Reinforced Concrete Software Suite RC-SOFTER software programs RC-TOOL / RC-BIAX / RC-SLAB1 / RC-WALL / RC-MODEL Developed by Professor Abdelhamid Charif (
[email protected]) King Saud University, Civil Engineering Department. Last edited in July 2013.
Part 1 General presentation 1.1.
Introduction
The software programs described in this document have been under continuous development for the last twelve years. They perform reinforced concrete analysis and design according to Saudi Building Code SBC 304 and American Code ACI. The author has a long past experience with development of software for structural analysis and design according to various codes of practice. While structural analysis programs are nowadays available in plethoric numbers, advanced design tools are however rather scarce, with limited options, and comply with few building codes only. SBC code provisions were integrated as the Saudi code was developed. It is believed that these programs are the first, and probably the only ones, to integrate the specific provisions of the new Saudi Building Code SBC 304. Four programs are currently available: 1. RC-TOOL: Analysis and design of RC beam and column sections in uniaxial bending combined with an axial force. Shear design of beams is also included. 2. RC-BIAX: Analysis and design of RC columns sections in biaxial bending combined with an axial force. 3. RC-SLAB1: Analysis and design of one-way slabs and continuous beams 4. RC-WALL: Analysis and design of retaining walls RC-TOOL deals with the design beam and column sections, subjected to a uniaxial bending moment and an axial force, and is thus limited to sections with a vertical axis of symmetry, and steel bars at the same levels are assembled in layers. RC-BIAX is extended to biaxial bending combined with an axial force for arbitrary shaped sections, and is more relevant for columns and 1
shear walls. Reinforcing bars are considered separately, not in layers. The two programs are complementary. Although uniaxial bending is a particular case included in RC-BIAX, some important design options are only available in RC-TOOL software. The latter is more suitable for beam design and shear design is not available in RC-BIAX. RC-SLAB1 performs analysis and design of continuous beams and one way slabs. Finite element and coefficient methods are used for the analysis. Both one way solid slabs and one way joist slabs are considered. Various code checks are performed and the design delivers bar cutoff along the members. RC-WALL software performs stability and structural analysis as well as design of retaining walls. Multi layer backfill soil, water table presence, ground inclination, as well as presence of surcharge, are all considered. Rankine theory is used for earth pressure analysis. All design computations are performed scrupulously according the SBC / ACI codes. No further simplification or approximation is used. Net material areas are used. The amount of concrete displaced by the embedded reinforcing bars is considered throughout. This document does not describe the theoretical background. Basic theory is available in textbooks [1] and lecture notes by the author [2]. More advanced topics are covered in references [3-5]. 1.2.
Original contributions
Apart from the integration of the Saudi Building Code SBC provisions, the programs have other original unmatched particularities such as the unique 3D representation of the interaction surface with animation and the extraction of various pertinent 2D scans, as well as the analytical integration of stress fields through arbitrary shaped sections. Five papers related to this work have been presented and published in international journals and seminars [4-8]. 1.3.
Research options
Despite being mainly devoted for design, the programs offer some research options such as :
Compare ACI and SBC codes and investigate the effects of their various parameters
Investigate the separate material (steel and concrete) contributions to RC section strength
Ductility and moment-curvature relationships for various concrete models in compression and tension for normal and high strength concrete
Investigate limitations of some code approximate methods such Bresler reciprocal equation and equivalent eccentricity method, in biaxial bending
2
1.4.
Software validation and use
The various options of these programs have been tested and validated over several years. They have served as a support for the author over long years of teaching reinforced concrete structures. The programs have been used by many students and engineers inside and outside KSU. Several final year projects were based on these programs. 1.5.
Graphical user interface
Use of RC-TOOL / RC-BIAX / RC-SLAB1 / RC-WALL does not require any data file. A powerful and friendly graphical user interface is provided to enter the various data. The main screen of each program is self explanatory. The software programs perform various data checks and deliver graphical echoes to help tracking errors. Data is input in various forms:
Selection from a proposed list of items (one choice is made)
Checking / un-checking items as desired (activation of some analysis or design options)
Entering numerical numbers in user edit cells
User edit cells can contain one or many numbers. If more than one number is required, then spaces (or commas) must be left between the successive values. Data input obey common sense logic. For instance, the software cannot read the reinforcement data (steel locations) before knowing the section geometry. Steel layer depths and areas cannot be read unless the number of layers is known. Help is provided at all stages to assist the user. Some data is entered directly in the main screen (such as loading values) while others are input by first pushing the corresponding button (such as section type and dimensions, material properties).
3
Part 2 RC-TOOL Manual 2.1.
Introduction
This program performs analysis and design of RC beams and columns at ultimate state under bending and axial force according to Saudi SBC and American ACI codes. Analysis option includes drawing of moment - axial force interaction curve at ultimate state, as well as moment - curvature stiffness relationships under an increasing moment, requiring numerical integration of material models. Various section types with one, two (equal or unequal) or more than two steel layers can be designed or analyzed. Positive or negative moments and axial forces as well as slenderness effects are all considered. Optimal (minimum steel) design is based on strain compatibility and equilibrium equations. For complex section shapes, an original and powerful fast-converging re-analysis algorithm is used to deliver optimum design. An independent module for shear design of beams is also included. It designs stirrups (number of legs and spacing) including spacing variation along the beam span. The software offers original and unmatched features and is presented in a compact form with one single executable file. 2.2.
RC-TOOL main screen
The main RC-TOOL screen is shown in Figure 2.1. One analysis option and two design options are available as shown in the top left of the main screen:
RC Section analysis
RC Column design
RC Beam design
The analysis option treats beams and columns and includes drawing of the P-M interaction curve. Beams are considered as simple particular cases with zero axial force. Columns may be tied or spiral. The other buttons in the screen are used to enter various required data such as the section geometry and dimensions, the material (concrete and steel) properties as well as the reinforcement data. A graphical echo of the section is produced and the selected code and material data are also displayed. Reinforcement data cannot be entered before selecting the section and entering its dimensions.
4
Figure 2.1: RC-TOOL main screen – Analysis case
The remaining buttons in the main screen are:
“Other Options” : Selecting the code of practice (ACI or SBC) and some secondary options
“View Models” : Graphical display of ACI / SBC ultimate state models used
“Info” / “Help” : Background information and help for users
“Slender Columns” : Complete implementation of ACI / SBC moment magnification method for slender columns (braced or sway). This option is detailed later.
“Bar calculator” : Useful tool to perform various steel area calculations such as delivering, for a given area, the required bar number knowing the diameter, or the required bar diameter knowing the bar number. It also computes the resulting area from any combination of bars.
2.3.
Concrete and steel properties
Figure 2.2 shows the material dialog box. The user can select any of the available choices or enter any other value. Concrete equivalent rectangular block factor β1 and steel yield strain εy are instantly computed and displayed. 5
Figure 2.2: Concrete and steel material data dialog box
2.4.
Section geometry and dimensions
A wide variety of sections can be used provided they present a vertical axis of symmetry to avoid biaxial bending. Un-symmetric sections are analyzed and designed using RC-BIAX software. Figure 2.3 shows typical sections that can be used. All sections have polygon forms except the circular and tubular sections. Complex shapes can be generated by appropriate combinations of trapezes and voids (Figure 2.3). Figure 2.4 shows the section selection dialog box and the dimensions dialog box as well as the graphical echo giving the shape, dimensions, area, centroid location and moment of inertia.
6
Figure 2.3: Various section shapes (with vertical axis of symmetry) analyzed and designed
2.5.
Reinforcement data in section analysis – Example of beam analysis
Figure 2.5 shows the reinforcement data dialog box for polygon sections. The user must enter the number of steel layers (up to 100 layers) and their corresponding depths and areas. Blanks or commas must be used to separate successive values in a single cell. Steel depth and steel area assistants located on the right hand side part can be very helpful. From the selected values of the concrete cover, bar diameter, and tie (stirrup) diameter, the steel depth assistant automatically delivers the top and bottom depth values. The bar area assistant gives the steel area from the selected bar diameter and bar number. The graphical display shows the steel layer locations and area values.
7
Figure 2.4: Section selection, dimensions input, and graphical echo
Figure 2.5: Steel data dialog box in case of polygon section analysis 8
Figure 2.6 shows the analysis output for the preceding beam. It delivers the axial force – bending moment interaction curve. Beam bending case is located on the horizontal axis (P = 0). It is clearly seen from this curve that the beam is tension controlled. Various display options are available, such as detailed calculations about any point on the curve, including the beam bending point on the horizontal axis.
Figure 2.6: RC Beam analysis output
9
Figure 2.7 shows detailed beam bending information:
Neutral axis depth
Depth and area of concrete compression block
Concrete compression force and moment
For each steel layer: Strain and stress Force and moment considering displaced concrete if the layer is inside the compression block
Total nominal moment
Total design moment
Various steel ratios (tension, compression, balanced, minimum)
Figure 2.7: Details for the beam bending point
10
2.6.
Reinforcement data in section analysis – Example of column analysis
Figure 2.8 shows the reinforcement data dialog box for circular and tubular sections as well as a graphical echo. The user enters the diameter and number of bars, the radial depth, and selects the adopted bar arrangement as shown.
Figure 2.8: Dialog box for analysis of RC circular sections
11
The delivered P-M interaction curve, shown in Figure 2.9, highlights the limits between the tension control, transition, and compression control zones. The software delivers automatically the values of the axial force and bending moment at the location of the moving mouse cursor. It also performs the safety check for any load combination (Pu , Mu). Detailed information on any point in the curve can be obtained.
Figure 2.9: RC column analysis output
12
Figure 2.10 shows detailed information related to the balanced point, including the values of neutral axis depth, compression block depth, concrete and steel layer contributions, as well as the total nominal and design values of the axial force and bending moment.
Figure 2.10: Dialog box for interaction point details – Balanced point details
13
2.7.
Beam and column design
Figures 2.11 and 2.12 show the main screens in case of beam design or column design with new buttons activated. Beam design requires the value of the ultimate moment. Column design activates the “Tied or Spiral” selection option and requires both the ultimate axial force and moment values. ACI / SBC minimum steel area is enforced by default but the user may deactivate this option. The “Slender Columns” button is used to obtain the magnified moment. This will be described later. 2.7.1. Common options of beam and column design Beam or column design can be performed using the following common options : 1) Standard design with one or two unknown steel layers As and As' 2) Two steel layers with a prefixed value for As' 3) Two equal steel layers As = As' 4) Many steel layers with prefixed ratios As' is the compression steel layer in beam bending. It is the top layer for a positive bending moment. For columns subjected to bending and axial force, As' is the top layer for positive moments. As' is therefore more generally either the most compressed or the least tensioned steel layer. Combination of a bending moment (positive or negative) with an axial force (compression or tension) is the general case. Beam bending is a particular case with zero axial force. The first design option is the standard case where one or two layers are required. The software determines whether the second layer is necessary or not. The second option with an imposed value of As' may be useful if for instance the latter was determined by a design for a negative moment or simply to account for construction top bars used to hold stirrups. Considering the existence of As' will in general reduce the required value of As unless a top layer As' greater than the imposed value is necessary. In this case the imposed value of As' will be discarded by the software. The third design option is particularly convenient for columns where symmetric reinforcement is frequently adopted. The fourth design option is an invaluable tool if many steel layers are to be used. This is often the case in columns, deep beams and shear walls. However, the steel layers cannot have any independent values. They must be related by ratio or proportionality values R(i) which are defined as :
R(i) = A(i) / Ast
where Ast is the total steel area.
Any proportionality ratio Ru(i) can be used. The effective computed ratio is R(i)
Ru (i)
R (i) u
i
For equal layers, use the same (any) number for all layers. For layers using the same bar diameter, use the bar number in each layer i as its ratio R(i). 14
Figure 2.11: RC-TOOL main screen – Beam design case
Figure 2.12: RC-TOOL main screen – Column design case
15
2.7.2. Beam design example Figure 2.13 shows the reinforcement data dialog box for design of polygon sections using the first option (one or two unknown steel layers). The two steel depths must be provided and if the second layer is not required, it will be discarded. The steel depth assistant described before can also be used here. Figure 2.14 shows the design output. It delivers the required steel, the strain and stress distributions across the section, as well as the P-M interaction highlighting the loading point on the border of the safe design zone on the horizontal M-axis, which corresponds to an optimum design. Detailed design information can also be viewed as will be seen later in a column design example.
Figure 2.13: Steel data in case of section design with one or two steel layers
16
Figure 2.14: Standard beam design output
17
2.7.3. Column design example Figure 2.15 shows the dialog box using the option of design with many layers. It corresponds to the design of a 500x500 column using eight bars in three layers. The second layer is at mid-height of the section (d = h/2). The layer ratio used is the number of bars in each layer.
Figure 2.15: Steel data in case of section design with many steel layers
For the design of circular or tubular sections, the reinforcement data dialog box is similar to that in Figure 2.8, except that the bar diameter is not required. A constant ratio is used. Figure 2.16 shows the design of a circular column using eight bars. Again the loading point is on the border of the safe design region.
18
Figure 2.16: Circular column design output
19
Figure 2.17 shows the numerical design details including neutral axis depth and material (concrete and steel layers) contributions. The optimum design is confirmed, as the total design capacity (force and moment. is equal to the ultimate loading (ΦPn = Pu and ΦMn = Mu).
Figure 2.17: Column Design details
20
2.8.
More beam design options
Beam design includes more options as shown by the activated buttons in Figure 2.11: 5) Beam design with fixed bar diameters checking bar / layer number and spacing. 6) Beam design with lumping of many tension steel layers at their centroid. 7) Shear design of beams In design option five, the software determines the required steel areas assuming one single steel layer in tension and also in compression (if any). It then deduces the required numbers of bars. If more than one layer is required, successive cycles of analysis-and-check are performed until the design capacity of the beam is equal to or greater than the ultimate moment. As shown in Figure 2.18, the required data in this case is limited to the values of concrete cover, stirrup diameter, and bar diameters in tension and compression (if any). Steel layer depths, bar and layer spacing are automatically computed. For T-sections, top steel bars in the flange can be arranged in two ways as shown in Figure 2.18.
Figure 2.18: Steel data in case of beam design with fixed bar diameters
21
Figure 2.19 shows the design output using fixed bar diameters. Beam symmetry is maintained when determining bar numbers. In this case the loading point is usually not on the border but inside the safe design zone because of the added safety when rounding up the values of the bar numbers. Figure 2.20 shows the design of a T-section for a negative moment with two different options of bar arrangement. Confining the bars within the web width required two layers and one extra bar.
Figure 2.19: Beam design output with fixed bar diameters 22
Figure 20 : Design of a T-section for a negative moment with two options for bar arrangement
The sixth beam design option is based on lumping of several tension steel layers at their centroid. This option is frequently used in practice and in textbooks. Some cautions specific to this option are however ignored by many:
Layer lumping is only justified if all the lumped layers have yielded and have therefore the same yield stress.
Yielding assumption must be checked at the least tensioned layer with the minimum steel depth (and not at the centroid).
Tension control check must be performed at the most strained bottom layer with the maximum steel depth (and not at the centroid).
Figure 2.21 shows the dialog box in case of beam design with lumping of tension steel layers. The data required includes compression steel depth as well as the following tension steel depths:
Steel centroid depth
Extreme bottom layer depth
Least tensioned layer depth
Figure 2.22 shows the corresponding output for beam design using the lumping option.
23
Figure 2.21: Data dialog box for beam design with lumping of tension steel layers
Figure 2.22: Output of beam design using the lumping option 24
2.9.
Shear design of beams
Shear design of beams can be performed either at a critical section delivering the required stirrup spacing, or along half span of the beam delivering the stirrup spacing distribution. The design is fully automatic. The user specifies the minimum stirrup diameter as well as the minimum and maximum leg numbers. Spacing increments are also user-controlled. The software performs shear design starting with minimum settings. If the number of legs is insufficient (required spacing less than 100 mm), then it is incremented. But if the maximum number of legs is exceeded, then the stirrup diameter is increased. Figure 2.23 shows the shear design dialog box using 10-mm stirrups, and the corresponding output is shown in Figure 2.24a. If, at the critical section, the required leg number is greater than the minimum value, RC-TOOL software allows the reduction of the number of legs at other locations with a reduced shear force. Figure 2.24b shows the same previous beam designed in shear starting with 8-mm stirrups and activating the leg reduction option. At the critical section the minimum number of legs (two) is not sufficient. Three legs are required. The software goes back to the two-legged stirrup when it becomes appropriate.
Figure 2.23: Shear design of a beam 25
Figure 2.24a: Shear design output using 10-mm stirrups
Figure 2.24b: Shear design output using 8-mm stirrups with the leg reduction option 26
2.10. Slender columns The moment magnification method used for slender columns is implemented in RC-TOOL for braced or sway columns. Figure 2.25 shows the slenderness dialog box. The effective length factor k may be user defined or computed as per ACI / SBC codes. Guidelines are provided in case of user determination. In case of ACI / SBC compliance, various member inertia calculations at both column ends (top and bottom), are automatically obtained from section dimensions. RC-TOOL performs the various slenderness checks and uses the moment magnification method if the column is slender. It also issues a warning if the slenderness ratio exceeds the maximum limit of 100.
Figure 2.25: Slender column dialog box 27
The various details of slenderness checks and calculations as well as the magnified moment are shown in Figure 2.26.
Figure 2.26: Slender column results
28
2.11. Moment – Curvature relationship RC-TOOL determines the moment-curvature relationships for beams and columns subjected to constant axial force (zero for beams) and an increasing bending moment until failure. The material constitutive stress-strain equations are integrated through the section at all stages. Concrete and steel models used in this option are of course different from the ultimate state models used in design. Figures 2.27 and 2.28 show the two sets of models. The recent analytical integration method used [4,5] is more efficient than numerical integration techniques, and is able to track and capture local failures and the spread of concrete crushing inside the cover.
Figure 2.27: ACI / SBC ultimate state models used in RC design and analysis
29
30
Figure 2.29 shows the dialog box to select the material models and various options for the moment curvature relationship.
Figure 2.29: Dialog box for moment-curvature
31
Figure 2.30 shows a typical moment curvature relationship for a beam. The software displays the values of the moment and curvature at the horizontal location of the mouse cursor as well the corresponding stress distribution across the section. The concrete model used is also shown. Many curves can be viewed in the same screen for different values of the axial force (Figure 2.31). Stiffness variation may also be generated. Figure 2.32 highlights the advantages of the analytical integration method used, as it allows the capture of local failures such as the spread of concrete crushing through the cover.
Figure 2.30: Moment Curvature at zero axial force showing the stress distribution at various levels
32
Figure 2.31: Typical Moment-Curvature relationships in a column at various values of axial force
Figure 2.32: Moment-Curvature showing the capture of concrete crushing through the cover 33
2.12. More design examples using RC-TOOL Figures 2.33 and 2.34 show some more design outputs. The strain compatibility based method allows the capture of non-convex parts in the design interaction curves (Figure 2.33). Negative moments (tension in top) and negative axial forces (tension) are equally treated (Figure 2.34).
Figure 2.33: Design of a beam with two steel layers (compression steel required)
Figure 2.34: Design of a square column subjected to negative moment and tensile axial force 34
2.13. Material contribution to RC section strength RC-TOOL can be used to decompose the strength capacity of a reinforced concrete section and determine the separate contributions of concrete and steel reinforcement. Appropriate techniques allow the capture and drawing of the P-M interaction curve for a plain concrete section, and setting concrete strength to zero delivers the reinforcement contribution. Figure 2.35 shows the interaction curve for a square section column 400x400 mm with eight 25-mm bars representing a steel ratio of 2.5%. Separate contributions of concrete and steel reinforcement are also shown. The various steel layers contributions may also be determined separately.
Figure 2.35: Separate contributions of concrete and steel to the strength capacity of an RC section
35
Part 3 RC-BIAX Software 3.1.
Introduction
This program performs analysis and design of RC tied and spiral columns, at ultimate state under biaxial bending and axial force, according to Saudi SBC and American ACI codes. Analysis option includes drawing of biaxial moment - axial force interaction surface at ultimate state. Various section types and shapes can be designed or analyzed. Positive or negative moments and axial forces as well as slenderness effects are all considered. Optimal (minimum steel) design is based on an original and powerful fast-converging re-analysis algorithm. The software offers original and unmatched features and is presented in a compact form with one single executable file. 3.2.
RC-BIAX main screen
The main RC-BIAX screen is shown in Figures 3.1 and 3.2, in analysis and design cases respectively. The analysis option includes drawing of the P-M-M interaction surface. Design option requires the ultimate values of the two bending moments and the axial force. The other buttons in the screen are used to enter various required data such as the section geometry and dimensions, the material (concrete and steel) properties as well as the reinforcing bar data. A graphical echo of the section is produced and the selected code and material data are also displayed. Reinforcing bar data cannot be entered before selecting the section and entering its dimensions. 3.3.
Concrete and steel properties
The material data input is the same as described in section 2.3 for RC-TOOL.
36
Figure 3.1: RC-BIAX main screen – Analysis case
Figure 3.2: RC-BIAX main screen – Design case
37
3.4.
Section geometry and dimensions
A wide variety of sections, including typical and irregular shapes, can be used. Figure 3.3 shows the section selection dialog box and the dimensions dialog box as well as the graphical echo giving the shape, dimensions, area, centroid location and moments of inertia.
Figure 3.3: Section selection, dimensions input, and graphical echo 3.5.
User defined section
The user defined section can be of any shape and may contain voids. User generation of the section geometry is illustrated through an example. The section shown in Figure 3.4 is composed of:
Main polygonal section with six vertices, whose coordinates are : (250 , 0) / (750 , 0) / (1000 , 500) / (750 , 1000) / (250 , 1000) / (0 , 500)
One polygonal hole (code = 1) with three vertices (400 , 150) / (600 , 150) / (500 , 350)
One circular hole (code = 2) with a 100-mm radius and (500 , 750) for center coordinates.
38
Figure 3.4 shows the corresponding dialog box data and the graphical echo. Any system of axes can be initially used. The software will then perform an origin shift for appropriate graphical display. The subsequent input of reinforcing bar data must then use the displayed system of axes.
Figure 3.4: User defined complex section and graphical echo
39
3.6.
Reinforcing bar data
Biaxial design is based on a known reinforcement pattern. In analysis or design, the number of bars must first be entered in the main screen before specifying their locations using the “Bar Data” button. Steel bars are identified by their coordinates. These can be entered separately or by using one of the numerous generation schemes. Figure 3.5 shows the generation of a regular reinforcing bar pattern in a rectangular section.
Figure 3.5: Dialog box for regular bar generation in a rectangular section
40
Figure 3.6 shows the use of linear bar generation scheme in an irregular section and the corresponding data dialog box. Coordinates are required for few bars only and the remaining bars are linearly generated. In RC analysis case, bar diameter is also required (Figure 3.5) whereas in design the bar area ratio is required (Figure 3.6). Any proportionality ratio can be used as described in section 2.7.1. “Bar Data” button may be used as many times as required to enter the required information, particularly if the bars do not have the same diameter or design ratio.
Figure 3.6: Linear bar generation in an irregular section 41
3.7.
P-M-M interaction surfaces
The biaxial bending-axial force interaction surface is generated in analysis and design cases. The original and powerful method used allows the capture of any complex 3D surface shape. Using five degree increments, an efficient strain compatibility algorithm is used to determine the P-M-M interaction meridian curve for any neutral axis angle,. These 3D meridian curves are then assembled to form the 3D surface which is drawn using an adequate hidden surface technique. Both nominal and design surfaces can be viewed (Figure 3.7) and rotated at will. The colors represent the three main parts (compression controlled, transition zone, tension controlled) and other sub-parts.
Figure 3.7: Nominal and design interaction surface for a square RC section 42
Figure 3.8 shows the interaction surface for an L-section with respect to the original axes and principal axes. The non-plane and irregular meridian curves with respect to the original axes are scrupulously tracked (Figure 3.8a), without however affecting the expected symmetry about the principal axes (Figure 3.8b).
Figure 3.8: Interaction surface for an L-section and views with respect to original and principal axes
3.8.
Extraction of 2D scans
This is another major originality in RC-BIAX software. There is no need to perform a new analysis for any desired plane interaction curve (Mx -My curve for fixed value of P, P-Mx curve for fixed value of My , …). The 3D P-M-M surface is generated and assembled in a way allowing the use of appropriate geometry intersection algorithms to perform scans and cuts and extract various 2D interaction views
43
. Figure 3.9 shows the dialog box and the various 2D scans available. It must be reminded that the P-M interaction meridian curve for a given neutral axis angle is in general not plane, especially for irregular sections (Figure 3.8).
Figure 3.9: Dialog box for extraction of 2D scans from 3D surface
Figure 3.10 shows an Mx -My interaction curve for a constant value of P. The resulting 2D scan is shown in plane and isometric views. The nominal and design parts are easily distinguished. Figure 3.11 shows the P--My interaction curve for a constant moment Mx. The sliced cut is isolated. This slice can be rotated and viewed from any angle. Detailed numerical information about any point on the interaction surface can be obtained. It gives the neutral axis angle, neutral axis and compression block depths, concrete and steel layer contributions, as well as nominal and design force and biaxial moments. Figure 3.12 shows the numerical details at the balanced point for a neutral axis angle of 30 degrees. In addition to the various display options and 2D scans, the user can check the safety of any ultimate loading combination (Pu , Mxu , Myu) and compare with established approximate methods, such as Bresler reciprocal equation and the equivalent eccentricity method. This topic will be described later.
44
Figure 10: 2D scan for P = constant, with plane view and isometric view
45
Figure 3.11: 2D scan with plane view and 3D slice view
46
Figure 3.12: Detailed information on a point of the interaction surface (balanced point for a neutral axis angle of 300)
47
3.9.
RC design in biaxial bending and axial force
An efficient fast converging re-analysis strategy is used to deliver the optimum design and the graphical output delivers the inclined neutral axis location as well as the strain and stress distributions across the rotated section (Figure 3.13). Detailed design information can be viewed as shown in Figure 3.14. The P-M-M interaction surface is also generated as in analysis case described before.
Figure 3.13: Design of a square column section with eight bars – Design output
48
Figure 3.14: Design details
Figures 3.15 and 3.16 show more design problems. In Figure 3.15, an L-shaped section is designed for a uniaxial moment only (Myu = 0) combined with an axial force. The dissymmetric section resulted in an inclined neutral axis. As mentioned previously, bending of dissymmetric sections can only be performed with biaxial methods. Figure 3.16 illustrates the capacity of RC-BIAX to design complex sections with holes.
49
Figure 3.15: Design of an L-section under uniaxial bending moment (Myu = 0)
Figure 3.16: Design of a complex section with holes in biaxial bending with 450 angle
3.10. Safety check and Bresler reciprocal equation Despite their limitations, Bresler reciprocal equation and, to a lesser extent, the equivalent eccentricity method, have been, and still remain, the only approximate tools used to check the section safety in biaxial bending. They are recommended by many building codes including ACI and SBC. RC-BIAX software can now perform this safety check more accurately. It can also display the ultimate loading point on any appropriate 2D scan with respect to the safe design zone. Figure 3.17 shows the safety check dialog box with the possibility to compare with the two approximate methods. The detailed check calculations are delivered as shown in Figure 3.18. With a ray ratio of 1.0105, the combination is very close to the border but safe. The two approximate methods produce wrong check results as they both conservatively return an unsafe verdict. RC-BIAX can therefore be used to investigate the limitations of these approximate methods and ways of improving them. 50
Figure 3.17: Dialog box for safety check including approximate methods
Figure 3.18: Safety check output and comparison with approximate methods 51
A 2D scan at the same axial force of 2000 kN, shows the loading point very close to the border (Figure 3.19) but inside confirming the previous safe check.
Figure 3.19: Safety check of a combination (loading point on border of safe design zone)
3.11. Slender columns in biaxial bending The moment magnification method, used and described in RC-TOOL software, is also applied in biaxial bending independently for each of the two bending planes (Figure 3.20). The appropriate value of the moment of inertia is automatically used for each selected plane.
52
Figure 3.20: Slenderness check about two planes in biaxial bending
53
Part 4 RC-SLAB1 Manual 4.1.
Introduction
This program performs analysis and design of RC one way slabs according to Saudi SBC and American ACI codes. Both one-way solid slabs and joist slabs are considered. The slab or the joist as well as the supporting beams can be analyzed and designed with automatic load transfer from the slab to the beams. Both ACI / SBC coefficient method and elastic finite element method can be used for the analysis. The code method can only be used if all its conditions are satisfied. But even if these conditions are satisfied, the user can still choose either method for comparison purposes. Load transfer from the solid slab or joist slab to the beams is performed by the software, according to the beam tributary width. Beam loading may include wall line load. In a joist slab, the inter-rib spaces may be void or contain “hourdis” blocks. The beams may be designed with the original rectangular section or with the effective T-section or L-section. The T-section (for internal beams) and L-section (for edge beams) results from the interaction between the slab and the beam. The T-section or L-section flange width is automatically determined by the software, according to SBC / ACI provisions. Checking, analysis and design steps can be performed separately or in one single operation. The output includes results of the various checks (minimum thickness, shear, flange, ...). Values of the clear lengths, shear and moment coefficients as well as the minimum thickness are displayed for each span. Shear force and bending moment diagrams are also displayed. When using the code coefficient method, envelope curves of the diagrams are generated. Design results include required steel areas as well as the number of bars and bar spacing. Code values for minimum steel and maximum spacing are enforced. The software delivers an optimum reinforcement pattern along the model by performing appropriate bar cutoff. Both demand and capacity moment diagrams are produced. Shear design is performed for beams or ribs requiring it. Variation of stirrup spacing is delivered for each span.
54
4.2.
RC-SLAB1 main screen
Figure 4.1 shows RC-SLAB1 main screen. Two types of one way slabs can be treated as shown in the top left of the main screen:
One way solid slab
One way joist slab
Figure 4.1: Main screen of RC-SLAB1 software The area loading (kN/m2) provided includes the super imposed dead load and live load. The self weight is automatically computed for all members. The software performs various code checks, analysis, and design for both the one way slab, or joist (left hand part of the main screen) as well as the supporting beams (right hand part). 55
The other buttons in the screen are used to enter various required data such as the span and support data, the material (concrete and steel) properties as well as the reinforcement data. A graphical echo of the model is produced and the selected code and material data are also displayed. 4.3.
Concrete and steel properties
The material data input is the same as described in section 2.3 for RC-TOOL. 4.4.
Analysis and design of a one way solid slab
The slab is supported by beams which rest on columns or girders (Figure 4.2). Analysis and design of 1-m strip is performed and design results are generalized to the whole slab. Shrinkage (temperature) steel is then provided in the other direction. The slab strip is modeled as a continuous beam with the transverse beams acting as supports.
Figure 4.2: One way solid slab with beams and girders The analysis can be performed by either the ACI / SBC coefficient method or the finite element method. The code method can only be used if all its conditions are satisfied. But even if these conditions are satisfied, the user can still choose either method for comparison purposes.
56
Figure 4.3 shows the span and support dialog box. The user must provide the span lengths as well as the widths of the supports. Special external support cases are catered for (free end for cantilever, hinge, fixed end).
Figure 4.3: Span and support data dialog box
The various checks performed are:
Minimum thickness
Conditions of the coefficient method
Shear check
Figure 4.4 shows the analysis results using the coefficient method. In addition to the envelope curves for bending moment and shear force diagrams, the various coefficients are also displayed. The design results shown in Figure 4.5, give the bar spacing while enforcing all code requirements (minimum steel, tension control, maximum spacing). The required shrinkage steel is also delivered. Figure 4.6 shows selected detailed numerical results including checking, analysis and design.
57
Figure 4.4: Analysis results for slab strip using ACI/SBC coefficient method
Figure 4.5: RC design results for slab strip showing bar spacing 58
Figure 4.6: Selected detailed numerical results for the slab Checking - Analysis - Design
59
4.5.
Analysis and design of a one way joist slab
A typical joist is analyzed and designed as a continuous beam supported by transverse beams (Figure 4.7). Shrinkage (temperature) steel is then provided in the other direction. The span and support data is provided as shown before in Figure 4.3. The joist dimensions are also entered through the dialog box shown in Figure 4.8. The presence of light hollow blocks (hourdis) is optional. In addition to the standard checks described before (minimum thickness, conditions of the coefficient method, shear), two more specific chic checks are performed:
ACI / SBC conditions on the joist dimensions
Flange check as a plain concrete member
Figure 4.9 shows a typical joist design output satisfying all code requirements with the appropriate bar cutoff. Shrinkage reinforcement is also delivered.
Figure 4.7: One way joist slab
60
Figure 4.8: Joist data dialog box
Figure 4.9: Design of a typical joist Figure 4.10 shows selected detailed numerical results including standard and flange checking, as well as analysis and design. 61
Figure 4.10: Selected detailed numerical results for the joist Checking - Analysis – Design (flexure, shear, shrinkage)
62
4.6.
Analysis and design of supporting beams
The load is transferred to the beams according to their tributary width. As shown in Figures 4.2 and 4.7, the beams may be supported either by girders (beams) or columns. The span and support data is entered in a similar way as shown in Figure 4.3. The beam section dimensions and the tributary width are also entered in the main screen. Load transfer to the beams is automatically computed and the possible direct wall loading may be included. The same methods are used for the beam analysis and typical results are shown in Figure 4.11. The dimensions of the effective section resulting from the beam-slab interaction are automatically computed.
Figure 4.11: Analysis of a supporting beam 63
The design settings can be edited as shown the dialog box of Figure 4.12 and the flexural design results are shown in Figure 4.13. Shear design is also performed for all beams and joists requiring it, delivering the stirrup spacing along each span. Detailed numerical results are shown in Figure 4.14.
Figure 4.12: Design data for supporting beams
Figure 4.13: Design of a supporting beam
64
Figure 4.14: Selected detailed numerical results for the beam Checking – Analysis – Design (Flexure and shear)
65
Part 5 RC-WALL Manual 5.1.
Introduction
This program performs analysis and design of RC retaining walls using Rankine theory for earth pressure analysis. Backfill soil may be constituted of one or many different layers. The ground may be either horizontal or inclined with a possible surcharge. Water table presence is also accounted for. Its effects on earth pressure and soil bearing capacity are both included. The wall thickness may be constant or vary linearly. The foundation with a constant thickness may possibly have a toe as well as a key. The presence of a key, which may be either below the wall or at the heel end, improves horizontal sliding stability. The graphical interface produces geometric echo of data and delivers various result presentations including pressure, shear force, and bending moment distributions. RC design according to ACI and SBC codes is performed at critical sections and along the wall and foundation. Design values of shear forces and bending moments, at the ultimate limit state, are multiplied by a user defined load factor. A value lying between 1.5 and 2.0 is recommended. RC design delivers the required steel along the wall and foundation while performing various checks for minimum steel, bar spacing, shrinkage steel and shear check of thickness. A detailed execution plan is also delivered. 5.2.
Earth pressure theory
Earth pressure is computed using Rankine theory which is best suited to mutli-layer soils. Rankine original theory for cohesionless soils has been extended to include cohesion effects. Cohesive and non-cohesive soils can therefore be considered. As earth pressure is reduced by cohesion it is therefore recommended to consider a low (if not zero) value for soil cohesion. It is recommended to use cohesive soils in bottom layers. An option transforming a cohesive soil to an equivalent cohesionless one, as described in reference [10], is also offered. It is limited to the top layer of backfill soil and is based on equal bottom pressures over the soil height at the plane of the heel end of the base foundation. Two Rankine models are used (Figure 5.1): a) The first determines the soil pressure acting directly on the wall in order to compute shear force and bending moment distributions on the wall for design purposes (Figure 5.1a).
66
b) The second considers soil pressure acting on a plane at the heel end of the base for stability analysis and check (overturning and sliding) as well as in the determination of pressure distribution in the foundation soil. (Figure 5.1b). In this second model (b), the top triangular soil above the wall limit is considered as part of the first layer, whereas the soil at the base face is an extension of the last layer.
Figure 5.1: Rankine earth pressure models used Figure 5.1 corresponds to the simple case of unilayer cohesionless backfill soil with no surcharge and no water table. This eases equation derivation. RC-WALL software treats however all other retaining wall types. The total active earth pressure force is (Figure 5.1):
Pa 0.5K a H 2
Case (a):
Case (b): Pa 0.5K a ( H ' ) 2
Horizontal and vertical components are: The passive force in case (b) is:
Pah Pa cos
Pav Pa sin
Pp 0.5 f K p D 2
Passive pressure is neglected in case (a), and considered only in sliding stability check in case (b). Total vertical forces acting on the foundation include the various weights, the possible top surcharge and the vertical component of the earth pressure:
F
v
Ww W f Ws Q Pav
Q is the possible surcharge
Subscripts w, f and s coreespond to wall, foundation and soil respectively. 67
5.3.
RC-WALL main screen
Figure 5.2 shows the main screen. It is self explanatory and the data terminology and actual graphical echo, shown in Figure 5.3, are also displayed with the main screen. Before performing analysis and design through the “Go Analysis” button, it is recommended to first check the stability. Any instability situation is detected by the software and adequate solutions proposed to the user.
Figure 5.2: RC-Wall main screen 68
Usual default values of safety factors are proposed but the user can change them. RC-WALL software issues other warnings in case of excessive cohesion resulting in some pressure reversal and when tension is generated in the foundation base. With the same total base length, the presence of a toe reduces the buried length and therefore reduces the stabilizing force and moment. Toe presence however reduces the eccentricity of the foundation axial force and tends therefore to flatten the pressure distribution in the soil under the foundation. The necessity of toe presence and its optimal length may be interactively investigated and determined before analysis. The foundation soil capacity is identified by its ultimate bearing value. It can be either user-defined or determined by one of three models to be described later.
Figure 5.3: Data terminology and actual graphical echo 5.4.
Stability analysis
Potential stability problems in a retaining wall include:
Overturning
Sliding
Soil bearing capacity
69
Figure 5.4: Stability model and forces Forces acting in the stability model are shown in Figures 5.1b and 5.4. Moments are computed about point O (Figure 5.1), which is the origin of X-Y axes, using appropriate lever arms for each part. Stability check (sliding and overturning) may be performed interactively before analysis and design. Overturning stability is checked using the ratio between resisting and overturning moments. Sliding stability is checked using the ratio of the resisting force (including vertical force multiplied by the soil friction, cohesion, and possible frontal passive force), to the driving horizontal force. The two preceding ratios must be greater or equal to the factor of safety. For the latter a default value of 2 is considered but the user may enter any other value. 5.4.1. Overturning Overturning moment is caused by the horizontal force:
M O Pah y a
At overturning failure, an extra vertical force Frv is activated. It is the soil-soil friction along line AB and is equal to:
Frv Pah tan(0.8 )
The resisting moment includes thereforce contribution of this extra vertical force Frv (instaed of of the vertical component of the earth pressure Pav): The overturning factor is:
NO
M R1 Ww x w W f
B Ws x s Qxq Frv B 2
M R1 MO
It must be greater than or equal to the factor of safety which varies in genearl between 1.5 to 2.0 70
5.4.2. Sliding Sliding is caused by the driving horizontal force Pah and resisted by base soil friction force Frh, (including internal friction and reduced cohesion c), as well as possible passive pressure. The horizontal friction force is:
2 Frh Fv tan f c f B 3
The sliding factor is therefore:
NS
Frh PP Pah
It must be greater than or equal to the factor of safety. Figure 5.5 shows detailed numerical stability calculations delivered by RC-WALL.
Figure 5.5: Detailed stability analysis results 71
5.4.3. Presence of a base key The are various opinions about the key effect and its location. It seems that it is more effective when loacted at the heel end. The key is recommended in case of sliding stability problems. There is no consensus on the resisting mechanism with the presence of a key. In RC-WALL software, presence of a key is assumed to guarantee sliding stability. 5.4.4. Bearing capacity Pressure in foundation soil is determined with rigid foundation assumption. The pressure is variable along the footing because of presence of a bending moment. It is recommended to design the foundation in such a way that soil tension is avoided. Maximum soil pressure must be less than or equal to the allowable bearing capacity. This latter is obtained as the ultimate bearing capacity divided by the factor of safety. Ultimate soil bearing capacity may be user-defined or computed according to any of Hansen / Jesic / Hannah models (Figure 5.6) [10]. Effect of water table on bearing capacity is included. Soil pressure along the base is caused by the total vertical force and the net moment. This forcecouple system can be replaced by a single eccentric vertical force. The resisting moment at normal state is: M R 2 Ww x w W f
B Ws x s Qxq Pav B 2
This expression excludes passive pressure and the extra vertical force, and is different from the previous overturning moment at failure. The resulting net moment acting on the footing is:
M net M R 2 M O
e
The eccentricity of the force on the foundation is therefore:
e
To avoid soil tension, eccentricity must satisffy:
B M R2 M O 2 Fv
B 6
Values of soil pressure at the ends of the foundation base are:
qtoe
F
6e 1 B B
q heel
v
F
6e 1 B B v
The maximum soil pressure must be less or equal to its allowable bearing capacity The bearing capacity factor is:
NB
qtoe qult
It must greater than or equal to the corresponding factor of safety (usually 2.0 to 3.0).
72
Figure 5.6: Models for soil ultimate pressure
73
5.5.
Structural Analysis
Weights of the wall and foundation are considered in the loading. The analysis determines the earth pressure on the wall, the pressure distribution in the foundation soil, as well as the shear force and bending moment distributions in both the wall and foundation, considering one meter strips. Possible counterforts are not considered in the model. The wall and foundation are modeled as cantilevers. In case of presence of a toe, the foundation has then two cantilevers. 5.5.1. Wall analysis Distribution of earth pressure on the wall, which is modeled as a cantilever, is determined from case (a) as shown in Figure 5.1a. RC-WALL software considers multi-layer backfill soil and possible water table and surcharge. Passive pressure is neglected. Shear force and bending moment distributions are deduced from horizontal pressure using numerical integration. The vertical component of earth pressure acts on the heel part of the foundation. 5.5.2. Foundation analysis The base foundation, modeled as a double cantilever (heel and toe), is subjected to bottom soil reaction and top pressure from various weights. The bottom soil reaction is usually trapezoidal and the end values are as given before. The top loading corresponds to the footing weight only for the toe and includes also for the heel the top soil weight and the possible vertical component of earth pressure of case (a). These unequal pressure distributions lead to different bending moments. The toe cantilever is usually subjected to net upward loading requiring bottom reinforcement whereas the heel cantilever is subjected to net downward loading requiring top reinforcement. Shear force and bending moment distributions are again obtained by numerical integration. Figure 5.7 shows the detailed analysis results including the loading, soil reaction, as well as the shear force and bending moment diagrams along the stem and the footing (heel and toe).
74
Figure 5.7: Structural analysis of retaining wall Pressure, shear force and bending moment along stem and footing
75
5.6.
RC design of wall and foundation
RC design is performed at ultimate state using factored forces and moments. The previous structural analysis is based on earth pressure loads determined in the serviceability limit state. Figure 5.8 shows the design dialog box with a recommended load factor from 1.5 to 2.0 (default is 1.7).
Figure 5.8: RC-Design dialog box
Standard RC methods are used to determine the required steel reinforcement and the corresponding spacing with various checks. RC design delivers the required vertical steel as well as the shrinkage horizontal reinforcement. RC checks include minimum, steel, maximum bar spacing and shear check for thickness. A detailed execution plan is delivered by the software. Figure 5.9 shows the required steel area along the stem (wall) and footing (heel and toe). In this case compression steel is not required. The thickness of the stem and footing are also checked for shear. Figure 5.10 shows the corresponding execution plan detailing the bar diameters and spacing used as well as shrinkage steel.
76
Figure 5.9: RC Design output of retaining wall (stem and footing)
77
Figure 5.10: Execution plan for retaining wall 5.7.
Retaining wall inundations
Some retaining walls such as urban and road subways have impermeable front surfaces which may collect water in case of inundations. This resulting front water table may cause unexpected stability problems and reverse the net horizontal force and overturning moment. RC-WALL software computes this frontal hydrostatic force and the resulting moment and delivers the critical values of the front water table not to be exceeded.
78
References 1.
J. G. MacGregor «Reinforced Concrete: Mechanics and Design, 4th edition » Prentice Hall 2001.
2.
A. Charif “Lecture notes – CE370 and CE472” , King Saud University
3.
A. Charif “Development of software for analysis and design of reinforced concrete sections subjected to bending and axial force according to Saudi building code SBC”, Res. Report KSU/CE/44/427, June 2007, 32p.
4.
A. Charif “Development of software for analysis and design of RC sections subjected to biaxial bending and axial force according to code SBC”, 7 th Saudi Engineering Conference, KSU, Riyadh 26-30 Nov.2007.
5.
A. Charif “Efficient re-analysis method for design of RC sections subjected to biaxial bending and axial force” , 3rd Int. Conf. on Modeling, Simulation and Applied Optimization , 20-22 Jan.2009, Sharjah,UAE
6.
A. Charif, M.J. Shannag, S.Dghaither “Analytical integration of material stress fields for section analysis and structure response”, Séminaire International Risques et Génie Civil, Université de Batna 26/27 Novembre 2012,
7.
A. Charif, M.J. Shannag, S.Dghaither “Analytical integration of stress fields for reinforced concrete analysis”, Submitted to ICE Structures and Buildings Journal, 22p, January 2013
8.
A. Charif , K.Demagh “Analysis model for biaxially loaded reinforced concrete columns”, ACI Int. Conf. on advances on cement based materials and applications in infrastructure, Lahore, December 12-14, 2007
9.
A. Charif , K.Demagh “Analysis model for biaxially loaded reinforced concrete columns”, ACI Int. Conf. on advances on cement based materials and applications in infrastructure, Lahore, December 12-14, 2007
10. J.E. Bowles “Foundation analysis and design” , McFraw Hill, 5 th edition, 1996
79
