Pyramid Structural Engineering Applications

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2 Pyramid Structural Engineering Applications Abstract Pyramid is a collection of structural engineering computer applications under Windows for the analysis and design of reinforced concrete framings and elements. It has been under continuous development by the author during the past 30 years for use in his own design work and is now being made available to all free of charge. The applications comply with 4 codes at the present. These are ACI-11, BS , Jordan JBC5-08 and the Saudi SBS304. They include simple calculations like the design of reinforced concrete sections subjected to flexure, shear and torsion; crack width determination; punching shear; free standing and helical stairs; and two way slabs. The applications also include more complex computations like beams and sub-frames of variable sections subjected to uniform, triangular and concentrated loads; columns of rectangular, circular and irregular sections subjected to biaxial bending; pad footings of various configurations; and earth retaining walls. The interface is user friendly and the calculations are interactive, permitting the user to explore design alternatives with relative ease. Keywords: reinforced concrete, beam design, column design, footing design, earth retaining wall design 1. Introduction The author has been a structural designer for more than 40 years and has been continuously developing computer programs to reduce calculation effort. With retirement approaching, these programs in their refined form are being made available to all free of charge under the name Pyramid. Powerful structural engineering software now exists where applications like ETABS and Robot can be employed to model a complete building in all its components and analyze the building under both gravity and lateral loading including P-delta effects and construction sequence. Pyramid cannot be compared to such applications. It can only model individual framings and only for gravity loading. Nonetheless Pyramid remains relevant. Its strength lies in its simplicity and user friendly interface. A structural designer is often faced with simple or small scale calculation needs which do not justify recourse to a super program. Furthermore, although designing a building for lateral loading can only be performed using a super program, Pyramid can be used for preliminary sizing of members and for design verification. Creating a complete building model requires a huge volume of data, is time consuming and not easy to perfect. Using a super program to size members can be an exasperating task. Furthermore the volume of data required for a complete building model permits a higher possibility for human error, requiring design verification by alternative means. In addition to the design of beams, slabs and columns, Pyramid tackles the design of simple members which are not addressed by today's super programs. These include corbels, freestanding stairs, column pads, combined and adjacent footings and retaining walls. The author is aware of one commercial program which is similar to Pyramid. The name of the program is PROKON. In fact Pyramid was checked against PROKON as well as against ETABS and SAP2000. It is the author's opinion that the interface of Pyramid is more user friendly and can serve as an effective calculation tool for structural engineers everywhere. The Codes covered by Pyramid at the present are ACI-11, BS , Jordan JBC5-08 and Saudi SBS304. Planned future developments include the addition of the Eurocode and other Arab codes as they become available. 2. Scope Pyramid comprises six main applications. These are: 2.1. Log Manager Register and delete Projects Scan and import Projects Copy data between archives 2.2. RC Calculator RC section flexural analysis and design RC section design for shear and torsion Page 1 of 19

3 Flexural crack width calculation Punching Shear design Design of corbels Design of freestanding stairs Design of helical stairs Design of two-way slabs 2.3. Subframes Design of cantilevers and simply supported beams Analysis and design of one-way and ribbed slabs Analysis and design of continuous beams and sub-frames subjected to uniform, triangular and concentrated loads Flat slab design by the equivalent frame method 2.4. Columns Design of single story or multi-story short or slender columns of rectangular, circular or irregular section under axial load and uniaxial or biaxial bending 2.5. Footings Centroidal and offset pad footings Adjacent column footings Combined footings Wall footings 2.6. Retaining Walls Gravity walls Sloped cantilever walls Stepped cantilever walls Boundary walls 3. Interface 3.1. First Window When launched, Pyramid displays its First Window on the screen (shown in figure 1). This permits the selection of the application required Main Window All Pyramid applications have a main window comprising three panes. These are: Title Pane, Output Pane and Input pane as shown in figure 2. The title pane shows the application name, the project name, the member mark and the author's name. The output pane shows all input and output as they would appear on an A4 printer in a scrolling display on a resizable window. The input pane is employed for simple input like the command buttons, the input of text or a limited number of numerical values. For more complex input requiring a large number of selections or numerical data Pyramid employs especially designed input forms which pop up over the main window in the logical sequence. 4. Log Manager Data for the two main applications Subframes and Columns is preserved in an internal archive organized by project and member mark. To commence the user must use Log Manager to register a Project specifying the project name, code, material properties, soil bearing capacity, unit weight of reinforced concrete and rebar list. See figure 3. Member data is saved separately under each project and is only accessible under that project. Values like material properties and soil bearing capacity are entered as default values for each member to simplify input but may be changed if required. Log Manager may be used at any time to revise project data or copy members between projects or projects between archives. Log Manager is also used to enter story names and story heights for each building or part of building in the project. Subfames analysis results may be employed as input within Subframes itself and by Columns, while Columns results may be employed as input for Footings. Page 2 of 19

4 Figure 1: Pyramid first window Title Pane Output Pane Input Pane Figure 2: Main Window Page 3 of 19

5 Figure 3: Project registration Form 5. RC Calculator RC Calculator is employed for simple reinforced concrete calculations. In can be most convenient in peripheral situations where an engineer directly calculates a moment or shear force for which a section needs to be calculated. It can also calculate the design moment for a given section and reinforcement when checking an existing design. The shear design component is fairly sophisticated providing shear design for beams, columns, shear walls and coupling beams. RC Calculator can also be employed for the calculation of crack width in a flexural section, as well as the design of corbels, freestanding stairs, helical stairs and two-way slabs. Because the modules in RC Calculator tend to be simple they all employ a single input form each and the calculation is interactive where the calculation result is shown on the input form itself. Figure 4 shows some of these modules. Figure 4: Some of the RC Calculator calculation forms Page 4 of 19

6 6. Subframes 6.1. Sub-frames All design codes agree that the moments, loads and shear forces employed in the design of individual columns and beams of a frame supporting gravity loads may be derived from an elastic analysis of a series of sub-frames. Each sub-frame is taken to comprise the beams at one level together with the columns above and below. The ends of the columns remote from the beams may generally be assumed to be fixed. The Pyramid Subframes and Pyramid Columns applications may therefore be employed for the design of monolithic frames not providing lateral stability. They may also be employed for the verification of design obtained from 3D building system applications like ETABS and Robot, particularly because such applications do not possess the facility for the pattern loading of building frames required by the Codes. Building design is often an interactive process between architect and structural designer, and the structural designer is usually permitted little time after the architect completes his work. Because of this the author has found that it is best to prepare the detailed structural design of all members employing Pyramid, easily coping with the inevitable changes and revisions by the architect and finalizing all member sizes. After the design is completed, the structural designer can create the 3D computer model and would only need to check the member reinforcement against the 3D model Loading Patterns Design codes require that the design of monolithic frames should provide for the most severe stresses resulting from possible loading patterns of factored loads. For this purpose Pyramid defines two types of gravity loads, namely: permanent loads and incidental loads. Incidental loads are loads which may or may not exist on a span, while permanent loads always exist. Maximum span moments are obtained when the full factored loading is applied to alternate spans while the other spans only support the permanent loads. Maximum support moment are obtained when adjacent spans and every alternate span to each side support the full factored load while the other spans are only subjected to the permanent load. The definition of permanent and incidental loads differ between codes. The ACI318 and Saudi SBC304 codes define the factored dead load as the permanent load, and the factored live load as the incidental load. On the other hand the BS8110 and Jordan 5-1/08 Codes define the unfactored dead load as the permanent load and the sum of the factored live load plus the adverse component of the factored dead load as the incidental load. The load factors for permanent and incidental loads are therefore as follows. Code Dead Load Live Load Permanent Incidental Permanent Incidental ACI Saudi SBC BS Jordan 5-1/ Load Cases BS8110 and Jordan 5-1/08 permit a simplification whereby the adjacent spans loaded case is replaced by an all spans loaded case. Thus for these two codes the load cases become simpler and only 3 load cases are required as shown in figure 5. For the ACI318 and Saudi SBC 304 additional load cases are needed where the number of additional load cases depends on the number of spans. The basic cases in the figure apply to all codes. Figure 5: Load Cases Page 5 of 19

7 The Pyramid Subframes application analyses sub-frames, beams on pinned supports, ribbed slabs or one-way slabs for all these load cases producing an envelope for maximum and minimum moments and shears over the length of each span while defining the moments and axial loads acting on the columns for each of these load cases. It then calculates the required flexural and shear reinforcement, producing an envelope of tension and compression flexural reinforcement as well as shear reinforcement. It also provides a list of suggested rebars for each span and support as well as suggested stirrups, leaving the choice to the user Load Types Load types can be uniform or triangular defined between any two points along the length of the span. They can also be concentrated (point) loads at any point along the span Frame Types Frame types can be: cantilever, sub-frame, simple beam, continuous beam, ribbed slab, solid one-way slab or flat slab Data Input Forms A large number of forms are employed for data input by Subframes. These pop up in sequence over the main window and all data input is reproduced on the output pane of the main window to permit the user to check input as he proceeds Basic Data Form The first input form is the Basic Data form. This permits the user to select the frame type and design code as well as the loading pattern and load factors to be employed. This form is shown in figure Analysis Data Beyond the basic data form, a series of forms are employed to input the beam and column size and beam loading. Beam loading can be as complex as necessary and beam sections can be uniform or variable. See figure Frame Analysis Figure 6: Basic Data form The results of the frame elastic analysis are produced both in tabular and graphical format on the main window. See figure 8. At this stage the user is given the opportunity to study all input and the analysis results, and may Page 6 of 19

8 then return to revise the input, redistribute the moments and delete or insert spans. Figure 7: Span loading and variable sections Figure 8: Bending moment and shear diagrams Beam Design Data Once the user chooses to proceed with beam design the beam design data form pops up. See figure 9. This form permits the user to either keep or change the default values for material properties. It also requires the input of the beam section type (rectangular, T or L), cover to the center of reinforcement, rebar choice for supports and beam ductility. Page 7 of 19

9 Figure 9: Beam design data form 6.11.Beam Design Results Beam design for flexure and shear is reproduced on the output pane of the main window in both tabular and graphical formats (figure 10). Beam deflection for the required reinforcement is also calculated (figure 11) Beam Deflections Deflections can be recalculated for any of the spans for the provided rebars. A special form is utilized for input of proposed rebars and their cut offs, permitting study of the influence of provided rebars in comparison with the required reinforcement as well as the effect of rebar cut offs. See figure Flat Slabs Most flat slabs at the present are designed employing grillage or finite element analysis software like SAFE. Pyramid provides an alternative in accordance with the code equivalent frame method. This can also be a powerful tool for verification of results obtained from SAFE. In Pyramid flat slabs are designed as sub-frames comprising slab strips on column lines taken in each of the two perpendicular directions (X and Y directions). Sub-frame stiffnesses are defined according to code specifications permitting the calculation of span and support moments in the usual manner. The code defined percentages of these moments are apportioned to column strips and middle strips for which the required flexural reinforcement is calculated. Punching shear reinforcement is also calculated where required to the code specified peripheries. Page 8 of 19

10 Figure 11: Reinforcement and deflection diagrams Page 9 of 19

11 Figure 11: Beam deflection calculation Page 10 of 19

12 7. Columns 7.1. General Pyramid Columns may be used for the design of a single story or a multistory column. Columns may be short or slender and loading may be axial (compressive or tensile) combined with biaxial bending. Combinations of axial load and out of balance moments which must be considered for column design include the effects of pattern loading on the beams they support and the beams at the level below. Comb 1 Comb 2 Comb 3 Comb 4 Comb 5 Comb 6 Comb 7 Comb 8 Comb 9 These general loading combinations must be considered for the design of columns of irregular sections. For square, rectangular and circular sections which are symmetrical about both axes, only 4 special load combinations need to be considered. Figure 12: Strain & Stress profiles in a column section 7.2. Column Data Input Input of column loading may be either manual or by importing the support reactions from sub-frame analyses. At the beginning the user selects the type of loading input and if frames exist in either the X and/or Y directions. Frames in the X direction result in moments about the Y axis and frames in Y result in moments about X. Loading is input for each level starting from the topmost story. When load is from a sub-frame, the user selects the sub-frame by name from the archive and then selects the support on that frame as shown in figure 14. Once all loading has been input the application collates the data, defines axial loads and moments for each load case at column top and column bottom for every story, adds self weight and proceeds with column design. Figure 13: Column main data input form Page 11 of 19

13 Figure 14: Form for selection of required support in a sub-frame 7.3. Column Section Design Form The same column section design form is employed for all column types whether for single story or multi-story and whether for slender or short. It is shown in figure 15. This form comprises two graphic panes in addition to the input, output and command panes. The panes change somewhat according to the section type. For example the loading pane changes to 9 combination for irregular sections, while the section size and rebar panes change to suit the section type. The graphic panes contain a plot of the column section and the interaction diagram for the critical load combination. The form is interactive as the column section design is performed by the application on the form itself, permitting the user to experiment with various section sizes and rebar configurations on the form before moving on to the next story Irregular Sections Pyramid Columns is capable of solving the design problem for any section shape. It is a requirement for such sections that the number and locations of rebars be defined to permit solution. To simplify input the application utilizes a set of customary section shapes with three preset categories of reinforcement: light, medium and heavy. The user would then simply select the shape, input its dimensions and define the category of reinforcement for the application to auto generate the section. The available section shapes are shown in figure 16. If a more complex shape is required the user would select the "Irregular Polygon" category. This would cause a pop up of a form for the graphic input of the section (shown in figure 17). The user would then be required to graphically input the vertex points which define the section periphery and to input the rebar locations one by one. The same form may be used to edit auto generated irregular column sections. 8. Footings 8.1. Centroidal Column Pad Pyramid performs the conventional calculation for a column pad on an interactive form (shown in figure 18) for up to three load combinations. A 3D graphic shows the soil contact stress and a second pane shows the footing in plan. The calculation is performed for a column size and factored loading which can either be input directly or imported from the results of the Columns application. Pad size and depth are either automatically determined by the application or input by the user. When the user is satisfied with the pad size and clicks [Done] the complete design calculations are output on the main window as shown in figure Offset Column Pad Pyramid permits the calculation of offset pads subjected to irreversible moments for cases such as edge columns or special freestanding columns supporting cantilevers. The interactive form is shown in figure 20. Only one load combination is permitted. Pad size and offset can be input or automatically calculated for a constant contact stress. Page 12 of 19

14 Figure 15: Column section interactive design form Figure16: Form for the selection of irregular section type Page 13 of 19

15 Figure17: Graphic Section Input 8.3.Combined Footings Pyramid calculates combined footings for two columns where each column can be subjected to 3 load combinations. Pad size can either be input or automatically calculated. The interactive form is shown in figure Adjacent Columns Pyramid calculates pads for adjacent columns where a small gap exists between two columns for an expansion joint. The calculation provides for 3 load combinations for each column Wall Footings Pyramid even provides an interactive form for the design of strip footings. Page 14 of 19

16 Figure 18: Centroidal Pad Interactive Form Figure 19: Centroidal Pad design output Page 15 of 19

17 Figure 20: Offset Pad Interactive Form Figure 21: Combined Footing Interactive Form Page 16 of 19

18 9. Earth Retaining Walls 9.1. General The Pyramid Retaining Walls application provides for the design of cantilever walls supporting drained cohesionless soils. A single interactive form is employed for data input and determination of wall size and configuration. This calculation is made for wall stability at the required factors of safety where the user may experiment with various alignments for the wall base plus the presence and location of a shear key. Once the user is satisfied with the wall size and configuration, a full design calculation plus reinforcement diagram are output on the main window Lateral Pressure Coefficients The application calculates lateral pressure according to either the Rankine or Coulomb theories. It also permits user input of these coefficients. However it automatically jumps from Coulomb to Rankine where it deems Coulomb unsuitable. The calculation of seismic earth pressure is performed for the specified earthquake zone according to the Jordan code Gravity Walls The interactive form for Gravity Walls is shown in figure 22. The user may experiment with the wall configuration, sloping either side or both to determine the stable wall size. The form provides the option where the resultant is permitted to fall outside the middle third although some codes prohibit this. It also provides for a reduction of the influence of the stabilizing passive pressure down to zero depending on the probability of future excavation behind the wall Sloped and Stepped Walls The interactive form for the input and sizing of cantilever earth retaining walls is shown in figure 23. All data input is performed on the form itself permitting an interactive determination of the wall base stable configuration. The wall may be with or without a shear key and the user may limit the length of either the toe or heel leaving the other base length to be determined by the application. Figure 22: Gravity retaining wall Page 17 of 19

19 Figure 22: Sloped cantilever wall 9.5. Boundary Walls The application also designs boundary walls and reverse boundary walls as shown in figure 23. Figure 23: Boundary and reverse boundary walls Page 18 of 19

20 Acknowledgments The author owes much to the Royal Scientific Society of Jordan where he first began writing code for Pyramid. He also owes a debt of gratitude to the renowned architect Jafar Tukan who granted him the opportunity to design many prestigious buildings during the next twelve years of his career. It was during those years that he developed and used Pyramid in his design work and has continued to expand, refine and utilize it till this day. The author would also like to mention his colleagues Rabab Haddad, Salma Ziadin, Farah Hiasat, Rawan Samara and many others who helped him over the years to test, correct and debug these programs. The Pyramid setup is a free installer obtained from jrsofrware.org References American Concrete Institute. Building Code Requirements for Structural Concrete (ACI ). British Standards. Structural Use of Concrete (BS ). British Standards. Code of Practice for Foundations (BS ). British Standards. Code of Practice for Earth Retaining Structures (BS ). National Building Codes of Jordan. Structural Concrete (JBC5-08). National Building Codes of Jordan. Foundations and Retaining Walls Code (JBC4-07). Saudi Building Code Requirements. Structural Concrete (SBS304). Copyrights Copyright for this article is retained by the author, with first publication rights granted to the journal. This is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license ( Page 19 of 19

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