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Introduction to the analysis of statically indeterminate structures
Published in A. Ghali, A. M. Neville, Structural Analysis: A unified classical and matrix approach, 2017
Two general methods of analysis of structures are available. One is the force (or flexibility) method, in which releases are introduced to render the structure statically determinate; the resulting displacements are computed and the inconsistencies in displacements are corrected by the application of additional forces in the direction of the releases. Hence, a set of compatibility equations is obtained: its solution gives the unknown forces.
Shear and Torsion
Published in L.H. Martin, J.A. Purkiss, Concrete Design to EN 1992, 2005
The elastic analysis of a hyperstatic, or statically indeterminate, structure involves the stiffness of the members. For a concrete member the bending stiffness is E/lL and the torsional stiffness is GC/L where G = elastic shear modulus and is equal to 0,42EL = length of a member between joint centresC = torsional constant equal to half the St. Venant value for plain concrete.
Fundamentals of structural analysis
Published in Eugene OBrien, Andrew Dixon, Emma Sheils, Reinforced and Prestressed Concrete Design to EC2, 2001
Eugene OBrien, Andrew Dixon, Emma Sheils
For the methods of analysis presented, a distinction is made between statically determinate and indeterminate structures. Statically determinate structures can be analysed solely by consideration of equilibrium. Statically indeterminate structures, on the other hand, require further information for their analysis and the solution process is more complex than for determinate structures. A distinction is also made between linear elastic and non-linear methods. Linear elastic methods of analysis are based on the assumption that deformation is proportional to the applied load (linear) and that deformation will go if the load is removed (elastic). Linear elastic methods are very important for studying the performance of structures under relatively small loads – in practice, serviceability limit state (SLS) loads. Non-linear methods of analysis, on the other hand, consider the performance of structures in which some of the members have yielded. In theory, therefore, we would expect people to use non-linear methods to analyse structures at the ultimate limit state (ULS). However, in practice, linear elastic methods are sometimes used for ULS as well as SLS conditions.
Exact closed-form solutions for nonlocal beams with loading discontinuities
Published in Mechanics of Advanced Materials and Structures, 2022
Andrea Caporale, Hossein Darban, Raimondo Luciano
Next application is a statically indeterminate nonlocal beam of length subject to a point force concentrated at the middle of the beam, as shown in Figure 10. Left end of the beam is clamped and right end is simply supported. This nanostructure is denoted with CS-F. The exact solution is obtained using the procedure described in Section 3. In Figure 11, the exact dimensionless transverse displacement is plotted against for different values of and for the beam characterized by Also for statically indeterminate structure, the adopted stress-driven model involves a stiffening behavior with the dimensionless displacement decreasing with increasing the scale parameter
An extension of Queiroz and Miyazawa's method for vertical stability in two-dimensional packing problems to deal with horizontal stability
Published in Engineering Optimization, 2019
Thiago A. Queiroz, Evandro C. Bracht, Flávio K. Miyazawa, Marco L. Bittencourt
In case (1) of transference of forces, the vertical resultant force of i is totally transferred to its adjacent item at i centre of mass. In cases (2) and (3), the vertical resultant force of i needs to be divided between/among its adjacent items in . For this, Queiroz and Miyazawa (2014) assumed that i is the structural element called beam, such that the adjacent items of i are assumed to be columns that support i. Then, reaction forces appear from the interaction between the beam and the columns that provide support to it, where the reaction forces represent the forces where i transfers to its adjacent items. In case of two supports (two adjacent items), the reaction forces and are directly obtained from the static equilibrium equations in (1). However, for case (3) (with three or more adjacent items), these equations are not enough to obtain all the reaction forces because there are more variables (reaction forces) than equations. Therefore, the system of forces is statically indeterminate (hyperstatic system), and then the three-moment equation method is used. This method is based on the Clapeyron's theorem of the three moments, which relates the bending moments on three consecutive supports of a horizontal beam. A complete review of the three-moment equation method is given in Gavin (2009).
From initial localized failures to collapse of structures in a probabilistic context
Published in European Journal of Environmental and Civil Engineering, 2018
El Hadji Boubacar Seck, Sophie Ortola, Luc Davenne
In this paper we present a probabilistic approach to characterize the different possibilities of progressive collapse of a structure. The suggested methodology is based on the construction of an event tree, considering all the potential initial damage events and their numerous combinations. Also, the resulting event tree contains exclusive paths, without intersection between branches stemming from the same node. It makes it possible to evaluate the robustness indexes of the structure according to each scenario and to identify the structural vulnerable zones that can cause a generalized collapse. The developed methodology is illustrated in the case of a clamped–clamped statically indeterminate beam. This simple structure allows setting up the used concepts and showing that not intuitive results can be highlighted with the developed principle. Moreover, since the proposed procedure of probability evaluation is analytical, the corresponding computations are faster than the usual Monte-Carlo simulations.