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# The Direct Stiﬀness Method I

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2 The Direct Sti ness Method I 2 1 Chapter 2 THE DIRECT STIFFNESS METHOD I TABLE OF CONTENTS Page 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 2 2 Foreword Why A Plane Truss Truss Structures Idealization The Example Truss Members Joints Forces and Displacements The Master Sti ness Equations The DSM Steps Breakdown Stage 2 9 1 Disconnection 2 9 2 Localization 2 9 3 Member Stiffness Equations Assembly and Solution Stage Globalization 2 10 1 Displacement and Force Transformations 2 10 2 Global Member Stiffness Equations Notes and Bibliography References Exercises 2 2 2 3 2 3 2 3 2 4 2 5 2 6 2 7 2 8 2 10 2 10 2 11 2 11 2 12 2 12 2 14 2 15 2 15 2 16 2 3 TRUSS STRUCTURES 2 1 Foreword This Chapter begins the exposition of the Direct Stiffness Method DSM of structural analysis The DSM is by far the most common implementation of the Finite Element Method FEM In particular all major commercial FEM codes are based on the DSM The exposition is done by following the DSM steps applied to a simple plane truss structure The method has two major stages breakdown and assembly solution This Chapter covers primarily the breakdown stage 2 2 Why A Plane Truss The simplest structural finite element is the two node bar also called linear spring element which is illustrated in Figure 2 1 a A six node triangle that models thin plates shown in Figure 2 1 b displays intermediate complexity Perhaps the most geometrically complex finite element at least as regards number of degrees of freedom is the curved three dimensional 64 node brick element depicted in Figure 2 1 c Yet the remarkable fact is that in the DSM all elements regardless of complexity are treated alike To illustrate the basic steps of this democratic method it makes educational sense to keep it simple and use a structure composed of bar elements a b c Figure 2 1 From the simplest through progressively more complex structural finite elements a two node bar element for trusses b six node triangle for thin plates b 64 node tricubic

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