SCAD is based on the finite element method (FEM) in the form of slope deflection method. This form has been selected due to its simple algorithmic implementation and physical interpretation; possibility of creating unified methods for the construction of stiffness matrices and load vectors for various types of finite elements; possibility of taking into account any types of boundary conditions and complex geometry of the designed structure. Detailed descriptions of the method with thorough proofs can be found in a lot of books (see for example, [6, 15]). This section only briefly covers the basic design relationships.
The stress and strain state of each material point x of a finite element with the volume V and the surface area S can be described by the vectors of stress σ(x) and strain ε(x). In the theory of linear elasticity the latter are expressed through the vector of displacements u(x) as follows:
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σ = Mε, ε = Bu, |
(1) |
where B is a linear matrix differential operator; M is a symmetric positive definite elasticity matrix of Hooke’s law that depends only on stiffness properties of the structural material.
The full potential energy of the element is determined by the following formula:
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\[ \Pi^{(e)}=\frac{1}{2}\int\limits_v {{\rm {\bf \varepsilon }}^{T}{\rm {\bf M\varepsilon }} dV} -\int\limits_v {{\rm {\bf u}}^{T}{\rm {\bf p}}dV} -\int\limits_s {{\rm {\bf u}}^{T}{\rm {\bf q}}\,dS} , \] |
(2) |
where p and q are vectors of volume and surface forces, respectively.
The displacements u(х) of any point of the considered element can be approximately expressed through the unknown nodal displacements Z:
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\[ {\rm {\bf u}}(x)=\sum\limits_i {\phi_{i} \mbox{(}x\mbox{)}\,{\rm {\bf Z}}_{i} } ={\rm {\bf \Phi }}(x)\,{\rm {\bf Z}}_{e} \] |
(3) |
where φi(x) are the interpolation functions usually referred to as shape functions, which should meet the smoothness requirements in order to ensure the convergence of the method [15]; Φ(x) is the matrix of interpolation functions; Ze is the vector of all unknown nodal displacements of the considered element (index “e”).
Substituting (1) and (3) to (2) we obtain
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\[ \Pi^{\mbox{(e)}} =1/2 {\rm {\bf Z}}_{e}^{T} \left( {\int\limits_v {({\rm {\bf B\Phi }})^{T}{\rm {\bf MB\Phi }}\,\,dV} } \right){\rm {\bf Z}}_{e} -\left( {\int\limits_v {{\rm {\bf p}}^{T}{\rm {\bf \Phi }}\,\,dV+\int\limits_s {{\rm {\bf q}}^{T}{\rm {\bf \Phi }}\,\,dS} } } \right){\rm {\bf Z}}_{e} \] |
(4) |
The expression (4) can be represented in the following form:
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П(е) = 1/2 ZeТK(е)Ze-feТZe, |
(5) |
where \( {\rm {\bf K}}_{(e)} \mbox{ = }\int\limits_v {({\rm {\bf B\Phi }})^{T}} {\rm {\bf MB\Phi }}dV \) is the stiffness matrix of the element;
\( {\rm {\bf f}}_{e}^{T} \mbox{=}\int\limits_v {{\rm {\bf p}}^{T}} {\rm {\bf \Phi }}dV\mbox{+}\int\limits_{\mbox{s}} {{\rm {\bf q}}^{T}} {\rm {\bf \Phi }}dS \) is the vector of reduced nodal forces.
The full potential energy of a system is a sum of the potential energies of all its elements:
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\[ \Pi \mbox{=}\sum\limits_e {\Pi^{\mbox{(e)}}} \] |
(6) |
and its minimization produces the system of governing equations of FEM:
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KZ = f |
(7) |
with the global stiffness matrix K and the vector of nodal forces f obtained by summing the respective members of the stiffness matrices K(e) and the vectors f(e) of particular finite elements, which is an advantage of the considered approach.
There are known conditions of convergence and error estimates for FEM in the form of slope deflection method. The conditions of convergence include the linear independence and the completeness of the system of basis functions, as well as their consistency (conformity) or other conditions compensating the inconsistency.
There are also some easily verified conditions for establishing the completeness of the basis functions, their consistency, or whether the conditions compensating the inconsistency are satisfied. In the case of polynomial approximations, these conditions come in the form of equalities which must be satisfied by the basis functions at each finite element. This theoretical basis enables us to check the correctness of the application of the known finite elements, to develop the principles for constructing new consistent and inconsistent elements and to obtain their error estimates as well.
SCAD finite element library contains a lot of elements for modeling the behavior of various types of structures. There are well-known bar elements, triangular and quadrangular elements for the analysis of the plane stress state, plates, shells, 3D elements such as tetrahedron, parallelepiped, and trihedral prism. The library includes the following elements: inconsistent triangular and rectangular elements of isotropic and orthotropic plates and shells, plates on the elastic subgrade, laminated plates and shells; consistent triangular and quadrangular elements for the analysis of the plane stress state, plates and shells with optional nodes on their sides, all constructed by the sub-area method. These elements are based on the elements for the analysis of the plane stress state with two degrees of freedom in a node, and plates with three degrees of freedom.
The library contains isoparametric elements for the analysis of the plane stress state and for solving spatial elastic problems; one- and two-dimensional (quadrangular and triangular) axisymmetric elements. Moreover, there are various special elements for modeling the constraints of finite rigidity, elastic compliance between nodes, null-elements of various types, elements specified by a numerical stiffness matrix. All finite elements contained in the library are theoretically justified and have error estimates of energy and displacements. The integral force error can be estimated by a value proportional to hτ, where h is the maximum size of finite elements, τ = 2 for rectangular plate elements, τ = 1 for all other elements. The displacement error can be estimated by a value proportional to hτ where τ = 4 for consistent rectangular and quadrangular plate elements, and τ = 2 for all other elements. The possibility of performing the analysis of curvilinear bars with rectilinear elements and of any shells with triangular and rectangular (for cylindrical shells) plane shell elements is also theoretically justified. The energy and displacement error in this case can be estimated by a value proportional to h.