STRESS SENSITIVITY ANALYSIS OF THE BEAM AND SHELL FINITE ELEMENTS STRESS SENSITIVITY ANALYSIS OF THE BEAM AND SHELL FINITE ELEMENTS

The stress sensitivity analysis in conjunction with finite elements method represents an important tool for the influence analysis of the structural parameters. This analysis plays a significant role in the decision process of the formulation of the structural optimizing or proba-bility analysis. The goal of the paper is to present theoretic and numerical aspects of the beam and shell element stress sensitivity analysis with the respect to the cross section parameters (cross section area, thickness, etc.). The whole computational procedure was inbuilt into Matlab’s software module MATFEM.


Introduction
Nowadays the sensitivity analysis is a significant tool helping to realize a structural parameters influence analysis. This analysis is usually very computer time consuming but the results are very innovative. This process is often applied to a structural analysis, i.e. in stress and strain analysis, modal and spectral or buckling analysis, stochastic analysis and so on [3,6].
Application of the sensitivity analysis is not associated only with the structural optimizing but also with the analysis of the mechanical systems with uncertain parameters, mainly in the usage of so-called perturbation methods based on differentiation of the response with respect to the uncertain system parameters (stiffness, mass, damping, etc.). Implementation of this computational process into the finite element method characterized mainly the era of development of structural optimizing techniques in the eighties.

Fig. 1 Stress distribution in beam
If the bending moment is [1,6] , (2) than the normal stress function is following (3) .
In agreement with points in Fig. 1 it is possible to write the relationship between normal stress in these marginal points of the cross-section and internal elements forces and moments, i.e.
or , The stress sensitivity centre is the derivative (5) with respect to a design variable X i , i.e.
. (6) Using the well-known finite element theory, the internal forces vector f j L in the local coordinate system is given by , where K G is the global stiffness matrix (in the global coordinate system), f G is the external nodal forces vector (in the global coordinate system), T j LG is a transformation matrix between the local and global coordinate systems, T j 01 is a Boolean matrix, i.e. the localization matrix determining the element position in the global stiffness matrix, it means .
Let's now realize the derivation of the internal nodal forces vector (7) with respect to X i : and n i is a number of all the elements containing X i . Further, the gradient of a global vector of the nodal displacements can be following Relationship between u j L and u G will be given by Substituting (13) into (9), we can obtain the derivation of the f j L with respect to X i as follows (14) Finally, after the substituting (14) into (6) we give the gradient of the j-th element stress vector (15) It should be noted that the derivation of the matrix C j with respect to X i , depends on the used cross-section. To realize the derivation means to find the following derivations , and . The last derivation usually equals to zero or is not significant for the sensitivity analysis.
Let the cross-section area be variable X i then other cross-section characteristics will be expressed as follows where parameters a, b, c and exponents p, q, r will be obtained exactly (a simple cross-section) or numerically (a more complicated cross section), using the least squares method. The values of these parameters are presented in Tab. 1. The derivation can be given by (17) where δ ij is Kronecker delta and matrices K j 1,2,3,4 are sub-matrices of the stiffness matrix corresponding axial, bending (about axes y and z) and torsion stiffness of the used cross section (More information about stiffness parameters is in [1,5,6]).
Let's now focus on the derivation . The transformations matrix C j L depends on coordinates of the marginal points 1, 2, 3, 4, which have to be expressed by the variable X i . Considering these conditions we can express matrix C j L for circular section as follows and the analyzed derivation of this matrix gets the following form It is possible to get exactly the previous matrices but, for example, C j L of the I-section has to be analyzed numerically [6].

Example 1
Let us consider the structural sensitivity analysis of normal stresses of the beam element from Fig. 2.

Fig. 2 Simple beam structure with 2 cross section
Applying previous relationships we can obtain the following derivations of matrices C 1 L and C 2 L and finally stress gradients in marginal points of the used crosssections are the following and [MPa/mm 2 ].
The graphic presentation of the stress gradients is in Fig. 3. The presented stress gradient analysis was confronted with a "classical" numerical computational approach (Δσj /ΔX i ) and it's possible to observe the absolute consensus.

Stress sensitivity analysis for a thin shell finite element
The finite element modeling of box, shell or thin-walled structures are usually realized using thin shell finite elements (Kirchhoff's or Mindlin's formulation) [1,4,9]. The stiffness parameters , , ,  depend on material constants and element geometry, mainly on its thickness. Therefore, the thickness t i will be the variable in the following theoretical and numerical stress sensitivity analysis of the shell finite element; the fundamental information about this analysis can be found in [1,4,9].
At first we have to prepare the stress calculation process. This process is based on the expression of the j-th element membrane forces and bending moments (without shear forces) [4,6], i.e. The auxiliary matrices I m and I b can be calculated only using the numerical approach. Further details about E m , E b , D, B m , B b , u el and t are presented in [6]. The extreme stress values can be expected at the top or at the bottom surface. Generally, it means or in compliance with the previous beam element (eq. 5) .
Let's build new material and auxiliary matrices , where the matrix I 3 is the classical unit matrix. Then (22) . A Assuming a relation between the local element displacements u j L and the global displacement vector u G , then (28a,b) may be rewritten as (31a) and (31b) where T LG is a classical transformation matrix between the local and the global coordinate systems, T 01 is again a Boolean matrix, i.e. the localization matrix determining the element position in the global stiffness matrix.
The derivative u with respect to ti may be expressed as (34) or, in more detail, . ( The relation is often zero and the derivative of the all the element components of the stiffness matrix can be realized as follows [6] .
The particular membrane, bending and shear matrices are presented in [1,7].
Finally, the derivative of the von Mises stress (at the top and at the bottom surfaces) with respect to the element thickness t i is the following  The results document the influence of both parameters on the stresses and the major signification of thickness t 1 . This information may be used for the next optimizing process.

Conclusion
The presented work deals with the theoretical aspects and numerical realization of the stress sensitivity analysis of the beam and shell finite elements focused on its cross section parameters (the area in the case of the beam element and the thickness in the case of the thin shell element). The whole computational procedure was inbuilt into Matlab's software module MATFEM. Testing examples support the authors' considerations about effectiveness of the proposed method.