CC BY
2
XOK 69.07
doi: 10.55287/22275398_2023_3_24
COMPARISON OF FINITE ELEMENT METHOD
AND FORCE METHOD IN ANALYSIS OF FRAME ELEMENTS
G. E. Okolnikova*1 ** S. Soumyadeep * F. Sh. Akoev*
* Peoples' Friendship University of Russia (RUDN University), Moscow
** Moscow State University of Civil Engineering (National Research University) (MGSU), Moscow
Abstract
This paper compares the finite element method and the force method of analysis. In this study, we looked at a continuous beam that is part of steel frame structures and compared it using the two methods of analysis. This analysis is performed to forecast errors in the results produced by the two methods. The results show that there are minor errors in the results that can be ignored. As a result, the finite element method results were discovered to be similar to the force method of analysis, and the results were also presented and discussed.
The Keywords
finite element method, force method, frame structure, continues beam, steel
Date of receipt in edition
10.06.2023
Date of acceptance for printing
15.06.2023
Introduction
The finite element method and force method analysis appear to be very different, but they both solve for indeterminate structures. Finite element analysis is a sophisticated problem-solving technique that has numerous applications in the field of complex engineering problems such as engine part analysis, cutting tools, and simulating actual working conditions in a virtual environment to obtain approximate results when analytical solutions are not available. Beginning with a variational statement of the problem, the finite-element method introduces piecewise definitions of functions defined by a set of mesh point values[1] - [4]. An indeterminate structure cannot be solved by using the equations of equilibrium alone. To solve an indeterminate structure, it is necessary to satisfy equilibrium, compatibility and force-displacement requirements of the structure[5] - [7]. The number of additional equations required to solve an indeterminate structure is known as degree of indeterminacy. For determinate structures, the force method allows us to find internal forces (using equilibrium i. e. based on Statics) irrespective of the material information. Material (stress-strain) relationships are needed only to calculate deflections[8] - [10]. A structure in the force method of analysis can be designated as structure (n, m), where (n, m) are the force and displacement degrees of freedoms (fof, dof), respectively. Equilibrium equations combined with compatibility conditions are fundamental to structural mechanics analysis methods. The equilibrium equations
are fundamentally about the balance of elemental forces. For advanced analysis of three-dimensional (3D) steel-framed structures, two common finite element approaches are used. the plastic zone methods (spread-of-plasticity) [11] - [15]and the plastic hinge methods (concentrated plasticity)[10], [12], [16] - [18]. To accurately predict the second-order effects and inelastic behaviour of steel structures, previous methods based on geometric stiffness matrices required members to be discretized into several elements. Because numerous discretization of elements are used in analysis modelling, it is widely acknowledged to be computationally expensive (computer resources, computational time) [19] - [22]. A historical review of the literature reveals some rather interesting findings [23] - [28]. Both the structural engineer and the applied mathematician can claim the finite-element method as their own, and both are correct to some extent. Examining the literature can help to explain how this happened. When a structure can be solved by using the equations of static equilibrium alone, it is known as determinate structure. An indeterminate structure cannot be solved by using the equations of equilibrium alone. To solve an indeterminate structure, it is necessary to satisfy equilibrium, compatibility and force-displacement requirements of the structure. The additional equations required to solve indeterminate structure are obtained by the conditions of compatibility and/or force-displacement relations. The number of additional equations required to solve an Q5 indeterminate structure is known as degree of indeterminacy. To solve indeterminate structure we use several Z methods but here I explained two of them ( Force (flexibility) method and Finite element (stiffness) method). I also O compared their methodological approaches and resulting values.
This study is an extension of previous studies comparing the finite element method to other methods of analysis frame elements. Many residential, social, and sports buildings are currently being constructed using various frame structures, and the application field of steel or composite frame is expanding. The findings of this study are applicable to software implementation, which gives it an advantage over similar jobs.
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Methods
A continuous beam is considered a case study to be compared using the finite element and force methods of analysis. The basic comparison for the two methods is presented in the table in this analysis.
Table 1
Differentiation between force method and FEM
Force (flexibility) method Finite element method
1. Convert the indeterminate structure into a determinate one by removing some unknown forces/ support reactions and replacing them with unit force. 1. Express member force-displacement relationship in term of unknown member displacements.
2. Using superposition calculate the force would be required to achieve the compatibility with the original structure. 2. Using equilibrium of assembled members, find unknown displacements.
3. Unknowns to be solved for are usually redundant forces. 3. Unknowns are usually displacements
4. Coefficients of the unknowns in equations to be solved are "flexibility" [A] coefficients. [A]*[X] = [B] 4. Coefficients of the unknowns are "Stiffness" [Ks] coefficients. [Ks]*[D] = [F]
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Considering an elastic uniform beam of total length L = 12 m and made of structural steel having Youngs modulus E = 2e11 N / m2, Density p = 7800 kg / m3 and Poisson's ratio | = 0.3 as shown in figure below. This is an isotropic, homogeneous and prismatic continuous beam with one end fixed and two hinges are at 9 m and 12 m from left support. The beam is divided into two parts AB and BC. An UDL load of 1 KN / m is acting downward along the span AB and a point load of10 KN downward is acting at middle of BC.
Fig. 1. Continuous beam element with external load
The continuous beam is discretized into a finite number of elements using the Finite Element method. Because the considered continuous beam is uniform, it is assumed that all elements used to mesh are identical. The beam is modelled using structural steel in this case. It is an engineering material with numerous applications in the manufacturing industry.
The force method analyses static indeterminacy by first determining the degree of indeterminacy, then selecting the redundant force, removing the redundant force, determining displacement at supports due to redundant forces, and generating a flexibility matrix. Create a compatibility equation using Maxwell's Theorem of Reciprocal Displacements (Betti's law), and then solve it. The virtual Work performed by a system of forces P2 while subjected to displacements caused by the system of forces P1 is equal to the virtual Work performed by the system of forces P1 while subjected to displacements caused by the system of forces P2.
In Finite element method, weighted residual concepts are used in finite element methods to solve partial differential equations. The finite element method works by dividing the spatial domain into simple geometric elements such as triangles or quadrilaterals. The weighted residual concept is then applied to each finite element domain to approximate the solution function. Moving from element to element, care must be taken to ensure the continuity of the dependent variables and their first partials. In time, partial differential equations are transformed into sets of ordinary differential equations. The method is best suited for problems with irregular geometries and steep gradients. The final goal is to interpret and analyse the result for use in the design and analysis process.
• Determination of location in the structure where large stress and deformation occur is generally important in making design and analysis decision.
• Post processor computer program helps the user to interpret the results by displaying them in graphical form.
• Basic comparison between two methods
Step I
Consider every span is fixed at both ends and draw free body diagram of the beam and find fixed end moments.
Fig. 2. Fixed end moment diagram
Step II
Displacement functions:
Moment in terms of angular displacement [M] = [K]*[0] Shear force in terms of vertical displacement [F] = [K]*[A]
Step III
Derive element stiffness matrix:
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Fig. 3. Free body diagram of beam
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Moment required for unit rotation is called stiffness (Kbb) = — & (KaA ) = — Assigning co-ordinates:
Fig. 4. Co-ordinate assign
Element stiffness(k) matrix for
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Consider every span is fixed at both ends and draw free body diagram of the beam and find fixed end moments.
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Equilibrium equation: M CB = 0
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Global equilibrium equation
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M = Unknown moments at supports; K = global stiffness matrix ; 6 = Angular displacement of beam at supports ; MF = Fixed end moments.
Step V
From the above matrix take linear eq. 2 & 3 and solve for 6B and 6C (MCB = 0 ; MBC + MBA = 0). Now substitute the values of 6B in elementary equation for element AB, 6A = 0, As End A is fixed.
By solving this linear equations the values of MAB & MBA are found and by the moment equilibrium equation at A we can find shear stresses RA & RB . See below figure.
Fig. 5. Fixed end moments of part AB
Now substitute the values of 0B & 0C in elementary equation for element BC.
By solving this linear equations the values of MBC & MCB are found and by the moment equilibrium equation at B we can find shear stresses RB & R C . See below figure:
Fig. 6. Fixed end moments of part BC
* Calculation and derivation of functions to find shear force and bending moments in every sections I used "Math", "Scipy" & "Numpy" library form python programming language.
Result and discussion
The figures below show a graphical comparison of shear force and bending moments calculated using the finite element method and the force method of analysis. For the finite element method, we considered a Math LAP to get the results. We divided the whole span into 10000 segments with the use of "arrange" function (numpy. arrange). Then two empty arrays were created with the help of "empty" function (numpy.empty) to store shear force and bending moments values of 10000 segments in each as an array. After this I derive a function within I input all data such: shear forces at supports, moment at fixed end, magnitude of UDL and point loads and their positions , then with the help of "for, enumerate and if" functions I derive the main formulas to find shear force and bending moments at every 10000 segments in the beam and append them into two empty arrays respectively and also represent them in a graphical form with the help of another library called "Matplotlib".
7.5
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Fig. 7. Shear force diagram for forced method
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Comparison of results for FEM and force method
Table 2
Parameter Support coordinate Finite element Force method
A -6.92 -6.917
Bending moments (KNm) B -6.40 -6.412
C 0 0
Deflection (m) A 0 -0.4E-3
B -1.1E-3 -1.17E-3
A 4.56 4.538
Shear force (KN) B 11.57 (upward) 11.564(upward)
C -2.87 -2.88
Conclusion 2
The results obtained using the finite element method and the force method are nearly identical. The absolute q percentage of divergence for moments, deflection, and shear forces are found to be around 0.043%, 0.059, and 0.048% for the continuous beam, respectively. When the results of the finite element method and the force method ^ are compared, the minimum error found is 0.001% for some case studies and the minimum error is 0.008%. EG The results show that the values obtained using the finite element method and the force method do not differ significantly. However, the FEA approach requires less computation time. Thus, for complex engineering problems, the FEA method outperforms the numerical or analytical methods.
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О
СРАВНЕНИЕ МЕТОДА КОНЕЧНЫХ ЭЛЕМЕНТОВ 5
И МЕТОДА СИЛ ПРИ РАСЧЕТЕ ЭЛЕМЕНТОВ РАМ 2
м Э
Г. Э. Окольникова * 1 ** СО
С. Сумьядип * Ф. Ш. Акоев *
* Российский университет дружбы народов (РУДН), г. Москва
** Национальный исследовательский Московский государственный строительный университет (НИУ МГСУ), г. Москва
Аннотация
В данной статье проводится сравнение метода конечных элементов и метода сил при расчете стальных рам. В этом исследовании мы рассмотрели неразрезную балку, являющуюся частью стальных каркасных конструкций, и сравнили ее, используя два метода расчета. Это исследование выполняется с целью прогнозирования ошибок в результатах расчета, полученных двумя методами. В ходе сравнения результатов расчета по двум методам были выявлены незначительные расхождения, которыми можно пренебречь. Было установлено, что результаты расчета, полученные при использовании метода конечных элементов, аналогичны результатам расчета по методу сил. В статье произведено детальное сравнение и анализ зовании двух расчетных методов.
Ключевые слова
метод конечных элементов, метод сил, рамная конструкция, неразрезная балка, сталь
Дата поступления в редакцию
10.06.2023
Дата принятия к печати
15.06.2023
результатов расчета при исполь-
Ссылка для цитирования:
G. E. Okolnikova, S. Soumyadeep, F. Sh. Akoev. Comparison of finite element method and force method in analysis of frame elements. — Системные технологии. — 2023. — № 3 (48). — С. 24 - 33.
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