2The DirectStiffness Method I2–1Chapter 2: THE DIRECT STIFFNESS METHOD ITABLE OF CONTENTSPage§2.1 Foreword ...................... 2–3§2.2 Why A Plane Truss? ................. 2–3§2.3 Truss Structures .................... 2–3§2.4 Idealization ..................... 2–4§2.5 The Example Truss ................... 2–5§2.6 Members, Joints, Forces and Displacements ......... 2–6§2.7 The Master Stiffness Equations .............. 2–7§2.8 The DSM Steps ................... 2–8§2.9 Breakdown Stage ................... 2–10§2.9.1 Disconnection ................ 2–10§2.9.2 Localization .................. 2–11§2.9.3 Member Stiffness Equations ............ 2–11§2.10 Assembly and Solution Stage: Globalization .......... 2–12§2.10.1 Displacement and Force Transformations ........ 2–12§2.10.2 Global Member Stiffness Equations .......... 2–14§2. Notes and Bibliography ................. 2–15§2. References ...................... 2–15§2. Exercises ...................... 2–162–2§2.3 TRUSS STRUCTURES§2.1. ForewordThis 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). Inparticular, 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. Themethod has two major stages: breakdown, and assembly+solution. This Chapter covers primarilythe breakdown stage.§2.2. Why A Plane Truss?The simplest structural finite element is the two-node bar (also called linear spring) element, whichis 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 leastas regardsnumberofdegrees of freedom) is the curved, three-dimensional, 64-node “brick” elementdepicted 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 simpleand use a structure composed of bar elements.(a)(c)(b)Figure 2.1. From thesimplest through progressively morecomplex structural finiteelements:(a) two-node bar element for trusses, (b) six-node triangle for thin plates, (b) 64-node tricubic,“brick” element for three-dimensional solid analysis.A simple yet nontrivial structure is the pin-jointed plane truss, whose members may be modeled astwo-node bars.1Using a plane truss to teach the stiffness method offers two additional advantages:(a) Computations can beentirely doneby handas long as the structure contains just a fewelements.This allows various steps of the solution procedure to be carefully examined and understood(learning by doing) before passing to the computer implementation. Doing hand computationson more complex finite element systems rapidly becomes impossible.(b) The computer implementation on any programming language is relatively simple and can beassigned as preparatory computer homework before reaching Part III.1A one dimensional bar assembly would be even simpler. That kind of structure would not adequately illustrate some ofthe DSM steps, however, notably the back-and-forth transformations from global to local coordinates.2–3Chapter 2: THE DIRECT STIFFNESS METHOD IjointsupportmemberFigure 2.2. An actual plane truss structure. That shown is typical of a rooftruss used in building construction for rather wide spans, say, over 10 meters. Forshorter spans, as in residential buildings, trusses are simpler, with fewer bays.§2.3. Truss StructuresPlane trusses, such as the one depicted in Figure 2.2, are often used in construction, particularlyfor roofing of residential and commercial buildings, and in short-span bridges. Trusses, whethertwo or three dimensional, belong to the class of skeletal structures. These structures consist ofelongated structural components called members, connected at joints. Another important subclassof skeletal structures are frame structures or frameworks, which are common in reinforced concreteconstruction of buildings and bridges.Skeletal structures can be analyzed by a variety of hand-oriented methods of structural analysistaught in beginning Mechanics of Materials courses: the Displacement and Force methods. Theycan also be analyzed by the computer-oriented FEM. That versatility makes those structures a goodchoicetoillustratethetransitionfromthehand-calculationmethodstaughtinundergraduatecourses,to the fully automated finite element analysis procedures available in commercial programs.§2.4. IdealizationThe first analysis step carried out by a structural engineer is to replace the actual physical structureby a mathematical model. This model represents an idealization of the actual structure. For trussstructures, by far the most common idealization is the pin-jointed truss, which directly maps to aFEM model. See Figure 2.3.The replacement of true by idealized is at the core of the physical interpretation of the finiteelement method discussed in Chapter 1. The axially-carrying-load members and frictionless pinsof the pin-jointed truss are only an approximation of the physical one. For example, building andbridge trusses usually have members joined to each other through the use of gusset plates, whichare attached by nails, bolts, rivets or welds. Consequently members will carry some bending aswell as direct axial loading.Experiencehasshown,however, thatstressesanddeformations calculated usingthesimpleidealizedmodel will often be satisfactory for preliminary design purposes; for example to select the crosssection of the members. Hence the engineer turns to the pin-jointed assemblage of axial-force-carrying elements and uses it to perform the structural analysis.2–4§2.5 THE EXAMPLE TRUSS(a) Physical System(b) Idealized Sytem: FEM-Discretized Mathematical ModelIDEALIZATION;;;;;;;;;;;;jointsupportmemberFigure 2.3. Idealization of roof truss: (a) physical system, (b) idealizationas FEM discretized mathematical model.In this and following Chapter we will go over the basic steps of the DSM in a “hand-computer” cal-culation mode. This means that although the steps are done by hand, whenever there is a proceduralchoice we shall either adopt the way that is better suited towards the computer implementation, orexplain the difference between hand and