{"history":[],"article":{"id":"b09e8c2f-490a-4767-b44c-73e9b8281901","title":"Finite element method in structural mechanics","slug":"finite-element-method-in-structural-mechanics","content":":::summary\n:::\n\n# \nThe **[finite element method](https://bliptext.com/articles/finite-element-method)** (FEM) is a powerful technique originally developed for numerical solution of complex problems in [structural mechanics](https://bliptext.com/articles/structural-mechanics), and it remains the method of choice for complex systems.\nIn the FEM, the structural system is modeled by a set of appropriate **finite elements** interconnected at discrete points called nodes.\nElements may have physical properties such as thickness, [coefficient of thermal expansion](https://bliptext.com/articles/coefficient-of-thermal-expansion), [density](https://bliptext.com/articles/density), [Young's modulus](https://bliptext.com/articles/young-s-modulus), [shear modulus](https://bliptext.com/articles/shear-modulus) and [Poisson's ratio](https://bliptext.com/articles/poisson-s-ratio).\n\n# History\nThe origin of finite method can be traced to the matrix analysis of structures where the concept of a displacement or stiffness matrix approach was introduced.\nFinite element concepts were developed based on engineering methods in 1950s.\nThe finite element method obtained its real impetus in the 1960s and 1970s by [John Argyris](https://bliptext.com/articles/john-argyris), and co-workers; at the [University of Stuttgart](https://bliptext.com/articles/university-of-stuttgart), by [Ray W. Clough](https://bliptext.com/articles/ray-w-clough); at the [University of California, Berkeley](https://bliptext.com/articles/university-of-california-berkeley), by [Olgierd Zienkiewicz](https://bliptext.com/articles/olgierd-zienkiewicz), and co-workers [Ernest Hinton](https://bliptext.com/articles/ernest-hinton), [Bruce Irons](https://bliptext.com/articles/bruce-irons-engineer); at the [University of Swansea](https://bliptext.com/articles/swansea-university), by [Philippe G. Ciarlet](https://bliptext.com/articles/philippe-g-ciarlet); at the [University of Paris](https://bliptext.com/articles/pierre-and-marie-curie-university); at [Cornell University](https://bliptext.com/articles/cornell-university), by Richard Gallagher and co-workers.\nThe original works such as those by Argyris and Clough became the foundation for today’s finite element structural analysis methods.\nStraight or curved one-dimensional elements with physical properties such as axial, bending, and torsional stiffnesses.\nThis type of element is suitable for modeling cables, braces, trusses, beams, stiffeners, grids and frames.\nStraight elements usually have two nodes, one at each end, while curved elements will need at least three nodes including the end-nodes.\nThe elements are positioned at the [centroidal](https://bliptext.com/articles/centroid) axis of the actual members.\n\n## Element interconnection and displacement\nThe elements are interconnected only at the exterior nodes, and altogether they should cover the entire domain as accurately as possible.\nNodes will have nodal [(vector) displacements](https://bliptext.com/articles/displacement-vector) or [degrees of freedom](https://bliptext.com/articles/degrees-of-freedom-engineering) which may include translations, rotations, and for special applications, higher order [derivatives](https://bliptext.com/articles/derivative) of displacements.\nWhen the nodes displace, they will drag the elements along in a certain manner dictated by the element formulation.\nIn other words, displacements of any points in the element will be [interpolated](https://bliptext.com/articles/interpolation) from the nodal displacements, and this is the main reason for the approximate nature of the solution.\n\n# Practical considerations\nFrom the application point of view, it is important to model the system such that:\nLarge scale commercial software packages often provide facilities for generating the mesh, and the graphical display of input and output, which greatly facilitate the verification of both input data and interpretation of the results.\n\n# Theoretical overview of FEM-Displacement Formulation: From elements, to system, to solution\nWhile the theory of FEM can be presented in different perspectives or emphases, its development for [structural analysis](https://bliptext.com/articles/structural-analysis) follows the more traditional approach via the [virtual work](https://bliptext.com/articles/virtual-work) principle or the [minimum total potential energy principle](https://bliptext.com/articles/minimum-total-potential-energy-principle).\nThe [virtual work](https://bliptext.com/articles/virtual-work) principle approach is more general as it is applicable to both linear and non-linear material behaviors.\nThe virtual work method is an expression of [conservation of energy](https://bliptext.com/articles/conservation-of-energy): for conservative systems, the work added to the system by a set of applied forces is equal to the energy stored in the system in the form of strain energy of the structure's components.\nThe principle of [virtual displacements](https://bliptext.com/articles/virtual-work) for the structural system expresses the mathematical identity of external and internal virtual work:\nIn other words, the summation of the work done 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.\nThe virtual internal work in the right-hand-side of the above equation may be found by summing the virtual work done on the individual elements.\nThe latter requires that force-displacement functions be used that describe the response for each individual element.\nHence, the displacement of the structure is described by the response of individual (discrete) elements collectively.\nThe equations are written only for the small domain of individual elements of the structure rather than a single equation that describes the response of the system as a whole (a continuum).\nThe latter would result in an intractable problem, hence the utility of the finite element method.\nAs shown in the subsequent sections, Eq. leads to the following governing equilibrium equation for the system:\nwhere\nOnce the supports' constraints are accounted for, the nodal displacements are found by solving the [system of linear equations](https://bliptext.com/articles/system-of-linear-equations), symbolically:\nSubsequently, the strains and stresses in individual elements may be found as follows:\nwhere\nBy applying the [virtual work](https://bliptext.com/articles/virtual-work) equation to the system, we can establish the element matrices \\mathbf{B}, as well as the technique of assembling the system matrices and \\mathbf{K}.\nOther matrices such as, , \\mathbf{R} and \\mathbf{E} are known values and can be directly set up from data input.\n\n# Interpolation or shape functions\nLet \\mathbf{q} be the vector of nodal displacements of a typical element.\nThe displacements at any other point of the element may be found by the use of [interpolation](https://bliptext.com/articles/interpolation) functions as, symbolically:\nwhere\nEquation gives rise to other quantities of great interest:\n

Virtual displacements that are a function of virtual nodal displacements:\n
Strains in the elements that result from displacements of the element's nodes:\nwhere \\mathbf{D} = matrix of [differential operators](https://bliptext.com/articles/strain-displacement-relations) that convert displacements to strains using [linear elasticity](https://bliptext.com/articles/linear-elasticity) theory.\nEq. shows that matrix **B** in is\n
Virtual strains consistent with element's virtual nodal displacements:\n

\n\n# Internal virtual work in a typical element\nFor a typical element of volume V^e, the internal virtual work due to virtual displacements is obtained by substitution of and into :\n\n## Element matrices\nPrimarily for the convenience of reference, the following matrices pertaining to a typical elements may now be defined:\nThese matrices are usually evaluated numerically using [Gaussian quadrature](https://bliptext.com/articles/gaussian-quadrature) for [numerical integration](https://bliptext.com/articles/numerical-integration).\nTheir use simplifies to the following:\n\n## Element virtual work in terms of system nodal displacements\nSince the nodal displacement vecto****r**** ****q**** is a subset of the system nodal displacements ****r**** (fo****r**** compatibility with adjacent elements), we can ****r****eplace ****q**** with ****r**** by expanding the size of the element mat****r****ices with new columns and ****r****ows of ze****r****os:\nwhere, for simplicity, we use the same symbols for the element matrices, which now have expanded size as well as suitably rearranged rows and columns.\n\n# System virtual work\nSumming the internal virtual work for all elements gives the right-hand-side of :\nConsidering now the left-hand-side of, the system external virtual work consists of:\n

The work done by the nodal forces **R**:\n
The work done by external forces on the part of the elements' edges or surfaces, and by the body forces\nSubstitution of gives:\nor\nwhere we have introduced additional element's matrices defined below:\nAgain, [numerical integration](https://bliptext.com/articles/numerical-integration) is convenient for their evaluation.\nA simila**r** **r**eplacement of **q** in with **r** gives, afte**r** **r**ea**r****r**anging and expanding the vecto**r**s :\n

\n\n# Assembly of system matrices\nAdding, and equating the sum to gives:\nSince the virtual displacements are arbitrary, the preceding equality reduces to:\nComparison with shows that:\nIn practice, the element matrices are neither expanded nor rearranged.\nInstead, the system stiffness matrix \\mathbf{K} is assembled by adding individual coefficients {k}_{ij}^e to {K}_{kl} where the subscripts ij, kl mean that the element's nodal displacements match respectively with the system's nodal displacements.\nSimilarly, is assembled by adding individual coefficients {Q}_{i}^e to {R}^o_{k} where {q}_{i}^e matches {r}_{k}.\nThis direct addition of {k}_{ij}^e into {K}_{kl} gives the procedure the name [Direct Stiffness Method](https://bliptext.com/articles/direct-stiffness-method).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T06:57:26.146Z","updated_at":"2025-01-26T06:57:26.146Z","search_vector":"'element':2 'finit':1 'mechan':6 'method':3 'structur':5","search_content":" \n \n\n# \nThe **[finite element method](https://bliptext.com/articles/finite-element-method)** (FEM) is a powerful technique originally developed for numerical solution of complex problems in [structural mechanics](https://bliptext.com/articles/structural-mechanics), and it remains the method of choice for complex systems.\nIn the FEM, the structural system is modeled by a set of appropriate **finite elements** interconnected at discrete points called nodes.\nElements may have physical properties such as thickness, [coefficient of thermal expansion](https://bliptext.com/articles/coefficient-of-thermal-expansion), [density](https://bliptext.com/articles/density), [Young's modulus](https://bliptext.com/articles/young-s-modulus), [shear modulus](https://bliptext.com/articles/shear-modulus) and [Poisson's ratio](https://bliptext.com/articles/poisson-s-ratio).\n\n# History\nThe origin of finite method can be traced to the matrix analysis of structures where the concept of a displacement or stiffness matrix approach was introduced.\nFinite element concepts were developed based on engineering methods in 1950s.\nThe finite element method obtained its real impetus in the 1960s and 1970s by [John Argyris](https://bliptext.com/articles/john-argyris), and co-workers; at the [University of Stuttgart](https://bliptext.com/articles/university-of-stuttgart), by [Ray W. Clough](https://bliptext.com/articles/ray-w-clough); at the [University of California, Berkeley](https://bliptext.com/articles/university-of-california-berkeley), by [Olgierd Zienkiewicz](https://bliptext.com/articles/olgierd-zienkiewicz), and co-workers [Ernest Hinton](https://bliptext.com/articles/ernest-hinton), [Bruce Irons](https://bliptext.com/articles/bruce-irons-engineer); at the [University of Swansea](https://bliptext.com/articles/swansea-university), by [Philippe G. Ciarlet](https://bliptext.com/articles/philippe-g-ciarlet); at the [University of Paris](https://bliptext.com/articles/pierre-and-marie-curie-university); at [Cornell University](https://bliptext.com/articles/cornell-university), by Richard Gallagher and co-workers.\nThe original works such as those by Argyris and Clough became the foundation for today’s finite element structural analysis methods.\nStraight or curved one-dimensional elements with physical properties such as axial, bending, and torsional stiffnesses.\nThis type of element is suitable for modeling cables, braces, trusses, beams, stiffeners, grids and frames.\nStraight elements usually have two nodes, one at each end, while curved elements will need at least three nodes including the end-nodes.\nThe elements are positioned at the [centroidal](https://bliptext.com/articles/centroid) axis of the actual members.\n\n## Element interconnection and displacement\nThe elements are interconnected only at the exterior nodes, and altogether they should cover the entire domain as accurately as possible.\nNodes will have nodal [(vector) displacements](https://bliptext.com/articles/displacement-vector) or [degrees of freedom](https://bliptext.com/articles/degrees-of-freedom-engineering) which may include translations, rotations, and for special applications, higher order [derivatives](https://bliptext.com/articles/derivative) of displacements.\nWhen the nodes displace, they will drag the elements along in a certain manner dictated by the element formulation.\nIn other words, displacements of any points in the element will be [interpolated](https://bliptext.com/articles/interpolation) from the nodal displacements, and this is the main reason for the approximate nature of the solution.\n\n# Practical considerations\nFrom the application point of view, it is important to model the system such that:\nLarge scale commercial software packages often provide facilities for generating the mesh, and the graphical display of input and output, which greatly facilitate the verification of both input data and interpretation of the results.\n\n# Theoretical overview of FEM-Displacement Formulation: From elements, to system, to solution\nWhile the theory of FEM can be presented in different perspectives or emphases, its development for [structural analysis](https://bliptext.com/articles/structural-analysis) follows the more traditional approach via the [virtual work](https://bliptext.com/articles/virtual-work) principle or the [minimum total potential energy principle](https://bliptext.com/articles/minimum-total-potential-energy-principle).\nThe [virtual work](https://bliptext.com/articles/virtual-work) principle approach is more general as it is applicable to both linear and non-linear material behaviors.\nThe virtual work method is an expression of [conservation of energy](https://bliptext.com/articles/conservation-of-energy): for conservative systems, the work added to the system by a set of applied forces is equal to the energy stored in the system in the form of strain energy of the structure's components.\nThe principle of [virtual displacements](https://bliptext.com/articles/virtual-work) for the structural system expresses the mathematical identity of external and internal virtual work:\nIn other words, the summation of the work done 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.\nThe virtual internal work in the right-hand-side of the above equation may be found by summing the virtual work done on the individual elements.\nThe latter requires that force-displacement functions be used that describe the response for each individual element.\nHence, the displacement of the structure is described by the response of individual (discrete) elements collectively.\nThe equations are written only for the small domain of individual elements of the structure rather than a single equation that describes the response of the system as a whole (a continuum).\nThe latter would result in an intractable problem, hence the utility of the finite element method.\nAs shown in the subsequent sections, Eq. leads to the following governing equilibrium equation for the system:\nwhere\nOnce the supports' constraints are accounted for, the nodal displacements are found by solving the [system of linear equations](https://bliptext.com/articles/system-of-linear-equations), symbolically:\nSubsequently, the strains and stresses in individual elements may be found as follows:\nwhere\nBy applying the [virtual work](https://bliptext.com/articles/virtual-work) equation to the system, we can establish the element matrices \\mathbf{B}, as well as the technique of assembling the system matrices and \\mathbf{K}.\nOther matrices such as, , \\mathbf{R} and \\mathbf{E} are known values and can be directly set up from data input.\n\n# Interpolation or shape functions\nLet \\mathbf{q} be the vector of nodal displacements of a typical element.\nThe displacements at any other point of the element may be found by the use of [interpolation](https://bliptext.com/articles/interpolation) functions as, symbolically:\nwhere\nEquation gives rise to other quantities of great interest:\n \n Virtual displacements that are a function of virtual nodal displacements:\n \n Strains in the elements that result from displacements of the element's nodes:\nwhere \\mathbf{D} = matrix of [differential operators](https://bliptext.com/articles/strain-displacement-relations) that convert displacements to strains using [linear elasticity](https://bliptext.com/articles/linear-elasticity) theory.\nEq. shows that matrix **B** in is\n \n Virtual strains consistent with element's virtual nodal displacements:\n \n \n\n# Internal virtual work in a typical element\nFor a typical element of volume V^e, the internal virtual work due to virtual displacements is obtained by substitution of and into :\n\n## Element matrices\nPrimarily for the convenience of reference, the following matrices pertaining to a typical elements may now be defined:\nThese matrices are usually evaluated numerically using [Gaussian quadrature](https://bliptext.com/articles/gaussian-quadrature) for [numerical integration](https://bliptext.com/articles/numerical-integration).\nTheir use simplifies to the following:\n\n## Element virtual work in terms of system nodal displacements\nSince the nodal displacement vecto****r**** ****q**** is a subset of the system nodal displacements ****r**** (fo****r**** compatibility with adjacent elements), we can ****r****eplace ****q**** with ****r**** by expanding the size of the element mat****r****ices with new columns and ****r****ows of ze****r****os:\nwhere, for simplicity, we use the same symbols for the element matrices, which now have expanded size as well as suitably rearranged rows and columns.\n\n# System virtual work\nSumming the internal virtual work for all elements gives the right-hand-side of :\nConsidering now the left-hand-side of, the system external virtual work consists of:\n \n The work done by the nodal forces **R**:\n \n The work done by external forces on the part of the elements' edges or surfaces, and by the body forces\nSubstitution of gives:\nor\nwhere we have introduced additional element's matrices defined below:\nAgain, [numerical integration](https://bliptext.com/articles/numerical-integration) is convenient for their evaluation.\nA simila**r** **r**eplacement of **q** in with **r** gives, afte**r** **r**ea**r****r**anging and expanding the vecto**r**s :\n \n \n\n# Assembly of system matrices\nAdding, and equating the sum to gives:\nSince the virtual displacements are arbitrary, the preceding equality reduces to:\nComparison with shows that:\nIn practice, the element matrices are neither expanded nor rearranged.\nInstead, the system stiffness matrix \\mathbf{K} is assembled by adding individual coefficients {k}_{ij}^e to {K}_{kl} where the subscripts ij, kl mean that the element's nodal displacements match respectively with the system's nodal displacements.\nSimilarly, is assembled by adding individual coefficients {Q}_{i}^e to {R}^o_{k} where {q}_{i}^e matches {r}_{k}.\nThis direct addition of {k}_{ij}^e into {K}_{kl} gives the procedure the name [Direct Stiffness Method](https://bliptext.com/articles/direct-stiffness-method).","revisionCount":0},"hasMore":false,"nextCursor":null}