#### Installation — business terrible - 1 part

September 8th, 2015

Civil Eng. Dept., Faculty of Eng., TarbiatModarres University Tehran, P. O. Box 14155-4838, Iran,

E-mail: hamid@modares. ac. ir and hamid. moharrami@gmail. com

Semirigid connections show nonlinear behavior even due to small loadings. Therefore linear analysis is not a proper solution algorithm for structures that have such connections; rather a nonlinear analysis should be done. The conventional methods of nonlinear analysis of frames are inherently iterative, and their final results include some small order of approximation. They usually are done through modification of the stiffness matrix of structure and/or load vector.

In this paper, a new method of nonlinear analysis has been presented that contrary to iterative methods, it is non-iterative. It does the analysis in one step without change in the initial model and stiffness matrix of the structure or its load vector. Theoretically it does not include approximation and gives exact results. In this method to force internal moments follow their nonlinear moment-rotation curves, some virtual moments (that are primarily unknown) are imposed to the structure at semirigid connections. To find the unknown virtual moments, a quadratic programming problem is formulated and solved. After finding the values of virtual moments, employing superposition principle, exact nonlinear response of structure is obtained and internal forces and moments of members are calculated.

The method is capable to model semirigid connections with multilinear moment-curvature relations. The formulation of the problem for bilinear and trilinear moment-curvature relations has been presented here. Two examples are presented to demonstrate the robustness, capability and validity of the method.

Keywords: Semirigid connection, multilinear moment-rotation relation, nonlinear analysis, mathematical programming.

In general connection of a beam to column can be categorized in three groups. The first group are rigid connections in which, theoretically saying, the angle between the two connected members does not change due to applied moments. The second group are hinged connections in which the connected members can have relative rotation without any resistance. In reality there is neither solid rigid connection, nor theoretical hinge connection, i. e. every rigid connection admits some rotation and every hinge connection tolerates some moment. Semi-rigid connections that have situation between the two groups constitute the third group. Their characteristics are determined by moment rotation relations that are usually nonlinear. To simplify nonlinear analysis, nonlinearity of materials is usually modelled as multilinear relations between stress and strain, the first part of which characterizes linear relations. The most simplified nonlinear relation is the bilinear elastic-plastic relation in which there is an elastic relation between stress and strain up to yield point.

If the structure is stressed up to this stress limit a linear analysis is sufficient for stress-strain calculations. However for further stress or strain a nonlinear analysis is necessary.

Literature is almost mature of nonlinear analysis methods. Crisfield (1991) and Owen and Hinton

(1980) have cited good summaries of classical nonlinear analysis techniques. Among the major

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M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 329-342.

© 2006 Springer. Printed in the Netherlands.

nonlinear analysis techniques the Incremental Scheme, Initial Stiffness method, Newton-Raphson method and combination of these methods can be mentioned

There are also some other techniques, that have been established for inelastic analysis of structures based on theorems of Structural Variation. Structural variation theory studies the effect of change of properties, or even removal, of a member on the entire structure. It takes advantage of linear analysis and sensitivity of structure to some self equilibrating unit loads that are applied at the end nodes of changing members. This technique has been applied to analysis of several types of inelastic skeletal structures including space trusses (Saka & Celik 1985), frames (Majid & Celik 1985) and grids (Saka 1997), etc. It has been also extended to nonlinear finite elements analysis (Abu Kassim & Topping 1985, and Saka 1991). Although this method takes advantage of initial stiffness matrix and does not require change in the stiffness matrix of structure during the analysis process, it is a historical and step by step method of analysis in which every step uses information from the previous step.

Nonlinear analysis of structures by mathematical programming is another field of research in this ground. De Donato (1977) presented fundamentals of this method for both holonomic (path independent) and nonholonomic material behaviours. In this method, it is assumed that displacement of nodes of an elastic-plastic structure comprises two parts namely elastic and plastic parts. Then, the problem of finding total displacement vector of a structure is formulated in the form of a quadratic programming (QP) problem with some complementarity yield constraints. These yield constraints state that individual members either are stressed within elastic limits and do not accept plastic deformations or they are stressed up to yield limit and, as a result, undergo some plastic deformations. The output of this sub-problem is linear and nonlinear deformation of structure. Despite its robustness, this method suffers from the considerable number of variables that enter in the QP sub-problem.

The goal of this research work is to bypass iterative techniques in the analysis of nonlinear structures and build up a method based on simple equilibrium relations to conduct analysis in one step. The idea of this technique has been initially examined by Moharrami et al (2000) for nonlinear analysis of structures including tension only and compression-only truss-type elements. Here in this paper it is extended to elastic-plastic flexural connections. This method holds simplicity of structural variation theorem, advantages and robustness of mathematical programming and precision of the results with less computation effort.