APPLICATIONS OF MATHEMATICAL SOFTWARE PACKAGES IN STRUCTURAL ENGINEERING EDUCATION AND PRACTICE Downloaded from ascelibrary.org by FLORIDA STATE UNIVERSITY on 08/22/13. Copyright ASCE. For personal use only; all rights reserved. By Primus V. Mtenga1 and Lisa K. Spainhour,2 Members, ASCE ABSTRACT: The use of mathematical software packages provides a number of benefits to an engineering user. In general, the packages provide a platform that supports iterative design and parametric analysis through a flexible, transparent interface combined with extensive computing power. This enables an engineering user to develop design equations that are based on fundamental mechanics theories, rather than relying on the ‘‘blackbox’’ approach of most commercial design packages. As an example, a closed-form solution for obtaining effective length factors for the design of stepped columns is presented. In the example a series of formula is used to demonstrate the transparency of Mathcad, including the ability of using real engineering units in the calculations, formulas as they may appear in textbooks or in codes, and ability to hide and password protect some areas. This facilitates easier automation of the design and design checking processes. Most commercial structural design packages can be classified as black-box packages. The analyst inputs data at one end only to receive results at the other without fully appreciating the process the input data have gone through. This phenomenon has the tendency of reducing the engineer to a technician, blindly implementing the ideas of the software designer. The Mathcad package discussed in this paper and similar mathematical packages returns the engineer to being in control of the design process. INTRODUCTION Advances in computing power have revolutionized the civil engineering field over the past few decades. Finite-element analysis programs have made it possible to solve complex systems with more degrees of freedom than imagined a few years past. Computer-aided drafting software has virtually eliminated hand drawing from the engineering office. Moreover, the ever-decreasing cost of the microcomputer has made them affordable to civil engineering firms of all sizes and specialties. This availability, coupled with the fast-growing library of engineering computer software, has greatly expanded the engineer’s effectiveness, thus allowing the engineer to be more competitive in the marketplace. However, with this increase in microcomputer and software availability, several ethical questions concerning a possible microcomputer misuse have arisen and need to be addressed (Gifford 1987). The ‘‘black-box’’ syndrome, infinite accuracy of the computer, data/input integrity, instant expertise of the engineer, and lack of judgment on the part of the engineer are identified as the microcomputer misuses of most concern. According to Gifford, these problems are best avoided through education in the ethical use of microcomputers. This education should be aimed at, among other things (1) understanding the software’s philosophy; (2) ascertaining the software’s suitability and design acceptability; and (3) realizing it is the professional responsibility of the engineer to use the microcomputer as a powerful design aid only, not allowing his judgment to be sacrificed. The increasing power of spreadsheet programs and mathematical software packages can be used as one of the tools to address the ethical problems raised by Gifford. The benefits of such programs are especially apparent in the classroom and small design office, where the ability to perform ‘‘what-if’’ type calculations has proven the most useful. This paper illustrates the application of a mathematical software package to 1 Civ. Engrg. Dept., Florida A&M Univ.-Florida State Univ. Coll. of Engrg., 2525 Pottsdamer St., Tallahassee, FL 32310. 2 Civ. Engrg. Dept., Florida A&M Univ.-Florida State Univ. Coll. of Engrg., 2525 Pottsdamer St., Tallahassee, FL 32310. Note. Editor: Sivand Lakmazaheri. Discussion open until March 1, 2001. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this technical note was submitted for review and possible publication on August 25, 1998. This paper is part of the Journal of Computing in Civil Engineering, Vol. 14, No. 4, October, 2000. qASCE, ISSN 0887-3801/00/ 0004-0273–0278/$8.00 1 $.50 per page. Technical Note No. 19098. the problem of stepped column design, where a closed-form solution, made feasible by the mathematical software package, is proposed. The advantages of using the closed-form approach are clear in comparison to the current method, which involves a 3D interpolation between values from several standard tables. The use of mathematical packages in the analysis and design process allows transparency without a significant adverse effect on the speed of the process. BENEFITS OF MATHEMATICS SOFTWARE PROGRAMS Mathematical software packages, including Mathcad, Mathematica, MathView, MATLAB, and Maple, provide powerful computational tools for both engineering education and practice. With these tools, it is possible to solve very complex mathematical problems with the click of a computer mouse. In addition, this capacity enables engineers to base their designs on fundamental mechanics theories, rather than empirical or tabular relationships. Although the capabilities and features, as well as the nature of the interface, vary from program to program, each provides an environment from which equations representing engineering problems can be easily and quickly solved. Most programs of this type provide the following capabilities: • • • • • Numeric and symbolic computations Algebraic, trigonometric, and matrix functions Graphics capabilities Conditional programming Flexible, easy to use interface Most design offices now use commercial software packages for their structural analysis and design. Most commercial packages can be classified as black-box packages. The analyst inputs data at one end only to receive results at the other without fully appreciating the process the input data has gone through. This phenomenon has the tendency of reducing the engineer to a technician, blindly implementing the ideas of the software designer. If this trend is allowed to continue, sooner or later the teaching of structural engineering will have to change to reflect this development. On the other hand, mathematical software packages provide a great deal of computing power and flexibility in an inexpensive package, making them ideal for educational and small office environments. The biggest benefit to this group of users, JOURNAL OF COMPUTING IN CIVIL ENGINEERING / OCTOBER 2000 / 273 J. Comput. Civ. Eng. 2000.14:273-278. Downloaded from ascelibrary.org by FLORIDA STATE UNIVERSITY on 08/22/13. Copyright ASCE. For personal use only; all rights reserved. however, is the transparency, which allows the user to vary the program inputs and see the effects at intermediate steps throughout the solution process. This ‘‘white-box’’ or ‘‘clearbox’’ approach can be contrasted with the black-box procedure inherent in many software systems, where only the input and output data are visible and all calculations are hidden to the user. The benefits of this approach in an academic environment are obvious. This class of software lets undergraduate students solve a larger number of more meaningful problems without having to learn the details of a programming language. Because of the white-box interface, the student can see the equations involved in the solution process, which makes it easier to find theoretical and programming errors. The student also can vary the input and see the results immediately, helping him or her to grasp the underlying theory, see the results of what-if type questions, and perform parametric studies (Pavlov and Mtenga 1997). A number of authors have described approaches to integrating mathematics software programs into the engineering curriculum (Agelidis 1994; DeLyser 1996; Myers 1992). Many of these benefits carry over into the engineering design office. Mathematics software packages can assist the design process by automating many routine calculations and allowing the designer to evaluate different alternatives instantly. The software is powerful enough to solve the large and cumbersome equations used to characterize system architectures. Further, the interactive nature of the interface supports iterative design and allows for rapid convergence to a final design. The white-box approach also provides designers and permitting officials efficient and reliable tools to check the results of more sophisticated software, where the logic and algorithms are often hidden in the source code, unavailable to the engineer (Magner 1995; Pavlov 1996; Pavlov and Mtenga 1997). The worksheetlike interface provided by some packages allows the user to add text blocks, which can be used to define variables and reference codes and to document the design procedure. STEPPED COLUMNS As indicated above, the increasing power of the available mathematical software packages can eliminate the need for tabulated information currently used in many codes and specifications, thus enabling the engineer to develop and utilize a complex system of equations that are grounded in basic mechanics theories. The problem of stepped-column design is used to illustrate this functionality aspect of these software packages. Stepwise uniform columns are a necessity in structural applications, especially in situations where intermediate loads are introduced into the column. A typical example of this scenario is a column supporting a crane loading, as shown in Fig. 1(a). The lower part of the column will have to support both the crane loading and the weight of the roof structure. As the roof loads are relatively small compared to crane loading, it may be economical to have a heavier section for the lower part and a lighter section for the upper portion of the column. Another situation where stepped columns may be seen is in a retrofit operation where, for example, cover plates may be added to the flanges of the column, resulting in a stepped column as shown in Fig. 1. The design of stepwise uniform columns is time consuming and complicated. This complexity has attracted a number of researchers to the study of stepped-column behavior. Examples of such studies include those reported by Lui and Sun (1995), Bendapudi (1994), Fraser (1990), Agrawal and Stafiej (1980), and Anderson and Woodward (1972). According to AISC’s Steel design guide series No. 7 (Fisher 1993), the best design approach for stepped columns is to design the upper and lower portions of the column as individual segments. As shown in FIG. 1. Examples of Stepwise Uniform Columns Fig. 1(a), the upper segment is designed to carry an axial load P1 and the associated upper column moment, whereas the lower portion is designed for P1 1 P2 and the associated lower column moments. This makes it necessary to determine effective length factors for both the upper and the lower column sections. These effective length factors for the column segments are not readily available. The AISC load and resistance factor design (LRFD) specifications provide no guidelines on the design of stepped columns. However, effective length factors for stepped columns are tabulated in its Steel design guide series No. 7 (Fisher 1993). These values were developed by Agrawal and Stafiej (1980) based on transcendental equations developed by Anderson and Woodward (1972). The following parameters were considered in developing the tabulated values: (1) The end fixity types at the column ends; (2) the ratio of the bottom segment length to the total column length; (3) the ratios of the segments’ moment of inertia; and (4) the ratio of the intermediate column load to the total axial load. These tabulated values and associated design approaches are not yet included in the AISC specifications (AISC 1995). In this paper, an equation-based approach to the problem of stepped columns will be used in a Mathcad example, eliminating the need for tabulated values. The formulas-based approach is easier to incorporate into mathematical software packages than the current approach based on tabulated values. If a designer were to use the tabulated K-factor values for a stepped column, advantages of automation would be minimized. In the example MathSoft’s Mathcad software (Mathcad 1991) was utilized for all of the computations. However, Mathcad was chosen only to illustrate the capabilities of this class of software packages. Any of the other programs mentioned in the ‘‘Introduction’’ could have been used instead. STEPWISE UNIFORM MEMBER In this paper, the above-described approach will be applied to stepwise uniform columns. The column will have two sections, with the top referred to as the reference section and the bottom as the modified section. In determining the effective length factors, the following ratios are used: • The ratio a of the moment of inertia of the lower segment to the upper segment 274 / JOURNAL OF COMPUTING IN CIVIL ENGINEERING / OCTOBER 2000 J. Comput. Civ. Eng. 2000.14:273-278. Q4 = 12mb(240b6 2 840b5 1 756b4 1 210b3 2 420b2 1 105) (6) Q5 = mb2(21,920b5 2 7,560b4 1 9,576b3 1 2,940b2 1 2,100b 2 1,260) 3 (7) 4 3 2 Downloaded from ascelibrary.org by FLORIDA STATE UNIVERSITY on 08/22/13. Copyright ASCE. For personal use only; all rights reserved. Q6 = mb (320b 2 1,400b 1 2,247b 21,575b 1 420) (8) What is now required is to determine a factor to modify the actual length of the column such that the buckling load will be the same as the buckling load of (2). For the reference section, that is, the top portion of the column, the critical buckling load is Pcr = Thus, solving for K1 yields FIG. 2. Model of Stepwise Uniform Column K1 = p • The ratio b of the length of the bottom segment to the total length of the column • The ratio m of the load at the stepped level to the load at the top of the column Thus, the modified section will be bL in length and have a moment of inertia equal to aI. The reference section will be (1 2 b)L long and have a moment of inertia I, as shown in Fig. 2(b). At one end, a rotational restraint with spring constant kR is provided. The value of kR is set equal to zero for a pinned end and infinity for a fixed end condition. To facilitate computation in Mathcad, a very large value, on the order of 1020 is assumed to be equivalent to infinity. A semirigid end condition is represented by an intermediate spring constant value. At the pinned end of the column (assumed to be the reference section) a compressive force P is applied. At the stepped level of the column, a force mP is applied. The application of the Rayleigh-Ritz method to the column described will yield the critical buckling load E E bL S( y0)2 dx 1 aS( y0)2 dx bL 0 Pcr = E E L (1) bL L 2 ( y9) dx 1 0 2 m( y9) dx 0 The external work done by load P will be as a result of the deformation of the entire length of the column. On the other hand, the external work done by the load at the stepped level mP will be due to deformation in the segment of length bL [Fig. 2(b)]. This explains the integration limits in the denominator of (1). Using a fourth-order polynomial for y(x) in (1) will yield Pcr = 252S [S 2(Q1 1 24) 1 DS(Q2 2 6) 1 D 2(Q3 1 1)] L2 [S 2(Q4 1 612) 1 DS(Q5 2 156) 1 D 2(Q6 1 12)] (2) where S = E ? I, where E = modulus of elasticity of the material and I = moment of inertia about the buckling axis; and D = kR ? L, where kR = rotational stiffness of the partial restraint at the end of the member Q1 = (a 2 1)b3(144b2 2 360b 1 240) 2 3 2 (3) Q2 = (a 2 1)b (296b 1 270b 2 240b 1 60) (4) Q3 = (a 2 1)b(16b4 2 50b3 1 55b2 2 25b 1 5) (5) p2S K 12L2 (9) SÎ D S L2Pcr (10) A substitution of (2) into (10) will yield K1 = pÏ[S 2(Q4 1 612) 1 DS(Q5 2 156) 1 D 2(Q6 1 12)] Ï252[S 2(Q1 1 24) 1 DS(Q2 2 6) 1 D 2(Q3 1 1)] (11) The axial load that will be acting in the modified section will be (1 1 m)P. The modification factor for this section will be such that (1 1 m)Pcr = p2aS K 22L2 (12) Solving for K2 yields K2 = p SÎ Î D Î S L Pcr 2 a 11m = K1 a 11m (13) Eqs. 21 and 23 were used to compute the effective length factors for the stepped column that were programmed in the Mathcad example. Mathcad is a Windows-based product that displays both text and equations in a ‘‘what you see is what you get’’ (WYSIWYG) format. New users find the program intuitive and easy to use. As changes are made to the files, the results are updated automatically, making parametric studies easy to formulate. Graphs are easy to create and can be frequently used to summarize and clarify results. Users can easily add notes, modify equations, and add supplemental checks to verify the program’s results. Mathcad and comparable mathematical software packages can be used to keep engineers and students in charge of their designs, as opposed to the black-box design software output. Mathcad is an excellent tool that can be used to bridge the gap between the traditional educational approach and the automated black-box industry approach. Unlike programming languages, Mathcad does not require extensive training in commands, syntax, or structured programming. For engineering applications, all that is needed is knowledge of how to solve the problem in a step-by-step flowchart manner, a classical classroom approach. In the classroom and in practice, students or engineers using Mathcad can solve problems, step by step, just as they would in a pencil and paper solution. Equations, notations, and symbols are used just as in the text and lectures. As problems become more complex, previous models and problems can be reused and combined to build more comprehensive problem JOURNAL OF COMPUTING IN CIVIL ENGINEERING / OCTOBER 2000 / 275 J. Comput. Civ. Eng. 2000.14:273-278. Downloaded from ascelibrary.org by FLORIDA STATE UNIVERSITY on 08/22/13. Copyright ASCE. For personal use only; all rights reserved. solutions. Equations and assumptions are visible in engineering notation, unlike ordinary spreadsheets. Text is frequently used to outline the design process as well as reference applicable theory, standards, specifications, and literature. The benefits of solving engineering problems in Mathcad are numerous. Users can easily build upon their earlier work by incorporating previous models into new projects. At the end of the project, the users will have a working model of the entire analysis and/or design. Students not only learn how to solve each piece of the design, they can understand the problem from an overall perspective. Showing the results both numerically and graphically in Mathcad reinforces the engineering concepts and theories. Once the model is complete, students have a working program. Written in engineering notation, they can reuse the model in other classes and as a verification tool to check other design programs. MATHCAD IMPLEMENTATION OF STEPPED-COLUMN PROBLEM Given below is a Mathcad implementation of the steppedcolumn problem discussed above. In the interest of saving space, some areas are hidden. However, unlike black-box programs, the contents of these hidden areas can be viewed by double clicking on the name icon of the area. For security reasons some areas may be password protected; thus, to view the contents of these areas, one may need to have the correct password. As seen in the example (Appendix I, notes 1–6), text can be inserted within the calculation, as is the case in a paper and pencil worksheet, thus creating a well-documented report. All this can be done without much impairment in the ability and sophistication of the developed application. Data can be imported from or exported to external files and input and output arrays can be created, thus creating an application capable of examining several design alternatives. APPENDIX I. STEPPED COLUMN: MATHCAD EXAMPLE OF MTENGA AND SPAINHOUR User Specified Units N [ 1 newton; KN [ 1,000 N; psf [ lbf/ft 2; MPa [ N/mm2; kip = 1,000 lbf; ksi [ 1,000 psi; plf [ lbf/ft Note 1: One of the most powerful characteristics of Mathcad is the ability to use units in the calculations (both built-in units and user-specified units). When the formulas’ noncompatible units are inputted, an error will appear, thus forcing the engineer or student to review the input information. Blackbox packages and even other mathematical packages permit ‘‘garbage in garbage out. Moreover, one can mix both metric and customary units (at appropriate magnitudes, e.g., Fy = 50 ksi entered as Fy = 345 MPa) in the same worksheet and end up with the same design results. Input Data Note 2: With Mathcad it is possible to hide and password protect some regions of the worksheet. Normally one will have the ‘‘Input Data’’ area visible, which is the case in this example. Material Data Modulus of elasticity E := 200 KN/mm2 Yield stress of material (A572 grade 50 steel) Fy := 345 N/mm2 Residual stress Fr := 69 N/mm2 CONCLUSIONS Bending strength reduction factor (per LRFD) The use of mathematical software packages provides a number of benefits to an engineering user. In general, the packages provide a platform that supports iterative design and parametric analysis through a flexible, transparent interface combined with extensive computing power. The following benefits need to be emphasized • Ability to incorporate the actual engineering units in the calculation • Ability to see the formulas on the worksheet in the exact form as they may appear in a textbook or a code • Ability to import and export data to external files These characteristics of Mathcad allow students, teachers, and engineers to develop sophisticated computer models that solve engineering problems in scientific notation. Complete solutions to engineering problems can be developed for a homework problem or to simply check the results of other programs. Mathcad allows students and teachers to focus on the engineering theory and procedures while letting the software handle the calculations along with the units of measure. As the engineering profession continues to move toward automated design software, engineers must find better tools that they can use to verify engineering calculations. Engineering graduates who possess portfolios of working engineering models, produced in Mathcad as part of their college course work, will be much better prepared to enter the automated paradigm now present in the engineering profession. fb := 0.90 Compression strength reduction factor (per LRFD) fc := 0.85 Geometry Data (Fig. 1) Length of top section (reference section) Ltop := 4 m Length of bottom section Lbot := 8 m Effective length of top section for buckling about weak axis L1y := 4 m Effective length of bottom section for buckling about weak axis L 2y := 4 m Maximum length of laterally unbraced segment in top portion Lbtop := 4 m Maximum length of laterally unbraced segment in bottom portion Lbbot := 8 m Rotational restraint spring constant 276 / JOURNAL OF COMPUTING IN CIVIL ENGINEERING / OCTOBER 2000 J. Comput. Civ. Eng. 2000.14:273-278. Q4i := 12 ? m ? b ? (240 ? b6 2 840 ? b5 1 756 ? b4 1 210 ? b3 k1 := 0.0 ? 1020 KN ? m Total length of column 2 420 ? b2 1 105) L := Ltop 1 Lbot Q5i := m ? b2 ? ((21,920 ? b5 1 7,560 ? b4 2 9,576 ? b3 1 2,940 ? b2 Fraction of column that is modified b := 1 2,100 ? b 2 1,260)) Lbot L Q6i := m ? b3 ? (320 ? b4 2 1,400 ? b3 1 2,247 ? b2 2 1,575 ? b 1 420) The equivalent effective length factor for the top portion of the column [(11)] Downloaded from ascelibrary.org by FLORIDA STATE UNIVERSITY on 08/22/13. Copyright ASCE. For personal use only; all rights reserved. Derived variable D := k1 ? L Kmtopi := Loading Data p ? Ï(Si)2 ? (Q4i 1 612) 1 D ? Si ? (Q5i 2 156) 1 D2 ? (Q6i 1 12) Ï252 ? [(Si)2 ? (Q1i 1 24) 1 D ? Si ? (Q2i 2 6) 1 D2 ? (Q3i 1 1)] The equivalent effective length factor for the bottom portion of the column [(13)] Factored compression force at top of column Putop := 80 KN Kmboti := Kmtopi ? Factored compression force at the stepped level of column Î ai (1 1 m) Pustp := 480 KN Ratio of the compression forces Note 5: The design of the column sections are in accordance with the LRFD approach. Again, in the interest of shortening this paper, the area is hidden. Pustp m := Putop Maximum factored bending moment in the top section Mutop := 160 KN ? m Note 6: The combined force index is shown for the various trial sections. The section is adequate only when both indices are equal or less than 1.0. Maximum factored bending moment in the top section Mubot := 220 KN ? m Top portion of column Moment modification factor for nonuniform moment (LRFD, Eq. F1-3: top section Cb o top := 1 Moment modification factor for nonuniform moment (LRFD, Eq. F1-3): bottom section Cb o bot := 1 Bottom portion of column Note 3: One first needs to determine the effective length factors for the top and bottom sections of the column. Before doing that, one needs to have trial column sections. Mathcad can be linked to section properties tables in which trial sections can be picked. In this example three sets of trial sections were picked and their section properties entered on the worksheet. In the interest of space, the section is compressed (hidden). A double click on the area name in active Mathcad application will reveal the contents of the area. For this example a top section of W10 3 33 and a bottom section of W21 3 62 will be chosen. APPENDIX II. Note 4: Using (3)–(13), the effective length factors are computed in the hidden area below. The clear formulas [(3)– (8), (11), and (13)] appear in the Mathcad worksheet as they would appear in a book or the code, clearly demonstrating the transparency of using this mathematical package. Derived values as per theory presented in this paper [(3)–(8)] Q1i := (144 ? b2 2 360 ? b 1 240) ? b3 ? (ai 2 1) Q2i := (296 ? b3 1 270 ? b2 2 240 ? b 1 60) ? (ai 2 1) Q3i := (16 ? b4 2 50 ? b3 1 55 ? b2 2 25 ? b 1 5) ? b ? (ai 2 1) REFERENCES Agelidis, V. G. (1994). ‘‘Incorporating software tools in electrical engineering laboratory experiments—an example.’’ Proc., 1994 IEEE 1st Int. Conf. on Multi-Media Engrg. Educ., IEEE, Piscataway, N.J., 319– 328. Agrawal, K. M., and Stafiej, A. P. (1980). ‘‘Calculation of effective lengths of stepped columns.’’ Engrg. J., 17(4). AISC. (1995). Manual of steel construction—load and resistance factor design, Chicago. Anderson, J. P., and Woodward, J. H. (1972). ‘‘Calculation of effective lengths and effective slenderness ratios of stepped columns.’’ Engrg. J., 4th Quarter, 157–166. Bendapudi, K. V. (1994). ‘‘Practical approaches in mill building columns subjected to heavy crane loads.’’ Engrg. J., 31(4), 125–140. Coates, R. C., Coutie, M. G., and Kong, F. K. (1988). Structural analysis, Van Nostrand Reinhold, New York, 331–335. DeLyser, R. R. (1996). ‘‘Using Mathcad in electromagnetics education.’’ IEEE Trans. on Educ., 39(2), 198–210. JOURNAL OF COMPUTING IN CIVIL ENGINEERING / OCTOBER 2000 / 277 J. Comput. Civ. Eng. 2000.14:273-278. Downloaded from ascelibrary.org by FLORIDA STATE UNIVERSITY on 08/22/13. Copyright ASCE. For personal use only; all rights reserved. Fisher, J. M. (1993). Industrial buildings: Roofs to column anchorage, Steel Des. Guide Ser. No. 7, AISC, Chicago. Fraser, D. J. (1990). ‘‘The in-plane stability of a frame containing pinbased stepped column.’’ Engrg. J., 27(2), 49–53. Gifford, J. B. (1987). ‘‘Microcomputers in civil engineering: Use and misuse.’’ J. Comp. in Civ. Engrg., ASCE, 1(1), 61–68. Lui, E. M., and Sun, M. (1995). ‘‘Effective lengths of uniform and stepped crane columns.’’ Engrg. J., 32(3), 98–106. Magner, T. (1995). ‘‘Automated structural design calculations using Mathcad.’’ Modern Steel Constr., 35(4), 42–47. Mathcad: User’s guide. (1991). Mathsoft, Inc., Cambridge, Mass. Myers, G. E. (1992). ‘‘Equilibrium via element potentials and computers.’’ Thermodynamics and the design, analysis, and improvement of energy systems, Vol. 27, ASME, New York, 205–212. Pavlov, A., and Mtenga, P. V. (1997). ‘‘Application of MathCAD in bridge design.’’ Proc., 4th Int. Congr. on Computing in Civ. Engrg., ASCE, New York, 666–671. Pavlov, A. V. (1996). ‘‘Florida Department of Transportation’s MastArm Program: Placing the engineer in control.’’ Proc., 3rd Congr. on Computing in Civ. Engrg., ASCE, New York, 473–479. APPENDIX II. NOTATION The following symbols are used in this paper: K1 = effective length factor for upper column segment; K2 = effective length factor for lower column segment; kR = stiffness of rotation spring (restraint) at bottom end of column; a = ratio of moment of inertia of bottom section to that of top section; b = ratio of bottom segment to total length of column; and m = ratio of load at stepped level to load at top. 278 / JOURNAL OF COMPUTING IN CIVIL ENGINEERING / OCTOBER 2000 J. Comput. Civ. Eng. 2000.14:273-278.