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Linear analyses of structural systems

Eduardo N. Manzi(1)

Amon Engenharia e Inc. Ltd, Brazil

The present programs are part of the research line Computational Mechanics fromAmon Ltd, which aims the application of numerical methods in t he determinationof structural systems answers.

The "CalcFEM" program executes linear analyses of st ructural systems using theFinite Elements Method, which consists of a division of the structure understudying in a finite number of small regions named “finite elements”.

The CalcFEM is able to carry out linear analyses of structural systems, formed byone-dimensional bars such as plain frames, plain or space trusses, frames withcircular or parabolic bending, etc. The main objective is to obtain thedisplacements and internal st ress of the struct ural system, due to external loadapplication, such as wind load, use load, tc. After the analysis is carrired out, theprogram generates text files to store the results obtained (displacements file andinternal stress file).

The draw2D program was presented at the WWDC2006, which uses theQuartz2D library. This program reads the file results and generates a PDF file,presenting the drawing of the shaped structure (deformed or in rest). The colors ofthe finite elements are specified in RGB space and calculated in function of theinternal stress.

With the objective of presenting the shaped structures in 3D, a Cocoa applicationwas created. This program reads the file results and presents the shapedstructure on the screen, using a NSOpenGLView object. As in the case of t hedraw2D program, the colors of the finite elements are specified in RGB space andcalculated in function of the internal stress.

Rotations of the shaped structure can be achieved, in relation to the initial positionor in relation to the previous one. Three NSSlider object are used to control therotations, one for each axis.

Abstract OpenGL

To achieve rotations with exact values, l ike 45°, 90°, 180°, etc, just fix the knobon the marks. Thus, it is possible to visualize the lateral view (see Picture 10, 90° rotation); Picture 10 also s hows that the values of the maximum axial st ress arelocated in the bars connected to the c olumns.

Picture 6  – Tension bar; source: PDF file; Draw2D program.

With the objective of presenting the shaped structures in 3D, a Cocoa applicationwas created. This program reads the file results and presents the shaped structureon the screen, using a NSOpenGLView object.

The colors of the finite elements are specified in RGB space and calculated infunction of the internal tensions. The maximum and minimum negative internaltensions are defined as Red (1., 0., 0.) and Yellow (1., 1., 0.), respectively. Thepositive ones are defined as Blue (0., 0., 1.) and Green(0., 1., 0.), respectively. Thetension bar in picture 6 i llustrates the colors of the intermediate internal tensions,calculated automatically according to the limi t values.

Picture 10  – Lateral view.

OpenGL

The CalcFEM program is able to accomplish linear analysis of structural syst ems composed of one-dimensional bars. Such analysis is made via Finite Element Method, whichinvolves the solutions of a system of algebraic linear equations. P icture 1 show a clamped arch, divided in 4 finite elements and subject to a concentrated load in the vortex.

The program’s first procedure is the reading of the entry data file. These data define the geometry of the s tructure, its physical properties, the boundary conditions and the externalload. The following procedure consists of assembling the global st tifness matrix of the st ructure and the external load vector. Picture 2 shows the equation system of t he structurein Picture 1.

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Figura 1 –Clampedarch; source: Manzi,2001.

Picture 2  – Globalsttifness  matrix;

source: Manzi, 2001.

The sttifness matrix of each finite element is obtained according to the deformation considered. In the case of t russes( plane or spatial), only the axial deformation is considered. In the case of frames, however, the effects of axial, flexionand shear deformation were considered, thus obtaining the relative internal stress.

After the global sttifness matrix is assembled, the equation system is solved, presenting the displacements of thenodal points, relative to the external load applied.

After the analysis i s carried out, t he program generates text files to store the results obtained (displacements file andinternal stress file).

Picture 3 shows a clampled parabolic arch subject to three types of l oad: (a) concentrated load at the vortex; (b) moment at the vortex; and (c) uniform distributed load. The arch is subdivided in ten one-dimensional finite elements,of parabolic curvature, and executed for each kind of external load. The deflection curves can be observed according toeach kind of load.

AutoCad

Picture 4  – Spatial metallic structure; source : DWG file.

 Picture 5 – Photo of the

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A routine in AutoLisp was developed in order to draw the structure under study inthe AutoCad. Picture 3 shows a draft of a spatial metallic structure. Number of finiteelements: 2928.

The routine establishes the value of each color for the finite elements following theminimum and maximum internal stress, associated with each basic color of theAutoCad. This way, each finite element with internal stress values within the

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Picture 3  – Clamped arch; source : Manzi et al., 2000 .

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Picture 11 shows the results of consecutive rotations of another structure; thevalues of the maximum axial stress, as in t he previous case, are located in thebars connected to the columns.

References

Picture 11 – Structure views; after c onsecutive rotations and a lateral view.

same range isidentified by the samecolor.

Apple Inc. Drawing with Quartz 2D. Inside Mac OS X, 2001.

Apple Inc. The Objective-C Programming Language, 2007.

NeXTComputer, Inc. OpenStep Specificati on, 1994.

Manzi, E.N. Curved finite elements formulations for arch analyses. Master ofScience Dissertation, 2001, Escola de Minas, UFOP, Ouro Preto, Brazil . (inportuguese )

Manzi, E.N., Si lveira, R.A.M., Araújo, E.C. Curved finite element for arch analyses.3rd International Seminar ”The use of Steel Structures in Civil Construction”,2000, Belo Horizonte, Brazil. (in portuguese )

Next works:

Linear analyses of spatial frames.

Non-linear analyses of st ructural sistems.

(1) Manzi, E.N., civil engineer, M. Sc., Escola de Minas, UFOP, 2001, Ouro Preto, Brasil.

Rotations of the shaped structure can be achieved, in relation t o the initial position orin relation to t he previous one. Three NSSlider objects are used t o control therotations, one for each axis. Pic ture 7 shows the Cocoa aplicaton controls.

Picture 7 – The Cocoa application controls.

Picture 9  – Structure view, after consec utive rotations.

Pictures 9 and 10 presents theresults of c onsecutive rotations ofthe structure in picture 8. The colorsof the finite elements represents theresult of an axial effort over thestructure. The compression is thenegative axial stress and thetraction is the positive axial stress.

Picture 8  – Inicial view.

AutoCad and AutoLisp are registred tr ademarks of AutoDesk Inc; Mac OS, Cocoa and Quartz 2D are trademarks of A pple Inc.; OpenGL is a registered trademark of SiliconGraphics, Inc.