The non-linear analysis of frames composed of semi-rigidly connected members with rigid end sections considering shear deformations

In present study, the geometrical nonlinear analysis of frames with linear prismatic members, which may have rigid end sections and rotational semi-rigid end connections, has been carried out taking into account the effects of shear deformations. In the analysis and design of reinforced precast concrete frames, and steel frames the real behaviour of beam-to-column connections are generally idealized either pinned or fully rigid. The notions of the two alternatives either pinned or fully rigid connections are simply the extreme cases of the actual connection behaviour, and experimental works up to date, many of which are referred to in section of previous studies of this paper show without doubt that true connections exhibit characteristics over a wide spectrum between these two extreme cases. The rigid connection idealization indicates that relative rotation of the connection does not exist and the end moment of the beam is entirely transferred to the columns. In contrast to the rigid connection assumption, the pinned connection idealization indicates that any restraint does exist for rotation of the connection and the connection moment is zero. Although these idealizations simplify the analysis and design process, the predicted response of the frame may be different from its real behaviour. Most connection normally regarded as fully rigid often display some flexural flexibility, while connections which are regarded as pinned, implying no moment transfer, posses some flexural stiffness. It would therefore seem more correct that to classify all the behaviour of beam-tocolumn connections in reinforced precast concrete frames, and steel frames under the heading of semirigid connections, while keeping in mind that pinned and fully rigid connections as extreme cases. In this study, the nonlinear analysis of planar frames with semi-rigid connections and rigid end sections is performed. Modified stiffness coefficients in the shear wall axis of the beam connected to the shear wall are obtained with assumption that the rigid strain would be at the part between the elastic beam end and the shear wall axis. The matrix stiffness method of analysis was modified to allow for flexible connections and rigid end parts. The stiffness matrices are modified by correction matrices and the resulting linear equations are solved for as in the normal stiffness method. The advantages of the matrix stiffness method are that it can be programmed so that relatively large frames can be analyzed with ease and that the use of interactive techniques would permit the inclusion of improvements in the end restraint representation. The current paper considers the nonlinear structural analysis of planar frames with flexibly connected members having rigid end sections taking into consideration the effect of shear deformations and a computer program has been prepared for the pertinent design purposes. To accomplish the foregoing goal, first the stiffness coefficients of the stiffness matrix of linear prismatic members has been found using pertinent differential equations assuming there are rigid sections of arbitrary lengths and rotational beam-to-column connections at the both ends. This kind of members, which may have rigid end sections, show up when there are lintel beams connecting shear walls and column connecting high beams. Then, the stiffness coefficients of the stiffness matrix of beams itself have been obtained, taking shear deformations into consideration. Finally, the fixed end forces have been found for a uniformly distributed load, a concentrated load, a linearly distributed load, a symmetric trapezoidal distributed load and a nonsymmetrical triangular distributed load. A computer program has been prepared for the numerical applications of the analytical results. This program carries out the second order structural analysis of planar frames and pierced shear walls using modified stiffness matrix method, taking shear deformations into considerations, not only the effects of bending and axial deformations, but also due to shear, as well. Solving some problems in different ways, the validity of the pertinent computer program has been proved by the close match of the results. Some examples from the literature have been treated by the present method and a perfect match has been observed between the corresponding numerical results. Bu çalışmada, kayma deformasyonlarının etkisi de göz önüne alınarak sonsuz rijit kısımları bulunan elastik bağlı çubuklardan oluşan düzlemsel çerçevelerin doğrusal olmayan analizi yapılmış ve bu konuda bir bilgisayar programı geliştirilmiştir. Önce, her iki ucunda rijit bölgeler ve dönel yaylar bulunan bir çubuk için rijitlik katsayıları elastik bölgesindekiler cinsinden elde edilmiştir. Daha sonra, ikinci mertebe teorisi kullanılarak ve kayma deformasyonları hesaba katılarak rijitlik katsayıları çıkarılmıştır. En sonunda, doğrusal prizmatik bir çubuğun üniform yayılı yük, tekil yük, doğrusal yayılı yük, simetrik yamuk şeklinde yayılı yük ve simetrik olmayan üçgen şeklinde yayılı ara yükler için değiştirilmiş ankastrelik uç kuvvetleri elde edilmiştir. Literatürde bulunan bazı sonuçlarla karşılaştırmalar yapılmış ve aralarında çok iyi uyum olduğu görülmüştür. Bu çalışmada, kayma deformasyonlarının etkisi de göz önüne alınarak sonsuz rijit kısımları bulunan elastik bağlı çubuklardan oluşan düzlemsel çerçevelerin doğrusal olmayan analizi yapılmış ve bu konuda bir bilgisayar programı geliştirilmiştir. Önce, her iki ucunda rijit bölgeler ve dönel yaylar bulunan bir çubuk için rijitlik katsayıları elastik bölgesindekiler cinsinden elde edilmiştir. Daha sonra, ikinci mertebe teorisi kullanılarak ve kayma deformasyonları hesaba katılarak rijitlik katsayıları çıkarılmıştır. En sonunda, doğrusal prizmatik bir çubuğun üniform yayılı yük, tekil yük, doğrusal yayılı yük, simetrik yamuk şeklinde yayılı yük ve simetrik olmayan üçgen şeklinde yayılı ara yükler için değiştirilmiş ankastrelik uç kuvvetleri elde edilmiştir. Literatürde bulunan bazı sonuçlarla karşılaştırmalar yapılmış ve aralarında çok iyi uyum olduğu görülmüştür.