example of qualitative analysis in structural engineering
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| qualitative example |
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Qualitative & Semi-Quantitative Reasoning Techniques |
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A Computer Application to Study Engineering |
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Decision Tools for the Engineering of Steel Structures by M. H. Gedig & S. F. Stiemer |
| about the example | |
This example illustrates some of the practical implications of qualitative analysis, as well as some refinements that may be made. |
The pin-jointed plane structure model shown in Figure 2 will be studied using qualitative analysis techniques. In the figure, the boxed numbers are the element labels and the other numbers are the node labels. The parameters of this model are the lengths (L1, L2, L3, L4) and axial stiffnesses (EA1, EA2, EA3, EA4 ) of each of the four members, as well as the nodal displacements at the upper left node (∆X2, ∆Y2), and the upper right node (∆X3, ∆Y3). The horizontal displacements ∆X2 and ∆X3 are considered positive if to the right, and the vertical displacements ∆Y2 and ∆Y3 are positive if upwards. A horizontal force which acts to the right is applied at the upper left node. One of the uses for such a model would be to estimate the deflected shape of the structure. |
| the problem statement | |
pin-jointed plane structure with horizontal load |
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| the qualitative solution | |
The qualitative equations which relate the nodal displacements to the material properties and geometry of the structure are given as follows: |
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| These qualitative equations are very similar to the usual quantitative equations. The difference is that positive numeric constants have been omitted. Positive constants may be eliminated from products, because the positive sign value acts as the identity for multiplication in the domain of signs. | |
flaws? |
Qualitative analysis produces four solutions to the problem, the last of which is the correct solution. The result of this analysis is a typical result in qualitative analysis: multiple solutions are generated but the ‘correct’ solution to the problem is always contained in the set of solutions. The other solutions are not, strictly speaking, incorrect, because they are a correct solutions to the qualitative model. A qualitative model is usually a generalization of an associated quantitative model. The process of generalizing a quantitative model allows solutions which do not satisfy the quantitative equations. One of the prime sources of weakness in qualitative predictions derives from the weakness of the qualitative addition function. The qualitative addition function introduces uncertainty which tends to propagate through systems of qualitative equations. Looking at equation (6), the assignment ∆X2 = -, ∆X3 = - causes the term (∆X2 - ∆X3) to evaluate to ?. Since the sign ? represents the interval [-∞, +∞], this combination of assignments satisfies equation (6), even though such values would not satisfy the corresponding qualitative equation. |
four solutions |
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| One way of strengthening the conclusions drawn by qualitative analysis is to make some use of ordinal relations between qualitative variables. Ordinal relations may be used to reduce the uncertainty caused by the qualitative addition operator. Whenever qualitative addition results in the sign ?, three new cases may be generated to reflect different possible ordinal relations between the arguments involved in the addition operation. | |
ordinal relations
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gives us better understanding |
Since qualitative analysis is troubled by combinatorial explosion, at first glance it may seem that considering ordinal relations between variables would exacerbate this problem. Without including ordinal relations, the number of combinations of two variables, each having three values, is nine, while there are thirteen possible combinations when ordinal relations are used. The advantage in using ordinal relations is that the relations provide additional information that may be used to filter inconsistent variable assignments. The ordinal relations are assertions about the relationship between two variables. This assertion may cause a contradiction at some stage of the solution process, which allows us to delete elements from the partial solution. In addition to information about the values of the variables, a solution obtained using ordinal relations provides insight into the relationship between the magnitudes of variables. The analysis yields the following additional information:
These inferred ordinal relations enable us to generate a fairly concise qualitative graphical representation of the behaviour of the structural model, as shown in below: |
graphical representation of final result |
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| findings | |
| In this example, qualitative analysis leads to a fairly clean, informative solution, considering the minimal amount of specific information that was furnished as input. It is important to note that, in general, a qualitative analysis produces more than one solution. Increasing the number of variables in a problem leads to a rapid increase in the number of combinations which must be considered, and also, an increase in the number of solutions. | |
| conclusion | |
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In the previous example from structural analysis, very little information was specified about the various quantities in the problems. In most problems in engineering, partial knowledge about quantities takes the form of partial numerical values. For example, the value of Young’s Modulus E was only specified as lying in the interval (0, +∞]. In reality, we may know that the structure is to be constructed of timber, so that the value of E ranges, say, from 6000 MPa for sawn timber to 14000 MPa for glued-laminated timber. Even if we have almost no idea of which material is to be used, a wide interval which includes values of E for timber, concrete and steel will provide some information. In this general case, E would range from about 6000 MPa for timber to 200000 MPa for steel, so that the range of E could be represented as the interval [6000, 200000] MPa. This observation suggests that, for an engineering application at least, it is more beneficial to pursue a formulation involving partial numeric information, rather than to seek refinements to the pure qualitative representation. |
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sponsor: |
NSERC |
conceptual design using qualitative analysis methods |
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Michael Gedig |
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