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Mr. A. Jennings, Lecturer in the Civil Engineering Department of the University of Manchester, writes: ‘The standard methods of solution of linear redundant structures are all formulated in terms either of the unknown forces or of the unknown displacements and use the flexibility or stiffness properties of the structure to effect a solution. The method of solution of free cable problems advocated in this paper is peculiar in that it does not formulate the analysis directly in terms of either the unknown displacements or the unknown forces, and no concept of stiffness or flexibility is introduced.
Mr. R. Park, Lecturer in Civil Engineering at the
University of Bristol, writes:-
‘In Mr. Christiansen’s paper an expression for the arching couple Ca is developed as a function of the central deflexion of the beam and the magnitude of the deflexion which makes Ca a maximum is determined, This maximum value of Ca is taken to be the arching
couple at the ultimate (maximum) load of the beam. It appears that in some cases the central deflexion of the beam at the ultimate load so found is extremely small and Mr. Christiansen points out that in the tests conducted the ultimate load was reached at central deflexions which were larger than those predicted by the theory. This discrepancy in deflexions may result from the assumption that the plastic hinges of the beams are fully developed at all stages. When a beam is loaded to failure it will only develop full plasticity at the critical sections when the deflexion is large enough to cause the required strains. At small deflexions the stresses at the critical sections are less than the plastic values. Hence if the maximum arching couple given by the theory occurs at too small a deflexion it may be outside the range for which the theory is applicable. The theory should overestimate the ultimate load in this case since the beam will reach ultimate load at a higher deflexion and the arching couple will be reduced. It is evident that a method for determining the central deflexion at which full plasticity at the hinges has just developed is required, since only when this deflexion is reached does the theory become applicable. For laterally restrained beams the tension steel reaches yield stress before the concrete reaches its ultimate value. Hence the central plastic deflexion when the last plastic hinge fully develops could be written in terms of the ultimate strain at the compressed edge of the concrete, the length of the region of the hinge, the depth to neutral axis and the position of the hinge in the beam. It is evident, however, that some other features of the theory are conservative since the failing loads found by Mr. Christiansen’s tests exceed the theoretical ultimates in spite of the difference in central deflexions.
‘ It is doubtful whether arching action can be
The method of complex potentials is applied to determine the internal forces in a stretched isotropic, circular ring-shaped plate subjected to tangential forces ôrè = -A sin è at its inner boundary and to an isolated point force at its outer boundary. Because of the double singularities (concentrated force and non-vanishing resultant on each boundary) the problem is solved by the sum of a closed expression and of a complex Fourier series. A numerical example is presented for the calculation of ó è at each boundary.
S. SARKADI SZABO