APPENDIX B MATRIX NOTATION. The Definition of Matrix Notation is the Definition of Matrix Multiplication B.1 INTRODUCTION

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1 APPENDIX B MAIX NOAION he Deinition o Matrix Notation is the Deinition o Matrix Mltiplication B. INODUCION { XE "Matrix Mltiplication" }{ XE "Matrix Notation" }he se o matrix notations is not necessary to solve problems in the static and dynamic analysis o complex strctral systems. However, it does allow engineers to write the ndamental eqation o mechanics in a compact orm. In addition, it prodces eqations in a orm that can be easily programmed or digital compters. Also, it allows the properties o the strctre to be separated rom the loading. hereore, dynamic analysis o strctres is a simple extension o static analysis. o nderstand and se matrix notation, it is not necessary to remember mathematical laws and theorems. Every term in a matrix has a physical meaning, sch as orce per nit o displacement. Many strctral analysis textbooks present the traditional techniqes o strctral analysis withot the se o matrix notation; then, near the end o the book MAIX MEHODS are presented as a dierent method o strctral analysis. he ndamental eqations o eqilibrim, compatibility and material properties, when written sing matrix notation, are not dierent rom those sed in traditional strctral analysis. hereore, in my opinion, the terminology matrix methods o strctral analysis shold never be sed.

2 APPENDIX B- SAIC AND DYNAMIC ANALYSIS B. DEFINIION OF MAIX NOAION o clearly illstrate the application o matrix notation, let s consider the joint eqilibrim o the simple trss strctre shown in Figre B.. 8 Figre B. Simple rss Strctre Positive external node loads and node displacements, shown in Figre B., are in the direction o the x and y reerence axes. Axial orces i and deormations d are positive i tension is prodced in the member. i,,, d,,, d, d, d, d, d,,, d, Figre B. Deinition o Positive Joint Forces and Node Displacements

3 SOLUION OF EQUAIONS APPENDIX B- For the trss strctre shown in Figre B., the joint eqilibrim eqations or the sign convention shown are:. (B.a). 8 (B.b). (B.c). 8 (B.d). (B.e). 8 + (B.) (B.g) We can write these seven eqilibrim eqations in matrix orm where each row is one joint eqilibrim eqation. he reslting matrix eqation is (B.) Or, symbolically A (B.) Now, i two load conditions exist, the eqilibrim eqations can be wrtten as one matrix eqation in the ollowing orm:

4 APPENDIX B- SAIC AND DYNAMIC ANALYSIS (B.) It is evident that one can extract the eqilibrim eqations rom the one matrix eqation. Also, it is apparent that the deinition o matrix notation can be written as: A (B.) il k, ik kl Eqation (B.) is also the deinition o matrix mltiplication. Note that we have deined that each load is actored to the right and stored as a colmn in the load matrix. hereore, there is no need to state the matrix analysis theorem that: A A (B.) Interchanging the order o matrix mltiplication indicates that one does not nderstand the basic deinition o matrix notation. B. MAIX ANSPOSE AND SCALA MULIPLICAION { XE "Matrix ranspose" }eerring to Figre B., the energy, or work, spplied to the strctre is given by: W i i (B.) i, We have deined:

5 SOLUION OF EQUAIONS APPENDIX B- and (B.8a and B.8b) he deinition o the transpose o a matrix is the colmns o the original matrix are stored as rows in the transposed matrix. hereore: [ ] (B.9a) [ ] (B.9b) It is now possible to express the external work, Eqation (B.), as the ollowing matrix eqation: or W W (B.) Also, the internal strain energy Ω, stored in the trss members, is deined by the ollowing: d d or Ω Ω (B.) hereore, the prpose o the transpose notation is to se a matrix that has been deined colmn-wise as a matrix that has been deined row-wise. he major se o the notation, in strctral analysis, is to deine work and energy. Note that the scalar / has been actored ot o the eqations and is applied to each term in the transposed matrix. From the above example, it is apparent that i: B C A BC A then (B.)

6 APPENDIX B- SAIC AND DYNAMIC ANALYSIS It important to point ot that within a compter program, it is not necessary to create a new transormed matrix within the compter storage. One can se the data rom the original matrix by interchanging the sbscripts. B. DEFINIION OF A NUMEICAL OPEAION { XE "Nmerical Operation, Deinition" }One o the most signiicant advantages o sing a digital compter is that one can predict the time that is reqired to perorm varios nmerical operations. It reqires compter time to move and store nmbers and perorm loating-point arithmetic, sch as addition and mltiplication. Within a strctral analysis program, a typically arithmetic statement is o the ollowing orm: A B + C x D (B.) he exection o this statement involves removing three nmbers rom storage, one mltiplication, one addition, and then moving the reslts back in high-speed storage. ather than obtaining the time reqired or each phase o the exection o the statement, it has been ond to be convenient and accrate to simply deine the evalation o this statement as one nmerical operation. In general, the nmber o operations per second a compter can perorm is directly proportional to the clock-speed o the compter. For example, or a MHz Pentim, sing Microsot Power FOAN, it is possible to perorm approximately,, nmerical operations each second. B. POGAMMING MAIX MULIPLICAION Programming matrix operations is very simple. For example, the FOAN-9 statements reqired to mltiply the N-by-M-matrix-A by the M-by-L-matrix-B to orm the N-by-L-matrix-C are given by: C. DO I,N DO J,L DO K,M C(I,J) C(I,J) + A(I,K)*B(K,L) ENDDO! end K do loop

7 SOLUION OF EQUAIONS APPENDIX B- ENDDO ENDDO! end J do loop! end I do loop Note that the nmber o times the basic arithmetic statement is exected is the prodct o the limits o the DO LOOPS. hereore, the nmber o nmerical operations reqired to mltiply two matrices is: Nop NML, or, i M L N, then Nop N (B.) It will later be shown that this is a large nmber o nmerical operations compared to the soltion o a set o N linear eqations. B. ODE OF MAIX MULIPLICAION Consider the case o a statically determinate strctre where the joint displacements can be calclated sing the ollowing matrix eqation: A CA [[ A C] A] A [ C[ A ]] (B.) I A and B are N by N matrices and is an N by matrix, the order in which the matrix mltiplication is condcted is very important. I the evalation is condcted rom let to right, the total nmber o nmerical operations is N + N. On the other hand, i the matrix eqation is evalated rom right to let, the total nmber o nmerical operations is N. his is very important or large matrices sch as those encontered in the dynamic response o strctral systems. B. SUMMAY Matrix notation, as sed in strctral analysis, is very logical and simple. here is no need to remember abstract mathematical theorems to se the notation. he mathematical properties o matrices are o academic interest; however, it is ar more important to nderstand the physical signiicance o each term within every matrix eqation sed in strctral analysis. here is no need to create a transpose o a matrix within a compter program. Becase no new inormation is created, it is only necessary to interchange the

8 APPENDIX B-8 SAIC AND DYNAMIC ANALYSIS sbscripts to access the inormation in transposed orm. here are a large nmber o comptational techniqes that exploit symmetry, sparseness and compact storage and eliminate the direct storage o large rectanglar matrices. Dierent methods o strctral analysis can be evalated by comparing the nmber o nmerical operations. However, very ew modern research papers on strctral analysis se this approach. here is a tendency o many researchers to make otrageos claims o nmerical eiciency withot an accrate scientiic evalation o compter eort reqired by their proposed new method.