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Direct Stiffness Method
Published in Kenneth Derucher, Chandrasekhar Putcha, Uksun Kim, Hota V.S. GangaRao, Static Analysis of Determinate and Indeterminate Structures, 2022
Kenneth Derucher, Chandrasekhar Putcha, Uksun Kim, Hota V.S. GangaRao
While the stiffness method and the direct stiffness method are essentially the same, well-known researchers have drawn a distinction between them (Weaver and Gere, 1990). Although the distinction is slight, it is important that this is explained, especially for the undergraduate students in civil engineering for whom this book is aimed at. In the stiffness method, the elements of a stiffness matrix are derived from the basic principles of engineering mechanics corresponding to the unknown displacements in the structure. In the case of the direct stiffness matrix, the standard stiffness matrix for each element (whether beam element, truss element, or frame element) is used to assemble a structure’s stiffness matrix. This matrix is then used to solve for displacements. Thus, the direct stiffness method is more mechanical, to put it in plain terms, and is very easy to use. For this reason, the direct stiffness method is very popular and widely used for the analysis of any type of structure. The details of the direct stiffness method will be discussed in the text and also applied to beam, frame, and truss structures with specific examples of each in this chapter, Chapter 10, and Chapter 11, respectively.
The displacement method-line element structures
Published in J.L. Meek, Computer Methods in Structural Analysis, 2017
This property of being able to add the member stiffness matrix directly into the structure stiffness matrix rather than the formal matrix multiplication leads to the definition of the direct stiffness method of analysis.
Introduction
Published in George G. Penelis, Andreas J. Kappos, P. E. Pinto, Earthquake-resistant Concrete Structures, 2014
George G. Penelis, Andreas J. Kappos, P. E. Pinto
forming the structure, as shown in Figure 3.22. For the analysis the direct stiffness method is used, according to which the stiffness matrix of the system is derived by appropriately adding the elements of the stiffness matrices of the individual members.
A multi-domain IBEM for the wave scattering and diffraction of P- and SV-waves by complex local sites
Published in Waves in Random and Complex Media, 2021
Zhenning Ba, Zhiying Yu, Ying Wang
The displacements for the incident P- and SV-waves from the bedrock can be written as in which lx=mx=cosθ and lz=mz=sinθ. AP and ASV are the amplitudes of the P- and SV-waves. And the superscript R stands for the bedrock. The free fields of the layered half-space can be calculated by the direct stiffness method [38] where PN+1 and RN+1 are the external forces at the surface of underlying half-space, which are formulated as in which ux0 and uz0 are the outcrop motions and the detailed solving procedures are referred to [38].
A new hybrid meta-heuristic algorithm for optimal design of large-scale dome structures
Published in Engineering Optimization, 2018
All the VPS parameters are set similar to the values proposed in Kaveh and Ilchi Ghazaan (2017). The effects of the population size and C parameter on the MDVC-UVPS results are investigated for the second example and these parameters are set to 30 and 10, respectively. The other parameters of the MDVC-UVPS algorithm are set similar to those for the VPS algorithm. Each example is solved 30 times independently, and 1500 and 1000 iterations are considered as terminal conditions for the VPS and MDVC-UVPS algorithms, respectively. The algorithms are coded in MATLAB and the structures are analyzed using the direct stiffness method by our own codes. All experiments are carried out on a PC with Intel Xeon 5680 running at 3.30 GHz with 4GB of RAM (note that only a single processor is used due to the sequential implementation of the algorithm). The operating system is CentOS release 6.3 (Final) and version of MATLAB is R2010a.
A closed-form solution for thermal stresses of structures using generalized variational principles
Published in Journal of Thermal Stresses, 2018
Zou-Qing Tan, Xue-Dong Jiang, Yun-Song He, Shu-Hao Ban, Ren-Qiang Xi, Yang-Chun Chen
Problems involving statically indeterminate trusses have been studied widely though numerical and analytical approaches [6]. Although numerical methods (such as finite difference method, finite element method, piecewise independent integral method) can analyze more complex and larger structures, analytical approaches are quite beneficial for providing physical insight. Recently, many analytical methods have been developed to obtain the mechanical behavior of statically indeterminate trusses, such as the direct stiffness method [7], Castigliano’s second theorem [8], two-stage method [9], angular displacement method [10], and generalized variational principles [11]. However, most methods are likely to be beneficial to trusses, but not to multilayer structures.