assumed to be zero. In this case, the centroid of the section in the vertical direction could be solved for using the following equation: A

Ad t h x (6)

where x is the depth of concrete in compression. In the spreadsheet for this project, x is solved for using the Excel Solver Add-in feature. Step 13 displays the appropriate centroid of the section as calculated above based on the previously determined location of the neutral axis. This location of the centroid is then used in determining the sections transformed moment of inertia and section modulus. Step 14: Transformed moment of inertia of the composite section Once the location of the centroid of the composite section has been determined, the transformed moment of inertia of section can be determined from first principals and parallel axis theorem as follows:

I I A d (7) When using equation (7), the transformed width of concrete and its depth in compression must be used. Also, note that di is the distance between the centroid of each area segment to the centroid of the section. Step 15: Transformed section modulus of the composite section In this step, the transformed section modulus is determined using the calculated transformed moment of inertia and the centroid of the section as follows: S I (8) The transformed section modulus is used to calculate the applied stresses on the composite section in the steps that follow. Step 16: Check of total stresses As mentioned previously, section 17.11 of CSA S16-01 requires that the sum of specified stresses during the three phases of construction be less than the specified yield strength of steel. In this step, the stresses during phases 1 and 2, where composite action between steel and concrete has not yet been achieved, are determined by dividing the applied moment resistance on the temporary structure by the elastic section modulus of steel. The stress on the composite action, which is achieved once the concrete slab has hardened, is determined by dividing the maximum total applied moment by the transformed section modulus. If the sum of stresses is greater than Fy, then a warning message is displayed for the user's consideration.



Steps 17 through 25: Check shear capacity Since the spreadsheet allows the user to specify wide flange beams or built-up sections, before determining the shear capacity of the section, its slenderness ratio must be checked. This is to ensure that the cross section does not buckle before reaching its shear capacity. To prevent this from happening, clause 14.3.1 of the CSA S16-01 specifies a maximum slenderness ratio of

F, for

webs of beams and girders, where Fy is the specified minimum yield point of the compression flange steel. In cases where the section is found to be slender, its shear capacity of can be improved and the onset of buckling delayed through the addition of stiffeners. Considering the above, the slenderness ratio of the web is calculated in Step 17 and a warning message is displayed if web buckling is determined to be an issue. The user may choose to change the design if the web slenderness becomes an issue. It must be kept in mind, however, that the web of a slender girder can carry loads even after it has buckled inelastically in shear. Shear buckling is characterized by diagonal tension strands in the web. The diagonal pattern of shear buckles allows the development of zones of tension called tension fields. The shear strength arising from the tension-field action in the web develops a band of tensile forces that occur after the web has buckled under diagonal compression. Equilibrium is maintained by the transfer of forces to the vertical stiffeners. As the girder load increases, the angle of tension field changes to accommodate the greatest carrying capacity. The longitudinal component of the tension field must be transmitted to the flange in the adjacent panel.

The load at which the web buckles in shear depends on both and the aspect ratio, , which is the ratio of stiffener spacing to web height. The aspect ratio of the section is calculated in Step 18. In Steps 19 and 20, the shear buckling coefficient, kv, and the aspect coefficient, ka, are determined based on the calculated value of aspect ratio, respectively given the following equations as provided in CSA S16-01 section 13.4.1.1:

k4 .

5.34 ifif

ah 1

ah 1

(9)

k (10)

As can be seen from equation (9), for unstiffened beams and girders the shear buckling coefficient is equal to 5.34 since the stiffener spacing is assumed to approach infinity. The shear buckling coefficient is then used to determine elastic and inelastic critical plate buckling stress in shear, Fcre and Fcri, respectively, in Step 21 according to the following two equations respectively, as set forth in CSA S16-01 section 13.4.1.1:

290 (11)

(12)



The above information is used in Step 22 to calculate the value of shear stress, Fs, based on equations (a) through (d) of S16-01 clause 13.4.1.1 depending on the range within which the slenderness ratio, , falls as follows:

F 0.66F if hw

439kF

F 290F kh w

if 439kF

hw

502kF

F F k 0.5F 0.866 F if 502kF

hw

621kF

F F k 0.5F 0.866 F if 621kF

hw

Finally, in Step 24, the shear resistance of the cross section is determined according to section 13.4.1 of S16-01 using the following equation: V A F (13) where Aw is the area of steel web calculated in Step 23. In Step 25, the shear resistance of the section is compared with the factored applied shear and a warning message is displayed if the applied shear exceeds shear resistance of the section. Steps 26 through 32: To ensure that composite action is achieved, shear flow must be transferred from the concrete slab to the steel section. This is why shear connectors, such as Nelson Studs are used to connect the concrete deck to the steel beams or girders. Therefore, in Steps 26

and 27 the area of shear studs and the concrete pull-out area are calculated based on user input values for stud diameter and height. In Step 28, the factored shear resistance per stud in concrete is determined based on type of slab specified. In Step 29, the factored shear resistance of each stud is determined based on their cross sectional area and ultimate capacity using the following equation given in CSA S16-01 clause 17.7.2.1:

q smaller of 0.5 A f E A F

(14)

Then, the number shear studs required is determined by dividing the horizontal shear force determined in step 6 by the stud shear capacity as determined above.

5.0 Visual Basic Analysis Tool Although the spreadsheet discussed in section 4 is a very useful analysis and design tool, it may not be very practical to use for very quick checks of cross section properties. When sectional moment resistance and properties are needed to be accessed very quickly, it is often more practical to refer to tables or other rapid access information tools. Tables with sectional properties of composite decks can be found in the Handbook of Steel Construction. However, it may be useful to have a tool that allows a more flexible selection of sectional dimensions and/or material properties. The complementary Visual Basic tool provided for this project allows quick calculation of important section properties for user specified section dimensions and material strengths. The proceeding sections provide a background on the assumptions based on which the complementary Visual Basic analysis tool was developed. Detailed descriptions of the methods used to create the



analysis tool are also given. Following similar procedures as in the previous section this software calculates the shear transfer across the interface of a user-specified composite steel-concrete deck section, determines the moment resistance of the section for 100, 70, and 40 percent shear transfer, and calculates the cross section's transformed moment of inertia and elastic section modulus.

5.1 Limitations on the Use of the Visual Basic Analysis Tool As is the case with any engineering software, the person using the composite section program for analysis must fully understand the fundamentals and methodologies used in the calculations. The results obtained from engineering software should never be taken to be flawless; the user must have at least an idea of what results are to be expected, and sample verifying calculations should always be completed.

The composite design program provided as part of this assignment is limited in that it assumes linear elastic-perfectly plastic behaviour of both the steel and concrete. It also calculates the transformed moment of inertia and section modulus based on 100 percent shear connection. Moreover, the area of steel is calculated based on the assumption that the steel section can be divided into perfectly rectangular segments. When using the program for analysis, the user must keep these limitations of the program in mind.

5.2 Program Set-Up The user interface for the complementary visual basic tool is a spreadsheet designed to receive the user input for section

dimensions and material properties of both steel and concrete components of the composite deck. The software operator, Visual Basic Application (VBA) subroutine, implemented as part of the spreadsheet, is called CompositeSection() and its main function consists of the following three components: 1. Extract the user-input data and assign them to appropriate

variables defined publicly in the subroutine 2. Call upon various functions to calculate all necessary values

to determine shear transfer, moment resistance for full and partial shear transfer, transformed moment of inertia and transformed section modulus

3. Output the results at the user interface The remainder of the VBA module created for this tool contains several functions that will appropriately be called upon by the operator subroutine CompositeSection(). The first of such functions in the VBA module is called AreaSteel(). This function simply calculates the total area of the steel section according to the dimensions specified by the user. The value of the area of steel returned by this function is used in the main subroutine, other functions and is also output at the user interface. Function EffecitveWidth() takes the values for the length and span of the concrete member, as provided by the user, and through the procedure described in section 4.2 determines the effective width of concrete in compression.



Function AreaConc() determines the total area of the concrete based on the provided thickness of concrete slab and the previously calculated effective width. Function Shear() compares the values calculated for tensile resistance of steel and compressive resistance of concrete and determines the necessary shear transfer based on the procedure described in step 4 of section 4.2. This function returns the value for Qr, and is later called upon by other functions as well as the main subroutine in determining horizontal shear and the location of neutral axis. The value of Qr then is also output at the user interface. If function Shear() determines that shear flow, Qr, is equal to Tr (i.e. the case in which TrCr) then, the main subroutine, CompositeSection(), calls upon function NAinSteel() to return the value of the sections moment resistance, assuming, this time. that the neutral axis is the steel. Function NAinConcrete() determines the location of the neutral axis in concrete through the following simplified equation: a = Tr / (Cr / thickness of concrete) NAinConcrete then simply calculates the moment resistance of the section as the couple force Cr and Tr separated by a lever arm that

extends from the centroid of the area of concrete in compression to the centroid of the steel section. Function NAinSteel() takes two parameters; the first parameter is the percentage of shear transfer and the second parameter is the total area of the steel cross section. This function is designed to calculate moment resistance for full or partial shear transfer. First, function NAinSteel() determines the area of steel in compression, Asc, in a similar fashion as described in section 4.2 step 8. Once the area of steel in compression is determined, the function NAinSteel() compares this value to the area of one steel flange. This will determine whether the neutral axis is in the steel flange or the steel web. This function will then call upon one of two functions NAinSteelWeb() or NAinSteelFlange() accordingly. Function NAinSteelFlange() takes the area of steel in compression, Asc, and the percentage of shear transfer as its parameters. It then determines the distance to the neutral axis from the top of the flange based on the area of steel in compression and the user input for flange width. At this point, Function NAinSteelFlange() calculates the moment arm for each compressive or tensile resistance with respect to the bottom of the section. This function will then calculate the compressive resistance in concrete as the full shear transferred, Qr, multiplied by the function parameter, percentage shear transferred. Once this is calculated, the moment due to each force can be determined. The value returned by this function is the sum of all moments calculated.



Function NAinSteelWeb() follows the exact same procedure used in function NAinSteelFlange(). However, to locate the neutral axis, only the web portion of the area of steel in compression is used. The location to the neutral axis is then determined as its distance from the bottom of the top flange. From here, the moment arms and moments are calculated as before and the sum of the moments is returned as the final moment resistance. Function MomentInertia() takes into consideration the location of neutral axis (for the transformed section) and the value returned by the function ybar() and determines the moment of inertia based on methodologies outlined in section 4.2 step 14.

Function ybar() calculates the location of the centroid of the section in the y-axis direction for the transformed section. This value is dependent on whether the neutral axis is in the concrete, or in the steel. If the neutral axis was determined to be in the concrete, the area of concrete not in compression has no effect on calculating the centroid or transformed moment of intertia.

Similarly, function SectionModulus() takes the values returned by functions MomentInertia() and function ybar() and returns the value of section modulus. Attached in Appendix A is the excel printout of the Visual Basic Code used in creating the composite section analysis tool.

6.0 Conclusion The spreadsheet and the complementary program developed for this project enable designers to quickly determine important cross-

sectional properties of a steel-concrete composite T-beam section for purposes of design and analysis. Among information output for the user are the effective width of the concrete slab, the bending moment and shear capacity of the composite section along with its transformed moment of inertia and elastic section modulus. These tools provide quick and accurate solutions to calculations that would normally be tedious to complete by hand. However, as is true with all engineering software, the user must be familiar with the fundamental concepts underlying design of composite sections and utilize the output of the spreadsheet and the software judiciously.



7.0 Bibliography Bradford, Mark A., Deric J. Oehlers. "Elementary Behaviour of Composite Steel and Concrete Structural Members." Butterworth-Heinemann, 1999. 1-15. Canadian Institute of Steel Construction. "Handbook of Steel Construction." Toronto, Ontario: Quadratone Graphics Ltd., 2006.



Appendix A: Visual Basic Code Option Explicit

'decalre public steel properties to be shared among various functions Public fi_s As Double Public fy As Double 'yield strength of steel Public tf As Double 'thickness of flange Public bf As Double 'flange width Public d As Double 'overall depth of steel Public tw As Double 'thickness of web Public Asteel As Double 'Area of Steel 'declare public concrete properties to be shared among various functions Public fc As Double 'compressive strength of concrete Public fi_c As Double Public alpha As Double 'usually equals 0.85 Public t As Double 'thickness of the concrete section Public S As Double Public L As Double Public beff As Double Public Aconc As Double 'Area of Concrete Public Cr As Double 'compression resistance Public Tr As Double 'tensile resistance Public Qr As Double 'shear flow Public A As Double 'distance from the top of the compression zone to the neutral axis Public h As Double 'deck thickness Public Es As Double 'Modulus of elasticity of steel

Sub CompositeSection() Dim Asteel As Double 'Asteel=total area of steel section

'Extract necessary data for concrete from the Input worksheet: Sheets("Composite Sections").Activate Range("C4").Select fi_c = ActiveCell.Value 'assign strength reduction factor of concrete Range("C5").Select alpha = ActiveCell.Value Range("C6").Select fc = ActiveCell.Value 'assign compressive strength of concrete (MPa) Range("E4").Select t = ActiveCell.Value 'assign thickness of concrete Range("E3").Select h = ActiveCell.Value 'assign thickness of concrete 'Extract necessary data for steel from the Input worksheet: Range("E7").Select tw = ActiveCell.Value Range("E8").Select bf = ActiveCell.Value Range("E9").Select tf = ActiveCell.Value Range("C7").Select fi_s = ActiveCell.Value 'assign strength reduction factor for steel Range("C8").Select fy = ActiveCell.Value 'assign Yield strength of steel Range("E10").Select d = ActiveCell.Value 'assign Total depth of steel section Range("C9").Select



Es = ActiveCell.Value 'assign Total depth of steel section 'Call function AreaSteel to calculate total area of Steel Asteel = AreaSteel() Sheets("Composite Sections").Activate ActiveSheet.Cells(17, 1) = Asteel 'Output the calculated area of steel on the worksheet 'Calculate the maximum tension developed in steel if all steel was in tension Tr = fi_s * fy * Asteel 'Sheets("Composite Sections").Activate 'ActiveSheet.Cells(11, 6) = Tr 'Output the calculated Tr on the worksheet; checked Range("E5").Select L = ActiveCell.Value 'Concrete Length Range("E6").Select S = ActiveCell.Value 'Concrete span 'calculate effective width of concrete according to S16-01 beff = EffectiveWidth() 'output the effective depth calculated in cell B2 Sheets("Composite Sections").Activate ActiveSheet.Cells(17, 2) = beff 'calculate the maximum compression developed in concrete if all concrete was in compression Cr = fi_c * alpha * fc * AreaConc() 'Sheets("Composite Sections").Activate 'ActiveSheet.Cells(11, 7) = Cr 'Output the calculated Tr on the worksheet; checked 'Call upon function Shear to compare value of Cr and Tr and return a value for shear transfer, Qr

Qr = shear() 'Output the shear transfer on cell F17 in kN Sheets("Composite Sections").Activate ActiveSheet.Cells(17, 6) = Qr / 1000 'Calculate the Moment resistance Mr for 100% shear transfer Dim Mrc100 As Double 'If the shear flow calculated earlier was equal to Tr then N.A. is in concrete If Qr = Tr Then Mrc100 = NAinConcrete() Else 'Otherwise the N.A. is in steel and Mrc should be calculated accordingly Mrc100 = NAinSteel(1, Asteel) End If 'Output the Mrc for 100% shear transfer Sheets("Composite Sections").Activate ActiveSheet.Cells(17, 3) = Mrc100 'checked Dim Mrc70 As Double 'Calculate the Moment resistance Mr for 70% shear transfer Mrc70 = NAinSteel(0.7, Asteel) 'Output the Mrc for 100% shear transfer Sheets("Composite Sections").Activate ActiveSheet.Cells(17, 4) = Mrc70 'checked Dim Mrc40 As Double 'Calculate the Moment resistance Mr for 70% shear transfer Mrc40 = NAinSteel(0.4, Asteel) 'Output the Mrc for 100% shear transfer Sheets("Composite Sections").Activate ActiveSheet.Cells(17, 5) = Mrc40



'Output the transformed moment of inertia and section modulus ActiveSheet.Cells(17, 7) = MomentInertia() / 1000000 ActiveSheet.Cells(17, 8) = SectionModulus() / 1000 End Sub

Public Function AreaSteel() 'calculate area of steel AreaSteel = (2 * tf * bf) + (tw) * (d - (2 * tf)) End Function

Public Function EffectiveWidth() As Double 'calculate effective width of concrete according to S16-01 If (L / 4) < S Then EffectiveWidth = L / 4 Else EffectiveWidth = S End If End Function

Public Function AreaConc() As Double 'calculate area of concrete AreaConc = beff * t End Function

Public Function shear() As Double 'Qr=min of (Tr & Cr) If Cr < Tr Then shear = Cr Else shear = Tr End If End Function

Function NAinConcrete() As Double 'calculate a

'a=(fi_s*fy*Asteel)/(0.85*fi_c*fc*beff) 'also (0.85*fi_c*fc*beff)=Cr/tc A = Tr / (Cr / t) Dim e As Double 'lever arm for the couple moment 'calculate e e = t + h + (d / 2) - (A / 2) 'Moment resistance = Tr * e (kN.m) NAinConcrete = Tr * e / 10 ^ 6 End Function

Function NAinSteel(PercentageShear As Double, Area As Double) As Double Dim Asc As Double Dim Af As Double If Qr = Tr Then 'if Qr=Tr then for the incomplete shear transfer the Area of steel in compression simplifies to Asc = 0.5 * (1 - PercentageShear) * Area 'Area of steel in compression for Qr=Tr 'Sheets("Composite Sections").Activate 'ActiveSheet.Cells(11, 9) = 1 - PercentageShear 'checked 'Sheets("Composite Sections").Activate 'ActiveSheet.Cells(11, 8) = Asc 'checked Else 'if Qr=Cr then the N.A. was in the steel to begin with and it remains in the steel

'simplified calculations for Asc = (Tr - Cr * PercentageShear) / (2 * fi_s * fy) 'Area of steel in compression for Qr = Cr



End If 'calculate area of one of the steel flanges Af = bf * tf 'Sheets("Composite Sections").Activate 'ActiveSheet.Cells(11, 10) = Af 'checked 'compare the area of steel in compression Asc 'if the area of steel in compression is less than the area of one flange, then N.A. is in the flange

If Asc



Dim Cw2 As Double 'centroid of web in tension Dim Af2 As Double 'Area of the part of top flange in tension Dim Cf2 As Double 'Centroid of tension part of top flange Dim Cf3 As Double 'Centroid of part of top flange in compression Dim V As Double 'Total shear force transferred Dim CC As Double 'Centroid of concrete in compression Af1 = tf * bf Cf1 = tf / 2 Af2 = tf * bf Cf2 = d - (tf / 2) x = (Area - Af1) / tw Aw2 = tw * x Cw2 = d - tf - (x / 2) Aw1 = tw * (d - (2 * tf) - x) Cw1 = tf + 0.5 * (d - (2 * tf) - x) V = Qr * Percentage 'shear transferred in concrete is the percentage of total Q CC = t / 2 + d + h Dim Mr1 As Double Dim Mr2 As Double Dim Mr3 As Double Dim Mr4 As Double Dim Mr5 As Double 'Calculate each individual moment Mr1 = (fi_s * fy * Af1 * Cf1) / 10 ^ 6 Mr2 = (fi_s * fy * Aw1 * Cw1) / 10 ^ 6 Mr3 = (fi_s * fy * Af2 * Cf2) / 10 ^ 6 Mr4 = (fi_s * fy * Aw2 * Cw2) / 10 ^ 6 Mr5 = (V * CC) / 10 ^ 6 'take the sum of moments acting on the cross section about the bottom of the steel NAinSteelWeb = Abs(Mr1 + Mr2 - Mr3 - Mr4 - Mr5)

End Function

Public Function MomentInertia() As Double Dim n As Double Dim btr As Double Dim dweb As Double Dim y As Double Dim c As Double Dim ybar As Double n = Es / 4500 / Sqr(fc) btr = beff / n dweb = d - 2 * tf y = (t * btr * (d + h + t / 2) + AreaSteel() * d / 2) / (AreaSteel() + btr * t) If y > d Then c = -AreaSteel() / btr / 3 + Sqr(AreaSteel() ^ 2 + 6 * btr * AreaSteel() * (d / 2 + h + t)) / btr / 3 ybar = (c * btr * (d + h + t - c / 2) + AreaSteel() * d / 2) / (AreaSteel() + btr * c) MomentInertia = 2 * bf * tf ^ 3 / 12 _ + bf * tf * (ybar - tf / 2) ^ 2 _ + tw * dweb ^ 3 / 12 _ + tw * dweb * (d / 2 - ybar) ^ 2 _ + bf * tf * (d - tf / 2 - ybar) ^ 2 _ + btr * c ^ 3 / 12 _ + btr * c * (d + t + h - c / 2 - ybar) ^ 2 Else ybar = (t * btr * (d + h + t / 2) + AreaSteel() * d / 2) / (AreaSteel() + btr * t) MomentInertia = 2 * bf * tf ^ 3 / 12 _ + bf * tf * (ybar - tf / 2) ^ 2 _



+ tw * dweb ^ 3 / 12 + tw * dweb * (d / 2 - ybar) ^ 2 _ + bf * tf * (d - tf / 2 - ybar) ^ 2 _ + btr * t ^ 3 / 12 _ + btr * t * (d + h + t / 2 - ybar) ^ 2 End If End Function Public Function ybar() As Double Dim n As Double Dim btr As Double Dim dweb As Double Dim y As Double Dim c As Double n = Es / 4500 / Sqr(fc) btr = beff / n y = (t * btr * (d + h + t / 2) + AreaSteel() * d / 2) / (AreaSteel() + btr * t) If y > d Then c = -AreaSteel() / btr / 3 + Sqr(AreaSteel() ^ 2 + 6 * btr * AreaSteel() * (d / 2 + h + t)) / btr / 3 ybar = (c * btr * (d + h + t - c / 2) + AreaSteel() * d / 2) / (AreaSteel() + btr * c) Else ybar = (t * btr * (d + h + t / 2) + AreaSteel() * d / 2) / (AreaSteel() + btr * t) End If

End Function Public Function SectionModulus() Dim c As Double c = d + h + t - ybar() If ybar() > c Then SectionModulus = MomentInertia() / ybar() Else SectionModulus = MomentInertia() / c End If End Function

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