United StatesDepartment ofAgriculture

Forest Service

ForestProductsLaboratory

GeneralTechnicalReport FPL-40

Purdue PlaneStructuresAnalyzer IIA Computerized WoodEngineering SystemStanley K. SuddarthandRonald W. Wolfe

Abstract

The Purdue Plane Structures Analyzer (PPSA) is a com-puter program developed specifically for the analysis ofwood structures. It uses recognized analysis pro-cedures, in conjunction with recommendations of the1982 National Design Specification for Wood Construc-tion, to determine stresses and deflections of woodtrusses and frames.

The program offers several options for the analysis ofmember capacity, depending on lateral support condi-tions, strength property variations, and critical loadassumptions. Tabulated output provides a summaryreport of the input analog as well as individual memberand total structure response to assumed load condi-tions. Program operation requires knowledge ofmaterial properties, member connections, boundaryconditions, and loads. The user also must have aknowledge of structural analysis to interpret the output.

This report provides guidelines for program use and in-terpretation of results and will be helpful to structuralengineers and designers.

Keywords: Matrix analysis, trusses, frames, computerprogram, structures.

May 1984

Suddartlh, Stanley K.; Wolfe, Ronald W. Purdue Plane StructuresAnalyzer II: A computerized wood engineering system. Gen. Tech. Rep.FPL-40. Madison, Wl: U.S. Department of Agriculture, Forest Service,Forest Products Laboratory; 1984. 38 p.

A limited number of free copies of this publication are available to thepublic from the Forest Products Laboratory, P.O. Box 5130, Madison, WI53705. Laboratory publications are sent to over 1,000 libraries in theUnited States and elsewhere.

The Laboratory is maintained in cooperation with the University ofWisconsin.

This report supersedes “A computerized wood engineering system: Pur-due Plane Structures Analyzer.” USDA Forest Serv. Res. Pap. FPL 168.Madison, Wl: Forest Products Laboratory; 1972.

PURDUE PLANE STRUCTURES ANALYZER 4.00, PPSA4

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PPSA is available under several licenses as given below. A special price is designated for singleinstallation copes for non-profit teaching and research organizations. Prices include shipment byfirst class mail.

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Mad order to: Business OfficePurdue UniversityDept. of Forestry and Natural Resources1500 Forestry BuildingWest Lafayette, IN 47907-1500

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Technical Information

Mr. Michael H. TricheDepartment of Civil EngineeringUniversity of AlabamaP.O. Box 1468Tuscaloosa, AL 35487

Phone: 205-348-5834

PPSA 4Purdue Plane Structures Analyzer Version 4.00

PPSA4 is a computer program for the analysis of wood structures. The executable program runs underDOS, however, source code is also available. The primary use of the program is for the analysis ofsimple structures such as continuous beams, or entire (plane) structures such as trusses and frames,Standard version allows up to 99 nodes and 99 members. Program runs in batch mode so that userscan add their own pre and post processing routines if desired. Program includes over 60 pages ofupdated documentation.

Acceptance

Over 20 years of use by practicing Engineers and Architects nationwide.

Preferred wood engineering software program of designers and researchers alike.

Accepted by the Truss Plate Institute for the analysis of metal plate connected wood trusses.

Added Versatility

Complies with NDS-91 or NDS-86. (8 options to choose from for Interaction Analysis)

Fixed or pinned member connections and available fictitious members for semi-rigid behavior.

Partially Distributed Loads (Triangular of Trapezoidal over entire member of any portion).

Up to 10 concentrated loads on each member.

Added flexibility for analyzing Glulam and other structural composites.

Analysis of non-rectangular sections.

Automatic nodal renumbering for complicated structures.

Includes shear deflection in all deflection calculations.

Automatic column length determination for most members.

Engineer Overrides are available for effective buckling and bending lengths if user desires tooverride automated calculations,

Faster Execution

State of the Art matrix analysis method with optimized code.

Utilizes math co-processor chip.

Preface Table of Contents

This manuscript was written as a guide to the use ofthe updated version of the Purdue Plane StructuresAnalyzer (PPSA II) computer program. For convenience,it has been divided into five sections. The first section,for those unfamiliar with PPSA, is a basic introductionto its background, use, and versatility. The second sec-tion provides instruction for the preparation of requireddata input. The third section presents a sample outputand the fourth discusses the significance of the in-teraction analysis. The final section, entitled SpecialTopics, provides guidelines for modeling more complexstructural systems.

For those interested in copying, updating, or modifyingthe program, Appendices A and B provide informationregarding array dimensions and input format.

Arrangements can be made to obtain a copy of PPSA IIby contacting

U.S.D.A. Forest ServiceState and Private ForestryC/O Forest Products LaboratoryP.O. Box 5130Madison, Wis. 53705(608)264-5735

PageIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Program Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 1Input Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . 2Structural Analog . . . . . . . . . . . . . . . . . . . . . . . . . . 2Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Program Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 4Preparation of Program input . . . . . . . . . . . . . . . . . . . 5Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Problem Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Member Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 6Stress Adjustment Factor . . . . . . . . . . . . . . . . . . . . 7Point Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Structural Assembly . . . . . . . . . . . . . . . . . . . . . . . . . 8Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Load Information and Interaction Interpretation . . 10Concentrated Loads . . . . . . . . . . . . . . . . . . . . . . . . . 10Uniform Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Nodal Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Example of Program Output 12Analog Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Analysis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Significance of Output in Table lX . . . . . . . . . . . . . . . 21Interaction Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 21Effective Column Length . . . . . . . . . . . . . . . . . . . . . 22Lateral Support of a Member . . . . . . . . . . . . . . . . . . 22Interaction Indices . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Special Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24FIRL Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Further Application Flexibility . . . . . . . . . . . . . . . . . 25

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Purdue Plane Structures Analyzer II1983 Version

This program is intended to facilitate the analysis ofwood structures. It gives an accurate report of struc-tural response of the input analog. Accuracy of theanalog and interpretation of structural adequacy arethe responsibility of the user.

i

Purdue PlaneStructuresAnalyzer II:A Computerized WoodEngineering System1

Stanley K. Suddarth, Professor of Wood Engineering,School of Forestry, Purdue UniversityandRonald W. Wolfe, Research EngineerForest Products Laboratory

Introduction

The Purdue Plane Structures Analyzer (PPSA) is acomputer program for the analysis of wood structures.It is a tool for accurate prediction of the behavior of anystable wood truss or frame that can be represented by atwo-dimensional analog. Introduced by Suddarth in1972, it has found repeated application In areas ofdesign and research. Its most frequest use has beenthe analysis of wood trusses.

Over the 10 years from 1972 to 1982, a number ofdesign changes have been implemented in the NationalDesign Specification (NDS) for Wood Construction(NFPA 1982). Design changes for member stability,combined compression-bending loads, and bucklingstiffness of sheathed compression membersnecessitated modification of PPSA, giving rise to thedevelopment of PPSA II. This program incorporatesthese NDS changes as well as providing addedversatility for more effective analog modeling.

Input and output formats for PPSA II are basically thesame as those used in the original version (Suddarth1972). They have undergone minor changes to provideadded versatility in modeling actual structures.

The program has been set up to use a single unit offorce--the pound--and a single unit of length--the inch.Any other units may be used if the same basic rule isfollowed in the input data. The heading labels in theprinted output indicate pounds and inches, however,and must be changed to reflect any change in unit.

The question of memory storage requirementsfrequently governs the usefulness of a computerprogram. PPSA II allows for changes to be made toaccommodate computer memory or problem sizelimitations. As the program is presently set up, it anddata storage require 27 K words of memory on aUNIVAC 1110 computer.

Program ObjectivesThe PPSA II program performs an analysis of an analogof a real structure. It must be recognized thatcomputers do not substitute for engineers; computersare only a tool. It is the engineer’s responsibility torecognize the limitations of the output to meet all theneeds of a total design and to judge the adequacy ofthe analog in modeling real performance.

Structural design requires awareness of various analogassumptions. These assumptions include both detailsof the analog configuration and the loadingarrangement. Analog configuration assumptions thatmay affect structural response include reactions andend fixity as well as elastic properties of individualmembers. Load assumptions include magnitude,location, and direction of applied loads. To vary theanalog configuration, the entire set of input data mustbe reread. Load variations, however, require only thatthe new loads, along with indices that control thestress adjustment factor and interaction analysisoption, be included as additional cases.

1Prepared in coorperation with Purdue University; the paper is JournalPaper 9632 of the Purdue Agricultural Experiment Station.

PPSA II can analyze joints as either rigid or pinnedconnections between individual members. Experiencewith a given type of structure may be sufficient forchoosing between the simple fixity types. In doubtfulcases more information can be obtained from a seriesof computer runs using alternate choices of rigidity.This will reveal the sensitivity of stresses anddisplacements in the structure to the fixityassumptions.

As a further refinement, the user can approximatepartial fixity through the insertion of short fictitiousmembers. These are rigidly connected to nodes at bothends but are made more flexible than the membersjoined so as to simulate axial and rotationaldisplacements in the connection. The use of fictitiousmembers requires knowledge of the load-displacementcharacteristics of the structural connection beingmodeled. Again, several computer runs with varied

stiffness parameters can provide useful information onthe sensitivity of stresses and deformation to thesevariations.

Input RequirementsThe matrix analysis methods used in this programrequire the engineer to form an analog model tosimulate a real-structures response to various loadconditions. A model such as that shown in figure 1consists of a structural analog and representativeloads.

Structural AnalogThe structural analog is read in as four data groups:member properties, node coordinates, structuralassembly, and reactions. Each group requires someengineering judgment regarding either expectedstructural performance or program limitations.

2

Analysis of members for design interpretation isdependent upon their use, grading, and mechanicalproperties, The engineer has a choice of severalmember categories to account for lateral supportconditions, effects of grading method, seasoning, andfunction in the structure. These categories are often ofgreat importance in the automated application of NDSrequirements within the program. If the analog memberrepresents a real member in the structure, the usermust specify dimensions, modulus of elasticity (MOE),and allowable stress in bending, compression, andtension.

The fictitious member is one that does not correspondto a real member in the structure but is used to provideversatility in modeling semirigid connections, flexiblereactions, or geometric features such as joint eccen-tricity. Three example applications are shown in figure2. Three sample cases using fictitious members inrelating analog models to structural test data havebeen reported (Suddarth and Percival 1972).

Nodes, defined as the intersection of two or moremembers, are read into the computer as a matrix of rec-tangular coordinates. In figure 1, the nodes are labeledby circled numbers. Members are shown as straightlines connecting two nodes and are labeled by numberswith arrows drawn through them. The arrow indicatesmember direction, negative end to positive end.Assignment of member direction is primarily aconvenience feature that facilitates understanding andinterpretation of the output. The sign convention thathas been found to work well defines positive memberdirection as being up or to the right. Thus member dataare input by designating their negative and positive endnodes. Member input must include the member’sattachment to its end nodes. Fixed or rigid nodeconnections are conceptualized as shown in figure 3. Inthis instance, the node is pictured as a disk to whichmembers A and B are rigidly connected and member Cis pinned. This simple representation of a joint wouldtransfer an in-plane bending moment from member A tomember B with no effect on member C. It will transferaxial and shear loads to all connected members.

Permissible reactions (shown in fig. 4) include fourgeneral categories: PIN, ROLL, FIX, and a special typecalled the FIRL reaction. The purposes of the first threecategories are implied by their names as traditionallyused in structural analysis: A PIN reaction resistshorizontal and vertical translation but offers noresistance to rotation; a ROLL reaction permits rotationand translation in one specified direction; the FIXreaction resists all movement. The FIRL reaction isidentical to the ROLL except for the further restrictionthat it does not permit node rotation. FIRL reactionshave special uses and are discussed more fully, withan example, in the Special Topics section.

Figure 2.--Applcations of fictitious members.Applications a and b can model jointeccentricity as well as flexibility. Variation ofdimensions and elastic properties enable afictitious member to model elastic supportconditions as shown in c. (ML83 5588)

3

Figure 3.-Joint mechanism representing anode as a flat disk with members A and Brigidly connected and member C pinconnected. (ML83 5589)

Figure 4.-Reaction options may be selectedto resist all or no rotation and all or partialtransition. PIN permits rotation withouttransition. FIX permits no movement. ROLLpermits rotation and translation in one direc-tion. FIRL permits transition in one directionwithout rotation. (M139 733)

LoadsApplied loads must be classified in one of threegeneral categories: concentrated, nodal, or uniform.Concentrated loads are point forces applied to amember; nodal loads are point forces or momentsapplied at a node; and uniform loads are appliedcontinuously over the length of an individual member.

The program interprets this input in terms of systemcoordinate axes. The loads are read in as componentsoriented parallel to the horizontal and/or vertical axesof the structure, commensurate with a standardCartesian coordinate system: positive up and to theright. For purposes of member stress analysis, theprogram converts these loads from a structureorientation to a member orientation, giving loadcomponents parallel and perpendicular to the loadedmember length.

In the case of uniform loads, only inclined memberscan accept both horizontal and vertical components.Horizontal members will not accept a horizontaluniform load and vertical members will not accept avertical uniform load.

Program OperationInput data are used to develop a stiffness matrix foreach member, which, in turn, is incorporated into asystem stiffness matrix. Derivation of the elements ofthis stiffness matrix assumes a shear modulus (G)equal to 1/16 of the MOE. The value of G is thenadjusted by a shear coefficient (called ALPHA in theprogram) of 6/5 (Orosz 1970) for rectangular sections.Both G and ALPHA have been programmed to facilitaterevision if the program user is analyzing other thanrectangular sections.

The matrix analysis of structures is a convenient pro-cedure that takes advantage of the capabilities ofdigital computers to perform structural engineeringtasks that were formerly considered economicallyprohibitive. Specifically, this program involves a linearstiffness method solution based on a virtual energyformulation. This topic is well established in textbookform (Przemieniecki 1968; Zienkiewicz 1967) andengineering journal articles. Because of the prolificsupply of background material, this report does notdiscuss theory or computer program details but ratherpresents the instructions for using this specific matrixanalysis program tailored to the needs of woodengineering.

in addition to echoing input, the standard programoutput inciudes reaction values, forces and moments atthe ends of each member, the critical axial and bendingstresses, maximum shear stresses, memberdeflections, and node displacements. Input optionspermit the user to eliminate unwanted items of output.

4

Preparation of Program Input

User-computer communications may make use of eithera keyboard computer terminal or cards for entering theanalog data. In the following discussions, the terminput line or line refers to the data recorded on onecard or one line of type. The PPSA II program Isformatted to read input data one line at a time and toidentify individual values by line sequence and locationIn the input line. Appendix A shows input data codingforms that may be used to facilitate the input process.

The user will note that this Input is quite detailed,requiring considerable preparation time. It has beenkept in this form to afford maximum flexibility in a widerange of applications. Some users may wish to writetheir own preliminary program that takes simplifiedinput instructions and develops this data file, utilizingtheir own rules for creating the analog of a specificclass of structures.

Integer values are often used to number or index anItem of input. Their values are expressed as wholenumbers and should always be right-justified in theirentry fields. A decimal point placed in an integer fieldwould not be recognized and would be Interpreted asan error. Real values, on the other hand, consist of bothwhole number and decimal portions. Input data formatsfor this program are written so that if no decimal issupplied by the user for a real variable, it isautomatically supplied by the program to the right ofthe last character In the specified data field, Realnumber Inputs must recognize these rules. Forinstance, if the real number 25 is to be entered incolumns 13-17 of an input data line, it may be read inas 25.0b or b25.b or bbb25. However, if it iS input asb25bb, the value read will be 2500. Real values mayalso be read using an exponential format. In thisinstance, the value is separated into a mantissa and anexponent by the letter E. For example, 1 million wouldbe read into a 7-character field as + 1.0E06. Anynumericai variables not labeled integer should beassumed to be real.

input data have been divided into eight categories:identification, problem size, member properties, stressfactor, point coordinates, structural assembly,reactions, and loads. The following discussiondescribes each of these categories and makesreference to the analog shown in figure 5 to illustratethe manner in which the categories are to be enteredfor program execution.

Figure 5.-Analog diagram for a single pitchedtruss supported at three points andsubjected to a uniform load, a concentratedload, a nodal load, and an end moment. Themoment (1657.7 lb-in.) and nodal load (128.8lb.) at node 4 represent the effects of a 2-footoverhang extension of the top chord. (M139730)

IdentificationAny keyboard characters can be used to Identify theparticular analysis being processed except thesequence ENDDCALC in columns 1-8. The identificationcharacter string may occupy up to 68 columns on oneline.

Example entry:

The sequence of characters read on this line is printedout as a heading for the program output.

5

Problem Size Member PropertiesThe Problem Size line tells the computer the size of thestructure being analyzed, the amount of output wanted,and the number of load situations to be considered.

Member Property lines describe the membermechanical properties, sizes, and type of use.

Line format:Line format:

Columns l-2--lnteger: Number of points (NP) used in theanalog.

Columns 3-4--lnteger: Number of members (NM) in theanalog.

Columns 5-6--lnteger: Number of Roll (NR) reactionsused in the analog,

Columns 7-8--lnteger: Number of Pin (NPIN) reactions inthe analog.

Columns 9-10--lnteger: Number of FIRL (NFIR) reactionsused in the analog.

Columns 11-12--lnteger: Number of FIX (NFX) reactionsused in the analog.

Column 130 or blank = List all input (tables I-VI).1 = Do not list input.

Column 140 or blank = List all output (tables VII-XII).1 = List only tables Vll and Vlll--reactions and

member end actions.2 = List only table lX--analysis.3 = List only tables IX and X--interaction and shear

stresses.4 = List only tables Xl and XIl--deformations and

displacements.5 = List only tables IX-XII.

Columns 15-16--lnteger: Number of load cases (NLOAD).If only one load case is to be considered, this may beleft blank. The purpose of this input is to permitrepeated analyses of a structural configuration underdifferent loads without the added time and expenseof reentering member and joint information andrecalculating the structural stiffness matrix. Subse-quent load cases operate at higher speed at a frac-tion of the computing time cost of the initial loadcase.

Columns 17-18--lnteger: Number of locations checkedfor stress and deformation along each member. If leftblank, 24 points will be checked along each member.

Example entry:

Columns 1-2--lnteger: Member group number beginningwith b1. Each variation in member size, strength, ortype of use creates a so-called member group andthus requires a separate input line.

Columns 3-7--Thickness, in inches, of the member in thedirection perpendicular to the plane of the structure.Columns 8-12--Depth, in inches, in the plane of thestructure.

Columns 13-18--MOE, in pounds per square inch, of themember material. Because of the magnitude of thisnumber, all MOE values are read in using an exeponential format. An MOE of 1,760,000 could be writ-ten 1.76 × 106: The computer recognizes this numberwritten as 1.76E6 in the six-column field.

Columns 19-22--Normal duration allowable bendingstress in pounds per square inch (ASIGM). This andthe following two entries come from lumber gradestress tables such as are found in the NDS (NFPA1982) and are entered as whole numbers with thedecimal supplied by the machine at the right-handside of the field. When a special stress adjustment ispertinent to only one category of stress, the adjustedvalue should be used in the appropriate field. An ex-ample for bending stress is the 15 percent Increasefor repetitive member use (table 4A of NDS). Sincethis factor is applicable only to bending stress (Fb),the adjustment should be made prior to entry in thisfield. The program estimates this adjustment factoron the basis of the effective bending length overrideentry discussed later under Structural Assemblyinput.

The ASIGM field is left blank if the member is fic-titious and not subject to shear deflection. If sheardeflection is to be included in a fictitious member,any nonzero entry will suffice, although it is convenient to use a number like 9,000, which is obviouslynot a wood stress value.

Columns 23-26--Normal duration allowable compressionstress, in pounds per square inch. This field is leftblank if the member is fictitious.

Columns 27-30--Normal duration allowable tensilestress, in pounds per square inch. This field is leftblank if the member is fictitious.

6

Column 31--lnteger: Type use classification for nonfic-titious members with the following code. For thisdiscussion, Member refers to main structuralmembers such as truss chords, and Interior Membersare secondary structural members such as trusswebs. This column is left blank for a fictitiousmember.0 o r b = Member, laterally supported as bysheathing, etc. (NDS 3.3.2) (S).1 = Member, not laterally supported (U).2 = Interior Member, laterally supported as by ade

quate bracing (SI).3 = Interior Member, not laterally supported (Ul).4 = Truss Chord, seasoned (NDS 3.10.5) (TD).5 = Truss Chord, green (NDS 3.10.5) (TG).6 = MSR Truss Chord, seasoned (NDS 3.10.5) (TDM).7 = MSR Truss Chord, green (NDS 3.10.5) (TGM).8 = MSR Member, laterally supported (SM).9 = MSR Member, not laterally supported (UM).A = MSR Interior Member, laterally supported (SIM).B = MSR Interior Member, not laterally supported(UIM).

Column 32--Last line indicator, which is 1 to indicatethat the next line begins a new type of data and isblank otherwise.

Entries 4 through A listed for column 31 were added tothe PPSA II program in response to new categorizationof compression members originating with the previousversion of the NDS. Special consideration is given inparagraph 3.10.5 of NDS to 2 x 4 trusses spaced 2 feeton center or less and suitably sheathed with plywoodas in residential construction. Also, reduced variationin MOE for MSR lumber leads to different columnformulas, as shown in Appendices G and O of the NDS.

When the member is classified as laterally supportedand is detected by the computer as a compression orbending-compression member, the interaction analysisis performed with column calculations referred only tomember depth in the plane of the structure. When themember is classified as not laterally supported,analyses are performed in the plane perpendicular tothat of the truss as well. The most critical interactionvalue is reported.

Example entry:

Stress Adjustment FactorThe Stress Adjustment Factor is used to adjust normalduration stress for design load in accordance withsection 2.2.5 of NDS. Any other general strengthalterations applicable to all elements in the entirestructure, such as the use of fire-retardant-treatedlumber, can also be combined into this one factor. Allallowable stresses from the previous line group aremultiplied by the factor prior to use within the program.

Up to three stress adjustment factor values may beread In to allow for treatment of different load cases.The first factor is entered in columns 1-5, the second incolumns 6-10, and the third in columns 11-15. An indexis read in later, along with load input, to specify whichfactor is to be used. If no index is given, the first factoris selected.

Example entry:

Point CoordinatesPoint Coordinate lines give the Cartesian coordinatesof each point in the analog. Member lengths anddirection cosines are derived by the program fromthese values. In setting up this coordinate system,careful selection of the origin may facilitate futurerevision of the analog. For example, if a complexsystem such as that shown in figure 1 were being used,it might be advantageous to place the origin at point 2,This would permit variations in the depth of the floortruss without affecting the node coordinates for therest of the structure.

Line format:

Columns 1-2-integer: Node number as shown on theanalog.

Columns 3-9 The x coordinate of the node, in inches.Columns 10-16 The y coordinate of the node, in inches.Column 17: Last card indicator, which is 1 for the last

card and blank otherwise.

Another important consideration in the assignment ofnode coordinates is the order in which nodes arenumbered. For a given structure, all numbering systemslead to the same size square symmetric stiffnessmatrix. Figure 6 compares the square stiffness matrixwhich requires N2/2 locations for storage of the upperor lower triangle to a banded form requiring only N x Bstorage locations. For a plane structure in whichmember end conditions consist of axial loads, shear,and moments:

N = 3 x (number of nodes) - (number of reactionconstraints)

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Figure 6.--A symmetric stiffness matrix thathas all nonzero terms within a distance Bfrom the diagonal may be stored in bandedform to conserve computer memory space.(ML83 5590)

The width (B) of the banded matrix is dependent uponthe member having the largest difference between endnode numbers:

B = 3 x [(maximum difference) + 1]

To minimize the number of steps required for theanalysls, the value of B should be kept as small aspossible by numbering nodes in such a manner as tominimize the difference between member end nodenumbers. For example, the analog shown in figure 1has a maximum difference of only 4. As the program iscurrently written, a difference greater than 11 willcause the error message:

BANDWIDTH OF STIFFNESS MATRIX IS GREATERTHAN ALLOWABLE STORAGE AREA

Example entry:

Structural AssemblyThe Structural Assembly lines identify the two endnodes to which each member is connected, designatethe connection as either fixed or pinned, and classifythe member according to member group as listed onthe member property input lines. In addition, theStructural Assembly lines give the user the option ofindependently specifying effective column lengths foreach member in planes parallel and perpendicular tothe plane of the structure and specifying effectivebending length for beams lacking continuous edgesupport and/or torsional restraint at their bearingpoints (NDS 3.3.3 and 4.4).

Line format:

Columns 1-2--integer: Member number.Columns 3-4--integer: Node location of the negative end

of the member.Columns 5-6--integer: Node location of the positive end

of the member.Columns 7-8--integer: Member group number to assign

appropriate member properties according to thenumber designated in the Member Properties line.

Column 9--Condition of negative member endconnection to the node:0 = Rigidly connected.1 = Pin connected.

Column 10--Condition of positive member endconnection to the node:0 = Rigidly connected.1 = Pin connected,

Column 11--Last line indicator, which is 1 for the lastline of member data and blank otherwise.

Columns 12-17--Effective column length, in inches, inthe plane of the structure (COLH) (optional override).

Columns 18-23--Effective column length, in inches,perpendicular to the plane of the structure (COLT)(optional override).

Columns 24-29--Effective unsupported bending length,in inches (BLE) (optional).

Fixing intersecting members to a node causes them torotate together and share moments with other membersthat may likewise be fixed to the same node,Pinning a member end at a node causes a zero momentat that member end and creates independence betweenrotation of the node and rotation of the member end.The original PPSA (Suddarth 1972) required that at leastone member end be fixed to each node in the analog toavoid an arithmetic error within the computer. PPSA II

8

has been structured to allow a node to be free offixation from any member, The node is then fixedagainst rotation in the analysis and a zero rotationaldisplacement will consequently be reported,

Specified values for effective column length can beused in the analysis of column buckling. These inputoptions override the automated effective column lengthdeterminations made by the computer. An example of ause for this option would be the analysis of trusschords laterally supported only at spaced intervals bypurlins. If a member is defined as being laterallysupported, no analysis is performed on columnbuckling in the perpendicular-to-plane direction. WhenIateral support is less than complete, the user mayenter the effective column length in the perpendicularplane due to purlin support and declare the memberIaterally unsupported. This will cause an analysis tooccur in both directions using the engineer’s specifiedeffective column length.

Structural cases can arise in which the end or ends ofa compression member are not confined to smalldisplacements in a direction perpendicular to thecritical column axis. This requires the engineer to usethe appropriate optional column length override (COLHor COLT) to input a suitable value for the effectivecolumn length.

Bending members with inadequate rotational or Iateraldisplacement restraint and a depth-to-thickness ratiogreater than 1 are subject to out-of-plane displacement(NDS 3.3.3 and 4.4). For these cases, the allowablebending stress should be adjusted by a slendernessfactor (NDS 3.3.3.4). PPSA II calculates this factor,using the BLE input (cols. 24-29), for any memberhaving a positive ‘effective unsupported bending length’and a ‘not laterally supported’ member groupdesignation (MTYPE = 1, 3, 9, or B on MemberProperties lines).

Member numbering has no effect on storage orcomputation time. A numbering system should beselected on the basis of convenience of output clarity.For the analog shown in figure 1, for instance,members are numbered consecutively with groups (i.e.,wall studs, truss chords, and truss web members). Analternative to this would be a floor, wall, roof division.The shed roof being used as a demonstration model(fig. 5) is simply numbered in a clockwise sequencebeginning at the lower left reaction.

Example entry:

Structural assembly entries are for the shed frame (fig.5). Note that no overrides on column or beam lengthhave been specified in this example.

ReactionsThese lines locate React ions at their respective nodes,give the types, and designate the direction of rollermovement when pertinent.

Line format:

Columns 1-2--lnteger: The node number at which thereaction occurs.

Columns 3-6--Name: The name of the kind of reactionfrom among the four types: PIN, ROLL, FIX, and FIRL.

Columns 7-11--Horizontal component of the vectordescribing the direction in which rollers are free tomove. This field is blank in the case of PIN or FIX.

Columns 12-16--Vertical component of the vectordescribing the direction in which rollers are free tomove. This field is blank in the case of PIN or FIX.

Column 17--Last card indicator, which is 1 for the lastcard and blank otherwise.

Example entry

9

In this shed frame example, the roller reaction at node2 permits rotation, moves only In the vertical direction,and resists movement due to any loads in thehorizontal direction. The roller at node 5 moves on a 2on 12 slope. The pin reaction at node 1 resistsmovement due to either horizontal or vertical loads butpermits rotation.

Load Information andInteraction interpretationFor each load case the user must specify load type,method of analyzing the Interaction of axial andbending stresses, and an allowable stress adjustmentfactor. The Load Information and Interaction inter-pretation line provides these specifications through theuse of option indices. The first three columnsdesignate yes/no options for concentrated, uniform,and nodal loads, respectively. Columns 4 and 5 eachpermit up to four options for interaction interpretationand allowable stress modification respectively.

Line format:

Column 1--Concentrated load indicator: Enter 1 if thereis at least one concentrated load on the structure;otherwise, enter 0 or leave blank.

Column 2--Uniform load indicator: Enter 1 if there is atleast one uniform load on the structure; otherwise,enter 0 or leave blank.

Column 3--Nodal load indicator: Enter 1 if there is atleast one force or moment applied directly to a nodein the structure.

Column 4--lnterpretation of interaction equation option(ITP) PPSA II allows for some engineering choice inthis matter with the optional entries 0, 1, 2, or 3. Eachreal member in the structure is treated the same wayaccording to this option. The choice relates primarilyto where the stresses for the interaction formula,NDS 3.10, are calculated. The consequences ofchoice appear in table IX of the output and arediscussed in more detail In a later section.

Option O.--This is the most conservative option andcalculates the interaction value using bending stress atthe point of maximum moment and axial stress at thepoint of maximum compression. The two maxima donot have to occur at the same point.

Option 1.--The interaction equation is calculated usingthe maximum bending stress and the axial stress at thesame location as the maximum bending stress. Thisoption is set up so that a constant or zero momentstress over the full length of the member will cause theinteraction value to be calculated at the point ofmaximum axial stress. Cases in which moment isconstant over only part of the member should becarefully examined for a more critical interaction valuethan that reported.

Option 2.--The interaction equation is calculated usingthe maximum axial stress and the bending stress at thesame location as the maximum axial stress.

This option is set up so that a constant axial stressover the full length of the member will cause theinteraction value to be calculated at the point ofmaximum bending stress. Cases in which axial force isconstant over only a part of the member should becarefully examined for a more critical interaction valuethan that reported.

Option 3.--This option closely follows the methodologyunderlying the development of the Truss Plate Institutedesign specification for 1978. It pertains only tostructures of the truss type with uniform loading. Loadsmay be imposed at nodes but concentrated loadsdirectly applied to members are not allowed. Theinteraction equation is calculated at each member endnode using the short column interpretation given inNDS 3.10.2.2. A third interaction value is calculated atthe point of maximum moment within the span usingthe compression stress at that point and anautomatically calculated effective column length. Thelargest of the three interaction values is used for themember.

Column 5--Stress factor index: This index correspondsto the order in which the stress factors are given on thestress adjustment factor line. A blank, 0, or 1 willcause the use of the first value. A 2 or 3 will result inthe use of the second or third value, respectively, andany index greater than 3 will result in a stress factor of1.0.

Example entry:

The example structure is loaded by at least one of eachload type, the interaction analysis will use themaximum bending moment and maximum compressionforce values in each member span, and the first stressfactor read in will be used.

Concentrated LoadsConcentrated Load lines identify the member carryingthe load, its magnitude and direction, and its locationalong the member. Each member may carry up to threeconcentrated loads. These are numbered sequentiallybeginning at the negative end of the member. Eachload requires a line of input.

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Line format:

Columns 1-2--lnteger: Member number carrying the load.Column 3--Integer: Load number 1, 2, or 3.Columns 4-9: Horizontal or x component of the load, in

pounds (positive to right).Columns 10-15: Vertical or y component of the load, in

pounds (positive up).Columns 16-21: Distance from the negative end of the

member to the load point.Column 22: Last card indicator, which is “1” for the

last line and blank otherwise.

Example entry:

Uniform LoadsUniform Load lines identify the loaded member andgive the horizontal and vertical uniform loadcomponents for that member (in pounds per unitlength) where length Is expressed in the same unitsused in defining node coordinates.

Line format:

Columns 1-2--lnteger: Member number.Columns 3-12: Horizontally applied uniform load, in

pounds per inch (positive to right).Columns 12-22: Vertically applied uniform load, in

pounds per inch (positive to up).Column 23: Last line indicator, which is 1 for the last

line and blank otherwise,

Example entry:

Nodal LoadsNodal Load lines Identify the node at which the load isapplied, the type of load, horizontal or vertical force (inpounds), or a moment (in inch-pounds), and themagnitude. One line is required for each load.

Columns 1-2--lnteger: Node number carrying the loadconcerned.

Column 3: The type of load which is coded as 1 for aforce in x direction, 2 for a force in the y direction,and 3 for a moment.

Columns 4-11: The force (in pounds) or moment (inpound-inches), whichever is the case. Forces arepositive to the right and upward to agree with the x,ysystem. Positive moments are counterclockwise.

Column 12: Last card indicator, which is “1” for thelast nodal load card and blank otherwise.

Example entry:

If more than one load case is to be used, as indicatedby the entry in NLOAD on the problem-size line, thenecessary information is entered after the last card ofthe previous load case. Required input for subsequentload cases begins with the Load Information andInteraction Interpretation line and continues through tothe end of each set of load input lines.

The last line of input data must contain ENDDCALC incolumns 1-8. This entry is required for normaltermination of program operation.

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Example of Program Output

Results of the computer analysis performed using the Tables II, Ill, IV, and V provide a description of analogPPSA II program are presented in the form of 12 tables physical properties. These include node coordinates,of output. The first six tables summarize the analog member layout, and the locations and types ofbeing considered and the last six provide an analysis of reactions. Table V provides a sequential listing ofstresses and deformations that would result from the members along with their Iength, optional overrides onloads imposed on the analog. column length;, and effective lenghts for beam

Analog Summaryslenderness calculations. Finally, table VI lists theloads carried, along with their locations and directions.

The computer report begins by Identifying the problem Example tables I-VI:(ID input line) and indicating basic statistics of theanalog. Table I shows properties of componentmaterials being considered for the structure along withthe classification of member use type.

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13

14

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Analysis SummaryThe results follow the data in three sections; the firstsection (table VIl) gives reactions; the second (tablesVlll, IX, and X) summarizes strength analysis; and thethird (tables Xl and XII) provides deflection information.

Table Vll Iists the reaction locations, magnitude, anddirection (horizontal, vertical, or moment). It alsochecks to see if the sum of forces and moments at thereactions are equal in magnitude and opposite indirection to the applied loads. The reactions aredetermined by summing end actions for all memberends intersecting each reaction node and thus wouldcontain any errors in the matrix analysis of the totalstructure. An excellent solution check is provided bysumming the resulting horizontal and vertical reactionforces as well as their moments about the x-y originand comparing these values with corresponding sumsfrom input loads. These sums and the differencebetween corresponding load and reaction values areprinted out immediately following table VII.

Example table VII:

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Member strength analysis begins with table Vlll. Thistable gives the axial and shear forces as well asbending moments occurring at the ends of eachmember. Table values are most conveniently utilized ifthey are referred to a standard member position thatinvolves a horizontal orientation with the negative endof the member located to the left. The sign conventionused for shear and axial forces is positive upward andto the right on a member in standard position. There-fore, axial load signs will be the same as the memberend for tensile members and opposite to member endfor compression members-. i.e., negative on the negativeend for tension member, etc. Bending moment signsfollow the common mathematical right-hand rule:counterclockwise is positive. Figure-7 provides agraphic representation of the member end actionanalysis of member 3 as interpreted from the member 3line in table Vlll.

Example table Vlll:

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Table IX of the output gives the results of theinteraction analysis. This analysis is dependent uponthe member type, the loads it carries, its length-to-depth ratios in or normal to the plane of the structure,and the interaction analysis option (ITP) chosen on theLoad Information and Interaction Interpretation inputline.

The first three columns provide preliminary information.Column 1 identifies the member by its number andgroup classification. Column 2 provides space for afootnote reference if additional information is neededto evaluate member design. The third column, labeledMAX INT VAL, gives the maximum value of theinteraction equation (combined stress index) within themember if both axial and bending stresses are present.If only axial or bending stress is present, theinteraction column contains the ratio of actual stressto allowable stress. The interaction concept is treatedin NDS 3.10 and is discussed in more detail in thefollowing section.

To find the maximum interaction value and themaximum shear stresses, bending, axial, and shearstresses are calculated at equally spaced locationsalong the length of each nonfictitious member. Thenumber of locations analyzed is determined by thevalue read into columns 15-16 of the Problem Size inputline. The input default in this case results in analysis at24 locations. Maximum values along with theirlocations are then printed out in tables IX and X of theoutput report. If a maximum stress value occurs morethan once, the location reported is usually that nearestthe positive end of the member.

The designer will note that many useful facts can bederived from table IX. For instance, the member 6portion of the lower chord (fig. 2) is more heavilyloaded, having an interaction value of 0.88 (table IX). Inthe upper chord, on the other hand, members 2 and 3have relatively low interaction values--0,505 --suggesting that a lower quality lumber may beused at these locations.

Stresses reported in the table are the values requiredfor the interaction option selected. They have algebraicsigns following the convention that axial tension ispositive and bending is positive when the tensile stressis along the bottom edge of a member placed instandard position (horizontal with the negative end tothe left).

Table IXA has been included to give supplementaryinformation Including analog length and the finaladjusted allowable stresses used in the calculation ofthe combined stress index reported in table IX.

Table X reports maximum shear stress in each memberfrom analyses made at the same points where bendingand axial stresses have been determined. if the samemaximum is repeated, the location reported is thatnearest the positive end of the member.

Footnotes such as that referenced for member 7 alsoprovide additional information. In the case of member7, the footnote tells the user that the critical stresscase occurs perpendicular to the plane of the structure.

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Example table lX:

Example table X:

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The last two tables of the computer report deal withdeformation. Table XI gives maximum memberdeflections and table XII lists horizontal, vertical, androtational displacements of nodes.

For table Xl, deflections are calculated at each of thelocations along the member for which interactionvalues and shear stresses have been calculated. Themaximum value and its location are reported. Althoughshear modulus is recognized in the formulation of thestructural stiffness matrix, it is not used in thecalculation of maximum member defections reportedin table Xl. The sign convention used for deflectiontabulations is mathematically standard, designating anupward deflection as positive when the member is instandard position.

Node displacements are reported in table XII. Positivetranslational displacements are upward and to theright. The rotational displacements have been reportedin exponential form to accommodate the wide rangesof values that may be encountered. For example, 1degree would be given as 1.745E-02 radians. The signconvention for rotational displacements interpretscounter-clockwise as positive. The calculation of nodedisplacements includes the influence of sheardeflection.

Example table Xl:

Example table XII:

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Significance of Output in Table IX

Data from the deflection tables can be combined toindicate how the member moves in space and deformsunder load. This has been done for member 3 in figure8. This illustration of member 3 before and afterloading uses an exaggerated scale to emphasize theeffects of the loads on node displacement and memberdeflection. If the deflection of a particular point in amember is needed, that information can be obtaineddirectly by constructing the analog with a node at thelocation in question and rigidly connecting it to theadjoining segments of the member. The desireddisplacement will then be given in table XII, butautomated equivalent column length calculationsunderlying values reported in table IX may be incorrectin such a case, and the stress information from thistable should be interpreted accordingly.

This completes the analysis of the example shed rooftruss. If the value of NLOAD in columns 13-14 on inputcard number 2 is set at a value greater than 1, theprogram will proceed to read in the next set of loaddata for the previously defined analog. Outputconsisting of tables Vll through IX or a selected sheetwill be produced for each load case. Once all loadcases have been processed, the program will try toread another complete set of analog data. When thereare no more data to be read, the line following the lastload line must have the letters ENDDCALC in the firsteight columns to terminate the program. This series ofletters is always used by the program for normaltermination of operations and must always be presentas the last line in any input data file.

Figure 8.-Member 3 from the analog of theexample truss is shown in the before andafter loading position with deflections drawnto an exaggerated scale. The nodedisplacement and maximum memberdefection (tables Xl and XII) provide the datafor this sketch. (ML83 5592)

Table IX provides information regarding the maximumstress interaction value (combined stress index) foreach member. The interaction analysis option (ITP),selected on the Load Information and InteractionInterpretation line, is printed out immediately below thetable title. It also provides additional information, in theform of footnotes, pertaining to members that mayrequire additional consideration beyond the scope ofthe PPSA II program.

Information presented in this table is useful for a widerange of applications, but its limitations must beunderstood. This discussion summarizes theassumptions and analysis procedures used to estimatethe values presented in table IX.

Interaction AnalysisThe values reported in column 4 of table IX arecalculated using one of three equations taken fromNDS 3.10:

Bending-Tension combined stress index

where fb, ft, and fc are imposed bending, tensile, andcompressive stresses, respectively, determinedaccording to the ITP options included in thediscussion of the Load Information and InteractionInterpretation input line, and

Ft and Fb are allowable tensile and bending,respectively, taken from Member Properties lines andadjusted by the value given in the Stress Factor line.Fb is further limited by the size factor given in NDS4.3.4 and 5.3.4 if its depth exceeds 12 inches.

Fc' depends on column length adjustments (NDS 3.7and 3.10) and may range from the allowablecompression stress Fc (adjusted by the Stress Factor)to an allowable stress for long columns.

The variable Fb' is the modified bending stressaccounting for a slenderness factor in members havingless than complete Iateral support on the compressionside. NDS 3.3.3 gives information to determine theeffective length, Ie, which is input as BLE in columns

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24-29 of the Structural Assembly lines. Fb' may beequal to or less than Fb depending on the value of theeffective length, beam cross-sectional dimensions, andMOE. A further requirement is that Fb' determined inthis manner Is also limited not to exceed Fb adjustedby the size factor given in NDS 4.3.4 and 5.3.4. This isautomated within the program utillzing the inputs ofmember width, depth, and BLE. An appropriate value ofFb' is produced for use in equations (2) and (3). If novalue IS entered for BLE, it is taken as zero.

The variable J (NDS 3.10.2) is an interpolation valueused to modify equation (3) for intermediate columns.Intermediate columns have I/d ratios between 11 and k,in which k is the transition ratio between intermediateand long columns (NDS 3.10.5). Within this range, Jvaries as a linear function of f/d ratio from zero forshort columns (f/d < 11) to 1 for long columns (l/d > k)(NDS 3.7,3.10.5 and appendix G).

Effective Column LengthThe effective column length, as defined in NDSappendix N, is the distance between two points alongthe length of a compression member between which itis assumed to buckle in the shape of a sine wave.

A value for effective column length is needed in theplane of the structures for all members subjected tocompression stress. It may be supplied as COLH in theStructural Assembly input by the user. If COLH isentered as zero or blank, an automated proceduredetermines an effective column length limited to theanalog length of a Member or 0.8 times the analoglength of an Interior Member, The procedure assumesthat the member ends are constrained againstsignificant translational displacement so as to restrictthe effective column length within the analog memberlength. Truss chord and web members usually fall intothis category. The method deals with each compressionmember in the example fashion given for member N infigure 9. The location of N within the structure Isshown in figure 9a along with its end rotationalstiffnesses, KN and Kp, which are quantitiesdetermined within the matrix structural analysis. Anequivalent beam structure on one pin and three rollersupports places member N in a similar end restraintcondition (fig. 9b). The values KN and Kp are duplicatedby suitable lengths for the end spans. An axial load, P,is placed at the roller end of N acting against the pinsupport at its other end. A stability analysis of thefigure 9b structure yields the buckling load, P. Finally,a simple column is created (fig. 9c) of such length, L,that P Is the buckling load. L is then utilized as theeffective column length for in-plane analysis ofmember N.

A value for effective column length may also be neededperpendicular to the plane of the structure. If themember is declared laterally supported, this columnlength is automatically set at 11 (short category). IfCOLT in the Structural Assembly input for a laterallyunsupported member is given a nonzero value, it isused for the effective length in the perpendicular plane.Otherwise, the analysis of a laterally unsupportedmember will use the analog length of a Member or 0.8times the analog length of an Interior Member.

PPSA II can perform analyses of a wide variety of planestructures providing useful values of member endactions and displacements but it cannot detect, withinitself, cases in which the automated treatments ofeffective member length are inappropriate. This isparticularly the case when effective member lengthsshould be longer than the analog member length. Insuch cases the user must supply suitable estimates ofeffective length through the use of COLH and COLT iftable IX values are to be utilized.

Lateral Support of a MemberLateral support conditions can affect the analysis ofany Ioad-carrying member. It is thus important that theIateral support members be given careful attention inthe column 31 entry in the Member Properties lines.This entry, labeled MTYPE1, offers 10 options.

The classification “Member” pertains to a mainmember in the structure such as a truss chord. It willlikely carry primary bending loads and will often beframed as continuing through a joint. An “InteriorMember” is secondary in the structure such as a trussweb framing into a chord. It will likely not carry primarybending loads and, since it is usually jointed to anedge of a member, will often have a longer analoglength than its actual length.

MTYPE1 classes 0 through 3 relate to visually gradedlumber or any other prismatic wood member that wouldrequire the use of column formulae as given in NDS 3.7and beam slenderness factors as given In NDS 3.3.3.MTYPE1 classes 8 through B are similar but pertain toMSR and herein use column formulae as given in NDSG.9 and beam slenderness formulae as given in NDS0.6. MTYPE1 classes 4 through 7 are for a specialclass of trusses as given in NDS 3.10.5. Of all of theseMTYPE1 classes, 1, 3, 9, and B are not laterallysupported. These may require entry of effective lengthBLE in columns 24-29 of the Structural Assembly lines.

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Figure 9.--Determination of effective column length. Member N (a) istreated as the center span of an axially loaded three-span column (b). Theload P is the critical buckling load for this configuration and the effectivebuckling length is the length of a simply supported column (c) for whichP is the critical buckling load. (M151 457)

Interaction IndicesValues reported in table IX are governed, in part, by theITP option selected. Interaction indices given arecalculated using either equation (1), (2), or (3). Theseequations will handle axial tension (eq. (1)), axialcompression (eq. (3)), bending (eq. (2)), tension-bending(eq. (1) or (2)), and compression-bending (eq. (3)) but arenot appropriate for a member that is subject to bothaxial tension and axial compression. In the latter mixedcase, a message MIXED TENSION-COMPRESSION,SPEClAL ANALYSIS REQUiED is printed in lieu of thetabl IX line of values. The engineer can then use tableVlll values for the member in constructing the shear,axial force, and moment diagrams necessary to makeappropriate design decisions. It is worth mentioningthat using PPSA II output values as input tomicrocomputer programs constructed for this purposeis a convenient way to handle such special cases.The complex arithmetic operations have already beenperformed within PPSA II.

For a laterally unsupported member, the interactionindex given is the maximum value from calculationsmade both within and perpendicular to the plane of thestructure. if a perpendicular plane analysis gives thelargest interaction index, a reference (’***’) to thefootnote INTERACTION INDEX ANALYZED FORPERPENDICULAR PLANE will appear in column 3.

If the interaction INDEX reported was calculated usingequation (2) (stress difference), the footnote MAXINTERACTION INDEX IS STRESS DIFFERENCE: NDS3.10.1 is referenced (‘#’) in column 3.

Values reported in columns 4-7 give the stress valuesused to calculate the interaction index and theirlocation within the member selected according to thespecified ITP option.

Column 8 lists the length/depth ratios calculated foreach member. These ratios are compared to limitingvalues of 50 (NDS 3.7.2, 3.8.2) for compressionmembers and 80 for tension members. if either of theselimits is exceeded, these members are flagged by afootnote reference in column 2.

When the length/depth ratio is reported as exactly 11,the effective column length may fail into a shortcategory, eliminating the necessity for calculating anactual value. For instance, in the uniformly loadedtruss option, ITP = 3, the column length/depth ratio isset at 11 (short column) to calculate combined stressindices at panel points.

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Special Topics

The PPSA II program provides options for handlinglarge symmetric structures or odd shapes. Thefollowing discussion describes techniques for handlingthese special cases.

FIRL ReactionThe FIRL reaction behaves like the ROLL reaction withthe added constraint that the reaction node isrestrained against rotation. Figure 10 illustrates thecomparative restraints of the two related reaction typesin a simple situation. The principal use of the FIRL is toreduce the problem size when the structure and loadsare symmetric except for minor reaction details.

Figure 11 provides an example of application. Thebeam at the top is symmetric about the verticalcenterline, and all vertical displacements in the left halfare identical to those in the same relative position inthe right half. Also, all horizontal displacements in thestructure are symmetric relative to the verticalcenterline even though they are asymmetric withrespect to the ground. The internal actions anddisplacements in half of this structure can beduplicated by using the analog containing a FIRLreaction as shown in the lower portion of figure 11. Theproblem size is reduced in terms of program prepara-tion, and less computing time is required whilecomplete information about the original structure canstill be obtained. The horizontal displacements in thesecond analog are given with respect to node 2, whichis the centerline in the original structure. Aside fromthis, all other necessary information is the same if thetotal structure is analyzed.

This example is intended to illustrate the principle anddoes not accurately portray the possible magnitude ofsavings in time and effort. Figure 12 shows a morerealistic application, in which the problem sizereduction is noteworthy. The symmetric truss andloading shown in the upper portion requires 13 nodes, 4node loads, and 23 members, of which 6 carry uniformloads. By splitting the structure at the centerline andusing 2 FIRL reactions at the cut, the problem nowrequires only 8 nodes, 2 node loads, and 12 memberswith 3 carrying uniform loads. The horizontal displace-ments must, of course, be corrected to relate the full tothe half structure in this regard. The latter problem isof such minor magnitude, however, that significantadvantage is realized by the use of the FIRL reactionsin symmetrical structures.

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Further Application FiexibilityApplication of this system is limited only by computercapacity and the user’s creativity in terms of thestructure configurations and combinations of loadingthat can be devised. It is only feasible at this point,therefore, to use three illustrations to stimulate aproper perspective in this regard.

An asymmetrical nonprismatic arched structure, suchas might be constructed by lamination, is shown infigure 13 with a possible analog. It is establishedpractice in structural engineering to approximatetapering members with a series of prismatic ones, anatural and easy procedure with the system describedhere. Once the appropriate number and size ofprismatic parts have been selected, the assembly ofthe analog is a simple matter. The right-hand side ofthe structure requires more detailed consideration,however, since the real member is curved as well astapered. Again, the matter of degree of approximationis decided by the engineer, and the actual member issimulated by a series of short, straight members ofsuitably varying section properties. The analogillustrated could be expanded or contracted in detailaccording to need. If many structures of this generaltype were to be designed, one of them could beanalyzed several times with increasing numbers ofparts in the analog to find that degree of complexityrequired to obtain satisfactory answers. In an analysissuch as this, table IX values would be largely ignored,since the breakup of the real structure into an analogdoes not result in members that can be adequatelytreated in the automated calculation of interactionvalues. Member forces, moments, shears, displace-ments, etc., from the other tables will, however, becorrect for the analog and can be used in achieving theend objective of the analysis.

The second illustration is an example. Many timeslumber members are scarf cut for jointing and the netsection is considerably reduced adjacent to the jointconnector assembly. Stresses at this critical sectioncan be obtained from table IX by placing a short realmember having reduced net section dimensions in theimmediate vicinity of the analog model of the joint.

An asymmetric truss bolted to a single pole rigidlyplanted in the ground, as shown in figure 14, is thethird illustration. An accurate analysis of this structureby traditional methods is a difficult and costly process.The computer system, on the other hand, can treat thisstructure with whatever degree of precision is built intothe analog. One possible analog is shown in the lowerportion of figure 14 with special details. The taperedpole is simulated by a series of four prismatic membersof rectangular section, having section propertiesrelated to average properties at comparable portions ofthe pole class used. The analog length of the pole canbe extended below the groundline to its fixed reactionto compensate for imperfect fixation near the surface

Figure 12. --A more complicated symmetricalstructure provides greater benefit to the useof the FIRL support. This figure shows aparallel chord truss and its reducedequivalent. (M139 738)

Figure 13.--Curved and tapered frames can beapproximated with prismatic segments asillustrated in the analog below the structure.The number of approximating segments canbe increased or decreased according toaccuracy requirements. (M139 739)

25

of the earth. The analog detail in the area where thepole and the central vertical web member of the trussare superimposed is shown in figure 15. Nodes 3 and 8are both junctions for rigid connection of trussmembers as shown in the upper sketch of the latterfigure. The pole section (fig. 15b) is pinned to node 3,and should be pinned to node 8 but continuous inpassing through this node. To accomplish this withinthe rules of the analytical system, node 10 is created ashort distance, say 2 inches, below 8 and the pole ismade up of analog members that join rigidly at 10 only.A short fictitious member is then rigidly connected to10 and extended to 8 where it is pinned. This

Figure 14.--A truss with all joints sufficientlyrestrained to be presumed rigid is fastenedto a pole with suitable timber connectors atthe upper and lower chord. These connectorsare presumed to act as pins. The analogdrawn below shows member 11 representingthe truss post, superimposed over member 16representing the pole. Member 11 fastens tonodes 3 and 8, whereas member 16 extendsfrom node 3 to node 10. (M139 736)

superimposed arrangement of members is acceptableto the computer system and reasonably simulates themechanical behavior of the truss pole assembly.

The length of the short fictitious member in theprevious example raises a question of importance tothe program user. Sometimes, as in this case,shortening a fictitious member increases the duplica-tion of reality until a limit of zero is reached. in thematrix solution used, an inversion process is performedthat makes the range of member lengths from longestto shortest a potentially critical matter and theinversion cannot be completed within a reasonabletime limit. in such a case, the computer keeps tryingbut cannot determine final answers for the structure.An increase in the length of the short fictitious memberor variation of some other parameter to decrease itsstiffness will correct the difficulty. Short membersapproximately 1/2 inch long have been successfully runwith long members of over 100 inches having the samecross section and MOE ratios.

Figure 15,-The central region of the trussanalog, shown with exaggerated nodes andlines at left, has all members rigidly attachedat nodes 3 and 8. Member 16, representingthe pole lying behind the center post of thetruss and shown at right, is pinned to node 3but passes by node 8 and is rigidlyconnected to node 10 a short distancebeyond. The pole immediately below thetruss is represented by member 18, which isalso rigidly attached to node 10. The pin-typetimber connection between the pole and thelower chord of the truss is simulated bymember 17 rigidly attached to node 10 andpinned to node 8. (M139 735)

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Acknowledgments Literature Cited

The PPSA II program is the result of contributions bymany people beginning with Gregory F. Reardon ofCSIRO, Melbourne, Australia. Initial formulation andprogramming began during his residence at PurdueUniversity, Lafayette, Ind., in 1966 as a visitingscientist. Since that time many others have contributedto system refinement. At Purdue Quentin B. Comus,Forrest E. Goodrick, Larry A. Beineke, Philip J.Przestrzelski, Michael H. Triche, and Frank E. Woesteare among those who have made significant contri-butions. Roy Adams, while a graduate student atWashington State University, performed a majorreorganization of the program which greatly increasedits speed of operation and made its functional under-standing easier through an improved reorganization ofsubroutines.

This reorganized program includes condensed versionsof subroutines developed at the Argonne NationalLaboratory, Chicago, Ill., to calculate eigenvalues andhas been the foundation for further development duringthe past several years. The authors are also grateful forthe cooperation of member companies of the TrussPlate Institute. Engineers and systems analysts inthese companies have made many trial runs of develop-mental versions of PPSA Il. Their findings andcomments have led to many refinements and improve-ments in the program.

National Forest Products Association. National designspecification for wood construction. Washington, DC:National Forest Products Association; 1982 edition.

Orosz, Ivan. Simplified method for calculating sheardeflection of beams. Res. Note FPL-0210. Madison, Wl:U.S. Department of Agriculture, Forest Service, ForestProducts Laboratory; 1970.

Przemienieckl, J. S. Theory of matrix structuralanalysis. New York: McGraw-Hill; 1968.

Suddarth, S. K. A computerized wood engineeringsystem: Purdue plane structures analyzer. Res. Pap.FPL 168. Madison, Wl: U.S. Department of Agriculture,Forest Service, Forest Products Laboratory; 1972.

Suddarth, S. K.; Percival, D. H.. Increasing theapplication efficiency of performance tests withanalytic procedures. Performance concept in buildings.Spec. Publ. 361, vol. 1. Washington, DC: U.S.Department of Commerce, National Bureau ofStandards; 1972.

Zlenkiewicz, O. C. The finite element method instructural and continuum mechanics. New York:McGraw-Hill; 1967.

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Appendix AInput Data

IdentificationAny 68 characters beginning in column 1 and extending to column 68 on a single lineor card.

Problem Size

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Member Properties

Note: For fictitious members, leave allowable stresses and MTYPE blank. For fictitiousmembers to include shear deflection, set allowable landing stress at 9000.

Stress Factor OptionsThree stress adjustment factors may be specified for use with current andsubsequent loading conditions placed on the same structural analogue.

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ReactionReaction node number is right justified.Reaction type is PIN, ROLL, FIX, or FIRL.LAST is last line indicator (1 for last line; blank otherwise).

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Load Information and Interaction InterpretationICON = 1 if concentrated loads anywhere; 0 otherwise.IUN = 1 if uniform loads anywhere; 0 otherwise.ISYST = 1 if loads at node anywhere; 0 otherwise.ITP = Interaction equation interpretations per text instructions.SFI = Stress Factor Index--value corresponding to order of stress factor input.

Concentrated LoadsMember number is right justified.NEXT is last line indicator (1 for last line; blank otherwise).

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Uniform LoadsMember number is right justified.NIX is last line indicator (1 for last line; blank otherwise).

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System Coordinate LoadsNode number is right justified.Load direction

1 = horizontal.2 = vertical.3 = moment.

NON is last line indicator (1 for last line; blank otherwise).

Note 1. For multiple load cases, repeat data input fromand including the Load Information and InteractionInterpretation line to this point.Note 2. Last line of data must contain ENDDCALCstarting in column 1.

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Appendix BProgram Modification For Problem Size

There are four primary variables used to determine thestorage capacity required by PPSA II to handle anygiven problem. They are the maximum number ofmembers (NM), and maximum number of nodes (NP),the maximum number of reactions (NR), and themaximum number of different member groups (MG) tobe assigned. Along with these primary values, there arealso two less important limiting values that may requiremodification: the maximum number of locations alongeach member checked for stress and deformation (ND),and the maximum matrix bandwidth (MBW). Table B-1lists these variables along with the maximum valuesallotted in the latest version of PPSA Il.

The maximum value currently allowed for MBW is 30.This value limits the spread between the end nodenumbers of any member to (MBW + 3)/3 or 11. Aproblem requiring a node spread greater than 11 willrarely be encountered. Therefore, there should be littleor no need to modify this value.

If any of the maximum values listed in table B-1must be changed, tables B-2 and B-3 list theaccompanying storage allotment changes required andtell where the affected variables may be found. Storageallocations are declared in common block statements(BEE1-BEE12) and dimension statements listed at thebeginning of each program subunit. Modifications listedin tables B-2 and B-3 must be made in each of thesubroutines indicated.

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