Stiffness Matrix Structural Analysis Educational Package by Mathcad

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1 Stiffness Matrix Structural Analysis Educational Package by Mathcad R. CEDEÑO-ROSEE Civil Engineering Department, Universidad de las Ame ricas Puebla, Puebla, Me xico Received 14 October 24; accepted 8 June 25 ABSRAC: his article presents an educational program designed to understand in a simple form, the stiffness structural analysis theory of two-dimensional framed structures. he program is written in spreadsheet form using the software Mathcad. wo examples are presented and they may be used as templates to solve different problems in which some interaction by the student is expected. his interaction contributes to the comprehension of the theory and helps the student to learn the theoretical basic concepts in less time. ß27 Wiley Periodicals, Inc. Comput Appl Eng Educ 15: , 27; Published online in Wiley InterScience ( DOI 1.12/cae.263 Keywords: matrix structural analysis; frames; education; computer program; Mathcad INRODUCION he stiffness matrix structural analysis theory has been a difficult subject to understand by civil engineering students. he fundamental theory concepts involved applies to trussed and framed structures as well as continues structures where the finite element approach is necessary to use in order to solve a structural problem. Many packages has been developed focused in spreading abilities in students to manage the structural behavior in different kind of structures by solving Correspondence to R. Cedeño-Rosete (rafael.cedeno@udlap.mx). ß 27 Wiley Periodicals Inc. quickly a large number of problems with different load conditions or supports [1 4]. However, students still have problems to understand the matrix structural analysis theory underneath the structural analysis packages they use. his article presents a spreadsheet developed in Mathcad (trademark of MathSoft Engineering & Education, Inc.) to solve 2D framed structures with the stiffness matrix method. wo examples are presented which can serve as templates to solve other problems. he student must make a few changes in the spreadsheet in order to solve a different problem. With this technique, the user is reviewing and practicing his theoretical concepts. he main objective is to present a tool which can contribute to reduce the learning curve of the stiffness matrix structural analysis theory. 17

18 CEDEÑO-ROSEE BACKGROUND A civil engineering student needs to acquire a very good understanding of structural behavior. he structural analysis method taught nowadays is the stiffness matrix method.

2 18 CEDEÑO-ROSEE BACKGROUND A civil engineering student needs to acquire a very good understanding of structural behavior. he structural analysis method taught nowadays is the stiffness matrix method. Some students have difficulties to grasp the main ideas of the stiffness method because it is necessary to invest a lot of time an effort to develop a structural analysis solution for a specific problem even for a small frame. Using Mathcad, an educational structural analysis packaged was developed to solve bidimensional frames with the stiffness matrix method. In this software, the student algebraically forms the force vector and the stiffness matrix of the structure in order to solve the proposed problem. he algebraic capabilities of Mathcad allows to solve the problem in an easy and straightforward way avoiding hard calculation and giving a transparent insight of the theoretical procedure. Using the programmed Mathcad sheets presented in this article, a student can build his own spreadsheet and use it to solve with a few changes, any small structural problem in order to gain a deep insight to the structural behavior necessary to be successful in dealing with structural problems. HEOREICA CONSIDERAIONS In order to solve a structure subjected to the action of applied forces, it is necessary to solve the algebraic matrix system of equations stated as: ffg ¼½KŠfdg ð1þ In this equation, {F} is the force vector applied to the structure nodes, [K] is the global structure stiffness matrix, and {d} is the unknown nodal vector displacement. Most of the work is dedicated to form the stiffness matrix [K] and solving the Eq. (1). Dealing Figure 2 with two-dimensional frames, every node has three degrees of freedom, two linear and one angular displacement. Accordingly, two forces and one moment can be applied to each node. he structural stiffness matrix is formed with the stiffness matrix of each element. he stiffness matrix of a two-dimensional frame element is a matrix. However, it is better to express it as a matrix formed by four submatrices 3 3. Eq. (2) represent the element stiffness matrix in a local coordinate system such as the one showed in Figure 1. he letter A represents the origin of the local coordinate system and the letter B the opposite end. ½kŠ ¼ k A k AB ð2þ k BA k B 2 ½k A Š¼ 6 4 wo floor frame. EA 6EI 12EI 3 6EI 2 2 4EI ð2aþ Figure 1 ocal coordinate system of a bar. Figure 3 Node and bar numbering.

3 SIFFNESS MARIX SRUCURA ANAYSIS 19 K := K B1 + K A2 + K A6 K BA2 K BA6 K AB2 K B2 + K A8 K BA8 K AB6 K B3 + K A4 + K B6 + K A7 K BA4 K BA7 K AB8 K AB4 K B4 + K B8 K AB7 K B5 + K B7 Figure 4 Stiffness structure matrix. 2 EA 3 ½k AB Š¼ 12EI 6EI EI 5 ð2bþ 2EI 2 ½k BA Š¼½k AB Š ð2cþ 2 3 EA ½k B Š¼ 12EI 3 6EI EI 5 ð2dþ 4EI 2 It is a common practice to form the global structure [K] matrix adding element by element of each of the global stiffness element matrix in the appropriate position of the structure matrix [5]. In this work, the stiffness structure matrix [K] is assembled using the 3 3 global submatrices k A, k AB, k BA ; k B of each element. his alternative approach results in a more convenient procedure to achieve this objective. EXAMPE ONE Figure 2 represents a two-dimensional frame with 8 bars, 5 nodes and three fixed supports. he bars have rectangular sections with a modulus of elasticity of 21,68, kn/m 2. o apply the stiffness matrix method to solve the structure shown in Figure 2, it is necessary to follow a few steps. he nodes, bars and supports must be numbered and the elements must be oriented defining an initial extreme A where the origin of its local coordinate system is assumed and a final extreme B. An arrow is drawn in the extreme B to indicate the position of the local coordinate system. Ib (, h) bh 3 := Abh (, ) := b h 12 he supports are numbered in sequence after all the free nodes have been numbered. Figure 3 depicts this procedure for the structure of Figure 2. he stiffness matrix of this example consists of 5 5 submatrices each one with a size of 3 3, being the full matrix size of Forming this matrix using the global submatrices k A, k AB, k BA, and k B of each bar is very simple following the orientation of each bar. For instance the position row 3, column 3 of the [K] structure matrix, is filled with the matrix k B3 þ k A4 þ k B6 þ k A7. Observing that the bar 6 joints nodes 3 and 1 of the structure and we go from node B to A of the bar 6 to joint nodes 3 1, the position row 3, column 1 of the global structure matrix [K] is filled with k BA of bar 6 (k BA6 ). In the same way, position row 3, column 4 is filled with k AB4 and position row 2 and column 3 is filled with the matrix because there is no bar joining the nodes 2 and 3. In this way, the stiffness matrix of the structure can be written as it is shown in Figure 4. SIFFNESS PROGRAM Mathcad has been chosen by many authors to develop learning tools due to its useful algebraic capabilities [6]. In the first part of the Mathcad spreadsheet, the user captures some basic information such as material properties, joint coordinates and member incidences. hen, many functions to obtain the moment of inertia, section area, local element stiffness matrix MB(case, P,, a) := P a 2 ( a) if case 1 2 P 2 if case 2 12 Pa 2 [2 ( a) a] if case 3 Figure 5 Section properties functions. Figure 6 Fixed end moment function.

4 11 CEDEÑO-ROSEE ka(e, A, I, ) := EA 12E I 3 6E I 2 6E I 2 4E I P A := V A := VA(1, 1, 5, 2) + VA(2, 2, 5, ) M A := MA(1, 1, 5, 2) + MA(2, 2, 5, ) P A P A6I := V A P B6I := M A P B V B M B Figure 7 Stiffness submatrix function. Figure 9 Fixed end forces in bar 6. and element transformation matrix were written in Mathcad. Figure 5 shows two simple functions to obtain the moment of inertia and section area. In the same way Figure 6 shows a function to obtain the fixed end moment in the extreme B of a bar for a three different types of transverse loads. Similar functions exist for the extreme A and for the vertical fixed end forces. he designed function to obtain the stiffness sub matrix function k A is shown in Figure 7. Once the area and moment of inertia are obtained for all the bars calling the function showed in Figure 5, the values are collected in two vectors as it is shown in Figure 8. For every bar with loads the fixed end forces are obtained using the corresponding functions. Figure 9 shows the fixed end moment calculated for the extreme A of beam 6. he following step is to obtain the stiffness matrix in a local coordinate system and transform it to a global coordinates. Figure 1 shows the procedure for bar number 2. Similar blocks are established for every bar in the structure. Forming the force vector and the stiffness structure matrix is the next step. he force vector is build adding the end fixed forces for every bar joining in each node and changing the sign. he stiffness structure matrix in Figure 4 is then formed using the intrinsic Mathcad augment and stack functions. Figure 11 shows the procedure. With a simple matrix operation the displacement vector is then determined in Mathcad as it is shown in Figure 12. Finally, the end forces for each bar in local coordinates are calculated. For the bar number 2 which does not have applied loads the solution is obtained as it is shown in Figure 13. For the bar number 6 which has intermediate applied loads, the fixed end moments previously calculated need to be added to the direct solution and the final end forces are shown in Figure 14. i:= IA n A := Area n x A := xy i1, x B := xy j1, Bar 2: n := 2 j:= IB n I:= Iner n y A := xy i2, y B := xy j2, A 1 A 2 A 3 I 1 I 2 I 3 := en x A, x B, y A, y B x B x A cos := k A2 := ka(e, A, I, ) y B y A sen := 2 := (sen, cos) k B2 := kb(e, A, I, ) A 4 Area := Iner := A 5 I 4 I 5 k AB2 := kab(e, A, I, ) k BA2 := k AB2 A 6 I 6 K A2 := 2 ka2 2 K B2 := 2 kb2 2 Figure 8 A 7 I 7 A 8 I 8 Area and moment of inertia. K AB2 := 2 kab2 2 K BA2 := K AB2 Figure 1 Matrix stiffness element in local and global coordinates.

5 SIFFNESS MARIX SRUCURA ANAYSIS 111 F := stack P A6I, P A8I, P B6I + P A7I, P B8I, P B7I K AB2 K1 := augment K B1 + K A2 + K A6,, K AB6,, K2 := augment K BA2, K B2 + K A8,, K AB8, K3 := augment K BA6,, K B3 + K A4 + K B6 + K A7, K AB4, K AB7 K4 := augment, K BA8, K BA4, K B4 + K B8, K5 := augment,, K BA7,, K B5 + K B7 P A6 := P A6 + P A6I P B6 := P B6 + P B6I P A6 = 95 P B6 = Figure 14 End bar forces for bar number 6. K := stack(k1, K2, K3, K4, K5) Figure 11 Force vector and stiffness matrix. Figure 15 hree bar frame. F:= stack P A2I, P B2I K AB2,( + ) K1 := augment K B1 + K A2, K2 := augment K BA2 K B2 K A3 K := stack ( K1, K2) Figure 12 Displacement solution. Figure 16 Stiffness matrix. n := 2 i:= IA j:= IB n n dx() i dx() j d A := dy() i d B := dy() j dz() i dz() j ( P A2 := 2 K A2 d A + K AB2 d B P B2 := 2 K BA2 d A + K B2 d B Figure 13 End bar forces for bar number 2. P A2 := P A2 + P A2I P B2 := P B2 + P B2I P A2 = 42 P B2 = Figure 17 End forces of bar 2.

112 CEDEÑO-ROSEE EXAMPE WO Appling small changes in the software presented, the user can solve different problems such as the one depicted in Figure 15.

6 112 CEDEÑO-ROSEE EXAMPE WO Appling small changes in the software presented, the user can solve different problems such as the one depicted in Figure 15. he main changes needed are in forming the force vector and the global stiffness structure matrix. hese changes are shown in Figure 16. he end forces in every bar do not require any change except for those bars which have intermediate loads. he corresponding change is shown in Figure 17. CONCUSIONS A civil engineering student can save a lot of time analyzing a structural problem with this software. Interaction with the software allows a student to acquire the basic stiffness analysis concepts. Using this problem as a template, a student can solve different problems with less time and effort. he programs allows to the user to focus on the procedure instead of wasting time in the large amount of calculations involved. ACKNOWEDGMENS he author gratefully acknowledges the support given by the Universidad de las Américas Puebla, for the elaboration of this article. REFERENCES [1] Dr. Beam Pro and Dr. Frame, UR: disoftware-home.com. [2] Ftool, UR: ftool.html. [3] X. F. Yuan and J. G. eng, Interactive web-based package for computer aided learning of structural behavior, Comp Appl Eng Educ 1 (22), [4] S. F. Almeida, R. Piazzalunga, and V. G. Ribero, A web-based 2D structural analysis educational software, Comp Appl Eng Educ 11 (23), [5] A. Kassimali, Strructural analysis, PWS Publishing Company, USA, [6] Ms. Al-Ansari and A. B. Senouci, Use of Mathcad as a teaching and learning tool for reinforced concrete design of footings, Comp Appl Eng Educ 7 (1999), BIOGRAPHY Rafael Cedeño-Rosete is a faculty member in the Universidad de las Américas Puebla, México. His teaching is concerned with structural analysis and design. His research interests include prestressed concrete elements and steel structures. He is also involved in developing educational software for civil engineering courses. He received the PhD degree in civil engineering from the exas ech University, in ubbock, exas.

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