ADAPT RC 2010€¦ · a closer agreement between the practice and code. 1.1.2 Requirements of Design Procedure The difficult part of automating concrete design is the floor slab

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ADAPT RC 2010

Theory, Examples and Verification v0410_0-1

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LIST OF CONTENTS Content

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LIST OF CONTENTS

THEORY ...........................................................................................................1 1.1 OVERVIEW............................................................................................................ 3

1.1.1 General Requirements ................................................................................. 3 1.1.2 Requirements of Design Procedure ............................................................. 3 1.1.3 Concrete Design in Relation to Other Materials.......................................... 4

A. Glass................................................................................................... 5 B. Steel 5 C. Concrete.............................................................................................. 6

1.1.4 Automation of Concrete Design................................................................ 13 1.1.5 Design Process .......................................................................................... 13

A. Structural Modeling and Analysis.................................................... 13 B. Analysis Methods............................................................................. 14 C. Load Path Designation ..................................................................... 16 D. Analysis Options.............................................................................. 20 E. FEM versus Frame Methods ............................................................ 26 F. Structural Detailing .......................................................................... 29 G. Waffle Slabs..................................................................................... 30 H. Summary .......................................................................................... 34

1.2 STRUCTURAL MODELING OF BEAMS AND ONE-WAY SLABS............... 34 1.3 TWO-WAY SLABS AND EQUIVALENT FRAME METHOD ......................... 35

1.3.1 Background ............................................................................................... 35 1.3.2 Equivalent Frame Method (EFM) ............................................................. 35

A. Equivalent Frame ............................................................................. 35 B. Torsional Members .......................................................................... 36 C. Stiffness of Slab-Beam..................................................................... 38 D. Stiffness of Columns........................................................................ 40

1.4 LOADING............................................................................................................. 41 1.5 BENDING MOMENTS, REACTIONS AND SHEARS ...................................... 42

1.5.1 Centerline Moments .................................................................................. 42 1.5.2 Reduction of Moments to the Face-of-Support ......................................... 43

1.6 DESIGN MOMENTS, SHEARS AND REACTIONS ......................................... 46 1.6.1 Design Actions .......................................................................................... 46

1.7 REINFORCEMENT (NONPRESTRESSED REINFORCEMENT) .................... 47 1.7.1 Minimum Rebar Criteria ........................................................................... 47 1.7.2 Limits of Reinforcement ........................................................................... 47

A. Maximum Tensile Reinforcement and Compression Rebar ............ 48 1.7.3 Ultimate Strength ...................................................................................... 48

A. Criteria ............................................................................................. 48


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B. Method of Ultimate Strength Calculation.........................................48 C. Determination of the Effective Depth of Sections............................51

1.7.4 Inflection Points and Length of Reinforcement .........................................51 1.8 PUNCHING SHEAR.............................................................................................55

1.8.1 Overview ...................................................................................................55 1.8.2 Calculation Procedure................................................................................55 1.8.3 Punching Shear Stress Ratio ......................................................................56 1.8.4 Locations of Punching Shear Check at a Joint...........................................58 1.8.5 Round Columns .........................................................................................58 5.8.6 Punching Shear Reinforcement Design .....................................................59

1.9 BEAM/ONE-WAY SLAB SHEAR.......................................................................61 1.9.1 General.......................................................................................................61 1.9.2 Calculation of Existing Shear Stress..........................................................61 1.9.3 Calculation of Permissible Shear ...............................................................61 1.9.4 Determination of Shear Reinforcement .....................................................62

1.10 DEFLECTIONS.....................................................................................................62 1.10.1 Overview ...................................................................................................62 1.10.2 Equivalent Moment of Inertia....................................................................63 1.10.3 Long-term Deflection (Creep and Shrinkage) ...........................................70

1.11 NONPRISMATIC BEAMS AND SLABS............................................................71 1.11.1 General.......................................................................................................71 1.11.2 Modeling and Analysis .............................................................................71 1.11.3 Strength Check...........................................................................................72

1.12 LAYOUT OF REINFORCEMENT IN TWO-WAY SLABS ...............................73 1.13 TREATMENT OF JOINTS...................................................................................74 1.14 REDISTRIBUTION OF MOMENTS ...................................................................77

1.14.1 General.......................................................................................................77 1.14.2 Redistribution Procedure ...........................................................................78 1.14.3 Redistribution Limits .................................................................................84

1.15 LATERAL ANALYSIS.........................................................................................84 1.15.1 Background................................................................................................84 1.15.2 Analysis Procedure ....................................................................................87 1.15.3 Input Screens .............................................................................................89 1.15.4 Description of Printout ..............................................................................90 1.15.5 Discussion of Lateral Loading Treatment..................................................91 1.15.6 Specific Features for Codes Other than ACI .............................................93

EXAMPLES.................................................................................................... 95 2.1 OVERVIEW ..........................................................................................................97 2.2 TWO-SPAN BEAM EXAMPLE...........................................................................97

2.2.1 Design Criteria and Standard Report .........................................................97 2.2.2 Investigation Mode – Option 1 ..................................................................98


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2.2.3 Investigation Mode – Option 2................................................................ 101 2.2.4 Investigation Mode – Option 3................................................................ 104

2.3 FOUR-SPAN, TWO-WAY FLOOR SLAB........................................................ 104

VERIFICATION...........................................................................................125 3.1 DEFLECTION .................................................................................................... 127

3.1.1 Background ............................................................................................. 127 3.1.2 Deflection Computation .......................................................................... 127

A. Dead Load Deflection (Deflection due to SW+SDL) .................... 131 B. Live Load Deflection ..................................................................... 133 C. Dead Load and Creep/Shrinkage ................................................... 135 D. Dead Load, Live Load and Creep/Shrinkage ................................. 136

3.1.3 Computation of Equivalent Moment of Inertia, Ie .................................. 136

REFERENCES..............................................................................................139

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Chapter 1

THEORY

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1.1 OVERVIEW

This Chapter describes the background to most of the formulation and computational assumptions used in ADAPT-RC. It starts with an outline of the concept of design of concrete floor systems and discusses briefly the analysis options. It demonstrates that regardless of the analysis method, designation of load paths is a prerequisite for concrete floor design. Integration of actions over design strips based on these load paths is shown to be a fundamental step in the interpretation of the solutions when using either the Equivalent Frame Method (EFM) or the Finite Element Method (FEM) of design.

The bulk of this section is an excerpt from the reference [Aalami, 2005].

1.1.1 General Requirements

The primary concerns of a structural engineer are the safety, serviceability, and economy of the structures he or she designs. Safety is understood as the structure’s ability to withstand code required loads without excessive damage. Serviceability is achieved if the structure performs as intended throughout its expected life span. Economy is taken to mean the structure’s owners feel that both its short- and long-term costs are reasonable.

Legality of the design procedure, defined as compliance with applicable building codes, is also important. It is not always easy to establish, however, particularly for post-tensioned structures. Codes tend to follow, rather than lead, practice with respect to post-tensioning. Much of what is currently considered appropriate practice for post-tensioned design is not incorporated into the codes. For nonprestressed concrete, however, there is a closer agreement between the practice and code.

1.1.2 Requirements of Design Procedure

The difficult part of automating concrete design is the floor slab. In most cases, skeletal members such as beams, columns, and frames made from them can be readily analyzed and designed. Concrete buildings are rarely limited to skeletal members; however, the floor slabs are typically a significant portion of the building and its design.

The following reviews concrete slab design concepts and presents a method for automating the design of both reinforced and post-tensioned concrete buildings. It is anticipated that this method will eliminate many

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of the problems associated with the integrated design of concrete buildings. A procedure for selecting load paths is also presented.

1.1.3 Concrete Design in Relation to Other Materials

The following example highlights the principal features that distinguish concrete design from other types of design. The example considers three materials: concrete, steel and glass, each of which has a distinctive feature in terms of design. Although the example is hypothetical, it illustrates how material properties affect design requirements and procedures.

Fig. 1.1-1(a) shows a partial plan of a plate or slab under uniform loading. The example reviews the design of the area surrounded by supports A, B and C, marked “Design Region.” The objective is to satisfy the serviceability and safety (strength) requirements of this region.

FIGURE 1.1-1

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A. Glass

Consider first a glass plate. The serviceability of the glass plate is determined by acceptable deflection; its safety is measured by the load that causes it to crack. Cracking occurs when the tensile stress at the surface reaches a value that is a material property of the glass. Glass is an extremely brittle material; once cracking is initiated, it will spread immediately and cause failure. Hence, the design procedure consists of:

• Estimating the deflection under service load; and

• Determining the load at which the maximum tensile stress reaches the cracking strength of glass.

For serviceability design, deflections can be estimated using approximate methods based on the plate’s geometry, support conditions, material properties and service loading. As noted above, however, failure occurs when the stress at any point on the plate reaches the cracking strength of the glass. In order to get a reliable estimate of the glass plate’s safety, both the location of the maximum stress and the stress value in relation to the applied loading need to be determined accurately.

The geometry and supports of the glass plate must be modeled accurately since they directly affect the magnitude of maximum stress. In most cases, the actual load path must be determined either analytically or experimentally. Approximate methods based on assumed load paths will not produce accurate results.

The need to accurately determine the stress at a point in order to ensure the safety of glass plate is what differentiates the response of glass under increasing load from that of the other materials. Local stresses calculated by finite element analysis are sensitive to the number of mesh divisions and the accuracy of the finite element formulation. In order to determine the value of stress at a point, a very fine mesh and an appropriate formulation must be used.

B. Steel

Assuming that the design region in Fig. 1.1-1(a) is made out of steel plate, its serviceability is governed by its deflection and permanent deformation under service loading. Its safety or ultimate strength is generally determined by excessive deformation under factored loading.

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Although approximate modeling can be used to estimate the deflection of the plate under service loading, permanent deformation will result if local yielding takes place. Local yielding is a function of the distribution of stress under service loading. Using the von Mises criterion, yielding occurs when the combination of stresses at a point reaches a characteristic value of the material. Either experimental techniques or a finely meshed finite element analysis must be used to evaluate local yielding. The reliability of the design depends on the accuracy with which the location and magnitude of the local stresses are calculated.

The design procedure is as follows:

• Assume an initial plate thickness and support conditions;

• Estimate deflection under service load using approximate methods;

• Use a rigorous method to determine local stresses and the likelihood of local yielding under service loads in order to avoid permanent deformation; and

• Determine the strength (safety) limit by using an approximate method to estimate the load at which overall plastification results in excessive deformation. (A rigorous method can be used for this but is generally not warranted.)

A central feature of steel design is that there is typically an initial assumption of plate thickness and support conditions. The calculations are then aimed at verifying the location and magnitude of the maximum von Mises stress. If the calculated stresses are less than allowable stress limits, the initial assumptions are regarded as an acceptable design. The fact that the design is essentially a verification of the initial assumptions is one of the key features that differentiate steel design from concrete design.

C. Concrete

Consider finally a concrete plate. Design criteria for the concrete plate are: (i) the deflections and crack widths must be within acceptable limits under service conditions, and (ii) the plate must not collapse under code stipulated factored loads.

Determination of local stresses is generally not meaningful when evaluating deflections and crack widths of concrete under service loading. Micro cracking and lack of material homogeneity make the

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use of simple linear-elastic analysis invalid; rigorous analysis is typically not warranted.

Moreover, unlike the design procedure for glass, the determination of local stresses does not need to be exact. A typical slab is usually modeled and designed following designated load paths as opposed to the analytically determined load paths used for the glass and steel plates.

The ability of a concrete section to crack and undergo a finite amount of rotation prior to failure is a reflection of the section’s ductility. In order to mobilize the assumed load paths and redistribute the load resistance in a floor, the slab must possess certain ductility. ACI-318 ensures a minimum ductility by limiting the amount of reinforcement in a section; this effectively limits the depth to the neutral axis. Other major codes impose similar limits.

FIGURE 1.1-2

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Fig. 1.1-2 is a schematic of a typical region of a post-tensioned beam designed according to ACI-318. The ratio of the ultimate curvature, φu , to the curvature at first yield, φy, gives a measure of the ductility of the section. Sections that comply with code restrictions on reinforcement and depth to the neutral axis are likely to have a ductility ratio in the range of 4 to 18.

For the example in Fig. 1.1-1(a), two load path options are shown (Fig.1.1-1(b) and 1.1-1(c)). In Fig.1.1-1(b) the slab is modeled as a strip spanning between walls A and B. In other words, the engineer designates a “load path” for the transfer of the design load to the two supports. For this load path, the reinforcement for safety against collapse will be bottom bars, referred to as primary reinforcement, between walls A and B.

Wall C is not a designated part of the load path shown in Fig. 1.1-1(b). Nevertheless, it will participate in supporting the load; the slab over it is thus likely to develop high tensile stresses. The design engineer must recognize this and place a nominal amount of top bars over the wall for crack control under service loading. This process of adding steel in selected locations is referred to as “Structural Detailing”. Structural Detailing is an essential step in concrete design and is highly dependent on the experience and engineering judgment of the design professional.

Structural Detailing is done after the amount and location of the primary reinforcement is determined. It fulfills the design concept by ensuring that:

• The load path envisaged by the engineer can develop at loadings equal to, or greater than, code stipulated values;

• The crack widths under service loading are within acceptable limits.

Referring back to the original plate design, in Fig. 1.1-1(c), the slab is modeled as a cantilever supported by wall C. This load path requires top bars over wall C as shown. These bars are supplemented with Structural Detailing. Top bars are placed over walls A and B for crack control.

The detailing involved in translating the bottom bars shown in Fig. 1.1-3(a-i) into the number, length and location of bars to be placed in the slab as shown in (a-ii) is referred to as “Construction Detailing”.

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FIGURE 1.1-3

Another example of Construction Detailing is the selection of the correct lap splices, hooks, and bar bending details as illustrated in Fig. 1.1-3(b).

In North America, Construction Detailing is shown on shop drawings generated by the materials suppliers. Structural Detailing, on the other hand, is done by the design engineer and is shown on the structural drawings. In many other parts of the world, however, there is no distinction between Structural Detailing and Construction Detailing. Unlike the practice in North America, the drawings generated by the design engineer also reflect the Construction Detailing.

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Fig. 1.1-4(a) shows an example of Structural Detailing for crack control.

FIGURE 1.1-4

Fig. 1.1-4(b) shows an example of Structural Detailing for development of the load path. In Fig. 1.1-4(b), a concentrated loading is distributed over the width of the assumed load path by distribution steel placed underneath the load. The added reinforcement ensures that the load path between walls A and B is able to materialize as envisaged by the designer. Note, however, that although this type of reinforcement is required for both safety and serviceability, it is not reflected in many design methods, including the one introduced in this article.

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The following general conclusions can be drawn about the design of concrete structures:

• When doing concrete design, the engineer must designate a load path in order to determine the reinforcement. This is unlike the glass alternative, where the load path must be determined by analysis, or the steel alternative, where local yielding can be determined without a designated load path. The load path designation is required for concrete because the layout of the reinforcing bars governs the orientation and magnitude of the resistance developed by the slab. Often, there is more than one possible load path. The load paths that are selected make up the skeleton of the “structural system” of the building.

• Concrete design is not sensitive to local stresses. The distribution of moments determined from elastic theory will be similar to the schematics of Fig. 1.1-5(a). The simplified, equivalent moment shown in Fig. 1.1-5(b) is generally used for reinforcement calculations, however. The reinforcement necessary for strength in each direction is that required to resist the total moment, i.e. the integral below the moment curve. The layout of the reinforcement is typically not critical as long as the bars are within the region corresponding to the moment they are designed to resist. This is based on the premise that failure follows the formation of a hinge line and the hinge line will mobilize all of the reinforcement that crosses it. This highlights another feature of concrete design, which is that the total moment is used for design. The distribution of the moment and local values of the moment are not critical. The total moment is considered to be resisted by a “design section” as opposed to glass and steel design where local moments are checked at design points. This feature places concrete at a great computational advantage since total (integral) values of the actions are not as sensitive to finite element discretization as local values. Finite element software is generally formulated to satisfy static equilibrium, regardless of the density of the mesh used to discretize the structure. A coarse mesh gives essentially the same “total” moment over a design section as a fine mesh. This observation is discussed further in [Aalami, 2001].

• As with glass and steel, stresses in prestressed concrete structures must be checked under service load. With glass, the objective of the stress check is to avoid breakage; with steel, the objective is to avoid local yielding and permanent deformation. With concrete, however, the objective is generally to control - not avoid - crack formation.

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FIGURE 1.1-5

The calculated concrete stresses used for design are “hypothetical,” since they are based on the application of the total moment to the entire design section. In reality, stresses over the supports will peak at much higher values. In most instances they will exceed the cracking limit of the concrete. The calculated stresses are thus an indication of the extent of crack formation over a region rather than true stresses.

In nonprestressed concrete, there is no requirement for stress check. The serviceability is achieved through (i) selection of member thickness within code specified limits, (ii) computation and limits on

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deflection based on properties of cracked sections, and (iii) control of crack width through selection of proper bar size and its judicial layout of rebar following the “column strip, middle strip” guideline as closely as is practical.

1.1.4 Automation of Concrete Design

In summary, the design of concrete floors as practiced today requires a designated “load path” for the determination of primary reinforcement. Designation of the load path and determination of the primary reinforcement must be followed by Structural Detailing. The following section presents a procedure, which automates much of the design procedure for a nonprestressed floor and the initial steps of the design of a post-tensioned floor.

1.1.5 Design Process

A. Structural Modeling and Analysis

The design process for concrete floors is summarized in the flow chart shown in Fig. 1.1-6. There are essentially four steps: a structural modeling step, an analysis step, a design step and a structural detailing step. The structural modeling step involves designating load paths. The analysis step determines the actions (moments and shears) that each load path must resist. The design step gives the area of reinforcement required to resist these actions. The Structural Detailing step determines the layout of the reinforcement; it also determines if additional steel is needed for crack control or load distribution.

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FIGURE 1.1-6

B. Analysis Methods

The three methods commonly used for analysis of concrete structures are: Simple Frame, Equivalent Frame, and Finite Element analysis. In the Simple Frame method, the slab is divided up into design strips. The geometry of the structure is modeled exactly, i.e., the frames are analyzed using the stiffnesses of the columns and associated slabs as calculated from their geometries. As a result, the analysis does not account for the influence of biaxial plate bending.

The Equivalent Frame Method (EFM) is a refinement of the Simple Frame method. It is somewhat more exact than the Simple Frame method since the relative column and slab stiffnesses are adjusted to account for the biaxial plate bending. In typical design, most column-supported floors are analyzed with the Equivalent Frame method.

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The information required as far as geometry, loading and boundary conditions is the same for both the Simple Frame and the Equivalent Frame methods. Although both methods are approximate they both yield lower bound (safe) solutions. The degree of approximation depends on the extent to which a floor system deviates from a uniform, orthogonal support layout and constant slab thickness.

The third method of analysis, the Finite Element method (FEM), is based on the division of the structure into small pieces (elements) whose behavior is formulated to capture the local behavior of the structure (Fig. 1.1-7). Each element’s definition is based on its material properties, geometry, location in the structure, and relationship with surrounding elements. The mathematical assemblage of these elements into the complete structure allows for automated computation of the response of the entire structure. FEM inherently incorporates the biaxial behavior of the floor system when determining the actions in the floor.

DISCRETIZATION OF FLOOR SLAB

FIGURE 1.1-7

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The following references give a general description of the Finite Element method [Zienkiewicz O.C., et al, 1989, Bathe, K.J. 1982]. Details of the procedure with specific application to concrete floor systems are given in [Aalami, 2001].

C. Load Path Designation

The focus of the structural modeling step is the designation of the load paths. The structural system is complete if the self-weight and applied loading at every location is assigned an explicit load path to a support.

Load path designation is based on the strip method. This requires dividing the floor into intersecting support lines, each of which has its own tributary area. The support lines indicate the assumed load paths; a support line, together with its tributary area, is referred to as a “design strip.” For nonprestressed floors, the load path is determined by the position and orientation of the reinforcement.

For most structures, selection of the load paths is essentially independent of the analysis method. Consider a typical floor from a multi-story building with columns and walls above and below as shown in Fig. 1.1-8. The following describes the structural modeling of the floor and illustrates the procedure for selecting load paths and design sections.

Define outline of floor slab and supports As a first step, the engineer defines the slab edge and any openings, steps or other discontinuities. Next, he or she identifies the location and dimensions of the walls and columns supporting the floor. The supports for this example are shown in Fig. 1.1-8. Note that beams are considered as part of the floor system rather than the support system. They are therefore modeled and designed in conjunction with the floor slab. Define support lines The engineer then determines a series of support lines in each of the two principal directions. Typically, these are lines joining adjacent supports along which an experienced engineer will intuitively place reinforcement. Fig. 1.1-9 shows the support lines, labeled A through G, in the X-direction (F is not a designated line of support).

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FIGURE 1.1-8

Fig. 1.1-10 shows the five support lines, labeled 1 through 5, in the Y-direction. If a floor system is highly irregular, i.e., the columns are significantly offset from one another, the support lines may be less obvious. The criteria for selection are the same as in a regular slab, however. The support lines are the lines along which an experienced structural engineer is likely to place the primary reinforcement for resisting the gravity load.

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FIGURE 1.1-9

Define tributaries and design strips Typically, the midpoints between support lines are used to designate the tributary areas for each support line. The midpoints are joined to identify the boundaries of the tributary. Fig. 1.1-11 shows the support lines in the X-direction. Points 8 and 9 would be used to determine the boundaries for the tributary of support line B, for example.

The tributaries for the design strips in the X-direction are hatched in Fig. 1.1-12. Fig. 1.1-13 shows the support lines in the Y-direction with their associated tributary areas.

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FIGURE 1.1-10

Design Sections Design sections are typically drawn across each design strip at the locations where the integrated actions on the design strip are greatest. There is no limit to the number of design sections that can be specified. Note that the maximum design actions in the field may not be at the midpoint of the spans.

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FIGURE 1.1-11

In addition, peak design actions for the strength and serviceability checks may not occur at the same location. Fig. 1.1-14 shows the design sections for two of the design strips in the X-direction. Across the width of the supports, sections can be chosen at the face of support to take advantage of the reduced actions away from the support centerlines.

D. Analysis Options

In the analysis step, the actions are distributed among the load paths (design strips) to satisfy the equilibrium of the applied loading.

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FIGURE 1.1-12

(i) Frame Analysis

In both the Simple and Equivalent frame options, each design strip is extracted from the floor and reconstructed with appropriate support conditions and loading to create an approximated frame model. Each design strip is analyzed as an independent structural system, isolated from the adjacent design strips.

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FIGURE 1.1-13

Consider design strip B, shown as a separate entity in plan (Fig. 1.1-15) and in elevation (Fig. 1.1-16). For plane frame analysis, the strip is straightened along its line of support as illustrated in Fig. 1.1-15(b).

The span length thus corresponds to the slant distance between adjacent supports. Note that the tributary widths may vary over a single span.

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FIGURE 1.1-14

To simplify the analysis, these varying tributary lines are typically idealized as straight boundaries (Fig. 1.1-15(c)).

Usually, the idealized tributary is chosen to be conservatively larger than the actual tributary. If the change in tributary width in any given span varies by more than 20%, it may be worthwhile to model the tributary as a series of steps to reduce the reinforcement required. Additional approximations may be necessary for other non-standard conditions.

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FIGURE 1.1-15

(ii) Finite Element Analysis

As noted above, if either the Simple Frame or the EFM is used, each design strip must be extracted from the floor system and analyzed as a plane frame. With the FEM, the entire floor can be analyzed at one time. The results of an FEM analysis must be processed as “design strips” and “design sections” for code stipulated serviceability and strength checks, however. As with the frame methods, the design strips are based on the assumed load paths. The design strips do not need to be selected before the analysis, though. This can sometimes be advantageous since the results of the FEM analysis can be used to select design

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strips which are more in line with the natural (assumed elastic) response of the slab.

FIGURE 1.1-16

Ideally, a design strip should be bounded by lines of zero shear transfer since this ensures that each design strip is designed to carry only the loading that is located directly on it. For the floor slab used in this example, the flow of the loading to the supports is shown by arrows in Fig. 1.1-17. The arrows are normal to the planes of maximum vertical shear; the length of each arrow indicates the magnitude of the shear. Fig. 1.1-18 shows the lines of zero vertical shear transfer in the Y-direction as determined using the flow of loading shown in Fig 1.1-17. The alternate hatched and clear regions indicate the natural load tributaries for the support lines A through G.

Displays such as the one in Fig. 1.1-18 that show the “natural tributaries” allow the calculated actions to be assigned to the design strips in accordance with the elastic response of the structure. A design based on these tributaries is likely to be more economical with respect to material usage, especially if the floor configuration is irregular. Design strips are typically based on the standard support lines and tributaries described before, however. In most cases, the increased design effort required to select natural tributaries outweighs the benefits of the refined design strips. Fig. 1.1-19 shows the design strips selected by the procedure outlined for the frame methods superimposed on the “natural tributary” lines of the floor.

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FIGURE 1.1-17

E. FEM versus Frame Methods

The design procedure for an FEM analysis is very similar to that of the frame methods. It consists of selecting load paths leading to design strips, and design sections. This is followed by the determination of demand actions for each design section. The design step consists of application of the demand actions to the respective design sections.

As with the frame methods, the objective of an FEM analysis is to provide information for a safe and serviceable design in accordance with the prevailing code(s). The amount of information

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obtained from a FEM analysis is generally more than that required by code for serviceability and safety checks, however. In particular, the FEM provides more accurate information on the floor system’s response to applied loading.

FIGURE 1.1-18

(i) Determination of Design Moment

To perform the strength evaluation of a design section, the design moment at the section is applied to the entire cross-sectional area of the section. With the FEM, design moments are determined by integrating the moment distributions across the design section. As an example, refer to Fig. 1.1-20, which is an enlargement of a corner of the floor slab example. Observe the design strip B, and the variation of moment M

y along the three

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design sections, one at the face of each support and one in the field of span 1-2. The moment used for the determination of reinforcement and stresses at each design section is the area (integral) of the moment distributions shown. For example, at the face of support at line 2, the design moment is My =559 kNm.

FIGURE 1.1-19

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FIGURE 1.1-20

F. Structural Detailing

The analysis and design steps discussed above determine the area of primary reinforcement required for each design strip. To ensure satisfactory performance under service conditions, it is essential that this reinforcement be properly distributed. Distribution of the primary reinforcement is considered to be Structural Detailing.

The ACI [ACI-318] recommendations for reinforcement layout are based on the column-strip, middle-strip concept. Although these recommendations are valuable guidelines, they cannot always be applied when the column plan is irregular or there are discontinuities such as large openings. The following are some of the underlying principles for determining reinforcement layout; they can be used as guidelines when a strict column-strip, middle-strip distribution will not work.

• Distribute the required reinforcement as close as practical to the distribution of the moment. It is not critical to follow strict

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percentages as long as the required area of reinforcement is accounted for. For the design strip shown in Fig. 1.1-20, for example, the reinforcement at the face of column at line 2 should be concentrated in the column region, while in the field; the distribution should be essentially uniform.

• At the exterior columns of a column-supported slab, place the entire amount of reinforcement within the column region.

• Place the majority of the reinforcement within the column region at interior columns. The width of the column region should be based on engineering judgment; one half of the tributary width is usually a reasonable value.

• The reinforcement everywhere along a design strip must be at least equal in both area and spacing to that required by code for shrinkage and temperature.

G. Waffle Slabs

Waffle slabs (Fig 1.1-21) provide added economy where waffle forms are readily available and concrete is relatively expensive. They are designed using the concept of load path designation discussed above. If a frame analysis method is used, each design strip is substituted by an idealized geometry as shown in Fig 1.1-21(b). The idealized geometry has the same area and moment of inertia as the actual structure in the direction of the frame; it has approximately the same torsional stiffness in the perpendicular direction.

When using Finite Element analysis there are two options. The floor system can be modeled faithfully with each waffle represented by its true geometry. Alternatively, the waffle stems can be lumped together and positioned along the lines of supports to represent the same area, moment of inertia and section modulii as the actual structure (Fig. 1.1-22).

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FIGURE 1.1-21

The cross-section of a waffle slab is shown in Fig. 1.1-23. For design, the waffle slab is idealized as illustrated in Figs. 1.1-21(b) and 1.1-22. The width of the beam in the direction of the analysis, B, is the sum of the waffle stems and any solid regions. In the transverse direction, a torsional member is created by lumping all of the stems joining the solid column block into a rectangle with effective width, be.

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FIGURE 1.1-22

The equivalent torsional member is assumed to consist of a relatively thin flange (c, hf) and a thin stem (be, h-hf). The effective width, be, is determined so as to provide a torsional stiffness equal to the sum of the individual stiffnesses of the stems meeting the solid block at the column.

Since the torsional constant of each of the constituent rectangles is one third of its length times its thickness cubed, the torsional stiffness, J, of the actual slab is given by:

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FIGURE 1.1-23

Depending on the aspect ratio of the structural model, one of two relationships is used for the calculation of the equivalent width, be:

J = (chf3/3) + ∑[(h-hf)bw

3]/3 (1)

(i) If be < (h-hf)

Je = (chf

3/3) + be

3(h-hf)/3 (2)

Setting Je equal to J from Eqn (1) and solving for be gives:

be = (∑bw

3)1/3 (3)

(ii) If be > (h-hf)

Je = (chf

3/3) + be (h-hf)

3/3 (4)

Setting Je equal to J from Eqn (1) and solving for be gives:

be = ∑bw3/ (h-hf)

2 (5)

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H. Summary

The key features of the concrete floor design process are as follows:

• There are three commonly used analysis methods: the Simple Frame, the Equivalent Frame, and the Finite Element methods. The Equivalent Frame method is a refinement of the Simple Frame method, which reduces the column stiffness to account for biaxial plate bending.

• Regardless of the analysis method used, the floor system must go through a structural modeling step, in order to designate “Design Load Paths”. Load paths must be designated before doing a Simple or Equivalent Frame analysis, but can be designated either before or after a Finite Element analysis.

• Design strips are selected in accordance with the designated load paths. The outcome of the analysis step is the moments and shears that must be resisted by each design strip.

• At the design stage, the entire (integral) moments at a design section are applied to the design section in order to calculate the reinforcement required.

• Calculation of the reinforcement is followed by “Structural Detailing,” which ensures the serviceability of the floor and the implementation of the design concept.

All three analysis methods are variations of the proven “strip method,” which requires choosing a statically admissible stress field that satisfies equilibrium. Thus, all three yield a lower bound (safe) solution. Recognition of common features in the three design methods, in particular the necessity of selecting “load paths,” is essential to the automation of concrete floor design.

1.2 STRUCTURAL MODELING OF BEAMS AND ONE-WAY SLABS

Beams and one-way slabs are handled as a single story beam frame with no side sway. Chapter 3 of the manual describes in detail the boundary conditions of columns and the end conditions of the slab/beam model.

In the one-way system the load is transferred primarily by a beam-like strip action to the strip supports, where the reaction is transferred to the adjoining structural elements [Aalami, 1993].

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Only bending is included in the formation of stiffness matrices for the spans and columns. Axial deformations of the members are not included. Hence, shortening of columns and settlement of supports do not enter the calculations directly. If their influence is expected to be significant, they should be accounted for subsequent to the calculations by ADAPT-RC.

1.3 TWO-WAY SLABS AND EQUIVALENT FRAME METHOD

1.3.1 Background

A two-way slab is one that biaxially resists applied loading. That is to say, a load applied on the slab causes moments and shears in two perpendicular directions.

There are several ways two-way slabs may be analyzed. The common procedure is to use variations of the strip method. In this method the slab is modeled as one-way design strips in intersecting directions. The one-way models are approximations. In some of the schemes the two-way action is accounted for by adjustments in the stiffnesses of the slabs and the columns of the one-way substitute. The Equivalent Frame Method described in ACI-318 is the one most widely used.

1.3.2 Equivalent Frame Method (EFM)

The structural model outlined by ACI-318 for the approximate analysis of two-way slabs is the equivalent frame method. The concepts and assumptions of EFM are described in Chapter 13 of ACI and are explained in its commentary on Building Code Requirements for reinforced concrete. ADAPT-RC uses EFM for the analysis of two-way slabs. The formulations and approximations employed in the algorithms of ADAPT-RC are directly those given in the ACI unless specifically noted herein.

In the present work the principal assumptions of EFM are reviewed with the objective of describing their implementation in ADAPT-RC. Some of the material used is quoted directly from the ACI publications.

A. Equivalent Frame

The structure is considered to be made up of equivalent frames on column lines taken longitudinally and transversely through the

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building. Fig. 1.3-1 shows one such slab frame between its upper and lower floors.

FIGURE 1.3-1

Each frame shall consist of a row of column or supports and slab-beam strips, bounded laterally by the centerline of panel on each side of the centerline of columns or supports.

The geometry of a typical interior span and its relating column supports are shown in Fig. 1.3-2. Fig. 1.3-2(b) shows the width of the frame to be used in the analysis.

B. Torsional Members

For the purpose of moment transfer, columns and supports are assumed to be attached to the slab-beam strips by torsional members transverse to the direction of the span. Fig. 1.3-3 illustrates schematically the moment transfer path assumed between the slab and the column. It is the sides (D) of the column that are assumed to receive the moments from the slab, as opposed to the faces (B), which in the regular beam/column models form the interfaces for moment transfer. The described model as applied to a two-span slab-beam strip (Fig. 1.3-4(a)) is shown in Fig. 1.3-4(b). For simplicity, in this figure it is assumed that the end supports rest freely with no ties or provision for moment transfer. It is only the central column for which the model would be applicable.

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FIGURE 1.3-2

The primary parameters defining the stiffness of the spring (Kt) are the geometry of the interface between the side of the column (D) and the slab, and the length E shown in the Fig. 1.3-3(a). For the frame analysis the stiffnesses of the spring (Kt) and the column (Kc) are combined into an equivalent stiffness (Kec) using the following relationship:

1/Kec = 1/Kc + 1/Kt

The consequence of this approximation may be viewed as the substitution of the original column, of height H, by a column of lesser rotational stiffness (longer height) as shown in Fig. 1.3-4(c).

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FIGURE 1.3-3

At each support, the values Kt, Kc and Kec calculated by ADAPT-RC are printed in an auxiliary file for use by the interested engineer.

C. Stiffness of Slab-Beam

The plan and longitudinal section of a typical slab-beam are shown in Fig. 1.3-2. The variation of the moment of inertia along a typical span is illustrated in Fig. 1.3-5. In the general case, there are seven regions of change in the moment of inertia along a span of the slab-beam model as described below:

I1, I7 = moments of inertia of slab sections over the left and right column (joints) respectively;

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FIGURE 1.3-4

I2, I6 = moments of inertia of column cap sections;

I3, I5 = moments of inertia of drop panel sections; and

I4 = moment of inertia of the slab proper.

The moments of inertia of the slab-beam model at any section outside the joints (columns) are based strictly on the gross area of concrete. That is to say, I2 through I6 are from the actual cross-sectional geometry of the slab and the drops at the respective sections. No considerations for the Effective Width are implemented, since in the EFM such effects are handled through the action of the torsional member. Cracking of sections and areas

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of reinforcement do not enter the computations of the moment of inertia at this stage. Moments of inertia over the joints are approximated using the ACI relationships.

The schematic in Fig. 1.3-5 is used by ADAPT-RC in the determination of the stiffness matrix of each slab-beam member. The stiffness matrices calculated are printed in a file for the review and use of the interested user.

FIGURE 1.3-5

D. Stiffness of Columns

Columns may be present above and below a support point. It is not mandatory to have an upper column.

Apart from the cross-sectional geometry, height, and modulus of elasticity of a column, two other factors affect its stiffness, namely (i) its boundary conditions, and (ii) the stiffening effect of the column cap/slab junction.

• ADAPT-RC handles fully free to fully fixed columns. For details refer to Chapter 3 of the manual.

• The increase in column stiffness due to the drops and the finite slab thickness is accounted for through the approximation recommended in the ACI 318 commentary and reproduced in Fig. 1.3-6.

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The approximation used for the increase in column moment of inertia is applied only to the lower columns. The upper columns are assumed uniform.

FIGURE 1.3-6

1.4 LOADING

The applied loading is assumed to be normal to the slab. Four classes of loading are treated, namely the DEAD LOADING, which is assumed to be continuously active, SUPERIMPOSED DEAD LOADING, LIVE LOADING, and OTHER LOADING. The effects due to applied loading are combined for serviceability and strength checks. ACI recommended combination factors are used as default values, but the user may override these during ADAPT-RC input.

A full description of different types of loading is given in Chapter 3, Section 3.3. Fig. 1.4-1 is a compilation of loading types that may be specified arbitrarily on each span. The dead and the live loads are independent from one another. For

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example, at a location where a concentrated dead load acts, it is not necessary to specify a concentrated live loading at the same time.

FIGURE 1.4-1

1.5 BENDING MOMENTS, REACTIONS AND SHEARS

1.5.1 Centerline Moments

The centerline moments are determined by forming the stiffness matrix of the design strip frame and solving for the joint rotations as given below:

[K]{r} = {UBM}

Where,

[K] = the system stiffness matrix; {r} = the vector of unknown joint rotations; and, {UBM} = the vector of unbalanced fixed end moments at

the joints.

The fixed end moments are calculated by accounting for the variation of moment of inertia along the length of slab as illustrated in Fig. 1.3-5.

The joint rotations obtained as a solution are employed to calculate the moments and shears at the end of spans using the previously computed span stiffness matrices.

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43

The system matrix [K] is inverted once and saved for subsequent use in the computation of live load moments.

Span moments and shears are calculated at 20 equal intervals along each span using the end moments obtained and the applied loading. In addition, moments are calculated at the face-of-support. The maximum field moment is selected from the array of moments calculated at these intervals.

1.5.2 Reduction of Moments to the Face-of-Support

The following describes the procedure used by ADAPT-RC to select representative critical moments for each span. The process is called Reduction of Moments. The representative maxima moments may be used for reinforcement determination in simplified designs. Note that these maxima, as described herein for simplified designs, are not used in ADAPT-RC's reinforcement computations. In ADAPT-RC the reinforcement for each section along a member is calculated based on the actions on that section. The representative moments are introduced in order to provide the design engineer with a quick overview of the intensity of actions in the member he/she is designing.

The Analysis of a problem using ADAPT-RC generally concludes with an envelope of moment distribution for each span giving the maximum and minimum bounds of moments at each point due to the specified combination of loading. Such a moment envelope for a typical span is illustrated in Fig. 1.5-1 (a). Observe that the moments extend from the support system line (mid-width of support) of one end of span to the support line at the other end. The moment envelope is the compilation of maxima of individual load combination conditions.

From a somewhat simplified standpoint, some engineers consider a span to be of uniform cross-section and detail each span in three distinct positions for reinforcement. These are reinforcement at center of span, and left and right supports. For each region, the reinforcement is usually selected uniform and constant. For this reason, and for the convenience of the engineers used to the simplified design in addition to the detailed fork, the actions and reinforcement are also summarized and expressed at three critical locations, namely left, center, and right. Obviously, for complex loading and/or members with variable cross-section, the detailed information describing the features of the entire span should be used.

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FIGURE 1.5-1

Fig. 1.5-1(a) describes the breakdown of a span into left, center and right regions. Note that, for example, the maximum positive moment selected for the left region (Fig. 4.6-1 (c)) is not located at the face-of-support. Whether a value refers to the face-of-support, midspan, or a region is clearly identified in the output of ADAPT-RC.

Fig. 1.5-1(b) is the elevation of an interior span with finite support widths. Note that the support width at left (Sl) is less than the left region. The support width at right (Sr) is selected larger than the right region.

Fig. 1.5-1(c) shows the representative moments selected for each of the three regions. For the left and right regions the representative values are

THEORY Chapter 1

45

the maxima of positive and negative values within the designated regions (0.15L), with two exceptions:

o When the support width is less than the designated region length (Sl or Sr < 0.15L), the selection is made outside the support at or between the face-of-support and the extent of the region (refer to left side of Fig. 1.5-1(c)).

o If the face-of-support falls outside the left or right region (Sl or Sr > 0.15L), the values at the face-of-support are selected as representative. This condition occurs for the right support in Fig. 1.5-1(c), where points E and F at face-of-support are selected.

For the central region, the maxima of positive and negative values are selected (points C and D in Fig. 1.5-1(c)).

In the general case, moments associated with points A through F in the envelope of Fig. 1.5-1(c) are likely to govern the reinforcement design. The system-line moments and the analytical minimum and maximum values of moments for the central region are at points A through F as identified in Fig. 1.5-2(b). Observe that these values do not generally match in location and value the reduced moments of the same span (Fig. 1.5-1(c)). The moment values for the central region, however, must be the same between the two presentations. Points A through F of Fig. 1.5-2(b) are referred to as system-line values.

The reduction of moment to the Face-of-Support is carried out strictly from the statics of each span. The moments at the face of support are determined from the equilibrium of the loads on the respective span, end moment and end shears calculated from the global analysis of the frame.

The presentation for reduced moments, where given in ADAPT-RC output, has no impact on the critical computations of ADAPT-RC. The reinforcement in ADAPT-RC is computed and reported from a detailed analysis that considers the specific conditions of each 20th point. The present discussion and the presentation of reduced moments are meant to serve the user who seeks to develop a sense for the values involved in the problem at hand and/or wants a speedy, but approximate, verification of the solution; or, the user who is accustomed to dealing with only three values of moments and reinforcement for each span.

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46

FIGURE 1.5-2

1.6 DESIGN MOMENTS, SHEARS AND REACTIONS

1.6.1 Design Actions

Design moments, shears and reactions are calculated for use in the ultimate Strength verification of the design. The design moments, shears and reactions are also referred to as factored or ultimate values. The relationship used for the calculation of factored actions is as follows:

U = Cd*D + Csd*SD + Cl*L + Cx*X

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47

Where,

U = Factored (design) action, moment, reaction, etc.; D = Contribution of dead load to the factored action;

SD = Contribution of superimposed dead load to the factored action;

L = Contribution of live load to the factored action; and X = Contribution of other load to the factored action.

Cd, Csd, Cl, and Cx are user-defined load factors. The load factor default value is the value of corresponding design code selected by user. For example, for ACI-318 the values are:

Cd = 1.2 Csd = 1.2 Cl = 1.6 Cx = 1.6

1.7 REINFORCEMENT (NONPRESTRESSED REINFORCEMENT)

Reinforcement has to be provided in concrete member on two grounds, (i) to enhance the serviceability performance as prescribed in the code; and, (ii) to meet the ultimate strength requirements. The former is referred to as minimum rebar.

1.7.1 Minimum Rebar Criteria

The ACI code requirements for minimum rebar, as well as the requirements of other codes supported by ADAPT-RC, are implemented in the program. For each case the applicable code stipulation and the resulting reinforcement, as calculated by ADAPT-RC, are printed in the output.

Changes in code requirements, as they occur, will be continuously implemented in ADAPT-RC and shown on the printout of the results.

1.7.2 Limits of Reinforcement

In addition to the requirements for minimum reinforcement described in the preceding, ACI-318 sets other consideration for the maximum and minimum amount of reinforcement in a section under bending, which is:

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48

A. Maximum Tensile Reinforcement and Compression Rebar

The total area and disposition of reinforcement used in any section shall not cause the section to fail in compression. ADAPT-RC checks this criterion. If the tensile reinforcement on its own does not provide adequate capacity, compression reinforcement is added.

1.7.3 Ultimate Strength

A. Criteria

All flexural sections should develop a moment capacity equal to or exceeding the design moment (demand moment) of the section. The check mentioned above is carried out by ADAPT-RC.

B. Method of Ultimate Strength Calculation

The ultimate strength of a section is developed from contributions of the concrete compression zone, the compression rebar (if present) and the tensile reinforcement (if present) using the equations of statics and strain compatibility for plane sections remaining plane. Fig. 1.7-1 shows the free body diagram of the force components enumerated.

ADAPT-RC uses an iterative algorithm to determine the depth of compression zone (a) in Fig. 1.7-1, to satisfy all the governing equations. Fig. 1.7-2 shows the different conditions encountered in the calculation of reinforcement for strength.

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FIGURE 1.7-1

FIGURE 1.7-2

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50

For ACI-318 design, the maximum depth of the compression zone is given by

amax = cmax / β1

Where,

cmax = 0.375 dt,

dt = distance of compression fiber to farthest tension reinforcement.

FIGURE 1.7-3

In the calculation of compression reinforcement the strain in compression bars is evaluated and used to determine the stress in compression reinforcement. The position of the compression reinforcement and the position tensile reinforcement are specified by the user as part of the input data. The calculation of such rebar

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51

in each instance is performed according to procedures outlined in the respective design code selected by the user.

C. Determination of the Effective Depth of Sections

Because ADAPT-RC accounts for the variation in depth of sections along the span, the lever arm for the development of internal moment will also vary at changes in cross-section. For this reason, the lengths of the caps, or drop panels (X3 and X4 shown in Fig. 1.7-3) might be too short to materialize an effective compression zone over the support. Furthermore, the compression reinforcement, if required, might not have the required development length. For these reasons, the selection of depth of a section in the computation of moment capacity is related to the length of the corresponding section along the span as described in Fig. 1.7-3.

If the option “conventional” input data is selected, ADAPT-RC will consider the “drop panel” definition of the ACI code. In doing so, the added thickness of regions around a column is included in the calculation of the resisting moment. Otherwise the lever arm for the resisting moment is not increased due to the added thickness. However, if the option “segmental” input data is selected, at any given section the reinforcement is calculated strictly based on the geometry of that section. In other words, the lever arm due to added thickness is accounted for.

1.7.4 Inflection Points and Length of Reinforcement

Inflection point along a member is defined as location of zero moment. At inflection point, tension zone moves from the top of a member to its soffit, or vice versa. For moment resistance, the primary reinforcement is required over the tension zone. However, in detailing and construction the reinforcement is extended beyond the inflection point for anchorage. The location of inflection point is an important factor in determining the position and length of the required reinforcement.

In actual structures, where members are subject to different combinations of loading, and where live loading is patterned, the point of zero moment is not unique. The following describes the procedure adopted in ADAPT-RC for selection of inflection point positions and their application in determination of reinforcement lengths.

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Consider a typical span of a continuous member. Loading combinations result in a moment envelope. Several variations of moment envelope patterns are shown in Fig. 1.7-4. Fig. 1.7-4(a) is the most common envelope for both the interior and exterior spans of continuous members.

FIGURE 1.7-4

In Fig. 1.7-4(a), which is referred to as common condition, two to four positions of zero moment are identified. In the upper part of the figure, two of the inflection points refer to the negative moment and are used to determine the length of top bars. The other two points identify the location of zero moment on the positive side and are used for the length computation of the bottom bars. The location of an inflection point is

THEORY Chapter 1

53

reported by its distance from the support centerline. These distances are referred to as Inflection Point Distances and are marked in the figure.

Refer to Fig. 1.7-5 and 1.7-6. For the common case, the inflection point distances reported are a, b, c and d. Solid lines in Fig. 1.7-5(b) and Fig. 1.7-6 show the reinforcement lengths required.

FIGURE 1.7-5

A development length must be added to the reinforcement lengths reported. The development length is shown as a broken line in the figures.

As far as position of inflection points are concerned, ADAPT-RC reports distances a and c for support (Fig. 1.7-6). But, for detailing, when the bar requirement on one side of support is longer than the other (a>c), the common practice is to select a length equal to twice the longer bar requirement. For this reason ADAPT-RC reports a bar length of 2a, as

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54

opposed to (a+c). Obviously the development length must be added by the user when detailing. For example, for the top bar at left support, the length reported is 2a, but the length for detailing is 2(a+e), where e is the development length (Fig. 1.7-5).

Consider the other moment envelopes of Fig. 1.7-4. In CASE 2, the negative moment envelope covers the entire span. Therefore, no inflection point is identifiable for the top bar placement. ADAPT-RC selects a top bar extending the entire length of span. A similar situation can occur for the positive moments, in which case reinforcement at bottom will be continuous. CASE 3 is an uncommon condition. It requires that the user examine either the moment envelope or the computed reinforcement at 1/20th points along the member, before a rational selection of rebar can be made. ADAPT-RC, where possible, conservatively combines the fragmented rebar requirements of the span into support and span bars. However, by reporting a CASE 3 condition, ADAPT-RC indicates that 1/20th points rebar computations need to be consulted by the user for an optimum bar selection.

FIGURE 1.7-6

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1.8 PUNCHING SHEAR

1.8.1 Overview

Punching shear is a biaxial phenomenon. Therefore, it is only calculated for the two-way systems.

1.8.2 Calculation Procedure

The evaluation of punching shear consists of the following steps:

o A critical section through the slab or the combination of slab/drop is selected. This hypothetical section is thought to define the surface along which the column region might punch through the slab. Fig. 1.8-1 shows a column region punched through the idealized critical surface.

o For punching shear, the critical surface in Fig. 1.8-1 is regarded to be subjected to the reaction of the column Vu and a fraction (γv) of the column moment Mu, that is to say (γv*Mu). Obviously, the balance of column moment (1 – γv)*Mu, as well as other actions which are generally present at a cut, as illustrated in the figure, must also be resisted by the critical section. But these are considered to be transferred through other paths not reflected in the punching shear formula, and need not be included in the present discussion. The variable v is a function of the geometry of the column/slab connection and is defined by the following ACI-318 equation:

γv = 1 – 1/[1 + (2/3)(b1/b2)1/2

]

Where,

b1 and b2 are dimensions of the critical section in the direction of the frame and the perpendicular direction respectively.

o The shear (Vu) and the moment (Mu) are assumed to result in distributions of stress as shown on the top right hand corner of Fig. 1.8-1. The two distributions are added and yield a maximum existing stress (vu), using the following relationship:

vu = (Vu/Ac) + v*Mu*c/Jc

Where,

Ac = Area of critical surface; c = Distance from centroid of the critical section to

the farthest fiber;

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56

Jc = Similar to the moment of inertia of the critical surface;

Mu = Factored moment; vu = Factored maximum shear stress; and Vu = Factored column reaction.

o Using ACI-318 or other applicable codes a permissible stress is calculated.

o The ratio of existing shear stress to permissible shear stress is calculated. If the ratio is less than unity, the design is acceptable.

FIGURE 1.8-1

1.8.3 Punching Shear Stress Ratio

The geometry of a joint is defined by the user as part of the input data. Based on the geometry, ADAPT-RC differentiates five conditions as listed below:

o Interior column;

o End column;

o Corner column;

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57

o Edge column; and

o Beam or wall support.

These conditions are shown in Fig. 1.8-2, and Fig. 1.8.3-2. For the beam or wall support, punching shear check is not applicable. These two conditions are designed using ACI One-way shear requirements, as presented in Section 1.9 of this chapter.

FIGURE 1.8-2

If the overhang of slab beyond the centerline of a support is less than seven times the slab thickness at that location, ADAPT-RC considers said extension not large enough to develop biaxial punching response. Depending on the location, the joint is then classified as an Edge or End Column. Fig. 1.8-2 illustrates the conditions of the Edge and End Columns. When the restrictions of an Edge and End Column both apply to a joint, the joint is treated as a corner condition.

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58

Note that once a joint is considered as an edge or end condition it is structurally modeled without the overhang, as shown in Fig. 1.8-2. This is a conservative assumption. For joints with an overhang for which the stresses computed by ADAPT-RC marginally exceed allowable values, the overstress condition may be regarded as acceptable, following a close evaluation by the user.

1.8.4 Locations of Punching Shear Check at a Joint

Punching shear is checked at two critical sections for each joint. Case 1 is the check at the Face-of-Support through the column cap (if one is specified). Case 2 is a check at the first change of cross section from the column face. It is normally in the slab at the face of column or drop cap.

Note that for Edge Columns, ADAPT-RC assumes that the slab extends to the outer face of column for punching shear checks (see Fig. 1.8-3).

FIGURE 1.8-3

1.8.5 Round Columns

Round Columns are treated as square columns having the same cross-sectional area. This approximation is conservative. Note the following

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59

quotation from the ACI Committee 352 report on recommendations for design of slab-column connections:

Punching loads for circular columns have been observed to exceed the punching loads of square columns of the same cross-sectional area. Thus, it is conservative and may be analytically simpler to represent circular columns by square columns having the same section.

5.8.6 Punching Shear Reinforcement Design

ADAPT-RC’s automatic punching shear design, reports the calculated punching shear stresses, and provides punching shear reinforcement, where required. You have the choice of specifying whether the punching shear reinforcement should be in the form of stirrups or studs.

Where punching shear reinforcement is required, the program continues checking the adequacy of the joint at code-specified successive layers (critical perimeters), until it reaches the perimeter, where no reinforcement is needed.

ADAPT-RC provides a clear summary of the actions on each column, the design values and the outcome of the code check. The program’s calculations and findings for the punching shear check are presented in Section 13 of the program’s report.

Refer to details for arrangement of punching shear reinforcement (Fig. 1.8-4 and Fig. 1.8-5).

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FIGURE 1.8-4

FIGURE 1.8-5

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61

1.9 BEAM/ONE-WAY SLAB SHEAR

1.9.1 General

ADAPT-RC conducts a check of the shear stresses in beams and one-way slabs using the one-way shear formulas as described herein. Based on the results of the stress checks, the required shear reinforcement is calculated and listed. If no shear reinforcement is required it will be so indicated on the printout. Per ACI, the shear check can start at a distance equal to depth of section from the Face-of-Support.

1.9.2 Calculation of Existing Shear Stress

The existing shear stress is calculated using the following relationship:

vu = Vu/(γ*b*dr)

Where,

b = Width of section (stem in the case of T-sections); dr = Depth of centroid of tension to extreme compression

fiber at the shear check location, but not less than 0.8 times total depth of section;

Vu = Factored shear. It contains the contributions of dead and live loading, as well as the secondary shear; and

γ = Design factor according to selected code.

1.9.3 Calculation of Permissible Shear

Permissible shear (vc) is determined using the applicable code.

Using ACI , the expression is:

vc = 2*(f’c)1/2

Where,

f’c is the concrete cylinder strength at 28 days.

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62

1.9.4 Determination of Shear Reinforcement

Three conditions govern the determination of shear reinforcement. These conditions are as follows:

A. vu < 0.5*φvc

No shear reinforcement required.

B. 0.5*φvc < vu < φvc

No reinforcement required for slabs. For beams, provide a minimum reinforcement as given in the code.

C. vu > φvc

Stirrups are required based on the equation in Chapter 11 of ACI:

Av = (vu - φ*vc)*s/(φ*fy*d)

Where,

Av = area of stirrups required per distances; fy = yield stress of shear reinforcement; and d = distance of compression fiber to centroid of tension rebar.

1.10 DEFLECTIONS

Deflections are calculated by ADAPT-RC due to dead loading, live loading, combinations of foregoing, and creep. Both elastic precracking and post-cracking conditions are treated.

1.10.1 Overview

In many instances, concrete cracks under service loading (cracked sections have a lesser bending stiffness). For the same loading, a cracked concrete member deflects more than if its sections were uncracked. For this reason, the deflection computation of nonprestressed concrete members must be based on a reduced moment of inertia, when the applied moment at a section exceeds its cracking capacity.

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1.10.2 Equivalent Moment of Inertia

The post-cracking reduced moment of inertia is represented through an Equivalent Moment of Inertia, Ie. The variation of the equivalent moment of inertia for a simply supported beam, in which the applied moment exceeds the cracking moment of the section, is shown in the schematic of Fig. 1.10-1. The equivalent moment of inertia is given by [ACI-318, 2005]:

Ie = (Mcr / Ma)3 Ig + [1-(Mcr / Ma)

3]Icr ≤ Ig (1.10.2-1)

Where,

Icr = Moment of inertia of cracked section; Ie = Effective moment of inertia; Ma = Maximum moment in member at stage deflection is

computed; and, Mcr = Cracking moment.

The applied moment, Ma, is calculated using elastic theory and the moment of inertia for the uncracked section – gross moment of inertia, Ig. The change in distribution of moment in indeterminate structures as a result of cracking in concrete is assumed to be generally small, and already accounted for in the empirical formula (1.10.2-1) for equivalent moment of inertia.

Mcr = frIg /yt (1.10.2-2)

Where,

fr = Modulus of rupture, flexural stress causing cracking. It is given by:

fr = 7.5 f’c

1/2 (1.10.2-3)

yt = distance of section centroid to farthest tension fiber

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FIGURE 1.10-1

For all-lightweight concrete, fr is modified as follows:

fr = 0.75 * 7.5 f’c1/2 (1.10.2-3a)

Values of Ig are based on the geometry of the concrete cross section, without accounting for the amount and location of reinforcement. For the common case of rectangular and flanged sections (Fig. 1.10-2), these values are:

For rectangular section:

Ig = bd3/12 (1.10.2-4)

For flanged section:

Ig = hf3(b – bw)/12 + bwh

3/12 + (1.10.2-5)

hf (b – bw)(h – (hf / 2) – yt)2 +

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bwh(yt – h/2)2

FIGURE 1.10-2

The cracked moment of inertia depends on the strain and force distributions on the cross-section typified in Fig. 1.10-3 for a rectangular section; concrete is assumed to take no tension. From the assumption of (a) plane sections remain plane, and (b) tensile force on section equals compression, the position of the neutral axis, c, is determined. For the simple case of rectangular section with tensile reinforcement only, the procedure is as follows:

Tension equals compression gives:

Asfs = bc(fc/2) (1.10.2-6)

Where fs and fc are stresses in steel and concrete respectively.

Since the steel stress fs = Esεs and concrete stress, fc = Ecεc can be rewritten as

AsEsεs = (bc/2)Ecεc (1.10.2-7)

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FIGURE 1.10-3

From similar triangles in Fig. 1.10-3b,

εc/c = εs/(d – c) (1.10.2-8)

or

εs = εc(d/c – 1) (1.10.2-9) From Eqs. 1.10.2-7 and 1.10.2-9, AsEsεc(d/c – 1) = (bc/2)Ecεc (1.10.2-10)

or

AsEs/Ec(d/c –1) = bc/2 (1.10.2-11)

Replacing the modular ratio Es/Ec by n, Eq. 1.10.2-11 can be rewritten as

(bc2)/2 + nAsc – nAsd = 0 (1.10.2-12)

The value of c can be obtained by solving the quadratic equation.

C = [(2dB + 1)1/2

– 1]/B (1.10.2-13)

Where,

B = b/(nAs) (1.10.2-14)

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Other cross-sectional shapes and reinforcement disposition can be treated in a similar manner. The outcome for the common case of rectangular and flanged sections with and without compression reinforcement is:


i. Without compression rebar:

c = kd = [(2dB+1)1/2 – 1]/B (1.10.2-13)

Where,

B = b/(nAs) (1.10.2-14)

ii. With compression rebar:

c = kd = {[2dB(1 + rd’/d) + (1.10.2-15) (1 + r)2]1/2 – (1 + r)}/B

Where,

B = b/(nAs) (1.10.2-14) r = (n – 1) As’/(nAs) (1.10.2-16)

For flanged section with compression zone exceeding the flange thickness:


c = kd = {[G(2d + hf f) + (1.10.2-17) (1 + f)2]1/2 – (1 + f)}/G

Where,

f = hf (b – bw)/(nAs) (1.10.2-18) G = bw/(nAs) (1.10.2-19)


c = kd = {[G(2d + hf f + 2rd’) + (1.10.2-20) (f + r + 1)2]1/2 –

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(f + r + 1)}/G

Where,

f = hf (b – bw)/(nAs) (1.10.2-18) G = bw/(nAs) (1.10.2-19) r = (n – 1) As’/(nAs) (1.10.2-16)

Once the position of the neutral axis is determined, the section is transformed as illustrated in Fig. 1.10-4.

The computed moment of inertia of the sections shown in this figure is:



Icr = (bk3d3)/3 + nAs(d-kd)2 (1.10.2-21)

Where, kd is given in Eq. 1.10.2-13.


Icr = bk3d3/3 + nAs(d – kd)2 + (1.10.2-22) As’(n-1)(kd-d’)2

Where, kd is given in Eq. 1.10.2-15.

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FIGURE 1.10-4

For flanged section with compression zone exceeding the flange thickness:


Icr = hf

3(b – bw)/12 + (bw k3 d3)/3 + (1.10.2-23) hf (b – bw)(kd – hf/2)2 + nAs(d – kd)2

Where, kd is given in Eq 1.10.2-17.


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Icr = hf3(b – bw)/12 + (bw k3 d3)/3 + (1.10.2-24)

hf (b – bw)(kd – hf/2)2 + nAs(d – kd)2 + As’(n –1)(kd-d’)2

Where,

kd is given in Eq 1.10.2-20.

For other sections, a similar procedure is used.

1.10.3 Long-term Deflection (Creep and Shrinkage)

Due to shrinkage and creep of concrete, a concrete member’s deformation changes with time. Shrinkage of concrete is due to loss of moisture. Creep is increase in displacement under stress. Under constant loading, such as self weight, the effect of creep diminishes with time. Likewise, under normal conditions, with loss of moisture, the effect of deformation due to shrinkage diminishes.

For building construction, the impact of creep and shrinkage is accounted for by a creep coefficient, T, given in ACI and shown in Fig. 1.10-5.

FIGURE 1.10-5

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1.11 NONPRISMATIC BEAMS AND SLABS

1.11.1 General

The following expands on the capability of ADAPT-RC in treatment of nonprismatic beams and slab systems. It also provides guidelines for its effective application.

1.11.2 Modeling and Analysis

The scope and limitations for modeling the geometry of a nonprismatic span are the same as described in User’s Manual. Each span may have up to seven segments. In the case of customary Input, the program generates these segments internally if drop caps and drop panels are present.

When the segmented input option is used, the user must enter as input the properties of each segment.

ADAPT-RC faithfully generates the input model internally and computes the associated moments at 1/20th points along each span. These moments are used for stress checks and strength calculations as is described next:

A model of nonprismatic beam is shown in Fig. 1.11-1.

FIGURE 1.11-1

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1.11.3 Strength Check

The factored moment at each 1/20th point is applied to the respective cross-sectional geometry at each location, except at the steps where two sections exist. At each step the cross-section closer to mid-span is selected, since sections closer to mid-span are generally weaker than those near the support. There is one exception to this case. ACI code specifies that drop caps and drop panels be considered only if they extend into the span beyond 1/6th of the respective span length. ADAPT-RC follows this stipulation only when customary input option is used. In this case, it considers drop caps and/or drop panels effective if they are larger than span/6; otherwise they are ignored.

Fig. 1.11-2 shows a section near a column. Note that the drop cap length X1 is smaller than L/6; therefore, it will not be considered in strength calculation and the section depth will be taken as d2 for the region within X1 from the column centerline. Drop panel with a length X2, however, extends beyond L/6 from the support centerline and will be considered in rebar calculations. The section depth for the region extending from X1 to X2 (i.e. drop panel) will be d2, which will be used for reinforcement calculations.

Again note that this consideration is invoked only when the customary input option is selected. In the segmental input option at each 1/20th point the strength is based strictly on the cross-sectional geometry available at that point.

FIGURE 1.11-2

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1.12 LAYOUT OF REINFORCEMENT IN TWO-WAY SLABS

For each design strip, the reinforcement calculated for the strength limit of two-way systems is based on the total design moment applied to the total cross-sectional area of the tributary of that design strip. From a safety standpoint and safeguard against collapse, the design strip will essentially develop its design capacity as long as the design strip contains the calculated reinforcement at the correct height within the slab thickness. The distribution of the reinforcement over the tributary is not critical to the strength capacity of the design section. The ability of a slab to re-distribute the applied moment in its post-elastic phase is the underlying reason for the flexibility in the disposition of reinforcement within a tributary. Concrete slabs fail by forming hinge lines such as the example illustrated in Fig. 1.12-1. A hinge line will mobilize the entire reinforcement along its length.

FIGURE 1.12-1

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Under service loading, the cracking in slab is limited. The distribution of moment resembles the elastic response of the slab. The elastic moments show a greater concentration over the columns. To reduce crack width and improve the in-service performance of a slab, it is beneficial to distribute the computed reinforcement to resemble the distribution of elastic moment. For this reason, in practice the calculated reinforcement is placed with a greater concentration around the columns and the lines joining adjacent columns.

As a guide for distribution of reinforcement in nonprestressed two-way systems, many engineers use the ACI-318’s recommendation reinforcement layout for slabs supported on a regular orthogonal array of columns. The ACI-318’s guidelines are reproduced in Table 1.12-1. The application of the recommended values in the table to actual conditions rests on the judgment of the design engineer. In highly irregular geometry, conditions detail the location judicially with added reinforcement, as opposed to an attempt to refine the coefficients in the table.

TABLE 1.12-1 GUIDELINES FOR PLACEMENT OF REINFORCEMENT IN COLUMN STRIPS

Location Reinforcement percentageExterior 100%

Top bars Interior 75%

Bottom bars All spans 60% Note: Distribute the remainder of design strip reinforcement uniformly in the middle strip

1.13 TREATMENT OF JOINTS

A joint is defined as the portion of beam or slab region that falls within the extended faces of support into the beam/slab (Fig. 1.13-1). A joint may have been cast monolithically with its support, or may be simply resting on the support.

A joint is not free to flex and displace in the same manner that the remainder of the beam/slab does. The added stiffness of the joint affects the magnitude and distribution of moments along the span.

There are two principal ways of treating a joint.

• The simple, but less realistic, way is to disregard the added stiffness of the joint and model the span supported at points of structural system line (middle of supports, Fig. 1.13-1(b)). It can be argued that the moments obtained are larger, and hence the modeling is generally more conservative.

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ADAPT-RC has this modeling as one of its options for users who wish to model the structure conservatively.

• The second option is to increase the moment of inertia of the span over the support, in order to simulate the increased stiffness of the joint (Fig. 1.13-1(c)). ACI-318 recommends an approximation for the increase in stiffness of span (Fig. 1.13-2) over the supports.

ADAPT-RC has incorporated this approximation as an option. For two-way systems this option is automatically invoked when the equivalent frame modeling is selected.

FIGURE 1.13-1

Consider Fig. 1.13-2. Part (a) shows the plan of a slab in the vicinity of a column support. The increase in moment of inertia of the span over the support for the region of span to the right of support is shown symbolically in Fig. 1.13-2(b).

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The increase in stiffness is a function of (c2/l2) [ACI-318]. The increased moment of inertia is used for stiffness calculation of the span. It does not affect the column geometry and its stiffness. The enlarged depth of span over the support is a symbolic representation to denote an increase in moment of inertia. The actual depth of span over at face support is used for other computations.

FIGURE 1.13-2

Fig. 1.13-3 shows two views from a column support of a two-way slab system with column capitals and drop panels. Part (a) of the figure shows the regular modeling. Modeling with increased moment of inertia and modeling used in the equivalent frame option are shown in part (b) of the figure. Note that two slab segments are added over the support region, one associated with each of the spans meeting at the joint. ADAPT-RC generates these segments and their contribution automatically.

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FIGURE 1.13-3

1.14 REDISTRIBUTION OF MOMENTS

1.14.1 General

Reinforced structures are analyzed using elastic theory. Moments, shears and reactions are computed on the premise of reinforced members acting as homogenous elastic materials. Each critical section of a reinforced structure is designed to withstand a factored moment. In resisting the factored moment, the section is considered cracked with full plastification.

It is recognized that from the onset of plasticity at a given section to the point of structural collapse, an elastically designed member can generally sustain additional loading. This is primarily due to the reserve of strength in a section past the stage of attaining first yield, and the ability of it to undergo additional rotation while sustaining the applied moments. Such

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post-elastic rotations result in a redistribution of the initially calculated elastic moments.

Extensive tests have confirmed that a controlled amount of plastification may be permitted to occur under working conditions at locations of maximum moments with deformations and the resulting cracks still within an acceptable range. Limited plastification generally results in the reduction of moments over the supports and a corresponding increase of positive moments in the spans.

The procedure of incorporating, in design, limited adjustments in the elastic moments for the purpose of usurping part of the plastic reserve of a beam/slab is referred to as redistribution of moments. ADAPT-RC is provided with a moment redistribution option, which may be invoked by the user when preparing input data.

A rigorous design method for moment redistribution is quite complex. However, recognition of moment redistribution can be accomplished using a simple method for permitting a reasonable adjustment of the elastically calculated factored gravity load moments as laid out in ACI-318. The amount of adjustment permitted by ACI is kept within predetermined limits.

The actual amount of redistribution allowed at each section depends on the ability of that section to deform inelastically.

If certain reinforcement conditions are met, negative moments at supports of continuous members calculated on the basis of elastic theory may be decreased or increased by an amount stipulated in the code.

1.14.2 Redistribution Procedure

For a beam-frame modeled by ADAPT-RC, the locations of permissible inelastic deformations are shown in Fig. 1.14-1. It is noteworthy that, from the ACI code standpoint, the negative moments at the end supports are not considered for redistribution.

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FIGURE 1.14-1

Figure 1.14-2 shows a typical interior support. Figure 1.14-2(b) illustrates the critical regions to the left and right of the support where the negative moments may be adjusted. The heavy line in Fig. 1.14-2(a) indicates the elastic moments. The hatched region on each side represents the range of adjustment in the elastic moments either upward or downward. The calculated permissible range of adjustment for one section is generally different from other sections of the same slab/beam. The permissible range depends on the geometry of each section and the corresponding amount of reinforcement.

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FIGURE 1.14-2

The final economy of a design depends on whether or not, and to what extent each moment is adjusted up or down. Reduction of all negative moments to their respective maximum permissible extents does not generally yield the economically optimum design.

ADAPT follows the steps outlined below for the redistribution of moments:

o Calculate the maximum permissible redistribution coefficient for each side of the support. These are listed in the output of ADAPT.

o In most cases the cross-sectional geometry of a beam/slab is the same on both sides of a support. Further, negative reinforcement

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required for one side of the support is extended to the other side. Therefore, the provided moment capacities of sections on each side of an interior support are generally the same, although they are subject to different elastic moments. For this reason, the redistribution procedure adopted aims at:

o readjusting the moments at each side of a support to become either equal, or as close to one another as permissible; and

o reducing the maximum negative moments at the supports.

Figure 1.14-3 illustrates this concept. Note that the moment at the left of support is reduced by its maximum permissible value. The moment at the right of support is adjusted to be equal to the redistributed value at left.

FIGURE 1.14-3

Different conditions may arise depending on the relative magnitudes of the moments at each side of a support. Figure 1.14-4 shows some of the common cases. Note that positive moment at left of support in Fig. 1.14-4(c) is not redistributed.

o Once the redistribution over the supports is finalized, the positive moments are adjusted on the basis of static equilibrium of individual spans under the adjusted support moments and the acting loads. Figure 1.14-5 displays two examples of adjusted moments in an interior span.

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FIGURE 1.14-4

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FIGURE 1.14-5

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1.14.3 Redistribution Limits

Maximum percentage of adjustment in negative moment allowed in continuous flexural members is given by ACI 318 -05 as:

1000*εt ≤ 20% (see ACI 8.4.1) (1.14.3-1)

Where,

εt = net tensile strain in extreme tension steel at nominal strength

Redistribution of a moment is permissible if the net tensile strain εt is not less than 0.0075:

εt ≥ 0.0075 (see ACI 8.4.3) (1.14.3-2)

1.15 LATERAL ANALYSIS

1.15.1 Background

In this document, “lateral forces” refers to the forces and effects that are generated in a slab/beam frame due to wind or earthquake. The forces and moments due to the lateral loading are additive to those due to gravity.

In high seismic or wind areas, however, buildings are usually provided with members that are specifically designated to resist the lateral forces. These members are called the primary lateral load resisting members. Collectively, they make up the lateral force resisting structural system of the building. The primary lateral load resisting system of a building must be designed for gravity and lateral forces. Common lateral load resisting systems are shear wall systems, braced frames, and moment frames.

Where relatively high lateral forces and possibly large horizontal displacements are anticipated, the integrity of the members, which are not part of the primary lateral load resisting system, must be checked against the displacements of the building. Such frames are not expected to contribute to the resistance of the lateral loads but are expected to remain serviceable after the lateral load-inducing event. The Uniform Building Code (UBC), for example, requires that all framing members that are not part of the lateral force-resisting system be shown adequate for a combination of gravity loads and induced moments due to a prescribed multiple of the displacements caused by the code-required lateral forces.

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Post-tensioned buildings that are not subjected to either high seismic or high wind loadings typically do not have a separate lateral load resisting system. The slabs and beams are designed to resist the wind or seismic forces in proportion to either their tributary area or the area of the façade they support.

The ACI requirements for the primary lateral force-resisting members are:

Mu = 1.20M

d + 1.00M

l + 1.00M

lat (1)

Mu = 0.90M

d + 0.00M

l + 1.00M

lat (2)

Where,

Mu = factored moment accounting for lateral effects;

Md = moment due to dead loading;

Ml = moment due to live loading; and

Mlat

= moment due to earthquake loading.

The intent of the two ACI relationships is to cover the most adverse combinations of dead, live and lateral loading to determine the factored moments that a given member should be designed to resist. Note that the factors 1.2, 1.0, etc, are quoted for illustration only, they may be different in other codes. The designer must determine the governing factors using the applicable building code.

Two-way Slab Systems

In addition to the check for the total combined moment to be resisted by the frame, there is a second requirement for two-way systems. At any joint of a two-way system, ACI requires that a fraction of column moment be resisted by a narrow strip of slab (referred to as the a strip) immediately over the column. This is referred to as transfer of unbalanced joint moment. The a strip extends 1.5 times the slab thickness, or the drop thickness if there is one, on either side of the column as illustrated in Fig. 1.15-1.

The fraction of unbalanced moment to be transferred by the a strip at each joint is calculated as:

γ = {1/(1 + (2/3)*[(c1+d)/(c

2+d)]1/2}

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FIGURE 1.15-1

Where,

c1 = size of rectangular or equivalent rectangular column,

capital, or bracket measured in the direction of the span for which moments are being determined;

c2 = size of rectangular or equivalent rectangular column,

capital, or bracket measured transverse to the direction of the span; and

d = distance of compression fiber to center of tension

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87

For two-way slab systems, the width of slab on either side of the column to be considered as making up the plane frame must also be determined. This is further discussed at the end of this chapter.

1.15.2 Analysis Procedure

For slab/beam frames subjected to lateral forces, either wind or earthquake, the following design procedure is commonly adopted:

o Design the frame for gravity loading.

o Combine the actions due to lateral loading with those from the gravity loading.

o Check the adequacy of each member for the combined actions. If necessary, add mild reinforcement to meet the requirements of the combined actions.

Figure 1.15-2 illustrates the loading and moments on a single span of a continuous frame. Figure 1.15-2(a) shows the dead loading, live loading, and lateral loading.

The procedure used by ADAPT in analysis and design for lateral moments is as follows:

o The moments due to lateral loading (Mlat

) at each end of a

beam/slab span are computed using an independent frame analysis. For this example, these moments are shown in Fig. 1.15-2(d). Note that the lateral moments vary linearly between adjacent support centerlines. Often, these moments can change sign due to reversal in direction of the lateral loading. The user specifies whether a sign change for the lateral moments should be accounted for when calculating the critical load combinations.

o Moments at 1/20th points due to dead loading are determined by ADAPT as part of the gravity analysis. These are shown for a typical case in Fig. 1.15-2(b).

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FIGURE 1.15-2

o There are two considerations for live loading:

Live loading may be completely absent. Since the absence of live loading may lead to more severe conditions at some points, the factor for the live loading is commonly zero in one of the moment combination equations.

When live loading is present, its arrangement on different spans should maximize the moments at given points. A typical moment envelope is shown in Fig. 1.15-2(c). The envelope lists the maximum positive and negative moments at 1/20th points along each span due to different live load patterns. This type of envelope is generated whenever skipped live loading is specified. If live loading is skipped, each of the two load combination equations must be evaluated twice at any given point along the span, once with the maximum positive live load moment and once with the maximum negative live load moment.

o The lateral load moments are combined with the moments at 1/20th points due to dead and live loading, derived from the gravity solution using the four specified equations. Up to sixteen moment

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combinations are evaluated for each of the 1/20th points along each span.

There are two relationships if the lateral moments change sign; each relationship is evaluated once for the positive direction of the lateral moments, and once for the reversed direction. If live load is skipped, each point is considered with both the maximum positive and maximum negative live load moment. The outcome of the combinations is listed as an envelope of factored moments at 1/20th points as shown in Fig. 1.15-2(e).

o For the purposes of a summary report, a set of six moment values is selected from the list of factored moments generated at 1/20th points. These are: two at the left support, two in the span, and two at the right support. At each location, the maximum and minimum moments are selected. For the left and right supports, the values are moments at the face-of-support, if the user has invoked this option in input data. Otherwise, they are centerline values. For the in-span moments, the midspan values are selected. The values of moments due to combinations of dead and live loads are listed in data block 8 of the Results Report. The values of the moments due to lateral load combinations are listed in data block 9 of the Results Report.

o The required mild reinforcement is calculated for each of the 1/20th points for each span. The maximum value of reinforcement for each of the three regions (left, center and right) of each span is selected and reported in data block 10 of the program output.

1.15.3 Input Screens

Figures 1.15-3 and 1.15-4 show the input screens for the Lateral Analysis.

The Lateral Load Combination tab shows the two underlying equations for the combination of moments where the user enters the combination coefficients. The user has an option to consider change of sign for the applied lateral moment.

The Lateral Moments tab contains a table for entering the applied lateral moments at the ends of each span.

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FIGURE 1.15-3

FIGURE 1.15-4

1.15.4 Description of Printout

A summary of the analysis and design for the combination of lateral and

gravity moments can be seen graphically if the user selects option . The file that lists the input lateral moments and moments due to lateral load combinations may be included in the Report by checking the Factored Lateral Moments on the Report Generator screen and the rebar required is incorporated with the gravity design and reported the envelope in data block 10.

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1.15.5 Discussion of Lateral Loading Treatment

The structural models used for gravity and lateral loads are different. For gravity loading, the Equivalent Frame approach is typically used. In the Equivalent Frame approach, each line of columns along with their respective slab tributaries is considered as a plane frame (Fig. 1.15-5(a)).

FIGURE 1.15-5

Each level is treated as a single story frame with the associated columns fixed at their far ends (Fig.1.15-5(b)). The basic difference between the Equivalent Frame approach and a regular plane frame is that the Equivalent Frame approach recognizes that a column in the actual three-dimensional structure is subject to a smaller moment than what is calculated from the plane frame. The reduced column stiffness used in the Equivalent Frame modeling greatly improves the accuracy of the frame approximation.

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For a lateral analysis (Fig. 1.15-6), vertical frames encompassing the horizontal extent of the building are handled as multistory plane frames. Due to concrete cracking and plate geometry, however, the entire tributary width of the slab does not participate in the frame behavior. With non-prestressed slabs, 25% of the tributary is generally assumed effective for lateral loading.

FIGURE 1.15-6

Figure 1.15-7 is an illustration of the frame stiffnesses for the gravity and lateral analyses as they relate to the slab tributaries. For the gravity loading condition, the equivalent frame approximation is expressed in terms of effective slab width, in order to afford comparison between the gravity and lateral frame behaviors. It is noted that for gravity loading a larger effective width is used. This nonconformity in modeling necessitates two independent analyses, one for the gravity and the other for the later loading, each having a different set of frame stiffnesses. The solutions obtained from the two analyses must be combined, as illustrated in Fig. 1.15-2, in order to design for critical combinations.

Beams and slabs are typically sized and designed on the basis of the gravity loading analysis. They are subsequently checked for combinations with lateral loading. Reinforcement is added if the initial gravity design is not adequate.

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FIGURE 1.15-7

1.15.6 Specific Features for Codes Other than ACI

Ultimate moments and unbalanced joint moments are calculated in the same manner for all design codes. The required reinforcement is determined according to the specific design code, however.

95

Chapter 2

EXAMPLES

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97

2.1 OVERVIEW

In this chapter, two numerical examples are presented. The objective is to illustrate the principal features of ADAPT-RC in a complete setting.

The examples also serve as a valuable reference material for the preparation of input data for ADAPT-RC. For this reason, the critical pages of the input data files for the two examples are reproduced.

For questions related to the input data interface, and for a full description of the ADAPT-RC program and its scope, reference should be made to User Manual, where the input generation features are laid out in great detail.

The examples handled in this chapter are:

• Two-span beam example demonstrates the design feature of the program, which is the program’s most common usage. This is followed by the three analysis (investigation) options of the program.

• The second example is a “design strip” from a two-way floor system using the Equivalent Frame Method.

The two examples are both based on the ACI Code. The modeling of the structure in all codes is identical. The provided examples, therefore, may be used as a guide for alternate codes or system of units.

2.2 TWO-SPAN BEAM EXAMPLE

2.2.1 Design Criteria and Standard Report

The following example has been solved for ACI code in US customary units. There are two spans with dimensions shown in Fig. 2.2-1. The dead and live loading is uniformly distributed. The self weight is entered as an applied dead load. The self weight load generation of the program is not used. Live loading is skipped.

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98

FIGURE 2.2-1

The remaining design criteria, such as concrete strength and cover, are reflected in Data Block 1 of this report. The first-time user is recommended to review Chapter 9 of the User Manual, where a detailed description of the output is given.

2.2.2 Investigation Mode – Option 1

In this investigation mode, in addition to the geometry, boundary conditions, and material properties, the input to the program is a list of reinforcement at the top and bottom of the beam. This list is reproduced in Table 2.2-1. The reinforcement provided in this example is purposely selected to be that determined in the design mode of the program and reported in Data Block 10.2 of the report. This selection serves as a

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verification of the program for its capability of reverse calculation, when the results of investigation Option 1 are compared with those of the Design mode.

TABLE 2.2-1 LISTING OF EXISTING (PROVIDED) REINFORCEMENT ALONG THE BEAM

Span 1 X/L X Ast top Cgs top Ast bot Cgs bot

1 0.00 0.00 2.37 2.50 0.00 2.50 FOS 0.02 0.75 2.37 2.50 1.58 2.50

2 0.05 2.00 0.79 2.50 1.58 2.50 3 0.10 4.00 0.79 2.50 1.58 2.50 4 0.15 6.00 0.00 2.50 3.16 2.50 5 0.20 8.00 0.00 2.50 3.16 2.50 6 0.25 10.00 0.00 2.50 3.16 2.50 7 0.30 12.00 0.00 2.50 3.16 2.50 8 0.35 14.00 0.00 2.50 3.16 2.50 9 0.40 16.00 0.00 2.50 3.16 2.50

10 0.45 18.00 0.00 2.50 3.16 2.50 11 0.50 20.00 0.00 2.50 3.16 2.50 12 0.55 22.00 0.00 2.50 3.16 2.50 13 0.60 24.00 0.00 2.50 3.16 2.50 14 0.65 26.00 0.00 2.50 3.16 2.50 15 0.70 28.00 0.00 2.50 3.16 2.50 16 0.75 30.00 2.37 2.50 1.58 2.50 17 0.80 32.00 2.37 2.50 1.58 2.50 18 0.85 34.00 2.37 2.50 1.58 2.50 19 0.90 36.00 4.74 2.50 0.00 2.50 20 0.95 38.00 4.74 2.50 0.00 2.50

FOS 0.98 39.25 4.74 2.50 0.00 2.50 21 1.00 40.00 4.74 2.50 0.00 2.50

Note: FOS = face-of-support X = distance from left support of each span L = span length Ast top = available area of steel at top Ast bot = available area of steel at bottom Cgs top, Cgs bot = distance of the centroid of steel (top steel

to top fiber & bottom steel to bottom fiber respectively

The result of the investigation is the tabular and graphical output giving the design capacity of the beam along its length. These results are reproduced in Table 2.2-2 and Fig. 2.2-1.

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TABLE 2.2-2 RESULTS OF INVESTIGATION USING OPTION 1

33 - Investigation Mode 33.1 Moment Capacity and Demand Moment SPAN 1

X/L X Demand Moment Pos

Demand Moment

Neg

Moment Capacity

Pos

Moment Capacity

Neg

Demand/Capacity Pos

Demand/Capacity Neg

ft k-ft k-ft k-ft k-ft 0.00 0.00 ---- ---- 252.42 -234.35 ---- ---- 0.05 2.00 ---- ---- 244.92 -123.06 ---- ---- 0.10 4.00 ---- ---- 479.40 -123.85 ---- ---- 0.15 6.00 ---- ---- 472.82 -12.88 ---- ---- 0.20 8.00 ---- ---- 472.82 -12.88 ---- ---- 0.25 10.00 ---- ---- 472.82 -12.88 ---- ---- 0.30 12.00 ---- ---- 472.82 -12.88 ---- ---- 0.35 14.00 ---- ---- 472.82 -12.88 ---- ---- 0.40 16.00 ---- ---- 472.82 -12.88 ---- ---- 0.45 18.00 ---- ---- 472.82 -12.88 ---- ---- 0.50 20.00 ---- ---- 472.82 -12.88 ---- ---- 0.55 22.00 ---- ---- 472.82 -12.88 ---- ---- 0.60 24.00 ---- ---- 472.82 -12.88 ---- ---- 0.65 26.00 ---- ---- 472.82 -12.88 ---- ---- 0.70 28.00 ---- ---- 472.82 -12.88 ---- ---- 0.75 30.00 ---- ---- 491.49 -345.45 ---- ---- 0.80 32.00 ---- ---- 259.00 -345.45 ---- ---- 0.85 34.00 ---- ---- 259.00 -345.45 ---- ---- 0.90 36.00 ---- ---- 38.60 -562.85 ---- ---- 0.95 38.00 ---- ---- 38.60 -562.85 ---- ---- 1.00 40.00 ---- ---- 38.60 -562.85 ---- ----

SPAN 2

X/L X Demand Moment Pos

Demand Moment

Neg

Moment Capacity

Pos

Moment Capacity

Neg

Demand/Capacity Pos

Demand/Capacity Neg

ft k-ft k-ft k-ft k-ft 0.00 0.00 ---- ---- 38.60 -562.85 ---- ---- 0.05 1.50 ---- ---- 38.60 -562.85 ---- ---- 0.10 3.00 ---- ---- 38.60 -562.85 ---- ---- 0.15 4.50 ---- ---- 24.52 -345.51 ---- ---- 0.20 6.00 ---- ---- 24.52 -345.51 ---- ---- 0.25 7.50 ---- ---- 259.00 -345.45 ---- ---- 0.30 9.00 ---- ---- 259.00 -345.45 ---- ---- 0.35 10.50 ---- ---- 259.00 -345.45 ---- ---- 0.40 12.00 ---- ---- 259.00 -345.45 ---- ---- 0.45 13.50 ---- ---- 259.00 -345.45 ---- ---- 0.50 15.00 ---- ---- 355.15 -12.45 ---- ---- 0.55 16.50 ---- ---- 355.15 -12.45 ---- ---- 0.60 18.00 ---- ---- 355.15 -12.45 ---- ---- 0.65 19.50 ---- ---- 355.15 -12.45 ---- ---- 0.70 21.00 ---- ---- 355.15 -12.45 ---- ---- 0.75 22.50 ---- ---- 355.15 -12.45 ---- ---- 0.80 24.00 ---- ---- 237.41 -11.66 ---- ---- 0.85 25.50 ---- ---- 237.41 -11.66 ---- ---- 0.90 27.00 ---- ---- 237.41 -11.66 ---- ---- 0.95 28.50 ---- ---- 244.92 -123.06 ---- ---- 1.00 30.00 ---- ---- 244.92 -123.06 ---- ----

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FIGURE 2.2-1 MOMENT CAPACITY FOR

EXISTING REINFORCEMENT


This option determines the demand as well as the design capacity of the member along its length and reports the outcome in both a tabular and graphical fashion. The input values selected for the applied load and the existing reinforcement are both from the design mode of the program referenced in Section 2.2.1. The program determines the demand (Mu, Vu) and compares it with the design capacity available from the reinforcement given.

The solution is reported in Table 2.2-3 and Fig. 2.2-2.

TABLE 2.2-3 RESULTS OF INVESTIGATION USING OPTION 2

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33 - Investigation Mode 33.1 Moment Capacity and Demand Moment SPAN 1

X/L X Demand Moment

Pos

Demand Moment

Neg

Moment Capacity

Pos

Moment Capacity

Neg

Demand/Capacity Pos

Demand/Capacity Neg

ft k-ft k-ft k-ft k-ft 0.00 0.00 0.00 -76.33 252.42 -234.35 0.000 0.326 0.05 2.00 1.81 -4.05 244.92 -123.06 0.007 0.033 0.10 4.00 98.00 0.00 479.40 -123.85 0.204 -0.000 0.15 6.00 184.42 0.00 472.82 -12.88 0.390 -0.000 0.20 8.00 257.25 0.00 472.82 -12.88 0.544 -0.000 0.25 10.00 315.83 0.00 472.82 -12.88 0.668 -0.000 0.30 12.00 360.08 0.00 472.82 -12.88 0.762 -0.000 0.35 14.00 390.08 0.00 472.82 -12.88 0.825 -0.000 0.40 16.00 405.75 0.00 472.82 -12.88 0.858 -0.000 0.45 18.00 407.25 0.00 472.82 -12.88 0.861 -0.000 0.50 20.00 394.42 0.00 472.82 -12.88 0.834 -0.000 0.55 22.00 367.33 0.00 472.82 -12.88 0.777 -0.000 0.60 24.00 325.92 0.00 472.82 -12.88 0.689 -0.000 0.65 26.00 270.25 0.00 472.82 -12.88 0.572 -0.000 0.70 28.00 200.33 0.00 472.82 -12.88 0.424 -0.000 0.75 30.00 116.17 0.00 491.49 -345.45 0.236 -0.000 0.80 32.00 17.67 -36.37 259.00 -345.45 0.068 0.105 0.85 34.00 0.00 -125.58 259.00 -345.45 0.000 0.364 0.90 36.00 0.00 -254.67 38.60 -562.85 0.000 0.452 0.95 38.00 0.00 -398.08 38.60 -562.85 0.000 0.707 1.00 40.00 0.00 -496.58 38.60 -562.85 0.000 0.882

SPAN 2

X/L X Demand Moment

Pos

Demand Moment

Neg

Moment Capacity

Pos

Moment Capacity

Neg

Demand/Capacity Pos

Demand/Capacity Neg

ft k-ft k-ft k-ft k-ft 0.00 0.00 0.00 -469.42 38.60 -562.85 0.000 0.834 0.05 1.50 0.00 -419.08 38.60 -562.85 0.000 0.745 0.10 3.00 0.00 -326.33 38.60 -562.85 0.000 0.580 0.15 4.50 0.00 -250.00 24.52 -345.51 0.000 0.724 0.20 6.00 0.00 -188.33 24.52 -345.51 0.000 0.545 0.25 7.50 0.00 -132.67 259.00 -345.45 0.000 0.384 0.30 9.00 20.85 -82.91 259.00 -345.45 0.081 0.240 0.35 10.50 68.78 -39.14 259.00 -345.45 0.266 0.113 0.40 12.00 108.67 -1.35 259.00 -345.45 0.420 0.004 0.45 13.50 140.58 0.00 259.00 -345.45 0.543 -0.000 0.50 15.00 164.42 0.00 355.15 -12.45 0.463 -0.000 0.55 16.50 180.17 0.00 355.15 -12.45 0.507 -0.000 0.60 18.00 188.00 0.00 355.15 -12.45 0.529 -0.000 0.65 19.50 187.75 0.00 355.15 -12.45 0.529 -0.000 0.70 21.00 179.42 0.00 355.15 -12.45 0.505 -0.000 0.75 22.50 163.17 0.00 355.15 -12.45 0.459 -0.000 0.80 24.00 138.75 0.00 237.41 -11.66 0.584 -0.000 0.85 25.50 106.42 0.00 237.41 -11.66 0.448 -0.000 0.90 27.00 66.02 0.00 237.41 -11.66 0.278 -0.000 0.95 28.50 21.86 0.00 244.92 -123.06 0.089 -0.000 1.00 30.00 4.28 -10.64 244.92 -123.06 0.017 0.086

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33.2 Rebar Used in Investigation SPAN 1

X/L X Top Rebar Bot Rebar ft in2 in2

0.00 0.00 1.58 0.00 0.05 2.00 0.79 1.58 0.10 4.00 0.79 3.16 0.15 6.00 0.00 3.16 0.20 8.00 0.00 3.16 0.25 10.00 0.00 3.16 0.30 12.00 0.00 3.16 0.35 14.00 0.00 3.16 0.40 16.00 0.00 3.16 0.45 18.00 0.00 3.16 0.50 20.00 0.00 3.16 0.55 22.00 0.00 3.16 0.60 24.00 0.00 3.16 0.65 26.00 0.00 3.16 0.70 28.00 0.00 3.16 0.75 30.00 2.37 3.16 0.80 32.00 2.37 1.58 0.85 33.99 2.37 1.58 0.90 35.99 3.95 0.00 0.95 37.99 3.95 0.00 1.00 39.99 3.95 0.00

SPAN 2

X/L X Top Rebar Bot Rebar ft in2 in2

0.00 0.00 3.95 0.00 0.05 1.50 3.95 0.00 0.10 3.00 3.95 0.00 0.15 4.50 2.37 0.00 0.20 6.00 2.37 0.00 0.25 7.50 2.37 1.58 0.30 9.00 2.37 1.58 0.35 10.50 2.37 1.58 0.40 12.00 2.37 1.58 0.45 13.50 2.37 1.58 0.50 15.00 0.00 2.37 0.55 16.50 0.00 2.37 0.60 18.00 0.00 2.37 0.65 19.50 0.00 2.37 0.70 21.00 0.00 2.37 0.75 22.50 0.00 2.37 0.80 24.00 0.00 1.58 0.85 25.50 0.00 1.58 0.90 27.00 0.00 1.58 0.95 28.50 0.79 1.58 1.00 30.00 0.79 1.58

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FIGURE 2.2-2


The printout for Option 3 is identical to Option 2.

2.3 FOUR-SPAN, TWO-WAY FLOOR SLAB

The FOUR-SPAN, TWO-WAY FLOOR SLAB (Fig. 2.3-1) is the same example as given in Chapter 1 (See Fig. 1.1-8 for plan view of floor slab). In this example, Support Line B has been selected for analysis (Fig. 1.1-11 demonstrates the general demarcation points for tributaries). Figures 1.1-15 and 1.1-16 give an isolated view of Support Line B in plan and elevation. The exact dimensions, material properties, and design criteria implemented in this example are shown in Data Blocks 1-3 of the following Standard Report shown in Fig. 2.3-2. The Detailed Report (Fig. 2.3-2) is also given, for a complete presentation of the program’s design output of a two-way floor system.

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FIGURE 2.3-1

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TWO-WAY FLOOR SYSTEM - NAHID BUILDING DESIGN STRIP B

Thursday, April 08, 2010

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TABLE OF CONTENT: Tabular Reports - Compact 1 - User Specified General Analysis and Design Parameters 2 - Input Geometry 2.1 - Principal Span Data of Uniform Spans 2.5 - Drop Cap and Drop Panel Data 2.7 - Support Width and Column Data 3 - Input Applied Loading 3.1 - Loading As Appears in User's Input Screen 4 - Calculated Section Properties 4.2 - Section Properties for Non-Uniform Spans 5 - Moments, Shears and Reactions 5.1 - Span Moments and Shears (Excluding Live Load) 5.2 - Reactions and Column Moments (Excluding Live Load) 5.3 - Span Moments and Shears (Live Load) 5.4 - Reactions and Column Moments (Live Load) 6 - Moments Reduced to Face of Support 6.1 - Reduced Moments at Face of Support (Excluding Live Load) 6.2 - Reduced Moments at Face of Support (Live Load) 8 - Factored Moments and Reactions Envelope 8.1 - Factored Design Moments (Not Redistributed) 8.2 - Reactions and Column Moments 10 - Mild Steel (No Redistribution) 10.1 - Required Rebar 10.1.1 - Total Strip Required Rebar 10.1.2 - Column Strip Required Rebar 10.1.3 - Middle Strip Required Rebar 10.2 - Provided Rebar 10.2.1 - Total Strip Provided Rebar 10.2.2 - Total Strip Steel Disposition 10.2.3 - Column Strip Provided Rebar 10.2.4 - Column Strip Steel Disposition 10.2.5 - Middle Strip Provided Rebar 10.2.6 - Middle Strip Steel Disposition 13 - Punching Shear Reinforcement 13.1 - Critical Section Geometry 13.2 - Critical Section Stresses 13.3 - Punching Shear Reinforcement 14 - Deflections 14.1 - Maximum Span Deflections 16 - Unbalanced Moment Reinforcement 16.1 - No Redistribution

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Tabular Reports - Detailed 23 - Detailed Moments 24 - Detailed Shears 29 - Detailed Rebar 1 - USER SPECIFIED GENERAL ANALYSIS AND DESIGN PARAMETERS

Parameter Value Parameter Value Concrete Fy (Shear reinforcement) 420.00 N/mm 2

F'c for BEAMS/SLABS 35.00 N/mm 2 Minimum Cover at TOP 25.00 mm For COLUMNS/WALLS 35.00 N/mm 2 Minimum Cover at BOTTOM 25.00 mm Ec for BEAMS/SLABS 27806.00 N/mm 2 Analysis and design options

For COLUMNS/WALLS 27806.00 N/mm 2 Structural system - Equiv Frame TWO-WAY CREEP factor 2.00 Moments reduced to face of

support YES

CONCRETE WEIGHT NORMAL Moment Redistribution NO Reinforcement DESIGN CODE SELECTED ACI-318 (1999)Fy (Main bars) 460.00 N/mm 2

2 - INPUT GEOMETRY 2.1 Principal Span Data of Uniform Spans Span Form Length Width Depth TF

Width TF

Thick. BF/MF Width

BF/MF Thick.

Rh Right Mult.

Left Mult.

m mm mm mm mm mm mm mm 1 1 9.00 1000 240 0 3.50 4.50 2 1 10.00 1000 240 0 4.10 5.25 3 1 10.60 1000 240 0 5.60 5.00 4 1 10.50 1000 240 0 5.60 4.75 C 1 0.80 1000 240 0 5.60 4.75

2.5 Drop Cap and Drop Panel Data Joint Cap T Cap B Cap DL Cap DR Drop TL Drop TR Drop B Drop L Drop R

mm mm mm mm mm mm mm mm mm 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 3 440 1500 750 750 0 0 0 0 0 4 440 3600 1800 1800 0 0 0 0 0 5 440 10350 300 800 0 0 0 0 0

2.7 Support Width and Column Data Joint Support

Width Length

LC B(DIA.)

LC D LC % LC CBC LC Length

UC B(DIA.)

UC D UC % UC CBC

UC mm m mm mm m mm mm

1 200.0 3.0 8000.0 200.0 100 (1) 3.0 8000.0 200.0 100 (1) 2 600.0 3.0 600.0 600.0 100 (1) 3.0 600.0 600.0 100 (1) 3 600.0 3.0 600.0 600.0 100 (1) 3.0 600.0 600.0 100 (1) 4 600.0 3.0 600.0 600.0 100 (1) 3.0 600.0 600.0 100 (1) 5 600.0 3.0 600.0 600.0 100 (1) 3.0 600.0 600.0 100 (1)

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3 - INPUT APPLIED LOADING 3.1 Loading As Appears in User's Input Screen

Span Class Type W P1 P2 A B C F M kN/m2 kN/m kN/m m m m kN kN-m 1 LL U 1.440 1 SDL U 6.939 2 LL U 1.440 2 SDL U 6.939 3 LL U 1.440 3 SDL U 6.939 4 LL U 1.440 4 SDL U 6.939

CANT LL U 2.400 CANT SDL U 6.939

4 - CALCULATED SECTION PROPERTIES 4.2 Section Properties for Non-Uniform Spans

Span Segment Area I Yb Yt mm2 mm4 mm mm

1 1 1920000.00 0.23E+12 120.00 120.00 1 2 1920000.00 0.92E+10 120.00 120.00 1 3 1920000.00 0.11E+11 120.00 120.00 2 1 2244000.00 0.12E+11 120.00 120.00 2 2 2244000.00 0.11E+11 120.00 120.00 2 3 2544000.00 0.25E+11 294.06 145.94 2 4 2544000.00 0.28E+11 294.06 145.94 3 1 2844000.00 0.29E+11 296.79 143.21 3 2 2844000.00 0.26E+11 296.79 143.21 3 3 2544000.00 0.12E+11 120.00 120.00 3 4 3264000.00 0.42E+11 271.47 168.53 3 5 3264000.00 0.47E+11 271.47 168.53 4 1 3204000.00 0.47E+11 270.56 169.44 4 2 3204000.00 0.41E+11 270.56 169.44 4 3 2484000.00 0.12E+11 120.00 120.00 4 4 4554000.00 0.73E+11 220.00 220.00 4 5 4554000.00 0.83E+11 220.00 220.00

CR 1 4554000.00 0.73E+11 220.00 220.00 5 - MOMENTS, SHEARS AND REACTIONS 5.1 Span Moments and Shears (Excluding Live Load)

Span Load Case Moment Left

Moment Midspan

Moment Right

Shear Left

Shear Right

kN-m kN-m kN-m kN kN 1 SW 0.00 0.00 0.00 0.00 0.00 2 SW 0.00 0.00 0.00 0.00 0.00 3 SW 0.00 0.00 0.00 0.00 0.00 4 SW 0.00 0.00 0.00 0.00 0.00

CANT SW 0.00 ----- ----- 0.00 ----- 1 SDL -278.68 198.59 -448.26 -230.96 268.65 2 SDL -477.94 253.07 -637.92 -308.40 340.40 3 SDL -638.56 249.71 -928.14 -362.51 417.15 4 SDL -971.80 292.24 -423.22 -429.29 324.80

CANT SDL -22.98 ----- ----- -57.45 ----- 1 XL 0.00 0.00 0.00 0.00 0.00

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2 XL 0.00 0.00 0.00 0.00 0.00 3 XL 0.00 0.00 0.00 0.00 0.00 4 XL 0.00 0.00 0.00 0.00 0.00

CANT XL 0.00 ----- ----- 0.00 ----- 5.2 Reactions and Column Moments (Excluding Live Load)

Joint Load Case Reaction Moment Lower Column

Moment Upper Column

kN kN-m kN-m 1 SW 0.00 0.00 0.00 2 SW 0.00 0.00 0.00 3 SW 0.00 0.00 0.00 4 SW 0.00 0.00 0.00 5 SW 0.00 0.00 0.00 1 SDL 230.96 -144.33 -134.36 2 SDL 577.05 -15.37 -14.31 3 SDL 702.91 -0.35 -0.29 4 SDL 846.45 -23.99 -19.67 5 SDL 382.26 213.49 186.75 1 XL 0.00 0.00 0.00 2 XL 0.00 0.00 0.00 3 XL 0.00 0.00 0.00 4 XL 0.00 0.00 0.00 5 XL 0.00 0.00 0.00

5.3 Span Moments and Shears (Live Load)

Span Moment Left Max

Moment Left Min

Moment Midspan

Max

Moment Midspan

Min

Moment Right Max

Moment Right Min

Shear Left

Shear Right

kN-m kN-m kN-m kN-m kN-m kN-m kN kN 1 -57.84 -57.84 41.21 41.21 -93.02 -93.02 -47.93 55.75 2 -99.17 -99.17 52.51 52.51 -132.42 -132.42 -64.00 70.64 3 -132.60 -132.60 51.89 51.89 -192.40 -192.40 -75.26 86.54 4 -201.18 -201.18 60.42 60.42 -88.77 -88.77 -88.95 67.54

CR -7.95 ----- ----- ----- ----- ----- -19.87 ----- 5.4 Reactions and Column Moments (Live Load)

Joint Reaction Max

Reaction Min

Moment Lower

Column Max

Moment Lower

Column Min

Moment Upper

Column Max

Moment Upper

Column Min kN kN kN-m kN-m kN-m kN-m

1 47.93 47.93 -29.95 -29.95 -27.88 -27.88 2 119.74 119.74 -3.19 -3.19 -2.97 -2.97 3 145.90 145.90 -0.10 -0.10 -0.08 -0.08 4 175.49 175.49 -4.83 -4.83 -3.96 -3.96 5 87.41 87.41 43.11 43.11 37.71 37.71

6 - MOMENTS REDUCED TO FACE OF SUPPORT 6.1 Reduced Moments at Face of Support (Excluding Live Load)

Span Load Case

Moment Left

Moment Midspan

Moment Right

kN-m kN-m kN-m 1 SDL -255.90 198.60 -370.20 2 SDL -388.30 253.10 -538.70 3 SDL -533.10 249.70 -806.30 4 SDL -846.20 292.20 -329.00

CANT SDL -8.98 ----- -----

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6.2 Reduced Moments at Face of Support (Live Load)

Span Moment Left Max

Moment Left Min

Moment Midspan

Max

Moment Midspan Min

Moment Right Max

Moment Right Min

kN-m kN-m kN-m kN-m kN-m kN-m 1 -53.10 -53.10 41.21 41.21 -76.81 -76.81 2 -80.58 -80.58 52.51 52.51 -111.80 -111.80 3 -110.70 -110.70 51.89 51.89 -167.10 -167.10 4 -175.20 -175.20 60.42 60.42 -69.18 -69.18

CR -3.11 ----- ----- ----- ----- ----- 8 - FACTORED MOMENTS AND REACTIONS ENVELOPE 8.1 Factored Design Moments (Not Redistributed)

Span Left Max

Left Min

Middle Max

Middle Min

Right Max

Right Min

kN-m kN-m kN-m kN-m kN-m kN-m 1 -448.53 -448.53 348.10 348.10 -648.86 -648.86 2 -680.61 -680.61 443.61 443.61 -944.24 -944.24 3 -934.53 -934.53 437.79 437.79 -1412.89 -1412.89 4 -1482.52 -1482.52 511.79 511.79 -578.21 -578.21

CR -17.85 ----- ----- ----- ----- ----- 8.2 Reactions and Column Moments

Joint Reaction Max

Reaction Min

Moment Lower

Column Max

Moment Lower

Column Min

Moment Upper

Column Max

Moment Upper

Column Min kN kN kN-m kN-m kN-m kN-m

1 404.88 404.88 -252.93 -252.93 -235.56 -235.56 2 1011.29 1011.29 -26.93 -26.93 -25.08 -25.08 3 1232.09 1232.09 -0.66 -0.66 -0.54 -0.54 4 1483.31 1483.31 -41.79 -41.79 -34.27 -34.27 5 683.82 683.82 372.19 372.19 325.63 325.63

Note: Moments are reported at face of support 10 - MILD STEEL - NO REDISTRIBUTION 10.1 Required Rebar 10.1.1 Total Strip Required Rebar

Span Location From To As Required Ultimate Minimum m m mm2 mm2 mm2

1 TOP 0.00 1.35 5522.00 5522.00 3108.00 1 TOP 7.20 9.00 8024.00 8024.00 3108.00 2 TOP 0.00 1.50 8469.00 8469.00 3632.00 2 TOP 8.00 10.00 11940.00 11940.00 3632.00 3 TOP 0.00 2.12 11750.00 11750.00 4118.00 3 TOP 7.95 10.60 8993.00 8993.00 5284.00 4 TOP 0.00 2.63 9466.00 9466.00 5186.00 4 TOP 8.93 10.50 7144.00 7144.00 4021.00

CR TOP 0.00 0.80 7372.00 108.20 7372.00 1 BOT 1.80 6.75 4330.00 4330.00 3108.00 2 BOT 2.00 7.50 5398.00 5398.00 3632.00 3 BOT 2.65 7.42 5469.00 5469.00 4118.00 4 BOT 3.15 8.40 6711.00 6711.00 4021.00

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10.1.2 Column Strip Required Rebar


1 TOP 0.00 1.35 5522.00 5522.00 3108.00 1 TOP 7.20 9.00 6018.00 6018.00 2331.00 2 TOP 0.00 1.50 6351.75 6351.75 2724.00 2 TOP 8.00 10.00 8955.00 8955.00 2724.00 3 TOP 0.00 2.12 8812.50 8812.50 3088.50 3 TOP 7.95 10.60 6744.75 6744.75 3963.00 4 TOP 0.00 2.63 7099.50 7099.50 3889.50 4 TOP 8.93 10.50 7144.00 7144.00 4021.00

CR TOP 0.00 0.80 7372.00 108.20 7372.00 1 BOT 1.80 6.75 2598.00 2598.00 1864.80 2 BOT 2.00 7.50 3238.80 3238.80 2179.20 3 BOT 2.65 7.42 3281.40 3281.40 2470.80 4 BOT 3.15 8.40 4026.60 4026.60 2412.60

10.1.3 Middle Strip Required Rebar


1 TOP 7.20 9.00 2006.00 2006.00 777.00 2 TOP 0.00 1.50 2117.25 2117.25 908.00 2 TOP 8.00 10.00 2985.00 2985.00 908.00 3 TOP 0.00 2.12 2937.50 2937.50 1029.50 3 TOP 7.95 10.60 2248.25 2248.25 1321.00 4 TOP 0.00 2.63 2366.50 2366.50 1296.50 1 BOT 1.80 6.75 1732.00 1732.00 1243.20 2 BOT 2.00 7.50 2159.20 2159.20 1452.80 3 BOT 2.65 7.42 2187.60 2187.60 1647.20 4 BOT 3.15 8.40 2684.40 2684.40 1608.40

10.2 Provided Rebar 10.2.1 Total Strip Provided Rebar

Span ID Location From Quantity Size Length Area m m mm2

1 1 TOP 0.00 15 22 2.10 5805.00 1 2 TOP 6.45 11 22 4.86 4257.00 2 3 TOP 7.20 16 22 5.76 6192.00 3 4 TOP 7.12 13 22 6.94 5031.00 4 5 TOP 8.10 20 22 3.20 7740.00 1 6 TOP 7.35 11 22 3.46 4257.00 2 7 TOP 8.20 15 22 3.70 5805.00 3 8 TOP 8.18 12 22 4.82 4644.00 1 9 BOT 1.05 12 22 6.46 4644.00 2 10 BOT 1.20 14 22 7.10 5418.00 3 11 BOT 1.82 15 22 6.44 5805.00 4 12 BOT 2.32 18 22 6.90 6966.00

10.2.2 Total Strip Steel Disposition

Span ID Location From Quantity Size Length m m

1 1 TOP 0.00 15 22 2.10 1 2 TOP 6.45 11 22 2.55 1 6 TOP 7.35 11 22 1.65 2 2 TOP 0.00 11 22 2.31 2 3 TOP 7.20 16 22 2.80

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2 6 TOP 0.00 11 22 1.81 2 7 TOP 8.20 15 22 1.80 3 3 TOP 0.00 16 22 2.96 3 4 TOP 7.12 13 22 3.48 3 7 TOP 0.00 15 22 1.90 3 8 TOP 8.18 12 22 2.42 4 4 TOP 0.00 13 22 3.46 4 5 TOP 8.10 20 22 2.40 4 8 TOP 0.00 12 22 2.40

CR 5 TOP 0.00 20 22 0.80 1 9 BOT 1.05 12 22 6.46 2 10 BOT 1.20 14 22 7.10 3 11 BOT 1.82 15 22 6.44 4 12 BOT 2.32 18 22 6.90

10.2.3 Column Strip Provided Rebar


1 1 TOP 0.00 15 22 2.10 5805.00 1 2 TOP 6.45 8 22 4.86 3096.00 2 3 TOP 7.20 12 22 5.76 4644.00 3 4 TOP 7.12 10 22 6.94 3870.00 4 5 TOP 8.10 20 22 3.20 7740.00 1 6 TOP 7.35 8 22 3.46 3096.00 2 7 TOP 8.20 11 22 3.70 4257.00 3 8 TOP 8.18 9 22 4.82 3483.00 1 9 BOT 1.05 7 22 6.46 2709.00 2 10 BOT 1.20 8 22 7.10 3096.00 3 11 BOT 1.82 9 22 6.44 3483.00 4 12 BOT 2.32 11 22 6.90 4257.00

10.2.4 Column Strip Steel Disposition


1 1 TOP 0.00 15 22 2.10 1 2 TOP 6.45 8 22 2.55 1 6 TOP 7.35 8 22 1.65 2 2 TOP 0.00 8 22 2.31 2 3 TOP 7.20 12 22 2.80 2 6 TOP 0.00 8 22 1.81 2 7 TOP 8.20 11 22 1.80 3 3 TOP 0.00 12 22 2.96 3 4 TOP 7.12 10 22 3.48 3 7 TOP 0.00 11 22 1.90 3 8 TOP 8.18 9 22 2.42 4 4 TOP 0.00 10 22 3.46 4 5 TOP 8.10 20 22 2.40 4 8 TOP 0.00 9 22 2.40

CR 5 TOP 0.00 20 22 0.80 1 9 BOT 1.05 7 22 6.46 2 10 BOT 1.20 8 22 7.10 3 11 BOT 1.82 9 22 6.44 4 12 BOT 2.32 11 22 6.90

10.2.5 Middle Strip Provided Rebar


1 2 TOP 6.45 3 22 4.86 1161.00 2 3 TOP 7.20 4 22 5.76 1548.00

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3 4 TOP 7.12 3 22 6.94 1161.00 1 6 TOP 7.35 3 22 3.46 1161.00 2 7 TOP 8.20 4 22 3.70 1548.00 3 8 TOP 8.18 3 22 4.82 1161.00 1 9 BOT 1.05 5 22 6.46 1935.00 2 10 BOT 1.20 6 22 7.10 2322.00 3 11 BOT 1.82 6 22 6.44 2322.00 4 12 BOT 2.32 7 22 6.90 2709.00

10.2.6 Middle Strip Steel Disposition


1 2 TOP 6.45 3 22 2.55 1 6 TOP 7.35 3 22 1.65 2 2 TOP 0.00 3 22 2.31 2 3 TOP 7.20 4 22 2.80 2 6 TOP 0.00 3 22 1.81 2 7 TOP 8.20 4 22 1.80 3 3 TOP 0.00 4 22 2.96 3 4 TOP 7.12 3 22 3.48 3 7 TOP 0.00 4 22 1.90 3 8 TOP 8.18 3 22 2.42 4 4 TOP 0.00 3 22 3.46 4 8 TOP 0.00 3 22 2.40 1 9 BOT 1.05 5 22 6.46 2 10 BOT 1.20 6 22 7.10 3 11 BOT 1.82 6 22 6.44 4 12 BOT 2.32 7 22 6.90

13 - PUNCHING SHEAR REINFORCEMENT 13.1 Critical Section Geometry

Column Layer Cond. a d b1 b2 mm mm mm mm 1 --- --- --- --- --- --- 2 1 1 96.40 192.80 792.80 792.80 3 3 1 96.40 192.80 1692.80 1692.80 4 1 1 196.40 392.80 992.80 992.80 5 1 2 96.40 192.80 696.40 792.80

13.2 Critical Section Stresses

Label Layer Cond. Factored shear

Factored moment

Stress due to shear

Stress due to moment

Total stress

Allowable stress

Stress ratio

kN kN-m MPa MPa MPa MPa 1 --- --- --- --- --- --- --- --- --- 2 1 1 -1011.40 +51.90 1.65 0.127 1.781 1.670 1.067 3 3 1 -1232.10 +1.32 0.94 0.001 0.945 1.310 0.721 4 1 1 -1483.30 +76.14 0.95 0.057 1.008 1.670 0.604 5 1 2 -683.76 -697.70 1.62 2.101 3.723 1.012 3.678

13.3 Punching Shear Reinforcement Reinforcement option: Shear Studs Stud diameter: 10 Number of rails per side: 2

Col. Dist Dist Dist Dist Dist Dist Dist Dist Dist Dist mm mm mm mm mm mm mm mm mm mm

1 --- --- 2 48.2 96.4

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3 4 5 *** ***

Dist. = Distance measured from the face of support Note: Columns with --- have not been checked for punching shear. Note: Columns with *** have exceeded the maximum allowable shear stress. 14 - DEFLECTIONS 14.1 Maximum Span Deflections

Span SW SW+SDL SW+SDL+Creep LL X Total mm mm mm mm mm mm

1 0.0 4.2 12.5(720) 0.8(11542) 0.0(*****) 13.3(678) 2 0.0 5.0 15.1(663) 1.0(10480) 0.0(*****) 16.0(624) 3 0.0 4.2 12.7(836) 0.9(12063) 0.0(*****) 13.6(782) 4 0.0 6.4 19.3(543) 1.3(7938) 0.0(*****) 20.6(509)

CR 0.0 -1.1 -3.4(236) -0.2(3502) 0.0(*****) -3.6(221) 16 - Unbalanced Moment Reinforcement 16.1 Unbalanced Moment Reinforcement - No Redistribution Joint Gamma

Left Gamma

Right Width Left

Width Right

Moment Left Neg

Moment Left Pos

Moment Right Neg

Moment Right Pos

As Top As Bot n Bar Top

n Bar Bot

m m kN-m kN-m kN-m kN-m mm2 mm2 1 0.00 0.87 0.00 8.00 0.00 0.00 -448.53 0.00 4703.00 0.00 13 0 2 0.60 0.60 1.32 1.32 0.00 0.00 -31.75 0.00 228.10 0.00 1 0 3 0.60 0.60 1.92 1.92 -9.71 0.00 0.00 0.00 70.41 0.00 1 0 4 0.60 0.60 1.92 1.92 0.00 0.00 -69.63 0.00 251.00 0.00 1 0 5 0.60 0.60 1.92 1.92 -560.36 0.00 0.00 0.00 4376.00 0.00 12 0

23 - DETAILED MOMENTS SPAN 1

X/L X SW SDL XL LL Min LL Max m kN-m kN-m kN-m kN-m kN-m

0.00 0.00 0.00 -278.68 0.00 -57.83 -57.83 0.05 0.45 0.00 -180.37 0.00 -37.43 -37.43 0.10 0.90 0.00 -93.30 0.00 -19.36 -19.36 0.15 1.35 0.00 -17.47 0.00 -3.63 -3.63 0.20 1.80 0.00 47.12 0.00 9.78 9.78 0.25 2.25 0.00 100.47 0.00 20.85 20.85 0.30 2.70 0.00 142.58 0.00 29.59 29.59 0.35 3.15 0.00 173.44 0.00 35.99 35.99 0.40 3.60 0.00 193.06 0.00 40.07 40.07 0.45 4.05 0.00 201.45 0.00 41.81 41.81 0.50 4.50 0.00 198.59 0.00 41.21 41.21 0.55 4.95 0.00 184.49 0.00 38.29 38.29 0.60 5.40 0.00 159.15 0.00 33.03 33.03 0.65 5.85 0.00 122.57 0.00 25.44 25.44 0.70 6.30 0.00 74.75 0.00 15.52 15.52 0.75 6.75 0.00 15.68 0.00 3.26 3.26 0.80 7.20 0.00 -54.63 0.00 -11.33 -11.33 0.85 7.65 0.00 -136.17 0.00 -28.25 -28.25 0.90 8.10 0.00 -228.96 0.00 -47.51 -47.51 0.95 8.55 0.00 -332.99 0.00 -69.10 -69.10 1.00 9.00 0.00 -448.26 0.00 -93.02 -93.02

EXAMPLES Chapter 2

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SPAN 2


0.00 0.00 0.00 -477.94 0.00 -99.17 -99.17 0.05 0.50 0.00 -331.85 0.00 -68.86 -68.86 0.10 1.00 0.00 -201.98 0.00 -41.91 -41.91 0.15 1.50 0.00 -88.33 0.00 -18.32 -18.32 0.20 2.00 0.00 9.10 0.00 1.89 1.89 0.25 2.50 0.00 90.31 0.00 18.74 18.74 0.30 3.00 0.00 155.30 0.00 32.23 32.23 0.35 3.50 0.00 204.07 0.00 42.35 42.35 0.40 4.00 0.00 236.63 0.00 49.10 49.10 0.45 4.50 0.00 252.96 0.00 52.49 52.49 0.50 5.00 0.00 253.07 0.00 52.51 52.51 0.55 5.50 0.00 236.96 0.00 49.16 49.16 0.60 6.00 0.00 204.63 0.00 42.45 42.45 0.65 6.50 0.00 156.08 0.00 32.37 32.37 0.70 7.00 0.00 91.31 0.00 18.93 18.93 0.75 7.50 0.00 10.32 0.00 2.12 2.12 0.80 8.00 0.00 -86.89 0.00 -18.06 -18.06 0.85 8.50 0.00 -200.32 0.00 -41.60 -41.60 0.90 9.00 0.00 -329.96 0.00 -68.51 -68.51 0.95 9.50 0.00 -475.83 0.00 -98.78 -98.78 1.00 10.00 0.00 -637.92 0.00 -132.42 -132.42

SPAN 3


0.00 0.00 0.00 -638.56 0.00 -132.60 -132.60 0.05 0.53 0.00 -456.76 0.00 -94.86 -94.86 0.10 1.06 0.00 -295.62 0.00 -61.40 -61.40 0.15 1.59 0.00 -155.14 0.00 -32.23 -32.23 0.20 2.12 0.00 -35.32 0.00 -7.35 -7.35 0.25 2.65 0.00 63.84 0.00 13.24 13.24 0.30 3.18 0.00 142.34 0.00 29.54 29.54 0.35 3.71 0.00 200.17 0.00 41.56 41.56 0.40 4.24 0.00 237.35 0.00 49.29 49.29 0.45 4.77 0.00 253.86 0.00 52.73 52.73 0.50 5.30 0.00 249.71 0.00 51.89 51.89 0.55 5.83 0.00 224.90 0.00 46.75 46.75 0.60 6.36 0.00 179.43 0.00 37.33 37.33 0.65 6.89 0.00 113.30 0.00 23.62 23.62 0.70 7.42 0.00 26.50 0.00 5.62 5.62 0.75 7.95 0.00 -80.95 0.00 -16.66 -16.66 0.80 8.48 0.00 -209.06 0.00 -43.23 -43.23 0.85 9.01 0.00 -357.84 0.00 -74.09 -74.09 0.90 9.54 0.00 -527.28 0.00 -109.24 -109.24 0.95 10.07 0.00 -717.38 0.00 -148.67 -148.67 1.00 10.60 0.00 -928.14 0.00 -192.40 -192.40

SPAN 4


0.00 0.00 0.00 -971.80 0.00 -201.18 -201.18 0.05 0.53 0.00 -756.32 0.00 -156.54 -156.54 0.10 1.05 0.00 -560.63 0.00 -116.00 -116.00 0.15 1.58 0.00 -384.74 0.00 -79.57 -79.57 0.20 2.10 0.00 -228.64 0.00 -47.25 -47.25

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0.25 2.63 0.00 -92.34 0.00 -19.03 -19.03 0.30 3.15 0.00 24.16 0.00 5.07 5.07 0.35 3.68 0.00 120.88 0.00 25.07 25.07 0.40 4.20 0.00 197.79 0.00 40.96 40.96 0.45 4.73 0.00 254.91 0.00 52.74 52.74 0.50 5.25 0.00 292.24 0.00 60.42 60.42 0.55 5.78 0.00 309.77 0.00 63.99 63.99 0.60 6.30 0.00 307.51 0.00 63.44 63.44 0.65 6.83 0.00 285.45 0.00 58.79 58.79 0.70 7.35 0.00 243.60 0.00 50.04 50.04 0.75 7.88 0.00 181.95 0.00 37.17 37.17 0.80 8.40 0.00 100.51 0.00 20.20 20.20 0.85 8.93 0.00 -0.73 0.00 -0.88 -0.88 0.90 9.45 0.00 -121.77 0.00 -26.07 -26.07 0.95 9.98 0.00 -262.59 0.00 -55.37 -55.37 1.00 10.50 0.00 -423.22 0.00 -88.77 -88.77

CR


0.00 0.00 0.00 -22.98 0.00 -7.95 -7.95 0.05 0.04 0.00 -20.74 0.00 -7.17 -7.17 0.10 0.08 0.00 -18.61 0.00 -6.44 -6.44 0.15 0.12 0.00 -16.60 0.00 -5.74 -5.74 0.20 0.16 0.00 -14.71 0.00 -5.09 -5.09 0.25 0.20 0.00 -12.93 0.00 -4.47 -4.47 0.30 0.24 0.00 -11.26 0.00 -3.89 -3.89 0.35 0.28 0.00 -9.71 0.00 -3.36 -3.36 0.40 0.32 0.00 -8.27 0.00 -2.86 -2.86 0.45 0.36 0.00 -6.95 0.00 -2.40 -2.40 0.50 0.40 0.00 -5.75 0.00 -1.99 -1.99 0.55 0.44 0.00 -4.65 0.00 -1.61 -1.61 0.60 0.48 0.00 -3.68 0.00 -1.27 -1.27 0.65 0.52 0.00 -2.82 0.00 -0.97 -0.97 0.70 0.56 0.00 -2.07 0.00 -0.72 -0.72 0.75 0.60 0.00 -1.44 0.00 -0.50 -0.50 0.80 0.64 0.00 -0.92 0.00 -0.32 -0.32 0.85 0.68 0.00 -0.52 0.00 -0.18 -0.18 0.90 0.72 0.00 -0.23 0.00 -0.08 -0.08 0.95 0.76 0.00 -0.06 0.00 -0.02 -0.02 1.00 0.80 0.00 0.00 0.00 0.00 0.00

24 - DETAILED SHEARS SPAN 1

X/L X SW SDL XL LL Min LL Max m kN kN kN kN kN

0.00 0.00 0.00 -230.96 0.00 0.00 -47.93 0.05 0.45 0.00 -205.98 0.00 0.00 -42.75 0.10 0.90 0.00 -181.00 0.00 0.00 -37.56 0.15 1.35 0.00 -156.02 0.00 0.00 -32.38 0.20 1.80 0.00 -131.04 0.00 0.00 -27.19 0.25 2.25 0.00 -106.06 0.00 0.00 -22.01 0.30 2.70 0.00 -81.08 0.00 0.00 -16.83 0.35 3.15 0.00 -56.10 0.00 0.00 -11.64 0.40 3.60 0.00 -31.12 0.00 0.00 -6.46 0.45 4.05 0.00 -6.14 0.00 0.00 -1.27 0.50 4.50 0.00 18.84 0.00 3.91 0.00

EXAMPLES Chapter 2

118

0.55 4.95 0.00 43.82 0.00 9.09 0.00 0.60 5.40 0.00 68.80 0.00 14.28 0.00 0.65 5.85 0.00 93.78 0.00 19.46 0.00 0.70 6.30 0.00 118.76 0.00 24.65 0.00 0.75 6.75 0.00 143.74 0.00 29.83 0.00 0.80 7.20 0.00 168.72 0.00 35.01 0.00 0.85 7.65 0.00 193.70 0.00 40.20 0.00 0.90 8.10 0.00 218.69 0.00 45.38 0.00 0.95 8.55 0.00 243.67 0.00 50.56 0.00 1.00 9.00 0.00 268.65 0.00 55.75 0.00

SPAN 2


0.00 0.00 0.00 -308.40 0.00 0.00 -63.99 0.05 0.50 0.00 -275.96 0.00 0.00 -57.26 0.10 1.00 0.00 -243.52 0.00 0.00 -50.53 0.15 1.50 0.00 -211.08 0.00 0.00 -43.80 0.20 2.00 0.00 -178.64 0.00 0.00 -37.07 0.25 2.50 0.00 -146.20 0.00 0.00 -30.33 0.30 3.00 0.00 -113.76 0.00 0.00 -23.60 0.35 3.50 0.00 -81.32 0.00 0.00 -16.87 0.40 4.00 0.00 -48.88 0.00 0.00 -10.14 0.45 4.50 0.00 -16.44 0.00 0.00 -3.41 0.50 5.00 0.00 16.00 0.00 3.32 0.00 0.55 5.50 0.00 48.44 0.00 10.06 0.00 0.60 6.00 0.00 80.88 0.00 16.79 0.00 0.65 6.50 0.00 113.32 0.00 23.52 0.00 0.70 7.00 0.00 145.76 0.00 30.25 0.00 0.75 7.50 0.00 178.20 0.00 36.99 0.00 0.80 8.00 0.00 210.64 0.00 43.72 0.00 0.85 8.50 0.00 243.08 0.00 50.45 0.00 0.90 9.00 0.00 275.52 0.00 57.18 0.00 0.95 9.50 0.00 307.96 0.00 63.91 0.00 1.00 10.00 0.00 340.40 0.00 70.64 0.00

SPAN 3


0.00 0.00 0.00 -362.51 0.00 0.00 -75.26 0.05 0.53 0.00 -323.53 0.00 0.00 -67.17 0.10 1.06 0.00 -284.55 0.00 0.00 -59.08 0.15 1.59 0.00 -245.56 0.00 0.00 -50.99 0.20 2.12 0.00 -206.58 0.00 0.00 -42.90 0.25 2.65 0.00 -167.60 0.00 0.00 -34.81 0.30 3.18 0.00 -128.61 0.00 0.00 -26.72 0.35 3.71 0.00 -89.63 0.00 0.00 -18.63 0.40 4.24 0.00 -50.65 0.00 0.00 -10.54 0.45 4.77 0.00 -11.66 0.00 0.00 -2.45 0.50 5.30 0.00 27.32 0.00 5.64 0.00 0.55 5.83 0.00 66.30 0.00 13.73 0.00 0.60 6.36 0.00 105.29 0.00 21.82 0.00 0.65 6.89 0.00 144.27 0.00 29.91 0.00 0.70 7.42 0.00 183.25 0.00 38.00 0.00 0.75 7.95 0.00 222.24 0.00 46.09 0.00 0.80 8.48 0.00 261.22 0.00 54.18 0.00 0.85 9.01 0.00 300.20 0.00 62.27 0.00 0.90 9.54 0.00 339.19 0.00 70.36 0.00 0.95 10.07 0.00 378.17 0.00 78.45 0.00

EXAMPLES Chapter 2

119

1.00 10.60 0.00 417.15 0.00 86.54 0.00 SPAN 4


0.00 0.00 0.00 -429.29 0.00 0.00 -88.95 0.05 0.53 0.00 -391.59 0.00 0.00 -81.13 0.10 1.05 0.00 -353.88 0.00 0.00 -73.30 0.15 1.58 0.00 -316.18 0.00 0.00 -65.48 0.20 2.10 0.00 -278.47 0.00 0.00 -57.65 0.25 2.63 0.00 -240.77 0.00 0.00 -49.83 0.30 3.15 0.00 -203.07 0.00 0.00 -42.00 0.35 3.68 0.00 -165.36 0.00 0.00 -34.18 0.40 4.20 0.00 -127.66 0.00 0.00 -26.35 0.45 4.73 0.00 -89.95 0.00 0.00 -18.53 0.50 5.25 0.00 -52.25 0.00 0.00 -10.71 0.55 5.78 0.00 -14.54 0.00 0.00 -2.88 0.60 6.30 0.00 23.16 0.00 4.94 0.00 0.65 6.83 0.00 60.87 0.00 12.77 0.00 0.70 7.35 0.00 98.57 0.00 20.59 0.00 0.75 7.88 0.00 136.28 0.00 28.42 0.00 0.80 8.40 0.00 173.98 0.00 36.24 0.00 0.85 8.93 0.00 211.69 0.00 44.07 0.00 0.90 9.45 0.00 249.39 0.00 51.89 0.00 0.95 9.98 0.00 287.10 0.00 59.72 0.00 1.00 10.50 0.00 324.80 0.00 67.54 0.00

CR


0.00 0.00 0.00 -57.46 0.00 0.00 -19.87 0.05 0.04 0.00 -54.58 0.00 0.00 -18.88 0.10 0.08 0.00 -51.71 0.00 0.00 -17.89 0.15 0.12 0.00 -48.84 0.00 0.00 -16.89 0.20 0.16 0.00 -45.96 0.00 0.00 -15.90 0.25 0.20 0.00 -43.09 0.00 0.00 -14.90 0.30 0.24 0.00 -40.22 0.00 0.00 -13.91 0.35 0.28 0.00 -37.35 0.00 0.00 -12.92 0.40 0.32 0.00 -34.47 0.00 0.00 -11.92 0.45 0.36 0.00 -31.60 0.00 0.00 -10.93 0.50 0.40 0.00 -28.73 0.00 0.00 -9.94 0.55 0.44 0.00 -25.85 0.00 0.00 -8.94 0.60 0.48 0.00 -22.98 0.00 0.00 -7.95 0.65 0.52 0.00 -20.11 0.00 0.00 -6.96 0.70 0.56 0.00 -17.24 0.00 0.00 -5.96 0.75 0.60 0.00 -14.36 0.00 0.00 -4.97 0.80 0.64 0.00 -11.49 0.00 0.00 -3.97 0.85 0.68 0.00 -8.62 0.00 0.00 -2.98 0.90 0.72 0.00 -5.75 0.00 0.00 -1.99 0.95 0.76 0.00 -2.87 0.00 0.00 -0.99 1.00 0.80 0.00 0.00 0.00 0.00 0.00

29 - DETAILED REBAR SPAN 1

X/L X Analysis Top

Analysis Bot

Minimum Top

Minimum Bot

Selected Top

Selected Bot

m mm2 mm2 mm2 mm2 mm2 mm2

EXAMPLES Chapter 2

120

0.00 0.00 5522.00 0.00 3108.00 0.00 5522.00 0.00 0.05 0.45 3848.00 0.00 3108.00 0.00 3848.00 0.00 0.10 0.90 1972.00 0.00 3108.00 0.00 3108.00 0.00 0.15 1.35 366.70 0.00 3108.00 0.00 3108.00 0.00 0.20 1.80 0.00 1001.00 0.00 3108.00 0.00 3108.00 0.25 2.25 0.00 2152.00 0.00 3108.00 0.00 3108.00 0.30 2.70 0.00 3055.00 0.00 3108.00 0.00 3108.00 0.35 3.15 0.00 3698.00 0.00 3108.00 0.00 3698.00 0.40 3.60 0.00 4128.00 0.00 3108.00 0.00 4128.00 0.45 4.05 0.00 4330.00 0.00 3108.00 0.00 4330.00 0.50 4.50 0.00 4258.00 0.00 3108.00 0.00 4258.00 0.55 4.95 0.00 3974.00 0.00 3108.00 0.00 3974.00 0.60 5.40 0.00 3370.00 0.00 3108.00 0.00 3370.00 0.65 5.85 0.00 2609.00 0.00 3108.00 0.00 3108.00 0.70 6.30 0.00 1584.00 0.00 3108.00 0.00 3108.00 0.75 6.75 0.00 332.30 0.00 3108.00 0.00 3108.00 0.80 7.20 1151.00 0.00 3108.00 0.00 3108.00 0.00 0.85 7.65 2881.00 0.00 3108.00 0.00 3108.00 0.00 0.90 8.10 4906.00 0.00 3108.00 0.00 4906.00 0.00 0.95 8.55 7246.00 0.00 3108.00 0.00 7246.00 0.00 1.00 9.00 8024.00 0.00 3108.00 0.00 8024.00 0.00

SPAN 2

X/L X Analysis Top

Analysis Bot

Minimum Top

Minimum Bot

Selected Top

Selected Bot

m mm2 mm2 mm2 mm2 mm2 mm2 0.00 0.00 8469.00 0.00 3632.00 0.00 8469.00 0.00 0.05 0.50 7150.00 0.00 3632.00 0.00 7150.00 0.00 0.10 1.00 4322.00 0.00 3632.00 0.00 4322.00 0.00 0.15 1.50 1851.00 0.00 3632.00 0.00 3632.00 0.00 0.20 2.00 0.00 190.20 0.00 3632.00 0.00 3632.00 0.25 2.50 0.00 1893.00 0.00 3632.00 0.00 3632.00 0.30 3.00 0.00 3307.00 0.00 3632.00 0.00 3632.00 0.35 3.50 0.00 4322.00 0.00 3632.00 0.00 4322.00 0.40 4.00 0.00 5060.00 0.00 3632.00 0.00 5060.00 0.45 4.50 0.00 5398.00 0.00 3632.00 0.00 5398.00 0.50 5.00 0.00 5398.00 0.00 3632.00 0.00 5398.00 0.55 5.50 0.00 5060.00 0.00 3632.00 0.00 5060.00 0.60 6.00 0.00 4402.00 0.00 3632.00 0.00 4402.00 0.65 6.50 0.00 3307.00 0.00 3632.00 0.00 3632.00 0.70 7.00 0.00 1934.00 0.00 3632.00 0.00 3632.00 0.75 7.50 0.00 218.40 0.00 3632.00 0.00 3632.00 0.80 8.00 1851.00 0.00 3632.00 0.00 3632.00 0.00 0.85 8.50 4247.00 0.00 3632.00 0.00 4247.00 0.00 0.90 9.00 7150.00 0.00 3632.00 0.00 7150.00 0.00 0.95 9.50 10390.00 0.00 3632.00 0.00 10390.00 0.00 1.00 10.00 11940.00 0.00 3632.00 0.00 11940.00 0.00

SPAN 3

X/L X Analysis Top

Analysis Bot

Minimum Top

Minimum Bot

Selected Top

Selected Bot

m mm2 mm2 mm2 mm2 mm2 mm2 0.00 0.00 11750.00 0.00 4118.00 0.00 11750.00 0.00 0.05 0.53 9951.00 0.00 4118.00 0.00 9951.00 0.00 0.10 1.06 6313.00 0.00 4118.00 0.00 6313.00 0.00 0.15 1.59 3306.00 0.00 4118.00 0.00 4118.00 0.00 0.20 2.12 737.00 0.00 4118.00 0.00 4118.00 0.00 0.25 2.65 0.00 1358.00 0.00 4118.00 0.00 4118.00 0.30 3.18 0.00 3039.00 0.00 4118.00 0.00 4118.00

EXAMPLES Chapter 2

121

0.35 3.71 0.00 4294.00 0.00 4118.00 0.00 4294.00 0.40 4.24 0.00 5098.00 0.00 4118.00 0.00 5098.00 0.45 4.77 0.00 5469.00 0.00 4118.00 0.00 5469.00 0.50 5.30 0.00 5360.00 0.00 4118.00 0.00 5360.00 0.55 5.83 0.00 4814.00 0.00 4118.00 0.00 4814.00 0.60 6.36 0.00 3817.00 0.00 4118.00 0.00 4118.00 0.65 6.89 0.00 2396.00 0.00 4118.00 0.00 4118.00 0.70 7.42 0.00 562.20 0.00 4118.00 0.00 4118.00 0.75 7.95 1718.00 0.00 4118.00 0.00 4118.00 0.00 0.80 8.48 4465.00 0.00 4118.00 0.00 4465.00 0.00 0.85 9.01 3859.00 0.00 5284.00 0.00 5284.00 0.00 0.90 9.54 5741.00 0.00 5284.00 0.00 5741.00 0.00 0.95 10.07 7927.00 0.00 5284.00 0.00 7927.00 0.00 1.00 10.60 8993.00 0.00 5284.00 0.00 8993.00 0.00

SPAN 4

X/L X Analysis Top

Analysis Bot

Minimum Top

Minimum Bot

Selected Top

Selected Bot

m mm2 mm2 mm2 mm2 mm2 mm2 0.00 0.00 9466.00 0.00 5186.00 0.00 9466.00 0.00 0.05 0.53 8345.00 0.00 5186.00 0.00 8345.00 0.00 0.10 1.05 6135.00 0.00 5186.00 0.00 6135.00 0.00 0.15 1.58 4192.00 0.00 5186.00 0.00 5186.00 0.00 0.20 2.10 4873.00 0.00 4021.00 0.00 4873.00 0.00 0.25 2.63 1957.00 0.00 4021.00 0.00 4021.00 0.00 0.30 3.15 0.00 510.70 0.00 4021.00 0.00 4021.00 0.35 3.67 0.00 2552.00 0.00 4021.00 0.00 4021.00 0.40 4.20 0.00 4193.00 0.00 4021.00 0.00 4193.00 0.45 4.72 0.00 5419.00 0.00 4021.00 0.00 5419.00 0.50 5.25 0.00 6256.00 0.00 4021.00 0.00 6256.00 0.55 5.78 0.00 6711.00 0.00 4021.00 0.00 6711.00 0.60 6.30 0.00 6623.00 0.00 4021.00 0.00 6623.00 0.65 6.83 0.00 6085.00 0.00 4021.00 0.00 6085.00 0.70 7.35 0.00 5234.00 0.00 4021.00 0.00 5234.00 0.75 7.88 0.00 3876.00 0.00 4021.00 0.00 4021.00 0.80 8.40 0.00 2095.00 0.00 4021.00 0.00 4021.00 0.85 8.93 30.12 0.00 4021.00 0.00 4021.00 0.00 0.90 9.45 2618.00 0.00 4021.00 0.00 4021.00 0.00 0.95 9.97 5693.00 0.00 4021.00 0.00 5693.00 0.00 1.00 10.50 7144.00 0.00 4021.00 0.00 7144.00 0.00

CR

X/L X Analysis Top

Analysis Bot

Minimum Top

Minimum Bot

Selected Top

Selected Bot

m mm2 mm2 mm2 mm2 mm2 mm2 0.00 0.00 108.20 0.00 7372.00 0.00 7372.00 0.00 0.05 0.04 108.20 0.00 7372.00 0.00 7372.00 0.00 0.10 0.08 108.20 0.00 7372.00 0.00 7372.00 0.00 0.15 0.12 108.20 0.00 7372.00 0.00 7372.00 0.00 0.20 0.16 108.20 0.00 7372.00 0.00 7372.00 0.00 0.25 0.20 108.20 0.00 7372.00 0.00 7372.00 0.00 0.30 0.24 108.20 0.00 7372.00 0.00 7372.00 0.00 0.35 0.28 108.20 0.00 7372.00 0.00 7372.00 0.00 0.40 0.32 99.41 0.00 7372.00 0.00 7372.00 0.00 0.45 0.36 83.47 0.00 7372.00 0.00 7372.00 0.00 0.50 0.40 68.44 0.00 7372.00 0.00 7372.00 0.00 0.55 0.44 56.13 0.00 7372.00 0.00 7372.00 0.00 0.60 0.48 43.97 0.00 7372.00 0.00 7372.00 0.00 0.65 0.52 33.76 0.00 7372.00 0.00 7372.00 0.00

EXAMPLES Chapter 2

122

0.70 0.56 25.09 0.00 7372.00 0.00 7372.00 0.00 0.75 0.60 17.21 0.00 7372.00 0.00 7372.00 0.00 0.80 0.64 11.04 0.00 7372.00 0.00 7372.00 0.00 0.85 0.68 6.17 0.00 7372.00 0.00 7372.00 0.00 0.90 0.72 2.78 0.00 7372.00 0.00 7372.00 0.00 0.95 0.76 0.69 0.00 7372.00 0.00 7372.00 0.00 1.00 0.80 0.00 0.00 7372.00 0.00 7372.00 0.00

Legend (2.1): Span C = Cantilever Form 1 = Rectangular, 2 = T or Inverted L, 3 = I, 4 = Extended T or L section Rh Elevation of top surface TF Top flange MF Middle flange BF Bottom flange Legend (2.5): Drop Cap Dimensions: Drop Panel Dimensions: CAP T = Total depth of cap DROP TL = Total depth left of joint CAP B = Transverse Width DROP TR = Total depth right of joint CAP DL = Extension left of joint DROP B = Transverse Width CAP DR = Extension right of joint DROP L = Extension left of joint --- DROP R = Extension right of joint Legend (2.7): The Column Boundary Condition (CBC): Fixed at both 1 Hinged at near end, fixed at far end 2 Fixed at near end, hinged at far end 3 Fixed at near end, roller with rotational fixity at far end 4 LC Lower Column UC Upper Column Legend (3.1): Class: SW: Selfweight, LL: Live Load, SDL: Superimposed Dead Load, X: Other Loading Type: U: Uniform, P: Partial Uniform, L: Line Load, M: Applied Moment C: Concentrated Load, R: Triangle, V: Variable, T: Trapezoidal Legend (4.1, 4.2): Yb: distance from centroid to bottom fiber Yt: distance from centroid to top fiber I: gross moment of inertia Legend (10.1, 11.1): From: Beginning of rebar measured from left support of the span To: End of rebar measured from left support of the span As Required: Envelope of minimum and ultimate rebar Ultimate: Required rebar for ultimate load combinations Minimum: Required minimum rebar Legend (10.2, 11_2): ID: ID number of the bar as shown on graph From: Beginning of rebar measured from left support of the span Quantity: Number of bars Size: Bar number Length: Total length of the bar Area: Area of reinforcement Legend (13): Layer : The layer of the reinforcement for each column

EXAMPLES Chapter 2

123

Cond. : 1 = Interior, 2 = End, 3 = Corner, 4 = Edge a : The distance between the layer and face of column or drop cap(*) d : Effective depth b1 : length of section parallel to span line b2 : length of section normal to span line Vu : Factored shear Mu : Factored moment Stress : Maximum stress Allow : Allowable stress Ratio : Ratio of calculated to allowable stress As : Required area of reinforcement Nstud : Number of shear studs between layers on each rail

FIGURE 2.3-2

125

Chapter 3

VERIFICATION

VERIFICATION Chapter 3

127

3.1 DEFLECTION

3.1.1 Background

The background theory for computation of deflection profile of concrete members, with due allowance to cracking of the section, is given in Chapter 2. The following is an example of longhand deflection calculation and its comparison with the solutions obtained from ADAPT-RC. It serves two purposes. First, it demonstrates that the software correctly computes deflections. Second, the detailed description of deflection calculation may be used as a guide for verification of users' problems.

The longhand computation is broken into two parts. The first assumes, as its entry value, the equivalent moment of inertia, Ie, computed by the software. The second illustrates how the software computes Ie.

3.1.2 Deflection Computation

Consider the two-span beam shown in Fig 3.1-1. The particulars of the beam are:

f'c = 28 MPa Ec = 24870 MPa Concrete; normal weight = 2400 kg/m3 Creep/shrinkage factor = 2.0 Steel yield strength = 460 MPa Distance from tension fiber to centroid of tension steel = 71 mm Distance from compression fiber to centroid of compression rebar = = 71 mm Cover = 60 mm top and bottom Loading: Applied dead load = 10 kN/m; Concentrated dead load, P = 25 kN at center of first span; Live load = 12 kN/m.


128

FIGURE 3.1-1

The computed deflections by ADAPT-RC are given in Fig 3.1-2 graphically. The graphs are a representation of the deflection computations at 1/20th points. The 1/20th point deflections for the dead loading are listed in file CDF_D.INT and the combined deflections for dead and live loading are given in file CDF_DL.INT. The values for the current problem are extracted from these files and listed in Table 3.1-1. In the regular output of ADAPT-RC, only the summary of maximum deflections of each span is given (data block 14). For the current problem, this summary is reproduced in Table 3.1-2. In the following, the deflection values of span 1 from this table are a summary of the longhand computations.

TABLE 3.1-1 ADAPT-RC COMPUTED DEFLECTIONS OF SPAN 1 X/L X Deflections, δ

(m) SW* SW+ SDL** (mm)

SW+SDL+LL**** (mm)

0.00 0.00 0.00 0.00 0.00 0.05 0.70 3.12 9.11 14.55 0.10 1.40 6.16 18.00 28.62 0.15 2.10 9.04 26.38 41.74 0.20 2.80 11.68 33.95 53.53 0.25 3.50 13.96 40.46 63.64 0.30 4.20 15.82 45.75 71.82 0.35 4.90 17.17 49.67 77.87 0.40 5.60 17.98 52.14 81.64


129

0.45 6.30 18.24 53.08 83.08 0.50 7.00 17.95 52.50 82.17 0.55 7.70 17.15 50.43 78.99 0.60 8.40 15.89 47.00 73.73 0.65 9.10 14.25 42.39 66.64 0.70 9.80 12.35 36.84 58.07 0.75 10.50 10.25 30.63 48.32 0.80 11.20 8.05 24.09 37.90 0.85 11.90 5.83 17.48 27.37 0.90 12.60 3.67 11.04 17.12 0.95 13.30 1.69 5.06 7.73 1.00 14.00 0.00 0.00 0.00

Note: Ec = Modulus elasticity of concrete = 24870.00 N/mm2 * = Reproduced from file CDF_D.INT ** = Reproduced from file CDF_DPS.INT ** = Reproduced from file CDF_DPSL.INT

TABLE 3.1-2: SUMMARY OF MAXIMUM DEFLECTIONS

14 – DEFLECTIONS 14.1 Maximum Span Deflections

Span SW SW+SDL

SW+SDL+Creep

LL X Total

mm mm mm mm mm mm 1 18.2 53.1 159.2(87) 30.0(466) 0.0(*****) 189.2(73) 2 -1.4 -4.2 -12.5(561) -3.6(1963) 0.0(*****) -16.0(437)

FIGURE 3.1-2

Service Combination: 1*SW+1*SDL +1*LL


130

The procedure adopted for deflection calculation is described next. Refer to Fig 3.1-3. The distance, t, offset to the tangent at A, from point B, is given by the second equation of the moment-area method.

t = (moment of the bending moment diagram about B)/(Ec*Ie) θa = t /AB (wmax + t') = θa * a

where, a, is the distance from support, A, the location at which the maximum deflection occurs (wmax). The distance, a, is read from the Table 3.1-1. For the current problem, this distance is 6300 mm. The user may verify that 6300 mm is the maximum deflection by opening the deflection graph and clicking on the maximum point. The X and Y coordinates of the point will display.

Likewise, the offset, t', at location of maximum deflection is given by:

t' = (moment of the bending moment diagram between A, and D, about, D)/(Ec* Ie)

wmax = θa * a - t'


131

FIGURE 3.1-3

A. Dead Load Deflection (Deflection due to SW+SDL)

The computation is summarized in Table 3.1-3. The values of the equivalent moment of inertia and the applied moment are extracted from the file, DPS_IE.INT, generated by ADAPT-RC. A printout of this file for the current problem is given in Table 3.1-4. Refer to Table 3.1-3 and Table 3.1-4 again:

t = ΣAi*Xi /(Ec*Ig) = 182.79 mm θa = 182.79 mm /14,000 mm = 1.3056E-02


132

(wmax + t' ) = 1.3056E-02*6,300 mm = 82.26 mm

t' = ΣA'i*X'i /(Ec*Ig) = 29.17 mm

wmax = 82.26 mm – 29.17 mm = 53.09 mm (ADAPT-RC 53.10 mm, data block 14, column 3)

TABLE 3.1-3 MOMENT-AREA INTERIM COMPUTATIONS X Moment* Ie* Ai Ci Xi Xi' AiXi/EcIe AiXi'/EcIe

(mm) (N.mm) (mm4) (N.mm2) (mm) (mm) (mm) (mm) (mm)

0 2.00E-02 7.93E+09 n/a n/a n/a n/a n/a n/a

700 7.86E+07 7.93E+09 2.75E+10 233.33 13533.33 5833.33 1.89 0.81

1400 1.48E+08 5.32E+09 7.93E+10 314.36 12914.36 5214.36 6.21 2.51

2100 2.08E+08 4.91E+09 1.24E+11 330.37 12230.37 4530.37 11.96 4.43

2800 2.58E+08 4.80E+09 1.63E+11 337.37 11537.37 3837.37 15.56 5.18

3500 2.99E+08 4.76E+09 1.95E+11 341.43 10841.43 3141.43 17.77 5.15

4200 3.30E+08 4.74E+09 2.20E+11 344.16 10144.16 2444.16 18.91 4.56

4900 3.53E+08 4.73E+09 2.39E+11 346.22 9446.22 1746.22 19.17 3.54

5600 3.65E+08 4.73E+09 2.51E+11 347.94 8747.94 1047.94 18.68 2.24

6300 3.68E+08 4.73E+09 2.57E+11 349.49 8049.49 349.49 17.58 0.76

7000 3.62E+08 4.73E+09 2.56E+11 350.99 7350.99 -349.01 15.99 n/a

7700 3.29E+08 4.74E+09 2.42E+11 355.59 6655.59 -1044.41 13.67 n/a

8400 2.87E+08 4.77E+09 2.15E+11 358.05 5958.05 -1741.95 10.85 n/a

9100 2.35E+08 4.84E+09 1.82E+11 361.62 5261.62 -2438.38 8.03 n/a

9800 1.73E+08 5.08E+09 1.43E+11 367.56 4567.56 -3132.44 5.29 n/a

10500 1.03E+08 5.72E+09 9.65E+10 379.95 3879.95 -3820.05 2.79 n/a

11200 2.23E+07 7.93E+09 4.37E+10 424.93 3224.93 -4475.07 0.83 n/a

11900 -6.73E+07 7.93E+09 -1.57E+10 117.47 2217.47 -5482.53 -0.18 n/a

12600 -1.66E+08 7.93E+09 -8.18E+10 300.53 1700.53 -5999.47 -0.70 n/a

13300 -2.75E+08 5.75E+09 -1.54E+11 321.34 1021.34 -6678.66 -0.93 n/a

14000 -3.93E+08 5.08E+09 -2.34E+11 329.39 329.39 -7370.61 -0.57 n/a

S = 182.79 29.17

Note: * Copied from file D_IE.INT

Ec = Modulus elasticity of concrete = 24870.00 N/mm2. Ai = Moment area each 1/20th subdivision (assumed as a trapezoid) (see Fig 3.1-3) C = Centroid of each 1/20th subdivision (assumed as a trapezoid) Xi = Moment arm of each 1/20th subdivision about point, B (see Fig 3.1-3) Xi' = Moment arm of each 1/20th subdivision about point, D (see Fig 3.1-3)


133

TABLE 3.1-4 ADAPT-RC MOMENTS AND MOMENT OF INERTIAS DUE TO DEAD LOAD

ADAPT STRUCTURAL CONCRETE SOFTWARE SYSTEM DATE: Apr 9,2010 TIME: 12:18 Data ID: Two-span Output File ID: DPS_IE.INT ==============================================================================

Applied moment (Ma), Cracked moment (Mcr), Gross Moment of Inertia (Ig) Cracked I (Icr) and Effective I (Ie)

S pts Ma Mcr Ig Icr Ie

------------------------------------------------------------------------------- 1 0 0.2000000E-01 0.8565768E+08 0.79334001E+10 0.14399999E+00 0.79334001E+10 1 1 0.7863000E+08 0.8565768E+08 0.79334001E+10 0.26143585E+10 0.79334001E+10 1 2 0.1478000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.53174912E+10 1 3 0.2076000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.49133686E+10 1 4 0.2580000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.48040709E+10 1 5 0.2989000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47616456E+10 1 6 0.3304000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47417984E+10 1 7 0.3525000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47318062E+10 1 8 0.3652000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47271112E+10 1 9 0.3684000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47260283E+10 1 10 0.3622000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47281613E+10 1 11 0.3291000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47424717E+10 1 12 0.2866000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47719168E+10 1 13 0.2347000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.48431053E+10 1 14 0.1733000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.50774328E+10 1 15 0.1025000E+09 0.8565768E+08 0.79334001E+10 0.26143585E+10 0.57186371E+10 1 16 0.2233000E+08 0.8565768E+08 0.79334001E+10 0.26165120E+10 0.79334001E+10 1 17 -.6730000E+08 -.1872418E+09 0.79334001E+10 0.28973284E+10 0.79334001E+10 1 18 -.1664000E+09 -.1872418E+09 0.79334001E+10 0.44938081E+10 0.79334001E+10 1 19 -.2748000E+09 -.1872418E+09 0.79334001E+10 0.47371648E+10 0.57482711E+10 1 20 -.3927000E+09 -.1872418E+09 0.79334001E+10 0.47371648E+10 0.50836337E+10

B. Live Load Deflection

Deflection calculation for live loading is followed in a similar manner, in which the applied moment and the associated equivalent moment of inertia are due to the simultaneous application of dead and live loading. The values of the parameters involved are given in Tables 3.1-5 and 3.1-6.

t = ΣAi*Xi /(Ec*Ig) = 292.28 mm

θa = 292.28 mm/14,000 mm = 2.088E-02

(wmax + t' ) = 2.088E-02*6,300 mm = 131.53 mm

t' = ΣA'i*X'i /(Ec*Ig) = 48.45 mm wmax due to dead and live loading = 131.53 mm – 48.45 mm = 83.08 mm

wmax due to live loading = 83.08 mm – 53.09 mm


134

= 29.99 mm (ADAPT-RC 30.0 mm, data block 14, column 5)

TABLE 3.1-5 MOMENT-AREA INTERIM COMPUTATIONS

X Moment* Ie* Ai Ci Xi Xi' AiXi/EcIe AiXi'/EcIe

(mm) (N.mm) (mm4) (N.mm2) (mm) (mm) (mm) (mm) (mm)

0 3.00E-02 7.93E+09 n/a n/a n/a n/a n/a n/a

700 1.24E+08 4.38E+09 4.33E+10 233.33 13533.33 5833.33 3.82 1.65

1400 2.32E+08 4.85E+09 1.24E+11 314.46 12914.46 5214.46 13.99 5.65

2100 3.25E+08 4.74E+09 1.95E+11 330.53 12230.53 4530.53 19.98 7.40

2800 4.03E+08 4.72E+09 2.55E+11 337.54 11537.54 3837.54 24.96 8.30

3500 4.65E+08 4.71E+09 3.04E+11 341.61 10841.61 3141.61 28.09 8.14

4200 5.12E+08 4.70E+09 3.42E+11 344.39 10144.39 2444.39 29.65 7.14

4900 5.44E+08 4.70E+09 3.69E+11 346.49 9446.49 1746.49 29.86 5.52

5600 5.60E+08 4.70E+09 3.86E+11 348.26 8748.26 1048.26 28.93 3.47

6300 5.61E+08 4.70E+09 3.93E+11 349.89 8049.89 349.89 27.05 1.18

7000 5.47E+08 4.70E+09 3.88E+11 351.48 7351.48 -348.52 24.42 n/a

7700 5.00E+08 4.70E+09 3.67E+11 355.24 6655.24 -1044.76 20.87 n/a

8400 4.38E+08 4.71E+09 3.28E+11 357.73 5957.73 -1742.27 16.72 n/a

9100 3.60E+08 4.73E+09 2.79E+11 361.34 5261.34 -2438.66 12.53 n/a

9800 2.68E+08 4.79E+09 2.20E+11 367.24 4567.24 -3132.76 8.48 n/a

10500 1.59E+08 3.44E+09 1.49E+11 379.56 3879.56 -3820.44 5.66 n/a

11200 3.59E+07 7.93E+09 6.84E+10 423.74 3223.74 -4476.26 1.56 n/a

11900 -1.03E+08 7.93E+09 -2.34E+10 107.91 2207.91 -5492.09 -0.26 n/a

12600 -2.57E+08 5.83E+09 -1.26E+11 300.02 1700.02 -5999.98 -1.25 n/a

13300 -4.26E+08 5.01E+09 -2.39E+11 321.09 1021.09 -6678.91 -1.81 n/a

14000 -6.11E+08 4.83E+09 -3.63E+11 329.22 329.22 -7370.78 -0.98 n/a

S = 292.28 48.45

Note: ** Copied from file DPSL_IE.INT Ec = Modulus elasticity of concrete = 24870.00 N/mm2. Ai = Moment area each 1/20th subdivision (assumed as a trapezoid) (see Fig 3.1-3) Ci = Centroid of each 1/20th subdivision (assumed as a trapezoid) Xi = Moment arm of each 1/20th subdivision about point B (see Fig 3.1-3) Xi' = Moment arm of each 1/20th subdivision about point D (see Fig 3.1-3)


135

TABLE 3.1-6 ADAPT-RC MOMENTS AND MOMENT OF INERTIAS DUE TO DEAD AND LIVE LOAD

ADAPT STRUCTURAL CONCRETE SOFTWARE SYSTEM DATE: Apr 9,2010 TIME: 12:18 Data ID: Two-span Output File ID: DPSL_IE.INT

==============================================================================

Applied moment (Ma), Cracked moment (Mcr), Gross Moment of Inertia (Ig) Cracked I (Icr) and Effective I (Ie)

S pts Ma Mcr Ig Icr Ie

------------------------------------------------------------------------------- 1 0 0.3000000E-01 0.8565768E+08 0.79334001E+10 0.14399999E+00 0.79334001E+10 1 1 0.1236000E+09 0.8565768E+08 0.79334001E+10 0.26143585E+10 0.43847813E+10 1 2 0.2319000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.48488945E+10 1 3 0.3248000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47447767E+10 1 4 0.4025000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47165051E+10 1 5 0.4649000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47055150E+10 1 6 0.5119000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47004170E+10 1 7 0.5437000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.46978995E+10 1 8 0.5602000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.46968100E+10 1 9 0.5613000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.46967419E+10 1 10 0.5472000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.46976573E+10 1 11 0.5002000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47015101E+10 1 12 0.4380000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47094927E+10 1 13 0.3604000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47288079E+10 1 14 0.2676000E+09 0.8565768E+08 0.79334001E+10 0.46851978E+10 0.47917307E+10 1 15 0.1594000E+09 0.8565768E+08 0.79334001E+10 0.26143585E+10 0.34397640E+10 1 16 0.3594000E+08 0.8565768E+08 0.79334001E+10 0.26165120E+10 0.79334001E+10 1 17 -.1028000E+09 -.1872418E+09 0.79334001E+10 0.28973284E+10 0.79334001E+10 1 18 -.2569000E+09 -.1872418E+09 0.79334001E+10 0.44938081E+10 0.58255585E+10 1 19 -.4262000E+09 -.1872418E+09 0.79334001E+10 0.47371648E+10 0.50081879E+10 1 20 -.6109000E+09 -.1872418E+09 0.79334001E+10 0.47371648E+10 0.48291963E+10

In cases in which the live loading is skipped, an envelope for the live load is calculated, and hence, the combined dead and live loading is determined. In the computation of the equivalent moment of inertia, the following conservative procedure is adopted. At a location where the combined dead and skipped live load moment does not change sign, the maximum value of the combined moment is used as Ma, since this produces maximum deflection. At locations where the combined moment changes sign, two values are computed for the equivalent moment of inertia, Ie: one value uses the maximum positive moment, and the other uses the maximum negative moment. Of the two values of computed Ie, the smaller is used for deflection computation.

C. Dead Load and Creep/Shrinkage

Creep/shrinkage factor = 2.0 wmax due to DL = 53.1 mm wmax due to creep/shrinkage and dead load =


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(2.0*53.1 mm) + (53.1 mm) = 159.3 mm

D. Dead Load, Live Load and Creep/Shrinkage

Creep/Shrinkage factor = 2.0 wmax due to LL = 30.0 mm wmax due to creep/shrinkage, live load, and dead load = 159.3 mm + 30.0 mm = 189.3 mm

3.1.3 Computation of Equivalent Moment of Inertia, Ie

In addition to the geometry of the section, the location and amount of both the tension and the compression reinforcement are necessary to compute Icr. Herein, Icr at the section next to the second support is hand calculated. This section contains both negative and positive reinforcement. Refer to Table 3.1-7.

Since tension occurs at the top fiber of the section, indicated by the negative moment in Table 3.1-6, the following are the primary equations used in determining the cracking moment of inertia, Icr.

Icr = bk3d3/3 + nAs(d - kd)2 + As'(n-1)(kd-d')2 (1.10.2-22)

Where,

c = kd = {[2dB(1 + rd'/d) + (1 + r)2]1/2 -(1 + r)}/B (1.10.2-15)

and,

B = b/(nAs) (1.10.2-14) r = (n - 1) As'/(nAs) (1.10.2-16)

For the current problem,

As = 7740 mm2 (Extracted from BAR_PT_N.INT file)

As' = 2322 mm2 (Extracted from BAR_PT_N.INT file) b = 400 mm d = 428.9 mm (Extracted from BAR_PT_N.INT file) d' = 71.1 mm (Extracted from BAR_PT_N.INT file) Ec = 24,870 N/mm2


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Es = Es' = 200,000 N/mm2 n = Es/Ec = 8.04

Solving for B and r, to obtain c,

B = 400/(8.04*7740)

= 6.428E-03 /mm

r = (8.04 - 1)*2322/8.04*7740 = 0.263

c = kd = {[2*428.9*6.428E-03*(1 + (0.263*71.1/428.9)) +

(1 + 0.263)2]1/2 - (1 + 0.263)}/ 6.428E-03 = 225.30 mm

(ADAPT-RC 225.34 mm, DPSL_CRP1.INT file)

And finally, solving for Icr,

Icr = 400*(225.30)3/3 + 8.04*7740*(428.9 - 225.30)2 + 2322(8.04 - 1)*( 225.30 – 71.1)2

= 0.4494E+10mm4 (ADAPT-RC .4494E+10 mm4, DPSL_IE.INT file)

From Chapter 1:

Ie = (Mcr / Ma)3 Ig + [1-(Mcr / Ma)3]Icr ≤ Ig (1.10.2-1)

where for the current problem,

Mcr = -0.1872418E+09 N.mm (Extracted from DPSL_IE.INT file) Ma = -0.2569000E+09 N.mm (Extracted from DPSL_IE.INT file) Ig = 0.7933400E+10 mm4 (Extracted from DPSL_IE.INT file)

After substitution,

Ie = (-0.1872418E+09 / -0.2569000E+09)3 *0.7933400E+10 +

[1-(-0.1872418E+09 / -0.2569000E+09)3]* 0.4494E+10 = 0.5826E+10 mm4 < Ig = 0.79334E+10 mm4

(ADAPT-RC 0. 5826E+10 mm4, DPSL_IE.INT file)


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TABLE 3.1-7 ADAPT-RC 1/20TH POINT REINFORCEMENT OUTPUTS

ADAPT STRUCTURAL CONCRETE SOFTWARE SYSTEM DATE: Apr 9,2010 TIME: 12:18 Data ID: Two-span Output File ID: BAR_PT_N.INT ============================================================================== Area of Tensile reinforcement, Compressive reinforcement, and Post-Tensioning, also their centroidal distances to extreme compression fiber This is for the case when moment is NEGATIVE

S pts Ast dr Asc dr` Apt1 dpt1 Apt2 dpt2 Apt3 dpt3 1 0 0 428.9 0 71.1 0 0 0 0 0 0 1 1 0 428.9 2322 71.1 0 0 0 0 0 0 1 2 0 428.9 4644 71.1 0 0 0 0 0 0 1 3 0 428.9 4644 71.1 0 0 0 0 0 0 1 4 0 428.9 4644 71.1 0 0 0 0 0 0 1 5 0 428.9 4644 71.1 0 0 0 0 0 0 1 6 0 428.9 4644 71.1 0 0 0 0 0 0 1 7 0 428.9 4644 71.1 0 0 0 0 0 0 1 8 0 428.9 4644 71.1 0 0 0 0 0 0 1 9 0 428.9 4644 71.1 0 0 0 0 0 0 1 10 0 428.9 4644 71.1 0 0 0 0 0 0 1 11 0 428.9 4644 71.1 0 0 0 0 0 0 1 12 0 428.9 4644 71.1 0 0 0 0 0 0 1 13 0 428.9 4644 71.1 0 0 0 0 0 0 1 14 0 428.9 4644 71.1 0 0 0 0 0 0 1 15 0 428.9 2322 71.1 0 0 0 0 0 0 1 16 3870 428.9 2322 71.1 0 0 0 0 0 0 1 17 3870 428.9 2322 71.1 0 0 0 0 0 0 1 18 7740 428.9 2322 71.1 0 0 0 0 0 0 1 19 7740 428.9 3870 71.1 0 0 0 0 0 0 1 20 7740 428.9 3870 71.1 0 0 0 0 0 0

Note:

S = Span pts = point along span Ast = area of tension steel dr = distance of compression fiber to center of tension steel Asc = area of compression steel dr' = distance of compression fiber to center of compression steel

Apt & dpt are not used herein.

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REFERENCES

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REFERENCES References

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Aalami, B. O. (2005), “Structural Modeling and Analysis of Concrete Floor Slabs,” ACI, Concrete International, December 2005, pp. 39-43.

Aalami, B. O. (2001) “ Software for the Design of Concrete Buildings,” ACI, Concrete International, December 2001, pp.28-35.

Aalami, B. O. (1993). “One-Way and Two-Way Post-Tensioned Floor Systems,” PTI Technical Note #3, October 1993, Post-Tensioning Institute, Phoenix, AZ, 10 pp.

ACI 318-05 (2005). “Building Code Requirements for Structural Concrete,” American Concrete Institute, Detroit, MI.

Bathe, K.J. (1982) “Finite Element Procedures in Engineering Analysis,” Prentice-Hall, Englewood Cliffs, N.J.

IBC2000, (2003) “International Building Code,” International Code Council, Inc., Falls Church, Va.

Vanderbilt, M. Daniel, and Corley, W. Gene, (1983), “Frame Analysis of Concrete Buildings,” Concrete International Design and Construction, ACI, V.5, No. 12, Dec, 1983, pp. 33-43.

Zienkiewicz, O.C. and Taylor, R.L. (1989) “The Finite Element Method,” vol. 1, 4th ed., McGraw-Hill, New York.

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ADAPT RC 2010€¦ · a closer agreement between the practice and code. 1.1.2 Requirements of Design Procedure The difficult part of automating concrete design is the floor slab