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Analysis of Beam using FINITE ELEMENT METHOD (in SMath Studio platform)
NOTE: This worksheet requires that the associated Plugin (Structural Beam Analysis Utility Functions by Redem Legaspi Jr.) be downloaded and enabled from SMath Studio Extension Manager tool.
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DATA INPUT SIMPLE GUIDE:
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a.) BEAM DIMENSION AND SUPPORT DATA:
1.) Beam Length Input Data
BEAM LENGTH DATA may take any desired 'distance unit', either in METRIC or in 'IMPERIAL' system. Unit system COMPATIBILITY is internally taken care of by the program. Hence, assigning any distance unit such as feet, meter, yard, mile, kilometer, cm, mm, inch, etc -- is absolutely permitted.
2.) Support Input Data
Support Input Data adapts the following syntax in 2-row matrix format:
┌ ┐ │ S1 S2 S3 ... Sn │ │ X1 X2 X3 ... Xn │ └ ┘
Syntax:
Where: S1,S2,S3... Sn are STRINGS (quoted words) that specify the type of support assigned at the given locations along the beam. Valid SUPPORT TYPES are as follows: a.) "PIN" or "PINNED" for pin support b.) "FIX" or "FIXED" for fixed support c.) "ROLLER" for roller support Note that in this application, ROLLER and PIN supports are treated as technically the same. X1, X2, X3... Xn are floating point values that correspond to the LOCATION of individual support. Each value MUST be associated with DISTANCE UNITS either in english or in metric system (e.g,. meter, ft, cm). An infinite number of supports maybe assigned to the beam; there is no limit so long as the computer's computing capacity can still handle the complexity of the resulting input matrices. LOCATION is defined as the distance from the left end of the beam to any point along the beam -- which is, by default, set at the ORIGIN or ZERO distance.
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b.) BEAM LOAD INPUT DATA:
Input data for 'Beam Loads' is specially formatted in a logical manner that is understood by SMath Studio. Like MS Excel software, SMath Studio is a special application that reads data (such that an input data should be written in a proper syntax or format), does calculations and returns results. The syntax of the input data is a programmer's discretion (and is decided based mainly on how Smath Studio paper-like interface picks and reads data) -- which for this application is defined in matrix format as below:
1.) Concentrated Load Data adapts the following syntax in 2-row matrix format:
┌ ┐ │ P1 P2 P3 ... Pn │ │ X1 X2 X3 ... Xn │ └ ┘
Syntax:
Where: P1,P2,P3... Pn are floating point values of concentrated loads. The units can be either in METRIC, ENGLISH or MIXED system (e.g., kip, kN, lbf). X1,X2,X3... Xn are the locations of concentrated loads measured from the origin or the left end of the beam. Units can be either in METRIC or ENGLISH system (e.g,. meter, ft, cm). Mixed unit system is also allowed.
2.) Moment Load Data adapts the following syntax in 2-row matrix format:
┌ ┐ │ M1 M2 M3 ... Mn │ │ X1 X2 X3 ... Xn │ └ ┘
Syntax:
Where: M1,M2,M3... Mn are floating point values of moment loads. The units can be either in METRIC, ENGLISH or MIXED system (e.g., kip-ft, kN-m, kN-ft, etc). X1,X2,X3... Xn are the locations of moment loads measured from the origin or the left end of the beam. Units can be either in METRIC or ENGLISH system (e.g,. meter, ft, cm). Mixed unit system is also allowed.
3.) Trapezoidal Load Data adapts the following syntax in 4-row matrix format:
┌ ┐ │ Ws1 Ws2 Ws3 ... Wsn │ │ We1 We2 We3 ... Wen │ │ Xs1 Xs2 Xs3 ... Xsn │ │ Xe1 Xe2 Xe3 ... Xen │ └ ┘
Syntax:
Where: Ws1,Ws2,Ws3... Wsn are floating point START values of trapezoidal loads. We1,We2,We3... Wen are floating point END values of trapezoidal loads. The units can be either in METRIC, ENGLISH or MIXED system (e.g., kip/ft, kN/m, kN/ft, etc). Xs1,Xs2,Xs3... Xsn are the locations of START values of trapezoidal loads measured from the origin or the left end of beam. Xe1,Xe2,Xe3... Xen, on the other hand, are the locations of END values of trapezoidal loads also measured from the origin or the left end of the beam. Units can be either in METRIC or ENGLISH system. Mixed unit system is also allowed (e.g,. meter, ft, cm).
4.) Uniform Load data adapts the following syntax in 3-row matrix format:
┌ ┐ │ W1 W2 W3 ... Wn │ │ Xs1 Xs2 Xs3 ... Xsn │ │ Xe1 Xe2 Xe3 ... Xen │ └ ┘
Syntax:
Where: W1,W2,W3... Wn are floating point values of uniform loads. The units can be either in METRIC, ENGLISH or MIXED system (e.g., kip/ft, kN/m, kN/ft, etc). Xs1,Xs2,Xs3... Xsn are the start locations of uniform loads measured from the origin or the left end of the beam. Xe1,Xe2,Xe3... Xen, on the other hand, are the end locations of uniform loads also measured from the origin or the left end of the beam. Units can be either in METRIC or ENGLISH system. Mixed unit system is also allowed (e.g,. meter, ft, cm).
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c.) BEAM GEOMETRIC AND DESIGN DATA:
1.) Modulus of Elasticity data adapts the following syntax in 3-row matrix format:
┌ ┐ │ E1 E2 E3 ... En │ │ Xs1 Xs2 Xs3 ... Xsn │ │ Xe1 Xe2 Xe3 ... Xen │ └ ┘
Syntax:
Where: E1,E2,E3... En are floating point values of moduli of elesticity. The units can be either in METRIC, ENGLISH or MIXED system (e.g., ksi, MPa, psf, psi, etc). Xs1,Xs2,Xs3... Xsn are the start locations of moduli of elasticity measured from the origin or the left end of the beam. Xe1,Xe2,Xe3... Xen, on the other hand, are the end locations of moduli of elasticity also measured from the origin or the left end of the beam. Units can be either in METRIC or ENGLISH system. Mixed unit system is also allowed (e.g,. meter, ft, cm).
Note: This example assumes that the beam is composed of two segments with different moduli of elasticity.
2.) Moments of Inertia data adapt the following syntax in 3-row matrix format:
┌ ┐ │ I1 I2 I3 ... In │ │ Xs1 Xs2 Xs3 ... Xsn │ │ Xe1 Xe2 Xe3 ... Xen │ └ ┘
Syntax:
Where: I1,I2,I3... In are floating point values of moments of inertia. The units can be either in METRIC, ENGLISH or MIXED system (e.g,. m^4, ft^4, in^4, cm^4, mm^4, etc). Xs1,Xs2,Xs3... Xsn are the start locations of moments of inertia measured from the origin or the left end of the beam. Xe1,Xe2,Xe3... Xen, on the other hand, are the end locations of moments of inertia also measured from the origin or the left end of the beam. Units can be either in METRIC or ENGLISH system. Mixed unit system is also allowed (e.g,. meter, ft, cm).
Note: This example assumes that the beam is composed of two segments with different moments of inertia.
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d.) PARSING DATA TO GRAPHICS MATRIX:
Before 'BEAM', 'BEAM LOADS' and 'SUPPORTS' data can be shown graphically or converted to drawing, each data should first undergo Data Parsing. The purpose of which is to convert each data into a matrix format that can be plotted onto SMath Studio 2D Chart. This part is what makes SMath Studio plugin programming challenging, interesting and enjoyable on the part of programming enthusiasts...
1.) Parse 'Beam Data' into Graphics Matrix Format:
Syntax: ParseBeam [BmSpan, BmThk]
Where: BmSpan = Span of Beam BmThk = Apparent Beam Thickness/Plot Scale
2.) Parse 'Support Data' into Graphics Matrix Format:
Syntax: ParseSupport [SptMtxData, BmSpan, Prefix, BmThk, Scale]
Where: SptMtxData = Support data definition in Matrix format BmSpan = Beam span Prefix = Support label prefix BmThk = Beam apparent thickness Scale = Graphics plot scale
3.) Parse 'Concentrated Load Data' into Graphics Matrix Format:
Syntax: ParsePLoad [BmPLoad, Prefix, Scale]
Where: BmPLoad = Concentrated load data definition in Matrix format Prefix = Concentrated load label prefix Scale = Graphics plot scale
4.) Parse 'Moment Load Data' into Graphics Matrix Format:
Syntax: ParseMLoad [BmMLoad, Prefix, Scale]
Where: BmMLoad = Moment load data definition in Matrix format Prefix = Moment load label prefix Scale = Graphics plot scale
5.) Parse 'Uniform Load Data' into Graphics Matrix Format:
Syntax: ParseQLoad [BmQLoad, Prefix, Scale]
Where: BmQLoad = Uniform load data definition in Matrix format Prefix = Uniform load label prefix Scale = Graphics plot scale
5.) Parse 'Trapezoidal Load Data' into Graphics Matrix Format:
Syntax: ParseTLoad = [BmTLoad, Prefix, Scale]
Where: BmTLoad = Trapezoidal load data definition in Matrix format Prefix = Trapezoidal load label prefix Scale = Graphics plot scale
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e.) PLOT BEAM, SUPPORTS, & LOADINGS:
Once DATA PARSING is done -- BEAM, together with SUPPORTS and LOADINGS, can now be plotted onto SMath Studio 2D chart using 'PlotBeam' command. The command, specially developed by Redem Legaspi Jr for Crystal Steel, takes an infinite number of arguments -- with syntax shown below:
Syntax: PlotBeam(Arg1, Arg2, Arg3... ArgN) Where: Arg1, Arg2, Arg3... ArgN are arguments that correspond to 'Parsed Beam Data', 'Parsed Support Data', 'Parsed Concentrated Load Data', 'Parsed Uniform Load Data', 'Parsed Trapezodial Load Data', 'Parsed Moment Load Data', and so on. Arguments, however, can be ARRANGED IN ANY ORDER... If supplied with correct parsed data, the'PlotBeam' command yields a realistic graphic representation of beam -- complete with supports and loadings as shown. From an engineering standpoint, we call it BEAM DIAGRAM...
Invalid Support DETECTION (a fool-proof feature): 'PlotBeam' command prompts a 'graphics error warning' if an invalid support definition is encountered. An example is as shown below:
The reason why support 'R2' is detected as invalid -- is that, by simple logic, FIXED support cannot be assigned anywhere along the beam span. It maybe assigned only to either end (or both ends) of the beam. Dectecting this sort of engineering lapses early on while doing beam analysis may help ease an engineer's burden and makes ones lovelife more colorful and more enjoyable. (a non sequitur?? he he he!!!) There are lots of other 'fool-proof' features embedded in this application which may take a lot of space and time if completely discussed here. You will eventually discover and encounter those features as soon as you start using the program.
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f.) PERFORM FINITE ELEMENT ANALYSIS:
NOTE: In this example I choose to apply 'CONCENTRATED LOAD' and 'UNIFORM LOAD' only due to my obvious preference for SIMPLICITY. (Simple is beautiful) One has the complete discretion over what loads he wanted to appear on the BEAM DIAGRAM and what loads he wanted to be included in the analysis. This is done, for example, by specifying only the 'CONCENTRATED LOAD' and 'UNIFORM LOAD' onto 'PlotBeam' and 'AnalyzeBeamXXXX' commands as shown below. This command call, for example, PlotBeam(plot_beam, plot_support, plot_uLoad, plot_pLoad) plots beam and supports with CONCENTRATED and UNIFORM loads only.
FEM Analysis of beam is performed using either of the two commands:
1.)
AnalyzeBeamCaprani This command bears the name of a well-known structural engineer, Dr. Colin Caprani -- so named to give him due credit for writing the original source code of the FEABeam (Finite Element Analysis) computer program from which this application is adapted from. The program source code, fortunately, was written in Visual Basic -- my favorite programming language. For those interested in programming, please visit http://www.colincaprani.com/. With the code upgraded to VB.NET and implemented in SMath Studio platform, the program still bears about 50 percent of the original code attributed to Dr. Colin Caprani. The other 50 percent comprised the implementation program code that belongs to Redem Legaspi Jr. The bottomline here is -- I just wanna say thanks to Dr. Caprani. Limitations: The program has a known bug that I just recently discovered. This bug has already been brought to the attention of Dr. Colin Caprani, but up to this writing no reply has been received yet from the kind engineer. Here is the bug: with TRAPEZOIDAL LOADING applied, the program returns wrong results for both shear and moment including the deflection. I have made several attempts modifying the code to correct the problem but with no apparent success. While this problem on trapezoidal loading is still being resolved, please refrain from applying trapezoidal load when doing beam analysis. The command adapts the following syntax:
Syntax: AnalyzeBeamCaprani(Span, sData, pData, mData, tData, uData, eData, iData)
Where: Span = Span of Beam sData = Support Data in Matrix Format pData = Concentrated Load Data in Matrix Format mData = Moment Load Data in Matrix Format tData = Trapezoidal Load Data in Matrix Format uData = Uniform Load Data in Matrix Format eData = Modulus of Elasticity Data in matrix Format iData = Moment of Inertia Data in Matrix Format If you opt not to apply any of the loadings, just supply the argument field with a pair of QUOTES ("") as shown below.
2.)
AnalyzeBeamYakpol Just like the 'AnalyzeBeamCaprani' command, this command is so named to give due credit to Yakov Polyakov -- a generous russian engineer and a very good programmer. This command has no problem dealing with trapezoidal loads. The command adapts the following syntax:
Syntax: AnalyzeBeamYakpol(Span, sData, pData, mData, tData, uData, eData, iData)
Where: Span = Span of Beam sData = Support Data in Matrix Format pData = Concentrated Load Data in Matrix Format mData = Moment Load Data in Matrix Format tData = Trapezoidal Load Data in Matrix Format uData = Uniform Load Data in Matrix Format eData = Modulus of Elasticity Data in SINGLE FLOATING VALUE iData = Moment of Inertia Data in SINGLE FLOATING VALUE To emphasize, the only difference that 'AnalyzeBeamCaprani' and 'AnalyzeBeamYakpol' commands have -- is the 'eData' and 'iData' inputs. In AnalyzeBeamCaprani command, the eData and iData inputs are both in MATRIX FORMAT. In AnalyzeBeamYakpol command, on the other hand, the eData and iData inputs are both in single FLOATING POINTS VALUES. In other words, the command does NOT take VARYING MODULUS OF ELASTICITY and VARYING MOMENT OF INERTIA throughout the beam. If you opt not to apply any of the loadings, just supply the argument field with a pair of QUOTES ("") as shown below.
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g.) SHEAR & MOMENT DIAGRAMS AND DEFLECTION:
BEAM LOADING DIAGRAM:
SHEAR DIAGRAM:
MOMENT DIAGRAM:
DEFLECTION:
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h.) SUPPORT REACTIONS:
Support Reactions may be determined using the following commands:
Where the first argument, AnLcaprani, is the variable that contains the analysis result matrix returned by either 'AnalyzeBeamCaprani' or 'AnalyzeBeamYakpol' command. The second argument is the distance along the beam that corresponds to the location of support.
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h.) QUERIES:
Shear, Bending Moment and Deflection along the beam span can be determined using the following commands:
Where the first argument, AnLcaprani,is the analysis result matrix returned by either 'AnalyzeBeamCaprani' or 'AnalyzeBeamYakpol' command. The second argument is the distance along the beam where Shear, Bending Moment, or Deflection is to be determined.
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i.) MAXIMUMS:
To determine the maximum values, use the following commands:
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j.) LOCATIONS OF MAXIMUMS:
Use the following commands determine the locations of the MAXIMUMS along the the beam span:
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k.) FEM RESULTS DATA STRUCTURE:
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WANT TO KNOW WHAT IS INSIDE THE VARIABLE AnLcaprani? And how the ANALYSIS RESULTS in matrix format are being STRUCTURED AND STORED in a variable? WELL, FIND IT BELOW... Please note that retrieving values from these results is far more complex than returning 'these results' itself. One of those complex things done in this program is the 'routine' that creates diagrams (shear, moment & deflection)& intelligently label values by itself.
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There are an endless list of possible upgrades to this program, which, as of the moment are beyond the author's grasp. Your comments and suggestions,therefore, are paramount to make these upgrades possible...