The original Tacoma suspension bridge was completed on 10 June 1940 and opened to traffic on 1 July 1940. The bridge was stable with respect to torsional oscillation until 7 November 1940. That day ...at 10 a.m. the diagonal tie attached to the midspan band of one main cable loosened and the cable began to slip through the band. Just after the loosening of the tie torsional oscillations appeared, lasted for more than 1h, and resulted in the collapse of the center span at 11:10 a.m. In this paper a continuous model of the original Tacoma suspension bridge is proposed. This model describes the mutual interaction of the main cables, central span, and hangers. The reaction of the ties attached to the midspan bands is included in the model, so it is possible to study the situation when only one midspan band loosens. The model is described by a system of variational equations which are derived from the Hamilton variational principle. Three different eigenvalue and eigenvector problems are formulated and analyzed. The problems correspond to the situations when the both midspans are loosened, the both midspan bands are fixed, and one midspan band is fixed and the other is loosened. The analysis of the three eigenvalue and eigenvector problems against flutter is carried out, which reveals possible reasons of the collapse.
A method for the detection of the initial stress tensor is proposed. The method is based on measuring distances between pairs of points located on the wall of underground opening in the excavation ...process. This methods is based on solving twelve auxiliary problems in the theory of elasticity with force boundary conditions, which is done using the least squares method. The optimal location of the pairs of points on the wall of underground openings is studied. The pairs must be located so that the condition number of the least square matrix has the minimal value, which guarantees a reliable estimation of initial stress tensor.
In this paper a dynamic linear model of suspension bridge center spans is formulated and three different ways of fixing the main cables are studied. The model describes vertical and torsional ...oscillations of the deck under the action of lateral wind. The mutual interactions of main cables, center span, and hangers are analyzed. Three variational evolutions are analyzed. The variational equations correspond to the way how the main cables are fixed. The existence, uniqueness, and continuous dependence on data are proved.
Some problems connected with 2D modeling of geosynthetic tubes on rigid foundations are studied. Basic equations are derived and analyzed. The analysis of the equations is based on the implicit ...function theorem. Geosynthetic tubes are made of a special fabric and then filled with water or slurry. After being filled tubes take certain shapes and tensions are induced in the geosynthetic. The modeling is based on the following hypotheses: the problem is two dimensional; the geosynthetic is flexible, inextensible and has negligible weight; the foundation is rigid; and there is no friction between the foundation and the geosynthetic.
Nonlinear models of suspension bridges MALIK, Josef
Journal of mathematical analysis and applications,
09/2006, Letnik:
321, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Nonlinear variational equations describing one type of suspension bridges are proposed and studied. The variational equations describe the behaviour of road bed, main cables and cable stays. The road ...bed is described by two functions connected with vertical and horizontal deformation of any cross section. The main cable is considered to be perfectly flexible and inextensible. The cable stays only resist tensile forces. The variational equations are derived from the principle of minimum potential energy. The existence of solution is based on the Brouwer Fixed Point Theorem. The local uniqueness and continuous dependence on the data represented by gravitational forces acting on the road bed are studied. The local results are based on the Implicit Function Theorem for Banach spaces. A certain stability criterion for suspension bridges is formulated and this criterion indicates how to influence the stability of suspension bridges.
This paper generalizes some results established in J. Malík, Nonlinear models of suspension bridges, J. Math. Anal. Appl., in press. The geometric nonlinearity connected with the torsion of a road ...bed is included in the generalized model. The basic variational equations are derived from the principle of minimum energy. The existence of a solution to the generalized problem is proved. The existence is based on the Brouwer fixed-point theorem.
Generalized G-convergence for a quasilinear elliptic differential equation is defined and studied. The equation describes heat conduction in the cores of large electric transformers. The coefficients ...of the equation depend on temperature and the corresponding differential operator is neither potential nor monotone. A theory which generalizes the classical G-convergence is proposed. The theory is applied to the homogenization of the quasilinear elliptic differential equation with periodic coefficients.
Background
Carotid endarterectomy (CEA) is accepted as a primary modality to treat carotid stenosis. The accuracy of measuring carotid stenosis is important for indication of the CEA procedure. ...Different diagnostic tools have been developed and used in the past 2 decades for the diagnosis of carotid stenosis. Only a few studies, however, have focused on the comparison of different diagnostic tools to histological findings of carotid plaque.
Method
Patients with internal carotid artery (ICA) stenosis were investigated primarily by computed tomography angiography (CTA). Digital subtraction angiography (DSA), Doppler ultrasonography (DUS) and magnetic resonance angiography (MRA) were performed as well. Atherosclerotic plaque specimens were transversally cut into smaller segments and histologically processed. The slides were scanned and specimens showing maximal stenosis were determined; the minimal diameter and the diameter of the whole plaque were measured. High quality histological specimen and histological measurement was considered to be the prerequisite for inclusion into the analysis. The preoperative findings were compared with histological measurement.
CTA and histological measurements were obtained from 152 patients. DSA measurements were available in 138 of these cases, MRA in 107 and DUS in 88. A comparison between preoperative and histological findings was performed. In addition, correlation coefficients were computed and tested.
Results
A significant correlation was found for each of the diagnostic procedures. The strongest correlation coefficient and the best allocation of stenosis into clinical significant groups (<50 %, 50–69 %, ≥70 %) was observed for CTA. Mean differences in the whole cohort between preoperative and histological measurements were as follows: CTA underestimated histological measurement by 2.4 % (based on European Carotid Surgery Trial ECST methodology) and 11.9 % (based on North American Symptomatic Carotid Endarterectomy Trial NASCET methodology). DSA underestimated the histological measurement by 7 % (ECST) and 12.2 % (NASCET). MRA overestimated the histological measurement by 2.6 % (ECST) and underestimated by 0.6 % (NASCET). DUS overestimated the stenosis by 1.8 %.
Conclusions
CTA yields the best accuracy in detection of carotid stenosis, provided that all axial slices of the stenosis are checked and carefully analysed. DSA underestimates moderate and mild ICA stenosis, whereas DUS overestimates high-grade ICA stenosis. For MRA, a relatively low correlation coefficient was observed with histological findings. We conclude that CTA-ecst technique is the most reliable technique for carotid stenosis measurement.
The
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then we obtain the List
k
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is
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does not contain
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as an induced subgraph. We continue an extensive study into the complexity of these two problems for
H
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+
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is the disjoint union of the
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We prove that List 3-Colouring is polynomial-time solvable for
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H
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up to seven vertices.
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