Understanding 2D Structural Analysis:

Transcription

1 Understanding 2D Structural Analysis: Learning Modules in the Modeling and Analysis of Framed Structures using GRASP Andres Winston C. Oreta Department of Civil Engineering De La Salle University Manila, Philippines This project was funded by the University Research Coordination Office (URCO) De La Salle University, Manila, Philippines 2004

2 Understanding 2D Structural Analysis: Learning Modules in the Modeling and Analysis of Framed Structures Using GRASP CONTENTS Preface 1 Understanding Structural Analysis 2 A Tour of GRASP 3 Loading Continuous Beams 4 Pattern Loading in Multistory Frames 5 Lateral Forces in Buildings 6 Pinned and Fixed Support Conditions 7 Soil Effects on Foundations 8 Support Settlements 9 Truss Analysis 10 Special Modeling Issues About the Author

3 PREFACE An exploratory-type of instructional and learning material consisting of ten modules about modeling and analysis of framed structures in 2D is presented. Each module focuses on a specific issue on structural modeling and analysis which is discussed with the aid of graphical and tabular results obtained from the 2D structural analysis software, GRASP. The set of learning modules is not a substitute to a textbook on structural analysis. The theory is not presented. No derivations or equations can be found. The student or reader must refer to the textbooks for definitions, equations and techniques. Each chapter begins with background information and a case study. The reader explores the issues raised in the case study through the Things to Do activities or by simply observing and analyzing the Observation and graphical and tabular results presented in the module. Included in the modules are Things to Try exercises and Things to Ponder comments on the analysis and design of structures. Using the set of learning modules, the reader or student with the aid of a structural analysis software like GRASP discovers important insights on the response and behavior of structures due to variations in the parameters of the model and configurations of the structure, changes in member and material properties, and also changes in the restraint and loading conditions. Through the graphical results, the student can visualize the phenomena and this would accelerate his understanding of concepts through the experience of seeing and interpreting solutions to various structural modeling and analysis problems. The implication and relevance of the case study to the safe and reliable design of structures are also discussed. Each chapter ends with a set of references and reading materials related to the issue presented in the module. The student is encouraged to perform the Things to Try exercises which are related to the case study. Since GRASP provides direct feedback graphically and numerically, the student can explore and have fun by simple modification of the configuration of the structural model or loading condition and will expand his knowledge and understanding about modeling, analysis and design of framed structures.

4 Understanding 2D Structural Analysis by A.W.C.Oreta 1-1 CHAPTER 1 UNDERSTANDING STRUCTURAL ANALYSIS ROLE OF STRUCTURAL ANALYSIS Structural analysis is an integral part of any structural engineering project. The structural design process of a typical structural engineering project may be divided in to three phases as shown in Figure 1.1. The conceptual design phase CONCEPTUAL DESIGN Architectural Functional Plans Final Design Structural System Trial Sections Detailing Connection Design Yes Modeling Revise Sections No Acceptable Analysis Member Design MODELING & ANALYSIS DESIGN & DETAILING Figure 1.1 The Structural Design Process (ACECOMS ISCAAD Workshop Notes)

5 Understanding 2D Structural Analysis by A.W.C.Oreta 1-2 usually involves the formulation of the functional requirements of the proposed structure, the preparation of the general layout and dimensions of the structures, and the consideration of the possible types of structural system to be used. In the modeling and analysis phase, a preliminary design of the structure is proposed using trial sections. A model of the structure is developed and the loads that may act on the structure are estimated. Structural analysis of the model is now performed to determine the stresses or stress resultants in the members and the deflections at various points of the structure. The results of the structural analysis are used in design and detailing phase where the structural members are designed to satisfy safety and serviceability requirements of the design codes. If the code requirements are not satisfied, then the member sizes are revised and a re-analysis of the model of the structure is carried out until all safety and serviceability requirements are satisfied. ANALYTICAL MODEL OF THE STRUCTURE To determine the behavior and performance of a real structure, the structure must be load tested. Testing and measurement of the real structure can only be done after the structure has been built. However, load testing is not possible at the planning and design stage of a new structure. Real structures can not be analyzed. We need to model the structure and analyze the model of the structure to determine approximately the response and behavior of the real structure due to external loads and excitation (Figure 1.2). The response quantities of the model which consists of internal forces and displacements are used in designing the members of the real structure. An analytical model is a simplified representation of a real structure. In developing a model of a structure, certain idealizations about the real structure must be made. How the members are supported and connected have to be represented by simple models. Loads expected to occur during the lifespan of the structure have to be estimated and applied to the model. The main objective

6 Understanding 2D Structural Analysis by A.W.C.Oreta 1-3 in modeling a structure is that the characteristics of the real structure must be represented as accurately as practically possible by a mathematical model so that the structural response predicted from the analysis of the model using computer tools may be relevant to the real structure. It is therefore imperative that the model represents the real structure with an appropriate likeness to capture the desired response. EXCITATION Loads Vibration Settlements Thermal Changes Structural Model RESPONSE Stresses Strains Displacements Stress Resultants Support Reactions STRUCTURE Figure 1.2 Structural Analysis The process of modeling is more of an art than a science. The engineer, through his practical experience and insight, must convert the real structure to an appropriate model (Figure 1.3) by making simplifying assumptions with regards to the type of structural model (3D or 2D), level of modeling (global or local), choice of model type (frame, grid, membrane, plate or solid), choice of elements (line, plate or solid), size and number of elements, type of restrains, properties of members and type of loads and excitations. Modeling of Structures in 2D If all the members of a structure and the loads acting on the structure lie on a single plane, the structure is modeled as a plane or two-dimensional (2D)

7 Understanding 2D Structural Analysis by A.W.C.Oreta 1-4 (a) Real Structure (b) Solid Model (c) 3D Plate-Frame (d) 3D Frame (e) 2D Frame Figure 1.3 Various Ways of Modeling a Structure (Anwar 2000) structure. Figure 1.4 shows two-dimensional models of plane structures. In these models, the members are represented as line elements with the line element corresponding to the centroid of the member. Beams are horizontal members used primarily to carry vertical loads distributed and concentrated loads. Beams resist external forces through bending moment and shear forces. Trusses are structural members made by assembling short, straight members connected by smooth pins at the joints and primarily designed to carry tensile and compressive axial forces. The loads in a truss are assumed to act at the joints which connect the members. Frames are composed of beams and columns that are either pinned or fixed connected at the joints. When the joints connecting the horizontal and vertical elements are fixed or rigid, the structure is said to be a rigid frame. The forces developed internally in a frame member consist of axial force, shear force and bending moment.

8 Understanding 2D Structural Analysis by A.W.C.Oreta 1-5 (a) Beam (b) Plane Truss (c) Plane Frame Figure 1.4 Two-Dimensional Models of Structures Structures, in general, are three-dimensional. However, there are many actual three-dimensional (3D) structures which can be divided into planar or twodimensional (2D) structures to simplify the analysis. A continuous bridge (Figure 1.5) is one example of a structure which can be modeled as a plane structure. The bridge deck supported by the piers and foundations and the traffic loads carried by the deck may be considered to lie on one plane and the bridge deck can be modeled as a beam. Figure 1.5 Continuous Bridge ( The truss of a bridge can be analyzed as a 2D structure (Figure 1.6). The bridge deck rests on beams called as stringers, which are then supported by floor

9 Understanding 2D Structural Analysis by A.W.C.Oreta 1-6 beams. The floor beams are connected at their ends to the joints on the bottom panels of the two longitudinal trusses. Thus, the weights of the vehicles, bridge deck, stringers and beams are transmitted to the supporting trusses at their joints; the trusses, in turn, Figure 1.6 Truss Bridge ( transfer the load to the foundation. Since the truss and the applied loads at the joints of the truss lie on one plane, the longitudinal truss can be treated as a plane truss. A multistory building (Figure 1.7) which consists of interconnected beams, columns, walls and footings may be modeled as a system consisting of several rigid plane frames (Figure 1.8). At each story of a building, the floor slab rests on floor beams, which transfer the floor loads including the weight of the slab and beams to the girders of the rigid frames. The loads are then transferred from the girders to the columns and then finally to the foundation. Since the applied loads and the rigid frame consisting of the girders, columns and foundations all lie on one plane, each frame can be analyzed separately as a plane structure. Buildings which are highly symmetrical in plan and framing system can ideally be represented as a system of 2D Figure 1.7 Multistory Building frames. (

10 Understanding 2D Structural Analysis by A.W.C.Oreta 1-7 Frame F1 Frame F2 F1 F2 F1 F2 F1 Building Plan 3D Building made from F1 and F2 Figure 1.8 A 3D Building as a System of Two Typical 2D Frames Although many three-dimensional structures can be subdivided into plane structures for the purpose of 2D structural analysis, some structures which are Figure 1.9 Space Structures (a) Tower (b) Dome (

11 Understanding 2D Structural Analysis by A.W.C.Oreta 1-8 referred to as space structures (Figure 1.9) such as domes, transmission towers, and highly unsymmetrical and irregular buildings are difficult to simplify into plane structures because of the complexity of the arrangement of the structural elements. A three-dimensional modeling and analysis has to be carried out for these types of structures to accurately predict their behavior. Modeling the Supports and Joints One of the most critical aspects in the modeling of a structure is the representation of the restraint conditions at the supports or at the joints. Depending on the type of restraint expected in the actual structure, the engineer has to decide on what appropriate model to use (Figure 1.10). The restraint conditions can be any of the following: Figure 1.10 Various Models for Supports o Roller - relative rotations at the joint and only translation parallel to the plane of the roller are allowed o Pinned - relative rotations at the joint are allowed but no translations o Fixed or Rigid - rotation and translation are not allowed o Flexible - spring models used to represent the relative stiffness of the joint

12 Understanding 2D Structural Analysis by A.W.C.Oreta 1-9 Modeling the External Loads In analyzing or designing a structure, it is necessary to determine the external forces that are expected to occur during its design life. Depending on the type of structure being analyzed, the following loads may have to be considered: (a) Dead loads are forces acting vertically downward that represent the weight of the structure and other permanent or fixed objects. (b) Live loads are vertical forces that may or may not be present on the structure at any given time. These loads are movable and can be applied anywhere on the structure. Occupancy loads in buildings and truck loading are examples of live loads. (c) Wind loads are cause by the pressure or suction due to wind at a point on a structure. These loads depend on various factors such as wind velocity, dimensions and orientation of the structure and geographical location of the structure. (d) Earthquake loads are developed when the structure vibrates due to ground excitation. Their magnitude depends on the type of ground accelerations, mass and stiffness of the structure, soil properties and location of structure with respect to seismic faults. (e) Other Environmental loads such as temperature changes, differential settlement of the foundation, vehicle loads, hydrostatic forces, soil pressure, etc. have to be considered depending on the type of the structure. The various loads mentioned are determined approximately. In most cases, these forces are modeled into two types of loads: (a) Nodal loads these are loads applied at the ends of the members or nodes. The loads can be vertical or horizontal forces or moments. (b) Member loads these loads are applied directly on the members or between the ends of the members. Various models of member loads are shown in (Figure 1.11) :

13 Understanding 2D Structural Analysis by A.W.C.Oreta 1-10 (1) Point or concentrated load (2) Distributed load (rectangular, triangular, trapezoidal) (3) Concentrated moment or couple (4) Temperature Figure 1.11 Types of Member Loads Load Combinations A great number of different types of loadings act on a structure. These loads do not act simultaneously on the structure. When these forces occur at the same time, the design loads are usually determined using load combinations. The combination which results to the worst condition is used in design. Load factors are multiplied on the basic loads and these factors depend on the design method being used. The basic load combinations can be found in the code. Examples of combination of factored loads from the NSCP 2001 Section 203.3, when Load and Resistance Factor Design (LRFD) is used, are: o 1.4 DL o 1.2 DL LL Lr o 0.9 DL ± (1.0 EQ or 1.3 W) On the other hand, when the Strength Design for concrete is used, NSCP 2001 Section provides these load combinations: o 1.4 DL LL o 0.75 (1.4 DL LL W)

14 Understanding 2D Structural Analysis by A.W.C.Oreta 1-11 o 1.3 DL LL EQ In these equations, the following notations were used: DL = dead load, LL = live load, Lr = roof live load, W = wind load and EQ = earthquake load. Results of Structural Analysis The main objective of structural analysis is to determine the behavior and response of the model of a structure. Various analytical methods are available - from approximate methods such as slope-deflection or moment distribution methods to the more refined finite element methods. The important structural response quantities that any structural analysis procedure must produce are: o Displacements at the nodes o Shear forces at various sections (Figure 1.12) o Bending moments at various sections (Figure 1.13) o Axial forces at the ends of members (Figure 1.14) o Reactions at the supports The results of the structural analysis are usually presented in tabular or graphical form (Figure 1.15). The engineer must be familiar with the sign convention used so that he can properly interpret the results. These results are used in the structural design of members assuring that safety and serviceability requirements of the design codes are satisfied. Figure 1.12 Shear Force Figure 1.13 Bending Moment

15 Understanding 2D Structural Analysis by A.W.C.Oreta 1-12 Figure 1.14 Axial Force Figure 1.15 Moment Diagram Using the Computer as a Learning Tool Software for structural analysis are now available commercially from simple to more sophisticated software and affordable to expensive ones (e.g. MicroFEAP, GRASP, BATS, STAAD, ETABS, SAP2000). Some textbooks in structural analysis (e.g., Kassimali 1999, Hibbeler 2000) also contain CD-ROM with software. In this notes, GRASP, a user-friendly software is introduced for twodimensional analysis of framed structures to enhance the learning and understanding of structural analysis. An advantage of using structural analysis software is that more complex and larger structures may be analyzed and designed by the students, which is not possible in the regular class in structural analysis where the calculator or general math solvers are used by students in their calculations. Another advantage of using software, especially those with graphics, is that students can visualize the behavior of complex systems. The software can be used to simulate a variety of structural and loading configurations and to determine cause and effect relationships between loading and various structural parameters, thereby increasing the students understanding on the behavior of structures. This develops the student s feel to real life problems.

16 Understanding 2D Structural Analysis by A.W.C.Oreta 1-13 An exploratory-type of instructional and learning material consisting of a set of modules are presented in the succeeding chapters. Each module focuses on a structural analysis issue which is presented through a case study. Included in the modules are hands-on exercises and problems on two-dimensional analysis of framed structures (beams, trusses and rigid frames). Using the set of learning modules, the student with the aid of GRASP discovers the behavior of structures due to variations in the parameters of the model and configurations of the structure, changes in member and material properties, and also changes in the restraint and loading conditions. Through the graphical results, students can visualize the phenomena and this would accelerate their understanding of concepts through the experience of seeing and interpreting solutions to many different problems. The set of learning modules is not a substitute to a textbook on structural analysis. The theory will not be presented. No derivations or equations can be found. The student must refer to the textbooks for definitions, equations and techniques. Each module will focus on a specific issue. A case study on the issue will be presented and walk through. The student by observing the graphical results and by interpreting the numerical output discovers important insight and can make conclusions. The implication and relevance of the structural analysis issue to the safe and reliable design of structures are also discussed. The module ends with a similar or related problem which the student has to solve using GRASP. Since GRASP provides direct feedback graphically and numerically, the student can explore and have fun by simple modification of the configuration of the structural model or loading condition and will discover new knowledge related to the structural analysis issue of the module. CASE STUDY: How do I represent a real structure as a line model? Structures can be modeled using line elements to represent the members. In modeling the structure, locate the centroids of the members and draw the lines

17 Understanding 2D Structural Analysis by A.W.C.Oreta 1-14 with respect to these centroids. For the portal frame shown, draw the model of the structure by representing the members as line elements. CASE STUDY 1 : Modeling a Portal Frame Reference: GRASP Help (Step by Step Examples) or User s Manual (1997)

18 Understanding 2D Structural Analysis by A.W.C.Oreta 1-15 References ACECOMS (AIT, Thailand) and Superior Software Solutions (Pakistan), GRASP Version 1.0 User s Manual, 1997 Anwar, N. (2000). Structural Modeling, ACECOMS News & Views, Jan-Jun 2000, pp. 8-10, AIT, Bangkok, Thailand Godden Structural Engineering Slide Library Hibbeler, R.C. (2000). Structural Analysis, 4th Edition, Chapter 1, Prentice Hall, New Jersey, USA Kassimali, A. (1999). Structural Analysis, 2 nd Edition, Chapters 1-2, Brooks-Cole Publishing Co., USA National Structural Code of the Philippines (NSCP 2001), Volume 1 : Buildings, Towers, and Other Vertical Structures, Chapter 2 and Sections 409, Association of Structural Engineers of the Philippines, Inc. (ASEP), Quezon City, Philippines Schodek, D.L. (1998). Structures. Chapters 1-3, Prentice-Hall, Inc. New Jersey, USA

19 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-1 CHAPTER 2 A TOUR OF GRASP INTRODUCING THE SOFTWARE GRASP stands for Graphical Rapid Analysis of Structures Program. It is a user-friendly software for two dimensional analysis of framed structures which includes beams, trusses and rigid frames. Especially developed for Windows, GRASP uses a Graphical User Interface(GUI) which provides an interactive, easy to use, graphical environment for modeling and analysis. GRASP is primarily based on a graphical means of interaction with the user and can provide direct feedback and effect of modifications. The major features of GRASP include: Modeling and analysis of multiple models in one file Presetting of default load cases and load factors Internal and automatic tracking of node numbers and member incidences Display the structural model at all times on the screen during analysis and superimposition of the analysis results on the model after analysis A Structure Wizard provides a step-by-step guideline for the generation of multistory structural models Supports SI, US and metric units and use of mixed units Apply loads on nodes and on members in multiple load cases Eight pre-defined types of cross-sections Set values of material properties Various restraint conditions including spring supports Apply member releases at the ends of members Diagram of results with values and tables View and print the analysis results for the full structure up to 20 sections for a member

20 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-2 CASE STUDY: How do you model and analyze a rigid frame using GRASP? In this example, you will model a two-story rigid frame shown in the figure. The frame will carry uniform dead and live loads which will be applied fully on the beams. Wind loads will also be applied at specified nodes. The basic loads will then be combined using the following load combination cases: o Dead Load (incl. Self Load) : 1.4 DL o Combined Dead and Live Loads : 1.2 DL LL o Combined Dead, Live and Wind Loads : 1.2 DL LL WL Things to Do Modeling and analysis using GRASP can be divided into five general steps. Follow the step-by-step procedure described by the figures for the following general steps. 1. Start analysis software and set basic parameters 2. Create geometry (in the figure assume dimensions are referred with respect to the centroids) 3. Apply basic loads 4. Define load combinations 5. Perfom analysis and view the results w DL = 15 kn/m, w LL = 7 kn/m 6 kn 10 kn Vertical Loading Wind Loading

21 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-3 CASE STUDY m 4.0 m 3.0 m 4.0 m 3.0 m Frame geometry 1200 mm 300 mm 100 mm 300 mm 300 mm Column cross-section Modulus of Elasticity = 21 kn/mm 2 Unit Weight = 24 kn/m 3 Coefficient of Thermal Expansion = 11 x 10-6 /C 250 mm Beam cross-section

22 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP Start analysis software and set basic parameters Step 1-1 : Select units option from Options menu to specify the working unit. To fix the working unit for future use Customize option and select the system that you prefer. Step 1-2 : Select the main unit system (SI, Metric or US) and other related measurement parameters (m, cm, mm; kg, k Not ton) from the options as shown in the dialog box above. Step 1-3: Select Structure => Materials and input material parameters. Change material properties of the default material if necessary. You may also add a new material by pressing Add Material.

23 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP Create geometry Step 2-1: There are two ways of creating the geometry in GRASP. You may draw the model using the mouse and GRASP graphical tools or use the Structure Wizard. Let us use the second method for step by step and quick creation of typical building models. Select Structure => Frame => Structure Wizard. Step 2-2: Select an appropriate typical frame based on bay width, story height and configuration. For this example, select type 2. If the frame that you want to model do belong to the four types, you can change the configuration of selected model later. Step 2-3: Specify the number of bays (3), number of stories (2) and typical values of bay width (3 m) and story height (4 m). You may also input the bay width (e.g., width = 4 m for bay 2) that is different from the default width. Step 2-4: Select the type and specify the dimensions for a typical column.

24 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-6 Step 2-5: Select the type and specify the dimensions for a typical beam. Step 2-6: Displaying the frame generated by Structure Wizard. Step 2-7: To define the type of supports, select Structure => Nodal Restraints and click at the node where restraints will be defined. Select the type of restraint. Step 2-8: To display the dimension line between selected nodes, select Structure => Add Dimensions and click any two nodes and double click at a position where the dimension line will be displayed. To delete, select Del Dimensions and click each line.

25 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-7 Step 2-9: To view the Node and Member numbers or labels, select View => Node Numbers and Member Numbers. Step 2-10: Here is the display of the frame with dimensions, member and node numbers. Observe that the height of the second story is 4.0 m. This must be changed to 3.0 m resulting to7.0 m as the total height of the frame. Step 2-11: To change the coordinates of the top most nodes, go to Structure => Change Nodal Coordinates and then click the node and enter the new Y-coordinate (7 m) Step 2-13: You may also view the outline of the members by selecting View => Member Outline. To remove the node numbers, member numbers and member outline, select View => Node Numbers, Member Numbers and Member Outline. Step 2-12: Display of the corrected model of the 2D frame.

26 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP Apply basic loads Step 3-1: Let us first apply the uniform loads in the horizontal members. You may apply the loads one member at a time by simply clicking the specific member or to all horizontal members. Let us select all horizontal members. Select Edit => Select Member and click on all horizontal members while pressing the shift key. Note the change of color of the selected members. Step 3-2: If you want to display only the horizontal members, select View => Members to Show => Horizontal. Let us now apply the Dead Load. Select Dead Load option from the load cases, combination and envelope list (rightmost-top). Step 3-4: Press Add, select the appropriate member load and enter the magnitude of the load. Step 3-3: Select Loading => Member Loads and click on the selected members.

27 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-9 Step 3-5: Displaying the Dead Load on horizontal members. Step 3-6: Select Live Load option from the load cases, combination and envelope list (rightmost-top). Step 3-7: Select Loading => Member Loads and click on the selected members. Press Add, select the appropriate member load and enter the magnitude of the load. Step 3-8: Displaying the Live Load on horizontal members.

28 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-10 Step 3-9: To apply the Wind Load, let us add a load case. Select Loading => Add Load Case. Step 3-10: Define the new load case Wind Load and select the option Basic Load Case. Step 3-11: : Select Wind Load option from the load cases, combination and envelope list. Select Loading => Nodal Loads and click on the node where the loads will be applied. Step 3-12: Input the magnitude and sign of the load. Follow the sign convention shown in the figures.to apply in opposite direction, use a negative value. Step 3-13: Displaying the Wind Loads

29 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP Define load combinations Step 4-2: Specify the name of the load and select Combination Load Case. Step 4-1: Define load combinations by pressing Loading => Add Load Case. Step 4-3: Input appropriate load factors for the defined load combination case. Step 4-4 : Repeat the same steps (4-1 to 4-3) for the load combination case combining dead load and live load.

30 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP Analysis and Results Step 4-5 : Repeat the same steps (4-1 to 4-3) for the load combination case combining dead load, Step 5-1: Carry-out the analysis using Perform => Self Load Calculation and Analysis in the menu option.

31 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-13 Step 5-2: To display graphical results, select the load case or load combination first. Step 5-3: Select the type of result from the menu option View => Bending Moment. Select View => Result Values if you want numerical values displayed in diagram.

32 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-14 Step 5-4 : Select the type of result from the menu option View => Shear Force. Step 5-5 : Select the type of result from the menu option View => Axial Force.

33 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-15 Step 5-6 : Select the type of result from the menu option View => Reactions. Step 5-7: You can view the nodal displacements by simply pointing the mouse at a node or the member results by pointing the mouse at a member.

34 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-16 Step 5-8: To view the displacements, select the type of result from the menu option View => Deflected Shape Step 5-9: Double click on any member to display the detailed results and diagrams for that member.

35 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-17 Step 5-10: Display the results in tabular form using Tables. Step 5-11: Prepare the report using File => Report Set-up.

36 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-18 Step 5-12: Select / Deselect the items to be included in the analysis report. Step 5-13: Select the File => Print Preview Report to preview the analysis report.

37 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-19 Step 5-14: Select the File => Print Preview Report to preview member results Step 5-15: Select the File => Print Preview Report to view graphical results You may print a hard copy of the report by selecting File => Print Report.

38 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-20 GRASP Toolbar GRASP has a toolbar which provides shortcuts in using the software. The toolbar buttons may be used instead of the commands in the menu.

39 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-21

40 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-22

41 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-23 GRASP Help and Manual If you want a step-by-step guide on the use of GRASP, you may refer to the document published by ACECOMS (AIT, Thailand) and Superior Software Solutions (Pakistan), GRASP Version 1.0 User s Manual, 1997 or click on the Help button and detailed information of various topics can be found.

42 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-24 Things to Try : Test Your GRASP Skills. Analyze the following planar structures using GRASP and fill in the blanks. A. BEAM: The beam has a rectangular cross-section of 250 mm x 400 mm. Assume the following material properties: Modulus of Elasticity, E = 21 kn/mm 2, Unit weight, γ = 24 kn/m 3 and coefficient of thermal expansion, α = 12E kn 20 kn/m A 12 m B 12 m 4m 4m C 1. The reactions at A are kn (vertical) and kn-m (moment). 2. The shear and moment at the left end in BC are kn and kn-m, respectively. 3. The maximum bending moment in member BC is about kn-m and is located at m from point B. B. PLANE TRUSS: All members are double angles 4 x 3 x 3/8, short legs back to back (A = 4.97 in 2 ; I = 3.84 in 4 and y top = in, modulus of elasticity, E = 29,000 ksi, unit weight, γ = 491 lb/ft 3 and coefficient of thermal expansion, α = 6.5 x 10-6 /F) A B C 2.0 kips 20 ft E F 15 ft = 60 ft D G 1.2 kips 1.5 kips 4. The reactions at D are kips (horizontal) and kips (vertical). 5. The axial force in bar AE is kips and bar BE is kips. 6. The nodal displacements at C are in (horizontal) and in (vertical).

43 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP 2-25 C. RIGID FRAME : Assume the following material properties: Modulus of Elasticity, E = 21 kn/mm 2, Unit weight, γ = 24 kn/m 3 and coefficient of thermal expansion, α = 12E-6. LOADS : DL = 31 kn/m and LL = 10 kn/m applied in all beams WL = 40 kn at the top level and 20 kn at the lower level LOAD COMBINATIONS: CASE NO. 1 = 1.4 D L CASE NO. 2 = 1.05 D L W SECTION PROPERTIES COLUMNS ( 400 mm x 400 mm) BEAMS (250 mm x 350 mm) 40 kn A B 4.0 m 20 kn C D E 5.0 m F G H 6.0 m 4.0 m 7. The reactions at the support F due to dead load (DL) are : Vertical Force = kn Horizontal Force = kn Moment = kn-m

44 Understanding 2D Structural Analysis by A.W.C. Oreta : A Tour of GRASP The end moments of the beam CD due to Combination Load Case 1 are : Moment at C = kn-m Moment at D = kn-m 9. The end moments of the beam CD due to Combination Load Case 2 are : Moment at C = kn-m Moment at D = kn-m 10. The axial force in the column CF for the different load cases are: P (WL only) = kn P (DL only) = kn P (Combination Load Case 2) = kn 11. The maximum left end moment for beam CD is kn-m and occurs at loading case:. 12. The maximum right end shear for beam AB is kn and occurs at loading case :. 13. The displacements at joint B due to wind load (WL) are : o Horizontal : mm o Vertical : mm o Rotation: rad 14. The maximum span moment for beam CD due to Load Case 2 is kn-m.

45 Understanding 2D Structural Analysis by A.W.C. Oreta : Loading Continuous Beams 3-1 CHAPTER 3 LOADING CONTINUOUS BEAMS BACKGROUND Beams and girders are straight horizontal members in structures which resist forces applied transversely to their lengths. This types of structural elements can be found in buildings supporting the floor slabs or resting on columns. Bridge decks which are frequently supported by piers and abutments are usually modeled as continuous beams. Beams are primarily designed to resist bending moments. Shear forces in beams must also be checked especially when the beams carry loads of large magnitude. The important basic variables which affect the behavior of beams include the magnitudes and arrangement of the loads, the nature of the support conditions and the section properties. This chapter explores the effect of loading conditions on the internal forces and moments in continuous beams. CASE STUDY : How should live loads be placed to produce the maximum and minimum bending moments and shear forces in continuous beams? The individual members of a structure must be designed for the worst combination of loads that can reasonably be expected to occur during its useful life. The internal forces developed in beams such as moments and shears are caused by the combined effect of two types of loads: dead loads and live loads. Dead loads, which include the weight of the beam, are constant and are placed fully on the beams. On the other hand, live loads such as floor loads from human occupancy or moving loads due to traffic can be placed on the beam in various ways. Is the positioning of the live loads critical in the design of beams and

46 Understanding 2D Structural Analysis by A.W.C. Oreta : Loading Continuous Beams 3-2 girders? What positions of live loads would produce maximum effects on a continuous beam? Things to Do 1. Model the three-span continuous beam shown in the figure using the given material and section properties. Using the GRASP toolbar, click the button for adding a member and draw graphically the geometry of the beam. Draw the continuous beam four times as shown in Figure Apply the dead load (W DL = 20 kn/m) on all spans. 3. Apply live load (W LL = 12 kn/m) for four basic load cases shown in Figure Combine the dead load and the corresponding basic live load: Service Load : DL + LL Ultimate Load : 1.4 DL LL 5. Perform analysis and view graphical and tabular results. CASE STUDY m = 12.0 m Three-span Continuous Beam Material Properties Modulus of elasticity = 20,500 N/mm 2 Unit weight = 24 kn/m mm Coefficient of thermal expansion = / o C 250 mm Beam cross-section

47 Understanding 2D Structural Analysis by A.W.C. Oreta : Loading Continuous Beams 3-3 (a) Case 1: Full Live Load (b) Case 2: Adjacent Spans (c) Case 3: Alternate Spans A (d) Case 4: Alternate Spans B Figure 3.1 Live Loading Cases and Deflection Curves (a) Case 1: Full Live Load (b) Case 2: Adjacent Spans (c) Case 3: Alternate Spans A (d) Case 4: Alternate Spans B Figure 3.2 Moment Diagrams due to Dead Load

48 Understanding 2D Structural Analysis by A.W.C. Oreta : Loading Continuous Beams 3-4 Observation One of the more interesting aspects of the behavior of statically indeterminate structures such as continuous beams is the structure s response under load. The response quantities which are affected by the loading conditions are the deflections, moments and shear forces. A three-span continuous beam when subjected to uniform load applied similarly at all spans will bend with a deflection curve similar to Figure 3.1 (a). The corresponding shape of the moment diagram for full loading conditions will have a shape similar to the diagrams in Figure 3.2. This figure shows the resulting moment diagram due to the dead load. Observe the location of the maximum (negative and positive) moments. The maximum negative moments occur at the internal pin supports while the maximum positive moments occur near the midspan. What is the effect of partially loading the spans of the continuous beam? Two types of partial loading conditions are shown in Figure 3.1. In Figure 3.1 (b), two adjacent spans are loaded with live load and the other span is not loaded. In Figures 3.1 (c) and (d), on the other hand, the live load is placed at alternate spans with the adjacent spans unloaded. These conditions reflect different loading patterns and each loading pattern will affect the internal forces at various sections (e.g. near the support or at midspan) of the beam. Figure 3.3 shows the resulting moment diagrams for the four cases of live loading. Maximum Negative Moment at a Support : Consider first the maximum negative moment at the second support. An inspection of the moments associated with the four cases reveals some curious results. The maximum negative moment at the second pin support does not occur when the structure is fully loaded (Case 1) but rather under a partial loading condition at adjacent spans (Case 2). The maximum negative moment under full live loading condition is 19.2 kn-m while the maximum negative moment when the adjacent spans are loaded is 22.4 kn-m. When the dead and live loads are now combined either under service load (Figure 3.4) or ultimate load (Figure 3.5), the same loading case produces the maximum negative

49 Understanding 2D Structural Analysis by A.W.C. Oreta : Loading Continuous Beams 3-5 moment at that support. This means that the maximum negative moment at a support will occur when the loads are placed on the two spans adjacent to that particular support and the next span unloaded Hence, if the maximum negative moment at the third pin support is desired, we must apply the live loads at the second and third spans which are adjacent to the support while the first span is unloaded. In case of more than three spans, the alternate spans must be loaded. (a) Case 1: Full Live Load (b) Case 2: Adjacent Spans (c) Case 3: Alternate Spans A (d) Case 4: Alternate Spans B Figure 3.3 Moment Diagram due to Live Load Maximum Span Moments : Observe the other two cases (Case 3 and 4) for alternate span loading. Figure 3.1 (c) and (d) shows the deflection curves due to alternate live loading. It can be seen that at the loaded spans, the curvatures are positive or concave upwards and since the bending moments are proportional to curvatures, the resulting span moment for the loaded spans are also positive. Figures 3.3 (c) and (d) show that the maximum span moments in the loaded spans do not also occur under full live loading condition (Case 1) but under partial loading

50 Understanding 2D Structural Analysis by A.W.C. Oreta : Loading Continuous Beams 3-6 (a) Case 1: Full Live Load (b) Case 2: Adjacent Spans (c) Case 3: Alternate Spans A (d) Case 4: Alternate Spans B Figure 3.4 Moment Diagram for Service Load (a) Case 1: Full Live Load (b) Case 2: Adjacent Spans (c) Case 3: Alternate Spans A (d) Case 4: Alternate Spans B Figure 3.5 Moment Diagram for Ultimate Load

51 Understanding 2D Structural Analysis by A.W.C. Oreta : Loading Continuous Beams 3-7 when alternate live loading is applied. A comparison of the magnitudes of the maximum positive moments for the first span is given in Figure 3.6 for Case 1 and Case 3. Under full live load, the maximum span moment of the beam is about kn-m. However, under alternate live load, the maximum span moment of the (a) Full Live Loading Condition (Case 1) (b) Alternate Live Loading Condition (Case 3) Figure 3.6 Moment Diagram of First Span due to Live Load

52 Understanding 2D Structural Analysis by A.W.C. Oreta : Loading Continuous Beams 3-8 same beam is about kn-m. This means that the maximum positive span moment in a continuous beam may occur at the loaded span for the alternate loading condition. Combining now the moments due to dead and live loads will produce the maximum positive span moments in the loaded spans as shown in Figures 3.4 and 3.5. Figure 3.7 shows member results for the Service Load Condition. Compare the results for B-1, B-4, B-7 and B-10 which correspond to the first span. The Maximum positive span moment is about 46 kn-m for B-7, a loaded span in the alternate loading condition (Case 3). Minimum Span Moments: Observe again the two cases (Case 3 and 4) for alternate span loading. This time observe the unloaded spans of the continuous beam. Notice that the curvatures in Figures 3.1 (c) and (d) are now negative or Figure 3.7 Member Results for Service Load Condition

53 Understanding 2D Structural Analysis by A.W.C. Oreta : Loading Continuous Beams 3-9 concave downwards meaning that the resulting span moments will be negative. Figures 3.3 (c) and (d) clearly show that the moments in the unloaded span are negative. If the moments due to dead and live loads are now combined for either service load (Figure 3.4) or ultimate load (Figure 3.5) conditions, the resulting span moments for the unloaded beams using the alternate span loading condition will be minimum (which may be negative) since the signs of the moments due to dead and live loads are not the same. This means that the minimum span moment in a continuous beam may occur at the unloaded span for the alternate loading condition. An inspection of the results for B-1, B-4, B-7 and B-10 in Figure 3.7 shows that the minimum span moment is about 22 kn-m for B-10, an unloaded span in the alternate loading condition (Case 4). Maximum Shear at a Support: Compare the shear diagrams under service load condition for the four cases as shown in Figure 3.8. Specifically, observe the shear forces at the second support. (a) Case 1: Full Live Load (b) Case 2: Adjacent Spans (c) Case 3: Alternate Spans A (d) Case 4: Alternate Spans B Figure 3.8 Shear Diagram for Service Load

54 Understanding 2D Structural Analysis by A.W.C. Oreta : Loading Continuous Beams 3-10 Which case produces the maximum shear? The maximum shear at the second support occurs under Case 2. It can be seen that the maximum shear at the support occurs when the support is between two spans which are loaded using the adjacent loading condition. Hence, placing the live load in the second and third spans will result to a maximum shear in the third pin support. It is also interesting to note here that the maximum shear at the end supports does not occur under full live load condition but under partial loading condition (Case 3). Things to Ponder Loading conditions that produce the maximum effects on a structure are called critical loading conditions. These conditions do not always occur when the live load is placed fully on the structure. Partial loading conditions may produce the critical moments and shear forces at a beam section. The occurrence of the critical values do not simultaneously occur under one loading condition. Hence, various loading arrangements have to be checked to determine what loading conditions are critical on a structure. In the design of beams, the code (e.g., NSCP 2001 sections 205) permits that the arrangement of live load may be limited and states that where uniform floor loads are involved, consideration maybe limited to full dead load on all spans in combination with full live load on adjacent spans and alternate spans. Things to Try 1. Analyze a continuous beam to obtain the maximum possible span and end moments. Model a continuous RC beam of 250 mm x 400 mm rectangular section consisting of four spans with pin supports. Each span has a distance of 5.0 m. The beams will carry uniformly distributed vertical loads consisting of the dead load (W DL = 20 kn/m) and live load (W LL = 15 kn/m). Consider various combination load cases for dead load and live load. Use the basic load combination factors for dead and live

55 Understanding 2D Structural Analysis by A.W.C. Oreta : Loading Continuous Beams 3-11 load specified in the NSCP 2001 section Obtain the possible maximum and minimum span and end moments. Observe also the effect of loading arrangement on the shear forces. 5.0 m = Repeat the same steps in Exercise No. 1 for the beam shown below which has fixed supports at both ends. 5.0 m = 20 m 3. For the same continuous beams above, try changing the magnitude of the live load and apply alternative live loadings. Observe if the span moments due to dead and live loads becomes positive or negative. What is the ratio of live load to dead load such that the combined effects produce a negative span moment? 4. For the same continuous beams above, change the distance between spans by moving the second and fourth pin supports one meter towards the end supports resulting to the span distances of 4.0 m 6.0 m 6.0 m 4.0 m. Apply the same

56 Understanding 2D Structural Analysis by A.W.C. Oreta : Loading Continuous Beams 3-12 loading conditions and observe the effect of spacing on the moments and shear forces. References and related readings National Structural Code of the Philippines (NSCP 2001), Volume 1 : Buildings, Towers, and Other Vertical Structures, Sections 203 and 205, Association of Structural Engineers of the Philippines, Inc. (ASEP), Quezon City, Philippines Nilson, A.H., Darwin, D. and Dolan, C.W. (2004). Design of Concrete Structures, 13 th Edition, Chapter 12, McGraw-Hill, Inc. NY, USA Schodek, D.L. (1998). Structures. Chapter 8, Prentice-Hall, Inc. New Jersey, USA

57 Understanding 2D Structural Analysis by A.W.C. Oreta : Pattern Loading 4-1 CHAPTER 4 PATTERN LOADING IN MULTISTORY FRAMES BACKGROUND Design codes specify that every building and every portion of the structure must be designed and constructed to sustain appropriate combinations of vertical loads and lateral forces. The individual members of a building frame which consists of beams and columns must be designed against loads which are reasonably expected to occur during the structure s useful life. The internal forces induced in the frame such as moments, shears and axial forces are caused by the combined effect of both vertical and lateral loads. Let us first consider the effect of vertical loads in a building frame. The vertical loads which consist of dead and live loads are carried by the horizontal members of the building. These loads are usually placed on the girders or beams when a model of the structure is analyzed. The dead loads are constant and are placed fully on the beams. On the other hand, live loads such as floor loads from human occupancy can be placed in various ways, some of which may result in larger effects than others. Chapter 3 demonstrated the effect of live load arrangement to the moments and shear forces in beams. These loading schemes also apply to beams of rigid frames. However, the live loading schemes must be extended to consider also the effect to the vertical elements or columns. This chapter explores the various schemes that the live load can be placed on the horizontal elements of a rigid frame and the corresponding effects on the internal forces in the beams and columns.