Are all members completed over EndDetermine unknown displacements. Compute forces in the membersRearrange global stiffness matrix as per boundary conditions or support conditionsCalculate nodal vector matrixStore global stiffness matrixNext memberAdd all members stiffness matrix to get total structure stiffness matrix using proper locationCalculate local stiffness matrix for the plane frame membersCompute / Give member propertiesInitialize global stiffness matrix to zeroRead nodal coordinates members connectivity ,supports , loadingSTART

m=2j+3T.I= E.R+I.R = 0+1=1. Hc=1 V0=1 RAC=1 1 0-0.8 0 0-0.6B0= 1 0 B1=-0.8 0.75 -1-0.6 -1.25 01 0 062161

0.8 = BD (4/5) . 1(4/5) = DC BD = 1. DC = 0.8 1(3/5) = BC = 0.6 AD = 0.6. AB = 0.8.

Ra(4) 1(3) = 0. DC = 1.Ra = -3/4 = -0.75. DB (4/5) =1 = 5/4 =1.25(T)Rb (4) - 1(3) = 0. AD = DB (3/5) = 0.75. Rb= 0.75. AB = DB (4/5) = 1. BC=0 Unassembled matrix f= 1/AE B1Tf= *f B1Tf= *1/ AEG1 = B1Tf B1=1/ AE

Static condensating and sub structuring :-Introduction :- In the previous chapters we established the relationship(through the system stiffness matrix) between forces applied at the nodal co-ordinates of the structure and the corresponding nodal displacements. These are instances ,such as the use of subtracting for the analysis of large structures ,in which it might be advantages to reduce the number of nodal co-ordinates . Subtracting requires the reduction of nodal co-ordinates to allow the independent analysis of portions of the structure. The process of reducing the number of free displacements or degrees of freedom is known as static condensation. The same process is also applied to dynamic problems although ,in that case, it is only approximate and in general may result in large errors. The static condensation method has recently been modified for applications to dynamic problems. This method is known as the dynamic condensation method, its application to dynamic problems gives solutions that are virtually exact.Static condensation :- A practical method of accomplishing the reduction of the number of degrees of freedom and hence the reduction of the stiffness matrix, is ti identify those degrees of freedom to be condensed as secondary degrees of freedom and to express them in terms of the remaining primary degrees of freedom. The relationship between secondary and primary degrees of freedom is found by establishing the {F}p = [K] {u}P (8.6) in which {F} = {F} P - [K]PS [K]-1SS {F}S (8.7)and[K] =[K]PP - [K]PS [K]PS [K]-1SS[K]SP (8.8.)Thus, equation (8.6) is the condensed stiffness equation relating through the condensed stiffness matrix [K], the condensed force vector {F} and the primary nodal coordinates {u}p .The solution of the condensed stiffness equation (8.6) gives the displacement vector {u}p at the primary nodal coordinates. The displacements {u}s at the secondary nodal coordinates are then calculated by equation (8.4) after substituting into this equation the displacement vector {u}p for the primary nodal degrees of freedom.The development yielding equation (8.4) and (8.5) may appear to indicate that the use of the condensation method requires the inconvenient calculation of the inverse matrix [K]-1 . However, the practical application of the static condensation method does not actually require a matrix inversion. Instead, the standard Gauss-Jordan elimination process is applied systematically to the system stiffness equation up to the reduction of secondary coordinates {u}s .At this stage of the elimination process the stiffness equation (8.1) has been reduced to

{F}s = [I] -[T]{u}s(8.9) . {F}P [O] [K]{u}p By expanding equation (8.9), if may be seen that{F}S = {u}S -[T] {u}P .(or) {u}S = {p}S + [T] {u}P (8.10) .and {F}P = [K] {u}P (8.11).are equivalent to equation (8.4) and (8.5). Therefore, the partial use of Gauss-Jordan elimination process yields both the relationship equation(8.10) between secondary nodal yields both the relationship equation (8.10) between secondary nodal coordinate {u}S as well as the stiffness equation for the condensed system equation (8.1) .The Following example illustrates the process of static condensation applied to sample structural system.Sub structuring :-sub structuring or the separation of structure into parts or substructures is a convenient way to design and analyze structures that have been modeled with a large number of elements, hence resulting in large number of equations.some examples of this type of structure are aircraft, naval vessels , large high-rise buildings and domes. In sub structuring, use is made of the static condensation method to establish the analytical relationship between the various parts (substructures) of structure.

fig 8.2To illustrates the use of sub structures consider the plane frame in fig 8.2 , which has been divided into three sub structures as shown in fig 8.3 where the connecting joints are indicated with dark dots.

Fig 8.3

Fig 8.3- Sub structures for the plane frame in fig 8.2 . In analyzing the plane frame shown in fig 8.2 we may select as primary degrees of freedom ,the displacements of the interface nodes between the substructures in fig8.3. we then proceed to condense in each of the sub structures the degrees of freedom not identified as primary selected degrees of freedom at the interface nodes. Each condensed sub structure will provide a condensed stiffness matrix and a condensed force vector, which then can be assembled to obtain the condensed stiffness equation of structure. For the plane frame in fig 8.3 the substructure-1 will be condensed, from 9 nodal coordinates to 3. Similarly, sub structures 2 and 3 will be reduced from 18 nodal coordinates to 9 and from 27 to 6, respectively. Then the subsequent combination of these condensed sub structures considered to be super-elements will result is the condensed system stiffness equation with only 9 nodal coordinates reduced from a total of 45 free nodal coordinates in the plane frame of fig-8.2.