# Hencky Bar-Chain/Net for Structural Analysis

**There are many books that have been published on buckling and vibration of structures. In these books, the method of analysis is either analytically based approach for exact solutions or numerical methods, such as the slope deflection method, the finite difference method, the differential quadrature method, and the finite element method. This book presents yet another method of analysis called the Hencky bar-chain/net model.**

**In 1920, Professor Heinrich Hencky of the Technische Hochschule**

**Darmstadt proposed a novel structural model comprising rigid bars of equal length that are connected by frictionless hinges with elastic rotational springs (each rotational spring having a stiffness equal to the flexural rigidity divided by the length of the rigid bar) for elastic buckling analysis of columns. He showed that by using this rigid bar-rotational spring model, the governing equation for buckling of columns reduces to a set of homogenous**

**equations in terms of discrete deflection values at the positions of the rotational springs. In this way, he avoids the need to solve the differential equation governing the buckling of columns. It turns out that Hencky’s set of equations are exactly the same as the first-order central finite difference equations if the latter nodal spacing is made equal to the length of the Hencky bar segment. So, the Hencky bar-chain model may be regarded as a physical structural model of the mathematical finite difference model for structures. As the adopted number of rigid segments of HBM increases, the buckling load and the vibration frequencies approach generally their continuum structure counterpart solutions from below, just like the finite difference solutions do as the nodal spacing decreases.**

**The authors find the Hencky bar-chain model absolutely fascinating**

**and started to investigate how this model may be applied for the buckling and vibration analyses of beams, frames, arches, rings and plates. In the course of this investigation, the authors discovered many interesting findings on the Hencky bar-chain model. For example, the authors found the equivalent Hencky bar-chain spring stiffnesses for elastically rotational restrained supports and for semi-rigid joints of frames with different member sizes. In addition, the authors are amazed at how easy it is to formulate the total potential energy of the Hencky bar-chain model and to write the computer code for analysis. The Hencky bar-chain model also allows one to readily modify the structure for local damage or local**

**stiffening by simply adjusting the rotational spring stiffnesses accordingly.**

**More interesting is the fact that the Hencky bar-chain model may be used to calibrate the small length coefficient of Eringen’s non-local model for buckling and vibration of nanostructures due to the phenomenological similarities between the two models.**

**This book compiles the recent research works of the authors on the**

**development of Hencky bar-chain model for buckling and vibration analyses of uniform beams (Chapter 2), non-uniform beams (Chapter 3), portal frames and multi-storey multi-bay frames (Chapter 4), arches and rings (Chapter 5) and plates (Chapter 6). Exercises are included at the end of each chapter to aid comprehension and guide learning. It is hoped that this book sheds insights and provides an understanding into yet another simple**

**structural model for buckling and vibration analyses of structures.**

**The authors gratefully acknowledge the support and encouragement of their respective universities in carrying out the collaborative research and writing of this book. It is a pleasure to acknowledge the help of the co-authors in the authors’ publications on the Hencky bar-chain/net model. These co-authors include Isaac Elishakoff (Florida Atlantic University, USA), Eugenio uocco (Seconda Universit`a Napoli, Italy), Yunpeng Zhang (The University of Queensland, Australia), Wenhui Duan (Monash**

**University, Australia) and Zhen Zhang (Huazhong University of Science and Technology, China).**

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