APPENDIX AFORMULATION OF THE STIFFNESS MATRIXA.1 IntroductionFor the purposes of analysis we define a structure as an assemble of joints inspace interconnected by structural elements. Joints are either pinned, rigid or semi-rigid.Structural elements are either one-dimensional (bar or beam or frame element) or twodimensional (plates and shells) or three dimensional (solids). For the sake of simplicity ,we will demonstrate the developed method on structures made up of one-dimensionalelements although the same principles can be applied to structures made up of two orthree dimensional elements called finite elements. One dimensional elements betweenjoints can be straight or curved and prismatic or non-prismatic. We will limit ourselvesby straight-prismatic elements, but the same principles hold for other types. We will alsolimit our discussion only to plane (two dimensional) structures.A.2 Formation of the element stiffness matrixFirst we will construct the stiffness matrix of a truss element which is the mostsimple finite element. The two dimensional truss element has four global degrees offreedom, i.e. two displacement components at each node, as shown in Figure A.1. Thisfigure also shows the relation between the local (element) coordinate system xy−andthe global (structure) reference system xy−. The element has a single deformationmode which is the axial extension along the local axis x. The constitutive relationbetween the axial extension δ and the internal axial force N can be derived from thedifferential equation of the axially loaded truss member and in the local reference systemxy− it isNkd=δwherekEALd=(A.1)(A.2)is the axial stiffness of the member.The relation between the extension δ and the end displacements in the local coordinatesystem represents the statement of geometric compatibilityδ=−uuxj xi(A.3)or in matrix notation[]δ=−=111uuxixjau(A.4)The relation between the internal axial force N and the end forces p in the localcoordinate system represents the statement of equilibrium yxαxylocal coordinate systemglobal coordinate systemjiuxjuxiuyjuyiLtruss elementFigure A.1 Two dimensional truss element: degrees of freedom and relation between local(element) and global (structure) reference systems.pb==−=−=ppNNNNxixj111(A.5)The sign convention in the above equation in shown in Figure A.2.Observe thatba11=T(A.6)By the application of the principle of virtual work it can be shown that (A.6) is not acoincidence and it always holds true. In fact, because of this relation, the displacementand corresponding force transformations are called contragradient transformations.[Ref.?].Substituting (A.4) into (A.1) we obtainNkd=au1(A.7)Premultiplying (A.7) by ba11=T and noting (A.5) we obtainpb a auku== =111NkTd(A.8)(A.8) expresses the relation between the end forces and the corresponding enddisplacements of two dimensional truss element in the local coordinate system. Thematrix k is the element stiffness matrix in the local coordinate system. It takes the formka a=11Tdk(A.9)iijjNNNNpxjpxitruss elementFigure A.2 Relation between internal forces and element end forces in local coordinatesystem xy−Before we can insert the element stiffness matrix into the structure stiffness, we need toexpress the element end displacements and corresponding end forces relative to theglobal coordinate system. Consider that the local coordinate system and the globalcoordinate system form an angle α, as shown in Figure A.1. It is easy to show that thefollowing transformation for the end displacements from the global to local referencesystem are trueuuuuuuxixjxiyixjyj=cos sincos sinαααα0000(A.10)oruau=2(A.11)As in the case of internal forces and deformations, it can be shown that the contragradiettransformation also holds true for the end forces and displacements. And therefore wecan write the transformation relation for the end forces from the local to global referencesystem aspap=2T(A.12)Substituting u from (A.11) into (A.8) we obtainpkukau==2(A.13)Premultiplying both sides of (A.13) by a2Tto transform the element end forces from localto global coordinates we obtainpapakauku== =222TT(A.14)We can see that the relation between element end displacements and corresponding endforces in the global coordinate system is given by the element stiffness matrix k given bythe relationkaka=22T(A.15)We can now substitute (A.9) into (A.15) to obtain in one step transformation from thestiffness matrix relative to element deformation to the stiffness matrix relative to degreesof freedom of the truss element in global coordinate systemkaa aa a a==21 12TTdTdkkwhereaaa=12(A.16)(A.17)The transformation matrix a can be constructed directly without first forming matricesa1and a2. For the two dimensional truss element a takes the following simple form[]a=− −cos sin cos sinαααα(A.18)The derivation described here for the two dimensional truss element can be extended toany type of finite element.A.3 Formation of the structure stiffness matrixAfter formulating the element stiffness matrix in global coordinates the next step is theassembly of the element stiffness matrix into the global structure stiffness matrix. Firstwe need to number the element and structure degrees of freedom. For example thenumbering conventions for two dimensional truss and frame elements is shown in FigureA.3. The numbering of the structural degrees of freedom are in a sense arbitrary,although it’s desirable that they have certain patterns which is out of the scope of thisproject.To construct the structure stiffness matrix we first construct the structure connectivitymatrix. Connectivity matrix of each element shows how are the degrees of freedom ofthat element related to the structural degrees of freedom. The element connectivitymatrix is a d by n matrix, where d is the number of element degrees of freedom and n isthe number of structure degrees of freedom. The connectivity matrix consists of 0’s and1’s. Each row has only one 1. If the 1 in the ith row is in the jth column it means the ithdegree of freedom of the element is connected to the jth degree of freedom of thestructure.Now to construct the structure stiffness matrix, first we combine all the element stiffnessmatrices into an l by l block diagonal matrix called Kd. Each block represents anelement stiffness matrix in