Finite element method in structural mechanics

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The finite element method (FEM) is a powerful technique originally developed for numerical solutions to complex problems in structural mechanics, and remains the method of choice for complex systems. In FEM, structural systems are modeled by an appropriate set of elements to that are interconnected at discrete points called nodes. Elements may have physical properties such as thickness, coefficient of thermal expansion, density, Young's modulus, shear modulus and Poisson's ratio.

Video Finite element method in structural mechanics

History

The origins of the method can be traced to structural matrix analysis in which the concept of the displacement or stiffness matrix approach is introduced. The finite element concept was developed based on engineering methods in the 1950s. The finite element method got a real boost in the 1960s and 1970s by John Argyris, and co-workers; at the University of Stuttgart, by Ray W. Clough; at the University of California, Berkeley, by Olgierd Zienkiewicz, and Ernest Hinton's co-worker Bruce Irons; at Swansea University, by Philippe G. Ciarlet; at the University of Paris; at Cornell University, by Richard Gallagher and colleagues. Original works such as those by Argyris and Clough form the basis for elemental structural analysis methods to this day. Earlier books such as by Zienkiewicz and more recent books such as those by Yang provide a comprehensive summary of developments in elemental-finite structural analysis. Applying methods in software are described in classical texts by Smith, Griffiths and Margetts.

Maps Finite element method in structural mechanics

Property element

• One-dimensional element that is straight or curved with physical properties such as axial rigidity, flexural, and torsional. This type of element is suitable for cable modeling, braces, frames, beams, stiffener, lattice and frame. The straight element usually has two nodes, one at each end, while the curved element will require at least three nodes including the final node. The elements are positioned on the central axis of the actual members.
• Two-dimensional elements that hold only in-air force with membrane action (field stress, field strain), and plates holding transverse loads with transverse and flexural movement (plates and shells). They may have different shapes such as flat or curved triangles and rectangles. Nodes are usually placed in element angles, and if required for higher accuracy, additional nodes may be placed along the edges of elements or even inside elements. The elements are positioned on the mid-surface of the actual coating thickness.
• Torus-shaped elements for axisymmetric problems such as membranes, thick plates, shells, and solids. The cross-section of elements is similar to the previously described type: one dimension for thin plates and shells, and two dimensions for solids, thick plates and shells.
• Three-dimensional elements for 3-D solid modeling such as engine components, dams, dikes or soil mass. Common elemental shapes include tetrahedral and hexahedral. Nodes are placed in dots and may be in the face of elements or inside elements.

Interconnection and moving elements

The elements are interconnected only on the exterior nodes, and they should cover the entire domain as accurately as possible. The node will have nodal displacement (vector) or degrees of freedom which may include translation, rotation, and for specific applications, high-order derivatives of displacements. When the nodes are replaced, they will drag the elements together in a certain way determined by the element formulation. In other words, the displacement of the points in the element will be interpolated from the nodal displacement, and this is the main reason for the approximate nature of the solution.

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Practical considerations

From the application point of view, it is important to model the system in such a way that:

• Symmetry or anti-symmetry conditions are exploited to reduce the size of the model.
• Transfer compatibility, including the required discontinuities, is confirmed at the node, and preferably, along the edges of the elements as well, especially when adjacent elements have different types, materials or thicknesses. The compatibility of multiple node movements can usually be imposed through constraint relationships.
• The behavior of elements must capture the dominant action of the actual system, both local and global.
• The mesh element should be good enough to produce an acceptable accuracy. To assess the accuracy, mesh is refined until important results show little change. For higher accuracy, elemental aspect ratios should be as close as possible to unity, and smaller elements are used above the higher voltage gradient section.
• Appropriate support limits are charged with special attention paid to nodes on the symmetry axis.

Large-scale commercial software packages often provide facilities to generate mesh, and graphical display of inputs and outputs, which greatly facilitates the verification of both data input and interpretation of results.

Theoretical_overview_of_FEM-Displacement_Formulation: _From_elements.2C_to_system.2C_to_solution "> Theoretical overview FEM-Displacement Formulation: Of the elements , to the system, to the solution

While FEM theory can be presented in different perspectives or emphases, its development for structural analysis follows a more traditional approach through the principle of virtual work or the principle of total potential minimum energy. The approach of the principle of virtual work is more common as it applies to both linear and non-linear material behavior. The method of virtual work is the expression of conservation of energy: for conservative systems, work added to the system by a set of forces applied equal to the energy stored in the system in the form of strain energy of the structure components.

Prinsip provides for virtual untuk system structural mengungkapkan identities mathematical to give virtual security virtual internally:

${\ displaystyle {\ mbox {External Kerja}} = \ int_ {V} \ delta {\ boldsymbol {\ epsilon}} ^ {T} {\ boldsymbol {\ sigma}} \, dV \ qquad \ mathrm {(1)}}   In other words, the sum of the work performed on the system by the set of external forces is equal to the work stored as strain energy in the elements that make up the system.The virtual internal work on the right side of the above equation can be found by summing up the virtual work performed on each element. The latter requires that force-transfer functions be used which describe the response for each individual element. Therefore, the displacement of structures is represented by the response of individual elements (discrete) collectively. The equations are written only for the small domains of each structural element rather than a single equation that describes the overall system response (a continuum). The latter will generate a hard problem, then the utility of the finite element method. As shown in the next section, Equation (1) leads to the following set of equilibrium equations for the system:                         R                 =                   K           r                                               R                           Â                                                             (           2          )                       {\ displaystyle \ mathbf {R} = \ mathbf {Kr} \ mathbf {R} ^ {o} \ qquad \ qquad \ qquad \ mathrm {(2 }}}   dimana                                    R                           {\ displaystyle \ mathbf {R}}    = vektor kekuatan nodal, mewakili kekuatan eksternal yang diterapkan ke node sistem.                                    K                           {\ displaystyle \ mathbf {K}}    = matriks kekakuan sistem, yang merupakan efek kolektif dari masing-masing matriks kekakuan elemen Â:                                                 k                                    e                                      {\ displaystyle \ mathbf {k} ^ {e}}    .                                    r                           {\ displaystyle \ mathbf {r}}    = vektor perpindahan nodal sistem.                                                 R                                    o                                      {\ displaystyle \ mathbf {R} ^ {o}}    = vektor gaya nodal yang setara, mewakili semua efek eksternal selain dari gaya nodal yang sudah termasuk dalam vektor gaya nodal sebelumnya R . Efek eksternal ini mungkin termasuk kekuatan permukaan terdistribusi atau terkonsentrasi, kekuatan tubuh, efek termal, tekanan awal dan strain. Setelah kendala dukungan 'diperhitungkan, perpindahan nodal ditemukan dengan menyelesaikan sistem persamaan linear (2), secara simbolis:                                    r                   =                                  K                                    -              1                              (                     R                   -                                  R                                    o                             )                                                (            3           )                           {\ displaystyle \ mathbf {r} = \ mathbf {K} ^ {- 1} (\ mathbf {R} - \ mathbf {R} ^ {o}) \ qquad \ qquad \ qquad \ mathrm {(3)}}    Selanjutnya, strain dan tekanan dalam elemen individu dapat ditemukan sebagai berikut:                              ?                =                 B         q                                                           (           4           )                     {\ displaystyle \ mathbf {\ epsilon} = \ mathbf {Bq} \ qquad \ qquad \ qquad \ qquad \ mathrm {(4)}}                                ?                =                 E                (                  ?                -                             ?                                or                        )                                  ?                                or                        =                 E                (                 B         q                -                             ?                                or                        )                                  ?                                or                                                           (          5           )                     {\ displaystyle \ mathbf {\ sigma} = \ mathbf {E} (\ mathbf {\ epsilon} - \ mathbf {\ epsilon} ^ o} \ mathbf {\ sigma} ^ {o} = \ mathbf {E} (\ mathbf {Bq} - \ mathbf {\ epsilon} ^ {o}) \ mathbf {\ sigma} ^ {o} \ qquad \ qquad \ qquad \ mathrm {(5)}}   dimana                                    q                           {\ displaystyle \ mathbf {q}}    = vektor perpindahan nodal - bagian dari vektor perpindahan sistem r yang berkaitan dengan elemen yang sedang dipertimbangkan.                                    B                           {\ displaystyle \ mathbf {B}}    = matriks regangan-perpindahan yang mengubah perpindahan nodal q menjadi strain pada titik mana pun dalam elemen.                                    E                           {\ displaystyle \ mathbf {E}}    = matriks elastisitas yang mengubah strain efektif menjadi penekanan pada titik mana pun di elemen.                                                ?                                    o                                      {\ displaystyle \ mathbf {\ epsilon} ^ {o}}    = vektor dari strain awal dalam elemen.                                                ?                                    o                                      {\ displaystyle \ mathbf {\ sigma} ^ {o}}    = vektor tekanan awal dalam elemen. Dengan menerapkan persamaan kerja virtual (1) ke sistem, kita dapat menetapkan matriks elemen                                    B                           {\ displaystyle \ mathbf {B}}    ,                                                 k                                    e                                      {\ displaystyle \ mathbf {k} ^ {e}}    serta teknik merakit matriks sistem                                                 R                                    o                                      {\ displaystyle \ mathbf {R} ^ {o}}    dan                                    K                           {\ displaystyle \ mathbf {K}}    . Matriks lain seperti                                                ?                                    o                                      {\ displaystyle \ mathbf {\ epsilon} ^ {o}}    ,                                                ?                                    o                                      {\ displaystyle \ mathbf {\ sigma} ^ {o}}    ,                                    R                           {\ displaystyle \ mathbf {R}}    dan                                    E                           {\ displaystyle \ mathbf {E}}    adalah nilai yang diketahui dan dapat langsung diatur dari input data. src: www.isr.umd.edu Fungsi interpolasi atau bentuk Biarkan                                    q                           {\ displaystyle \ mathbf {q}}    menjadi vektor perpindahan nodal dari elemen yang khas. Perpindahan pada titik lain dari elemen dapat ditemukan dengan menggunakan fungsi interpolasi sebagai, secara simbolis:                                    u                   =                     N                              q                                                         (            6           )                           {\ displaystyle \ mathbf {u} = \ mathbf {N} \ mathbf {q} \ qquad \ qquad \ qquad \ mathrm {(6)}}    dimana                                    u                           {\ displaystyle \ mathbf {u}}    = vektor perpindahan pada titik mana pun {x, y, z} dari elemen.                                    N                           {\ displaystyle \ mathbf {N}}    = matriks fungsi bentuk berfungsi sebagai fungsi interpolasi. Matriks ini biasanya dievaluasi secara numerik menggunakan Gaussian quadrature untuk integrasi numerik. Penggunaannya menyederhanakan (10) ke hal-hal berikut:                                                 Pekerjaan virtual internal                              =         ?                                            q                                    T                                                      (                                                      K                                    e                                         q                                                    Q                                    o              e                                                     )                                                  (            13           )                           {\ displaystyle {\ mbox {Internal virtual work}} = \ delta \ \ mathbf {q} ^ {T} {\ big (} \ mathbf {K} ^ {e } \ mathbf {q} \ mathbf {Q} ^ {oe} {\ big)} \ qquad \ mathrm {(13)}}    Kerja virtual elemen dalam hal pemindahan nodal sistem Karena vektor perpindahan nodal q adalah subset dari pemindahan nodal sistem r (untuk kompatibilitas dengan elemen yang berdekatan), kita dapat mengganti q dengan < b> r dengan memperluas ukuran matriks elemen dengan kolom baru dan baris nol:                                                 Pekerjaan virtual internal                              =         ?                                            r                                    T                                                      (                                                      K                                    e                                         r                                                    Q                                    o              e                                                     )                                                  (            14           )                           {\ displaystyle {\ mbox {Internal virtual work}} = \ delta \ \ mathbf {r} ^ {T} {\ big (} \ mathbf {K} ^ {e } \ mathbf {r} \ mathbf {Q} ^ {oe} {\ big)} \ qquad \ mathrm {(14)}}    where, for simplicity, we use the same symbol for the element matrix, which has now been expanded in size as well as the row and column rearranged accordingly. src: www.ethz.ch Working virtual system Menjumlahkan kerja virtual internal (14) untuk semua elemen memberikan sisi kanan dari (1):                                                 Kerja virtual internal sistem                              =                    ?                         e                             ?                                            r                                    T                                                      (                                                      k                                    e                                         r                                                    Q                                    o              e                                                     )                              =         ?                                            r                                    T                                                      (                                        ?                         e                                                      k                                    e                                                     )                                         r                           ?                                            r                                    T                                        ?                         e                                                      Q                                    o              e                                                  (            15           )                           {\ displaystyle {\ mbox {Sistem kerja virtual internal}} = \ jumlah _ {e} \ delta \ \ mathbf {r} ^ {T} {\ big (} \ mathbf {k} ^ {e} \ mathbf {r} \ mathbf {Q} ^ {oe} {\ big)} = \ delta \ \ mathbf {r} ^ {T} {\ big (} \ sum _ { e} \ mathbf {k} ^ {e} {\ big)} \ mathbf {r} \ delta \ \ mathbf {r} ^ {T} \ jumlah _ {e} \ mathbf {Q} ^ {oe} \ qquad \ mathrm {(15)}}    Source of the article : Wikipedia$