Sensitivity Calculation
Introduction
Chapter 8 of the book introduces a number of methods and procedures for estimating the impact that uncertainty and error have on the predictions of in-service structural behaviour made by a finite element analysis. In essence these are attempting to estimate by how much the output from the finite element analysis deviates from the measured in-service behaviour due to the presence of error or uncertainty.
The first method presented relies on directly evaluating the sensitivity of the finite element output to changes in material properties; boundary conditions etc. and demonstrates the process using the simple framework example introduced in the framework page. The second employs artificial sensitivities to estimate the impact of uncertainties in the boundary conditions either at the edge of the analysis domain or at interior edges or boundaries within the domain. The purpose of this page is to make available some of the examples which a reader of the book can use to become familiar with the methodology.
Direct Evaluation of Sensitivities
Direct evaluation of sensitivities can be employed in an analysis where the structure behaves according to the linear theory of elasticity and in the absence of complex time dependent histories. It exploits an approach employed by structural optimisation algorithms and allows the analyst to compute the effect of error or uncertainty in analyses involving static, dynamic and buckling problems. The approach is applicable in cases where the errors or uncertainties relate to a single finite element within the total system or to clusters of elements combined into superelements. It is also applicable to cases where substructures are employed in a multi-company environment where several teams are responsbile for the finite element analysis of a large and complex structure. In this situation an analysis team, responsible for a substructure or structural component, may have concerns about the quality of the models provided by other groups. The methodology presented in this section provides techiques that can be used to estimate the impact of potential errors and uncertainties originating in an imported substructure.
Calculation of Sensitvities for Static Analyses
In the case of a structural finite element analysis subjected to static loads only the formulae for computing sensitivities is relatively straight forward as demonstrated in the book. Consider the case where we wish to find the sensitivity of a specific displacement to a change in a property that changes the stiffness properties of a given finite element. This sensitivity can be calculated using the following formula:
where:
duj is the rate of change in displacement 'j' due to changes in the stiffness properties of the element'i'.
vi is the component of the vector of nodal displacement due to the application of a unit load at nodal displacement 'j' that act on the nodes of element 'i'.
dki is the element stiffness matrix for the 'i' element with the terms in that matrix replaced by a set of terms corresponding to the assumed or known values of the changes associated with errors or uncertainties. It is called, in Chapter 8 the 'static modification matrix'.
ui is the component of the vector of displacements due to the application of the real loads that act on the nodes of element 'i'.
Definition of the Static Analysis Problems
In this section the method for generating sensitivities for a structure loaded by static loads only is demonstrated using the examples from chapter 8 of "A Practical Guide to Reliable Finite Element Analysis". The first problem is shown in the figure to the right of this text where it is seen to be a repeat of the frame work example used in the MathCad Examples page. Once more it involves the analysis of a pin-jointed framework modelled by 2-noded bar elements subjected to the two loads shown, fixed to a rigid boundary at the nodes 0, 5 and 8. As before all the bars have the same density and Modulus of Elasticity but these values can be varied so that each element could have different values for these quantities. The problem being demonstrated is that of finding the rate of change of the nodal displacement number 14 for a 10% change in the stiffness value of element number 3.
The second is a beam analysis that exploits the example in the Roark text and is shown diagramatically in mathcad examples page and involves a structure that is encastre at one end and simply supported at the other with a distributed load over part of its length. In this case several elements are subjected to change. Refering to the element layout diagram shown on the right; the sensitvity analysis is undertaken assuming that an uncertainty is present that inputs a variability to the stiffness matrices of elements 0 through 3.
Framework Sensitivities with Deflated Matrix
In this case the boundary conditions are set up on the assumption that the structure is rigidly fixed at specified nodal positions. The data is presented to the MathCad program through a set of vectors and matrices described in the 'Initial Information' block in the program. It should be noted that there are now additional terms to those used in the 'Framework' page that provide the information related to the required sensitvities. The solution process takes into account the boundary conditions by deflating the stiffness matrix by removing appropriate rows and columns from the global stiffness matrix.
In order to see this example click on the 'Sensitivity deflated' symbol below. If you have a version of MathCad working on your computer the example will open and you can then copy the contents of the MathCad file and paste it into you own MathCad folder so that you can modify it and work on it at your leisure to incorporate you own 2-dimensional statically loaded pin-jointed framework If you do not have MathCad available then follow the instructions given in the Introduction section above. Once you have opened the MathCad example you can return to this page by clicking on the back button at the top of the MathCad web page
Framework Sensitivities with the Penalty Method
This example is the same problem as that in the Deflated Stiffness Matrix section above and again the relevant data is presented to the MathCad program through a set of vectors and matrices described in the program. Although the boundary conditions are set-up on the assumption that the structure is attached to a rigid support at specified nodal boundary positions because the solution process employs the penalty method solution procedure these specified displacements need not have zero value. The sensitivities are now calculated for this second solution process using the same formulae as for the 'Deflated Matrix' approach given above.
Both of these sensitivity calculations show that a specified change in the value of the stiffness of element number 1 gives rise to a very small estimated incremental change in the displacement at node 7.
Why not try a simple modification and assess the impact of making a 10% change in the terms of the stiffness matrix for element 4 on displacement (degree of freedom) number 15? In order to undertake this task the MathCad program has to be modified so that the input data correctly identifies that the modification matrix relates to changes in the properties of element 4 and the target displacement is degree of freedom 15.The required sensitivity output data can be generated using either the defalted or penalty solution methods.
Beam Sensitivities
This part of the page calculates the rate of change of the vertical displacment at node 9 of the cantilever beam for a 10% change in the width of the elements 0 through3 i.e. for one half of that part of the beam carrying the distributed load. It illustrates how sensitivies can be calculated when several elements have their values altered. In this case all elements experience the same change but this is obviously not a neccessary requirement. For this example a cut-and-paste approach has been adopted which corresponds to how a practicing engineer might operate. Thus, the modification matrix is constructed using the beam MathCad program displayed in the mathcad examples page with modified values for the element widths and the appropriate components cut from the overall stiffness matrix. Similarly, the displacements at the nodes of elements 0 through 3 for both the specified applied loads and for a unit load applied verically at node 9 are found using the same beam program. These are then pasted into a new MathCad program that can be seen by clicking on the Beam Sensitivity icon below.
The result of perfoming the calculation shown in the program indicates that this 10% change in width generates at change of 0.00002113 metres from the original value of 0.0005022 metres.
The final line in this short MathCad program illustrates the difference between the above estimated value of change in this displacement, for the 10% change, with that obtained by repeating the complete analysis with the modified values directly incoporated in the structural stiffness matrix.
Sensitivities for a Framework Dynamic Analysis
This section follows the approach described in chapter 8 of the book where the sensitivity of natural frequencies and mode shapes to variations in structural stiffness and mass is considered for a structure undergoing free vibration. In this web page the example presented considers the sensitivity of the structure's natural frequency only and does not include any reference to the mode shape sensitivity. The formula which gives the rate of change for the natural frequency 'i' for changes is mass and stiffness is derived in the book and found to be:
where:
uij is the that part of the 'i' modes shape vector operating on the nodes of element 'j'.
dkj and dmj are the element stiffness and mass matrices for the 'j' element with the terms in that matrices replaced by a set of terms corresponding to the assumed or known values of the changes associated with errors or uncertainties. These are called, in Chapter 8, the 'dynamic modification matrix'.
The Dynamic Problem
This dynamic problem employs the same basic framework as the static example above. The task is to find by how much the first natural frequency, computed in the mathcad examples page, changes due to a 10% rise in the values of the of stiffness and mass of element number 0 for the 18-bar frame shown on the right. The numbering of the nodes has been selected to both reduce the bandwidth and to make applying the boundary conditions relatively easy. The bar is fully restrained at the nodes 0, 1 and 2 as shown. As in the static case all the bars in this example have the same density and Modulus of Elasticity but these values can be varied so that each element could have different values for these quantities.
Sensitivity Calculation
The input to the dynamic analysis problem defined in the mathcad examples page has to be changed in order to incorporate the information telling the progam the element in which there is a change in stiffness and/or mass. These are explained in the MathCad example which can be opened by clicking on the Dynamic sensitivity icon below. Having opened the MathCad example a return to this page is achieved by clicking on the back button. Alternatively the reader can copy the example to their own MathCad files so that the input data can be changed to allow different sensitivity studies to be undertaken.
By opening the example the reader will note that the original data defining the analysis problem gave a natural frequency of 438.68 radians/sec and performing the sensitivity study for the 10% change in the designated mass and stiffness matrices resulted in a new natural frequency estimated to be 441.95 radians/sec.
Artificial Sensitivities
The calculation of sensitivities thus far has assumed that all the sources of uncertainty or error relate to structural entities and properties that can be directly related to the properties of the finite elements used in the structural model. But in certain situations this may not be appropriate and an alternative is to have recourse to experimental elements that can adequately represent a structural behaviour or property without attempting to be a true mathematical model. With these elements in place sensitivity studies can be undertaken using either a direct or indirect methodology but the sensitivities obtained are artificial as they have not been found using elements that purport to represent the true mathematical model of the component under examination. For example the analyst may have decided on the region that constitutes the Domain of Analysis but may feel that there is still some uncertainty in the selected outer limit but does not wish to continue adding additional complex finite elements to reanalyse an extended domain. In this case multi-dimensional springs could be used as appropriate experimental elements and distributed along the edge of the Domain of Analysis to permit a rapid assessment of the likely uncertainties associated with the selected domain. This approach is useful when the analysis involves structural components, joints, supports etc where the properties are not known with exactitude. The impact on the qualification parameters can be placed within upper and lower assessment bounds using appropriate experimental elements and rapid re-analyses. Of course, the results emerging from this process might lead to the conclusion that the information on these properties must be known with greater exactness and that this requires additional tests.
In order to illustrate this approach consider the simple problem shown in the figure on the right and take the situation where the support at nodes 1, 6 and 9 are known to have some flexibility in the vertical direction. It is assumed that the analyst wishes to explore the influence of this flexibility on the qualification parameter which is taken to be the displacement of node 7 in the horizontal direction i.e. dof 14. This exploration can be undertaken by adding springs so that the structure is now attached to ground in a flexible manner as shown in the figure. Additional horizontal constraints must be added at nodes 1, 6 and 9 to remove rigid body movement. The indirect method,described in chapter 8 of the book, can now be used to assess the sensitivity of the qualification parameter to variations in the support stiffness. This is done by repeatedly re-analysing the structure using a range of stiffness values that bracket the actual value. These spring stiffness values are derived from estimates made by the analyst.
A specific example where the value for the spring stiffnesses is twice that given to the elements used to model the frame is shown in the MathCad example which can be opened by clicking on the the heading artificial sensitivities below. Different values for the support stiffnesses can be examined by changing the relevant stiffness values in this model as shown in the book. Note, in this case the penalty method solution only is provided.