Saikat Basak
Tips and Tricks for
Computer Aided
Structural Analysis
Saikat Basak
M.Eng (Structural), BCE, CIC, AIE (Ind.), A.ASCE
Struc...
© Saikat Basak
The author and publisher of this book have used their best efforts in preparing
this book. These efforts in...
Computer Aided Structural Analysis
- 4 -
CONTENTS
ABBREVIATION...............................................................
Computer Aided Structural Analysis
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23. FINITE ELEMENT ANALYSIS (FEA) METHOD IS APPROACHING… ...........................
Computer Aided Structural Analysis
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Abbreviation
Several abbreviations have been used throughout this book. They have...
Computer Aided Structural Analysis
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1. Introduction (before you begin…)
In this book I shall tell you some practical ...
Computer Aided Structural Analysis
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otherwise cannot be produced with a ton of computer output”. You should paste
thi...
Computer Aided Structural Analysis
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2. What is Computer Aided Structural Analysis?
This section is a head start for t...
Computer Aided Structural Analysis
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3. Analysis types
In this section, you will learn various analysis options those...
Computer Aided Structural Analysis
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This functionality offers a linearized solution to a nonlinear problem. Sounds
c...
Computer Aided Structural Analysis
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self-weight, is always present and can cause considerable changes in the
structu...
Computer Aided Structural Analysis
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Response spectrum analysis is used extensively by Civil Engineers who must
desig...
Computer Aided Structural Analysis
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Vibration Analysis (Modal Analysis)
All things vibrate. Think of musical instrum...
Computer Aided Structural Analysis
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Modal Analysis to deal with this type of more complex situation. These are
calle...
Computer Aided Structural Analysis
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Buckling analysis calculates the smallest (critical) loading required buckling a...
Computer Aided Structural Analysis
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Thermal radiation is the thermal energy emitted by bodies in the form of
electro...
Computer Aided Structural Analysis
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sitting on the table and a weight is placed on top of the coffee cup, then the t...
Computer Aided Structural Analysis
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4. Sign convention (mind your signs)
In structural analysis, sign convention is ...
Computer Aided Structural Analysis
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When dealing with 3D structures, the program will generally consider y-axis as
e...
Computer Aided Structural Analysis
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Now please solve the following problems using your program and check the
result ...
Computer Aided Structural Analysis
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Now solve the following truss.
All members are made of
3 kN steel (E=200 GPa) wi...
Computer Aided Structural Analysis
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5. Numbering of joints and members
Proper node and joint numbering is very impor...
Computer Aided Structural Analysis
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6. Specifying moment of inertia
New users of structural analysis programs often ...
Computer Aided Structural Analysis
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Figure 6-2
Sometimes you may need to specify inertia directly especially for irr...
Computer Aided Structural Analysis
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Y
Z
X
Y
X
Z
Figure 6-4
The above figure shows orientation of local axes for the ...
Computer Aided Structural Analysis
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7. Specifying Loads
All programs have the option for specifying concentrated and...
Computer Aided Structural Analysis
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α 0 0.2 0.375 0.5
r 1 1.02 1.05 1.068
It is seen that maximum difference of mid ...
Computer Aided Structural Analysis
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Figure 7-4
To specify you generally supply fluid height, axis and density. Now t...
Computer Aided Structural Analysis
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Figure 7-7 Figure 7-8
In figure 7-8, the load is projected on horizontal axis. T...
Computer Aided Structural Analysis
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8. Column Buckling test
Solve the following problem: A steel (E=200 GPa) column ...
Computer Aided Structural Analysis
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such structure using a standard PC and inexpensive program, chances are that
you...
Computer Aided Structural Analysis
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9. Portal and Cantilever method
You may have been taught to use portal and canti...
Computer Aided Structural Analysis
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10. Deflection of Reinforced Concrete member
Consider a simply supported beam ma...
Computer Aided Structural Analysis
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For a T-beam with k>hf – use same equation as that for a rectangular beam.
In al...
Computer Aided Structural Analysis
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11. Shear deformation
Most programs do not take into account deformation due to ...
Computer Aided Structural Analysis
- 37 -
12. Inclined support
If your program supports specifying inclined local axes for...
Computer Aided Structural Analysis
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13. Maximum bending moment, shear force and reaction in
building frame: Substitu...
Computer Aided Structural Analysis
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floor. (Though it is customary to use reduced live load in roof level). Similarl...
Computer Aided Structural Analysis
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14. Support settlement
Take any statically determinate structure, for example a ...
Computer Aided Structural Analysis
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15. 2D versus 3D
For symmetrical structures, often it is possible to convert the...
Computer Aided Structural Analysis
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Figure 15-2
Look, in fig. 15-1, the loads are towards X direction. If there were...
Computer Aided Structural Analysis
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16. Curved member
Most frame analysis programs do not have curve element. You wi...
Computer Aided Structural Analysis
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17. Tapered section
Many programs have the option of specifying tapered or varia...
Computer Aided Structural Analysis
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18. Nodes connected by a spring
Many programs allow you to define a spring suppo...
Computer Aided Structural Analysis
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19. Sub-structuring technique and symmetry (break them into
pieces…)
In the anal...
Computer Aided Structural Analysis
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11.42 kNm 11.42 kNm -5 kN/m
25 kN
4.28 kN 5 kN 4.28 kN
A B C D
Figure 19-3
Obser...
Computer Aided Structural Analysis
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Continuous beams, with even number of spans.
Actual beam
Figure 19-4
Symmetry ut...
Computer Aided Structural Analysis
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Figure 19-8
Fixed
Fixed
Fixed
Figure 19-9
Computer Aided Structural Analysis
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Figure 19-10
X translation & Z rotation fixed
X translation & Z rotation fixed
X...
Computer Aided Structural Analysis
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20. Staircase analysis
A staircase is actually a folded plate structure. But in ...
Computer Aided Structural Analysis
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Figure 20-2
This analysis was done in Visual Analysis 3.5. An interesting point ...
Computer Aided Structural Analysis
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What result do you see? The bending moment (and reactions as well) is same in
al...
Computer Aided Structural Analysis
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21. Cables
It is possible to analyze cables with a mere frame analysis program. ...
Computer Aided Structural Analysis
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Figure 21-1
Computer Aided Structural Analysis
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Vertical reaction at A = 104 kN (down), at B = 250 kN (up). Moments: MAB =
0, MB...
Computer Aided Structural Analysis
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22. Pre-stressed cable profile
Does your program offer specifying pre-stressed c...
Computer Aided Structural Analysis
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With reference to the figure 22-3, the calculation is
shown below.
For left span...
Computer Aided Structural Analysis
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shown in green color. The ultimate equivalent load will be that of as shown in
f...
Computer Aided Structural Analysis
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23. Finite Element Analysis (FEA) Method is approaching…
We now come to the most...
Computer Aided Structural Analysis
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loaded by a uniformly distributed load. We like to find out the stresses at
vari...
Computer Aided Structural Analysis
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Figure 23-4
As a crude rule, when you use triangular you will normally need much...
Computer Aided Structural Analysis
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One distinguishing feature of finite element method is that it does not provide
...
Computer Aided Structural Analysis
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Sometimes even the most expensive finite element analysis programs produce
wrong...
Computer Aided Structural Analysis
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Exercise
A 5-m steel (E=200GPa) beam has width 200 mm and depth 500 mm. It is
lo...
Computer Aided Structural Analysis
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Figure 23-8
The figure 23-9 shows one of mid plane stresses, local σx distributi...
Computer Aided Structural Analysis
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To analyze the beam as 2D, you should not face any difficulty. However, you
shou...
Computer Aided Structural Analysis
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Did you see that finite element analysis programs normally give you output in
th...
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  1. 1. Saikat Basak
  2. 2. Tips and Tricks for Computer Aided Structural Analysis Saikat Basak M.Eng (Structural), BCE, CIC, AIE (Ind.), A.ASCE Structural Engineer PUBLISHED BY ENSEL SOFTWARE
  3. 3. © Saikat Basak The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research and testing of the theories and programs to determine their effectiveness. The author and publisher shall not be liable in any event for the incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs. All rights reserved. No part of this book may be reproduced, in any form or by any means, without the permission in writing from the author. 1st Edition 2001 Published on the web Price: FREE for online viewing
  4. 4. Computer Aided Structural Analysis - 4 - CONTENTS ABBREVIATION....................................................................................................................................................6 1. INTRODUCTION (BEFORE YOU BEGIN…).........................................................................................7 2. WHAT IS COMPUTER AIDED STRUCTURAL ANALYSIS? .............................................................9 3. ANALYSIS TYPES ....................................................................................................................................10 LINEAR STATIC STRESS ANALYSIS....................................................................................................................10 DYNAMIC ANALYSIS..........................................................................................................................................11 RANDOM VIBRATION.........................................................................................................................................12 RESPONSE SPECTRUM ANALYSIS.......................................................................................................................12 TIME HISTORY ANALYSIS..................................................................................................................................13 TRANSIENT VIBRATION ANALYSIS....................................................................................................................13 VIBRATION ANALYSIS (MODAL ANALYSIS)......................................................................................................14 BUCKLING ANALYSIS.........................................................................................................................................15 THERMAL ANALYSIS..........................................................................................................................................16 BOUNDARY ELEMENT .......................................................................................................................................17 4. SIGN CONVENTION (MIND YOUR SIGNS)........................................................................................19 5. NUMBERING OF JOINTS AND MEMBERS........................................................................................23 6. SPECIFYING MOMENT OF INERTIA .................................................................................................24 7. SPECIFYING LOADS...............................................................................................................................27 8. COLUMN BUCKLING TEST ..................................................................................................................31 9. PORTAL AND CANTILEVER METHOD .............................................................................................33 10. DEFLECTION OF REINFORCED CONCRETE MEMBER...........................................................34 11. SHEAR DEFORMATION.....................................................................................................................36 12. INCLINED SUPPORT...........................................................................................................................37 13. MAXIMUM BENDING MOMENT, SHEAR FORCE AND REACTION IN BUILDING FRAME: SUBSTITUTE (EQUIVALENT) FRAME .......................................................................................................38 14. SUPPORT SETTLEMENT ...................................................................................................................40 15. 2D VERSUS 3D.......................................................................................................................................41 16. CURVED MEMBER..............................................................................................................................43 17. TAPERED SECTION ............................................................................................................................44 18. NODES CONNECTED BY A SPRING................................................................................................45 19. SUB-STRUCTURING TECHNIQUE AND SYMMETRY (BREAK THEM INTO PIECES…)...46 20. STAIRCASE ANALYSIS ......................................................................................................................51 21. CABLES ..................................................................................................................................................54 22. PRE-STRESSED CABLE PROFILE ...................................................................................................57
  5. 5. Computer Aided Structural Analysis - 5 - 23. FINITE ELEMENT ANALYSIS (FEA) METHOD IS APPROACHING… ....................................60 24. A TYPICAL WORKED OUT PROBLEM OF FEA...........................................................................69 25. PLATES BY FEM ..................................................................................................................................76 26. INTERPRETING FEA RESULT..........................................................................................................79 27. TIPS FOR CREATING BETTER MESH............................................................................................83 28. COMMON FINITE ELEMENTS LIBRARY FOR LINEAR STATIC AND DYNAMIC STRESS ANALYSIS..........................................................................................................................................................89 29. SHEAR WALL .......................................................................................................................................92 30. FOLDED PLATE ...................................................................................................................................94 31. SHELLS.................................................................................................................................................102 32. A FIRST STEP IN STRUCTURAL DYNAMICS.............................................................................105 33. AN EXAMPLE OF A SINGLE DEGREE OF FREEDOM PROBLEM.........................................108 34. WHAT DYNAMIC ANALYSIS YOU SHOULD PERFORM? .......................................................112 35. NON-LINEAR ANALYSIS (NLA) – AN INTRODUCTION FOR BEGINNERS .........................115 MATERIAL NON-LINEARITY ............................................................................................................................115 GEOMETRIC NON-LINEARITY ..........................................................................................................................116 36. MECHANICAL EVENT SIMULATION ..........................................................................................119 37. IMPORTING MODEL FROM CAD PROGRAMS .........................................................................121 38. VIRTUAL REALITY IN ENGINEERING (VRML)........................................................................123 39. LINEAR PROGRAMMING IN SPREADSHEET............................................................................124 40. REINFORCEMENT DETAILING IN CONTINUOUS BEAMS ....................................................128 41. A GUIDE TO SOME STRUCTURAL ENGINEERING & FINITE ELEMENT ANALYSIS PROGRAMS.....................................................................................................................................................130 CIVIL ENGINEERING PROGRAMS......................................................................................................................131 MECHANICAL ENGINEERING PROGRAMS .........................................................................................................134 SOME CAD PROGRAMS… ...............................................................................................................................137 42. HOW TO SELECT THE MOST APPROPRIATE PROGRAM FOR YOUR NEED?.................139 43. HOW TO CHECK THE RESULT FOR ACCURACY? ..................................................................142 44. FILE NAME EXTENSION GUIDE (FOR SOME CAD/CAE PROGRAMS) ...............................143 45. COMMON ERROR MESSAGES AND SOLUTIONS.....................................................................144 OPERATING SYSTEM RELATED.........................................................................................................................144 ANALYSIS PROGRAM RELATED........................................................................................................................145 46. REFERENCES .....................................................................................................................................148
  6. 6. Computer Aided Structural Analysis - 6 - Abbreviation Several abbreviations have been used throughout this book. They have been defined in respective sections, but here is a list of them at a glance. [k] – Stiffness BM – bending moment C - damping CAD – computer aided design/drawing CAE – computer aided engineering CAM – computer aided manufacturing E – modulus of elasticity FE – finite element FEA – finite element analysis FEM – finite element method fy – yield strength of steel G – shear modulus I – 2nd moment of inertia IS – Indian Standard code LRFD – load and resistance factor design LSSA – linear static stress analysis M, m – mass MDF – multi degree freedom MES – mechanical event simulation NLA – non-linear analysis RSA – response spectra analysis SDF – single degree freedom SF – shear force T – time period of vibration THA – time history analysis UDL – uniformly distributed load VRML – virtual reality markup language x – displacement x’ – velocity x” – acceleration ε – Strain µ, ν – Poisson’s ratio σ – Normal stress τ – Shear stress ωn – natural frequency of the structure ξ – Damping ratio
  7. 7. Computer Aided Structural Analysis - 7 - 1. Introduction (before you begin…) In this book I shall tell you some practical tips for structural analysis using computer. Most structural engineering books are written to tell you how you will perform the calculation by hand. But even sometimes analysis using computer can be very tricky. You may need to manipulate computer input to solve a problem, which may at first appear to be unsolvable by that program. Finite element programs and structural analysis programs tend to be very expensive. Most small-scale engineering firms keep only one analysis program. Even for a large corporate companies it is seldom possible to maintain more than two standard analysis packages. Therefore it is essential that you use your present analysis program to its full extent. This is not a textbook. I make no attempt to teach you theory of structural analysis to score good marks in the exam! But it can help you to earn more money by enabling you to analyze some structures more easily and accurately, which you were previously thought too difficult to deal with your existing analysis program. Also, I am not going to teach you any particular structural analysis computer program. However, the techniques of analysis discussed here are applicable to most standard analysis packages. I presented the whole thing in an informative yet informal manner. I confined the boring theory and calculation to minimum level. No special knowledge is required to get the most out of this book. Only Bachelor Degree knowledge in Civil/Mechanical Engineering is assumed. However some parts of the book do discuss some topics which are normally covered in Master’s degree level in detail. Also, I expect that you are familiar with at least one standard structural analysis package otherwise you may find the contents of this book quite terse! This book does not contain listing of any computer program; because I know that most readers will not bother to type them or to even read them. But remember the most important advice: A structure will not behave as the computer program tells it should regardless of how accurate the program seems or how expensive it is! Thus goes the famous proverb “With good engineering judgement you can produce on the back of an envelop that which
  8. 8. Computer Aided Structural Analysis - 8 - otherwise cannot be produced with a ton of computer output”. You should paste this in front of your computer so that you see it everyday. (I did it!) Before you accuse me by complaining that my tips do not work with your program, I like to mention following important points. • I did not work with all the structural analysis programs available in the market. • Some features I discussed here may not be available in your program. It can even happen that the program you are using has better option to handle a particular problem compared to what I discussed in this book. • I am only providing you some “clues” for more effective use of structural analysis programs. However, every analysis problem is unique depending on type of project, cost, client’s requirement etc. Those specific criteria you have to solve yourself. • Documentation of the program you are using is very important. The program manuals are the best source of help always. The sections of this book are arranged in somewhat haphazard manner deliberately so that you don’t feel bored. The paragraphs are small and to the point. We have often returned to same topics in several sections from different viewpoints. Wherever necessary, numerical examples have been presented. There are also some exercises. Please try to solve them with your structural/FE analysis programs. I like to see your comments and suggestions. You can reach me at www.enselsoftware. com in World Wide Web. I shall be more than happy to answer your queries. Now sit back, relax and enjoy the book. Have a nice reading!
  9. 9. Computer Aided Structural Analysis - 9 - 2. What is Computer Aided Structural Analysis? This section is a head start for those who are using structural analysis programs for the first time. As the name suggests, Computer Aided Structural Analysis is the method of solving your structural analysis problem with the help of computer software. In earlier generation analysis programs, you had to supply the programs the nodal co-ordinates, member incidence (i.e. between what nodal points a particular member is connected), material properties, sectional properties of the all members and the loads (nodal force/moment/distributed member loads etc.). You also had to supply how the structure was supported, fixed, hinged or roller. The program then calculated the member forces, nodal reactions and joint displacements and presented in a tabular format. This type of structural analysis programs is still used in junior years in the university as a first learning tool. However, the commercial structural analysis programs of modern days are far more powerful and easy to use. Here, you can actually ‘draw’ your model on screen (as if you’re drawing in a paper with a pencil!) with the mouse and keyboard! Everything is graphical. You draw models graphically, apply loads and boundary conditions graphically and visualize the shear force, bending moment and even deflected shape diagram graphically. For the first time users, it seems rather like a magic! The availability of these programs has completely changed the way we analyze structures compared to we did the same just a decade ago! Now it is a child’s play to analyze structures having more than 10,000 degrees of freedom! However, analyzing structures using computers has created many other new problems. First, you must be very familiarize with the programs you are using. You must clearly understand its limitation and assumptions. All programs can’t be applied for analyzing all types of structures. Most programs solve the structures by stiffness method, though solution algorithm may differ from one program to another. What is most important is that you must interpret the output result accurately. This book will show you how to perform quickly, accurately and proper interpretation of data in easiest way. You will also learn to analyze many new kinds of structures without learning theoretical calculations! Sounds interesting? At the end of this book, you will also learn about some very recent concepts of structural analysis. Bon Voyage!
  10. 10. Computer Aided Structural Analysis - 10 - 3. Analysis types In this section, you will learn various analysis options those are offered by FEA programs. You are already familiar with most of the types of analyses, and some are new to you. (References 8 and 15 were considered for this section.) Linear Static Stress Analysis This is the most common type of analysis. When loads are applied to a body, the body deforms and the effects of the loads are transmitted throughout the body. To absorb the effect of loads, the body generates internal forces and reactions at the supports to balance the applied external loads. Linear Static analysis refers to the calculation of displacements, strains, and stresses under the effect of external loads, based on some assumptions. They are discussed below. 1. All loads are applied slowly and gradually until they reach their full magnitudes. After reaching their full magnitudes, load will remain constant (i.e. load will not vary against time). This assumption lets us disregard insignificant inertial and damping forces due to negligibly small accelerations and velocities. Time-variant loads that induce considerable inertial and/or damping forces may warrant dynamic analysis. Dynamic loads change with time and in many cases induces considerable inertial and damping forces that cannot be neglected. 2. Linearity assumption: The relationship between loads and resulting responses is linear. If you double the magnitude of loads, for example, the response of the model (displacements, strains and stresses) will also double. You can make linearity assumption if a. All materials in the model comply with Hooke’s Law that is stress is directly proportional to strain. b. The induced displacements are small enough to ignore the change is stiffness caused by loading. c. Boundary conditions do not vary during the application of loads. Loads must be constant in magnitude, direction and distribution. They should not change while the model is deforming. If the above assumptions are not valid, then we shall have to treat the problem as non-linear analysis. I shall devote a few sections on non-linear analysis later. Some FEA programs offer contact/gap elements. With this option, available during meshing, contacting mating faces may separate during loading and hence the load distribution in the model will change based on the gap forces generated.
  11. 11. Computer Aided Structural Analysis - 11 - This functionality offers a linearized solution to a nonlinear problem. Sounds crazy? Calculation of stresses Stress results are first calculated at special points, known ‘Gaussian’ or ‘Quadrature’ points, located inside each element. (See you FEA textbook for details) These points are selected to give optimal results. The program then calculates stresses at the nodes of each element by extrapolating the results available at the ‘Gaussian’ points. After a successful run, multiple results are available at nodes common to two or more elements. These results will not be identical because the finite element method is an approximate method. For example, if a node is common to three elements, there can be three slightly different values for every stress component at that node. During result visualization, you may ask for element stresses or nodal stresses. In calculating element stresses, the program averages the corresponding nodal stresses for each element. In calculating nodal stresses at a node, the program averages the corresponding results from all elements contributing to the stresses at that node. Dynamic analysis In general, we have to perform dynamic analysis on a structure when the load applied to it varies with time. The most common case of dynamic analysis is the evaluation of responses of a building due to earthquake acceleration at its base. Every structure has a tendency to vibrate at certain frequencies, called natural frequencies. Each natural frequency is associated with a certain shape, called mode shape that the model tends to assume when vibrating at that frequency. When a structure is excited by a dynamic load that coincides with one of its natural frequencies, the structure undergoes large displacements. This phenomenon is known as ‘resonance’. Damping prevents the response of the structures to resonant loads. In reality, a continuous model has an infinite number of natural frequencies. However, a finite element model has a finite number of natural frequencies that is equal to the number of degrees of freedom considered in the model. The first few modes of a model (those with the lowest natural frequencies), are normally important. The natural frequencies and corresponding mode shapes depend on the geometry of the structure, its material properties, as well as its support conditions and static loads. The computation of natural frequencies and mode shapes is known as modal analysis. When building the geometry of a model, you usually create it based on the original (undeformed) shape of the model. Some loading, like a structure’s
  12. 12. Computer Aided Structural Analysis - 12 - self-weight, is always present and can cause considerable changes in the structure’s original geometry. These geometric changes may have, in some cases, significant impact on the structure’s modal properties. In many cases, this effect can be ignored because the induced deflections are small. This is just a prelude to dynamic analysis. You will find several topics on dynamic analysis later in this book. However, since I shall not discuss theory of structural dynamics here, I strongly recommend that you read a structural dynamic textbook if you haven’t done so already. The following few topics – Random Vibration, Response Spectrum analysis, Time History analysis, Transient vibration analysis and Vibration modal analysis are extensions of dynamic analysis. Random Vibration Engineers use this type of analysis to find out how a device or structure responds to steady shaking of the kind you would feel riding in a truck, rail car, rocket (when the motor is on), and so on. Also, things that are riding in the vehicle, such as on-board electronics or cargo of any kind, may need Random Vibration Analysis. The vibration generated in vehicles from the motors, road conditions, etc. is a combination of a great many frequencies from a variety of sources and has a certain "random" nature. Random Vibration Analysis is used by mechanical engineers who design various kinds of transportation equipment. Engineers provide input to the processor in the form of a ‘Power Spectral Density’ (PSD), which is a representation of the vibration frequencies and energy in a statistical form. When an engineer uses Random Vibration he is looking to determine the maximum stresses resulting from the vibration. These stresses are important in determining the lifetime of a structure of a transportation vehicle. Also, it would be important to know if things being transported in vehicles will survive until they reach the destination. Response Spectrum Analysis Engineers use this type of analysis to find out how a device or structure responds to sudden forces or shocks. It is assumed that these shocks or forces occur at boundary points, which are normally fixed. An example would be a building, dam or nuclear reactor when an earthquake strikes. During an earthquake, violent shaking occurs. This shaking transmits into the structure or device at the points where they are attached to the ground (boundary points).
  13. 13. Computer Aided Structural Analysis - 13 - Response spectrum analysis is used extensively by Civil Engineers who must design structures in earthquake-prone areas of the world. The quantities describing many of the great earthquakes of the recent past have been captured with instruments and can now be fed into a response spectrum program to determine how a structure would react to a past real-world earthquake. Mechanical engineers who design components for nuclear power plants must use response spectrum analysis as well. Such components might include nuclear reactor parts, pumps, valves, piping, condensers, etc. When an engineer uses response spectrum analysis, he is looking for the maximum stresses or acceleration, velocity and displacements that occur after the shock. These in turn lead to maximum stresses. You will find an example of response spectrum analysis later. Time History Analysis This analysis plots response (displacements, velocities, accelerations, internal forces etc.) of the structure against time due to dynamic excitation applied on the structure. You will find more stuff on this particular type of analysis in later sections. Transient Vibration Analysis When you strike a guitar string or a tuning fork, it goes from a state of inactivity into a vibration to make a musical tone. This tone seems loudest at first, then gradually dies out. Conditions are changing from the first moment the note is struck. When an electric motor is started up, it eventually reaches a steady state of operation. But to get there, it starts from zero RPM and passes through an infinite number of speeds until it attains the operating speed. Every time you rev the motor in your car, you are creating transient vibration. When things vibrate, internal stresses are created by the vibration. These stresses can be devastating if resonance occurs between a device producing vibration and a structure responding to. A bridge may vibrate in the wind or when cars and trucks go across it. Very complex vibration patters can occur. Because things are constantly changing, engineers must know what the frequencies and stresses are at all moments in time. Sometimes transient vibrations are extremely violent and short-lived. Imagine a torpedo striking the side of a ship and exploding, or a car slamming into a concrete abutment or dropping a coffeepot on a hard floor. Such vibrations are called "shock, " which is just what you would imagine. In real life, shock is rarely a good thing and almost always unplanned. But shocks occur anyhow. Because of vibration, shock is always more devastating than if the same force were applied gradually.
  14. 14. Computer Aided Structural Analysis - 14 - Vibration Analysis (Modal Analysis) All things vibrate. Think of musical instruments, think of riding in a car, think of the tires being out of balance, think of the rattles in an airplane when they are revving up the engines, or the vibration under your feet when a train goes by. Sometimes vibration is good. Our ears enable us to hear because they respond to the vibrations of sound waves. Many times things are made to vibrate for a purpose. For example, a special shaking device is used in foundries to loosen a mold placed in sand. Or, in the food and bulk materials industries, conveyors frequently work by vibration. Usually, however, vibration is bad and frequently unavoidable. It may cause gradual weakening of structures and the deterioration of metals (fatigue) in cars and airplanes. Rotating machines from small electric motors to giant generators and turbines will self destruct if the parts are not well balanced. Engineers have to design things to withstand vibration when it cannot be avoided. For example, tyres and shock absorbers (dampers) help reduce vibration in automobiles. Similarly, flexible couplings help isolate vibrations produced by the engines. Vibration is about frequencies. By its very nature, vibration involves repetitive motion. Each occurrence of a complete motion sequence is called a "cycle." Frequency is defined as so many cycles in a given time period. "Cycles per seconds” or "Hertz”. Individual parts have what engineers call "natural" frequencies. For example, a violin string at a certain tension will vibrate only at a set number of frequencies, which is why you can produce specific musical tones. There is a base frequency in which the entire string is going back and forth in a simple bow shape. Harmonics and overtones occur because individual sections of the string can vibrate independently within the larger vibration. These various shapes are called "modes". The base frequency is said to vibrate in the first mode, and so on up the ladder. Each mode shape will have an associated frequency. Higher mode shapes have higher frequencies. The most disastrous kinds of consequences occur when a power-driven device such as a motor for example, produces a frequency at which an attached structure naturally vibrates. This event is called "resonance." If sufficient power is applied, the attached structure will be destroyed. Note that ancient armies, which normally marched "in step," were taken out of step when crossing bridges. Should the beat of the marching feet align with a natural frequency of the bridge, it could fall down. Engineers must design so that resonance does not occur during regular operation of machines. This is a major purpose of Modal Analysis. Ideally, the first mode has a frequency higher than any potential driving frequency. Frequently, resonance cannot be avoided, especially for short periods of time. For example, when a motor comes up to speed it produces a variety of frequencies. So it may pass through a resonant frequency. Other vibration processes such as Time History, Response Spectrum, Random Vibration, etc. are used in addition to
  15. 15. Computer Aided Structural Analysis - 15 - Modal Analysis to deal with this type of more complex situation. These are called Transient Natural Frequency Processors. Buckling analysis If you press down on an empty soft drink can with your hand, not much will seem to happen. If you put the can on the floor and gradually increase the force by stepping down on it with your foot, at some point it will suddenly squash. This sudden scrunching is known as "buckling." Models with thin parts tend to buckle under axial loading. Buckling can be defined as the sudden deformation, which occurs when the stored membrane (axial) energy is converted into bending energy with no change in the externally applied loads. Mathematically, when buckling occurs, the total stiffness matrix becomes singular (see section 8). In the normal use of most products, buckling can be catastrophic if it occurs. The failure is not one because of stress but geometric stability. Once the geometry of the part starts to deform, it can no longer support even a fraction of the force initially applied. The worst part about buckling for engineers is that buckling usually occurs at relatively low stress values for what the material can withstand. So they have to make a separate check to see if a product or part thereof is okay with respect to buckling. Slender structures and structures with slender parts loaded in the axial direction buckle under relatively small axial loads. Such structures may fail in buckling while their stresses are far below critical levels. For such structures, the buckling load becomes a critical design factor. Stocky structures, on the other hand, require large loads to buckle, therefore buckling analysis is usually not required. Buckling almost always involves compression. In civil engineering, buckling is to be avoided when designing support columns, load bearing walls and sections of bridges which may flex under load. For example an I-beam may be perfectly "safe" when considering only the maximum stress, but fail disastrously if just one local spot of a flange should buckle! In mechanical engineering, designs involving thin parts in flexible structures like airplanes and automobiles are susceptible to buckling. Even though stress can be very low, buckling of local areas can cause the whole structure to collapse by a rapid series of ‘propagating buckling’.
  16. 16. Computer Aided Structural Analysis - 16 - Buckling analysis calculates the smallest (critical) loading required buckling a model. Buckling loads are associated with buckling modes. Designers are usually interested in the lowest mode because it is associated with the lowest critical load. When buckling is the critical design factor, calculating multiple buckling modes helps in locating the weak areas of the model. This may prevent the occurrence of lower buckling modes by simple modifications. Thermal analysis There are three mechanisms of heat transfer. These mechanisms are Conduction, Convection and Radiation. Thermal analysis calculates the temperature distribution in a body due to some or all of these mechanisms. In all three mechanisms, heat flows from a higher-temperature medium to a lower- temperature one. Heat transfer by conduction and convection requires the presence of an intervening medium while heat transfer by radiation does not. I include a brief discussion on thermal analysis here. You must have read all these in high school. In this book, I shall not discuss anything more about thermal analysis. Conduction Thermal energy transfers from one point to another through the interaction between the atoms or molecules of the matter. Conduction occurs in solids, liquids, and gasses. For example, a hot cup of coffee on your desk will eventually cool down to the room-temperature mainly by conduction from the coffee directly to the air and through the body of the cup. There is no bulk motion of matter when heat transfers by conduction. The rate of heat conduction through a plane layer of thickness X is proportional to the heat transfer area and the temperature gradient, and inversely proportional to the thickness of the layer. Rate of Heat Conduction = (K) (Area) (Difference in Temperature / Thickness) Convection Convection is the heat transfer mode in which heat transfers between a solid face and an adjacent moving fluid (liquid or gas). Convection involves the combined effects of conduction and the moving fluid. The fluid particles act as carriers of thermal energy. Radiation
  17. 17. Computer Aided Structural Analysis - 17 - Thermal radiation is the thermal energy emitted by bodies in the form of electromagnetic waves because of their temperature. All bodies with temperatures above the absolute zero emit thermal energy. Because electromagnetic waves travel in vacuum, no medium is necessary for radiation to take place. The thermal energy of the sun reaches earth by radiation. Because electromagnetic waves travel at the speed of light, radiation is the fastest heat transfer mechanism. Generally, heat transfer by radiation becomes significant only at high temperatures. Types of Heat Transfer Analysis There are two modes of heat transfer analysis based on whether or not we are interested in the time domain. Steady State Thermal Analysis In this type of analysis, we are only interested in the thermal conditions of the body when it reaches thermal equilibrium, but we are not interested in the time it takes to reach this status. The temperature of each point in the model will remain unchanged until a change occurs in the system. At equilibrium, the thermal energy entering the system is equal to the thermal energy leaving it. Generally, the only material property that is needed for steady state analysis is the thermal conductivity. Transient Thermal Analysis In this type of analysis, we are interested in knowing the thermal status of the model at different instances of time. A thermos designer, for example, knows that the temperature of the fluid inside will eventually be equal to the room- temperature (steady state), but he is interested in finding out the temperature of the fluid as a function of time. In addition to the thermal conductivity, we also need to specify density, specific heat, initial temperature profile, and the period of time for which solutions are desired. Boundary Element A type of finite element sometimes used to connect the finite element model to fixed points in space. Typically this fixity is set with global boundary conditions, in which the fixity is totally rigid. A boundary element, on the other hand, allows for a flexible connection to the fixed space. Boundary elements and boundary points are normally used to simulate the constraints that actually occur when an object is used in the real world. For example, if a coffee cup is
  18. 18. Computer Aided Structural Analysis - 18 - sitting on the table and a weight is placed on top of the coffee cup, then the table is the boundary. Boundary points would be points on the plane of the table that are defined as being fixed in space and to which nodes of a finite element model of the coffee cup are attached. If the table has a spongy surface, you might want to use boundary elements to account for the flexibility. With many FEA software, boundary elements have an additional capability of imposing and enforced displacement upon a model. The force created by this imposed displacement would be calculated automatically. Additionally, the forces generated at a boundary by forces on the model can be obtained as output using boundary elements. There are another very powerful types of analysis offered by high-end FEA programs, known as Mechanical Event Simulation or Virtual Prototyping. You will find this in section 36.
  19. 19. Computer Aided Structural Analysis - 19 - 4. Sign convention (mind your signs) In structural analysis, sign convention is very important. You must follow same sign convention throughout your life! Normally, the force towards right is taken as positive and force acting upwards is considered positive. Anti-clockwise moment is taken as positive. This has been shown in following figure for 2D plane. Fy All positive Mz Fx Figure 4-1 Most standard analysis programs follow this sign convention. Although you can use any convention of your own, but I strongly advise you against that. You will always be fine with this convention. Please note that, because of taking y positive upwards, when specifying gravity loads, you often need to use “minus” sign to do so. For 3D structures, the sign convention will be of same type but somewhat complicate. This is shown below. Y My All positive Fy Mx X Fz Fx Z Mz Figure 4-2 When you see bending moment diagrams, remember that some programs draw them in tension side or some may do the opposite. Also note that the “sign” of bending moment diagrams indicate the “direction” (as shown in figure 4-1 and 4-2), they do not indicate whether the bending moment is sagging or hogging. Axial forces are normally considered positive for tensile forces and negative for compressive forces.
  20. 20. Computer Aided Structural Analysis - 20 - When dealing with 3D structures, the program will generally consider y-axis as elevation. This is as expected, because when dealing with 2D structures, you will normally use x-y plane. But it has exceptions as well. Some programs, by default use x-z plane for 2D analysis. Of course you can direct every analysis program to consider z-axis (or even x-axis) as elevation. My main point here to make you understand that co-ordinate system is very flexible. But you must follow same sign convention throughout. Different programs may follow slight different sign conventions. Before using the program, you should be familiar with that program’s sign convention. Solve some basic problems with them first and consult the user guide. As an example, the following figures show how SAP90/2000 describes frame member internal forces. AXIS 2 T P AXIS 1 T AXIS 3 P Positive Axial Force and Torque Figure 4-3 Compression face V2 Tension face Compression face M2 V3 M3 Tension face Positive Moment and Shear [1-2 plane] Positive Moment and Shear [1-3 plane] Figure 4-4
  21. 21. Computer Aided Structural Analysis - 21 - Now please solve the following problems using your program and check the result with the answer given. -3 kN/m (case 2) 5 kN (case 1) Node 4 Node 3 5 m 6 m 6 m All members are of 250-mm side square Cross section made of concrete E=20GPa Node 1 Node 2 fixed hinged Figure 4-5 Figure 4-6 The above figure shows the bending moment diagram and the free body diagram of each member. Now check the result and the sign with your analysis program. Please note that your program may draw the bending moment diagram on opposite side compared to what shown here! Observe the sign convention.
  22. 22. Computer Aided Structural Analysis - 22 - Now solve the following truss. All members are made of 3 kN steel (E=200 GPa) with 100-mm side square section 3 m 0.208 -3.642 2.0 2.833 2.833 -2 kN 4 m 4 m Figure 4-7 The axial forces are shown as italics in the above figure. Note that the left end is hinged and right end is roller. It is interesting to know that with some programs, you may need to “tell” the program that the structure is a ‘truss’ by specifying ‘moment releases’ in the truss members. Otherwise, you may wonder why the program result shows bending moment diagram in truss! Different programs have different options for specifying moment (or axial force, torsion etc.) releases. Some programs, which allow you to draw plate elements on screen, you should draw them in counter clockwise fashion. Otherwise you may get awkward result.
  23. 23. Computer Aided Structural Analysis - 23 - 5. Numbering of joints and members Proper node and joint numbering is very important for large models. Those programs, which allow you to “draw” the model on screen, apply joint and member numbers automatically. This default scheme may not always be convenient for you, especially if you are analyzing a multi-story building. Fortunately, most programs offer re-labeling option and you can even use alphanumeric labels. Since it is impractical to re-number hundreds of members manually, you should do it automatically. Generally, beams, columns and slabs are numbered on the story or floor level they reside. In that case, you can direct the program to use X-Z-Y re-labeling pattern (assuming Y-axis is the elevation). You may number all beams in the B- 5-10 or B05010 fashion where “B” indicates beam, next number indicates “floor” and the last number stands for serial number of beam on that floor. Similar procedure may be adopted for numbering columns, slabs and other structural members. You can also create ‘group’ for same type of members whose design will be same such as all columns in a particular floor. Improper node numbering may increase bandwidth of global stiffness matrix. However, most programs automatically re-number nodes internally while solving and again display the result in user specified numbering. Wondering what is ‘bandwidth minimization’? It is a technique for assembling global stiffness matrix so that non-zero terms in the matrix tend to become ‘closer’ rather than getting ‘dispersed’. Generally, the non-zero elements of global stiffness matrix are limited to a band adjacent to its diagonal. Lower bandwidth means less time necessary for solving equations. For example, in a multistory frame (assuming the height is more than the length); if you number nodes row wise (horizontally or more precisely along smaller dimension), bandwidth will be less compared to column wise (vertical i.e. larger dimension) node numbering.
  24. 24. Computer Aided Structural Analysis - 24 - 6. Specifying moment of inertia New users of structural analysis programs often find it confusing to define section properties of the members, particularly for 3D structures. You may get help from the following examples. Y X Plan of columns Figure 6-1 In the above figure, the beams are of 200 x 300 mm and columns 200 x 400 mm oriented as shown in plan. Beams can be specified as 200x300 mm without any problem. But for columns, you have to be careful. Generally, the programs will ask you to specify ‘depth’ and ‘width’ of the member. If you specify depth = 400 mm and width = 200 mm then you will get exact section as shown in figure 6-2. If you specify the dimension in opposite manner, then you will get wrongly oriented section for the columns. The above figure is taken from real time view of SAP2000. If your program does not offer real time view (i.e. the members should look like in the real structure in 3 dimension) option, you’re out of luck! Many programs, however, have the option for specifying sectional dimension using ‘tx’ and ‘ty’ (or it might be ‘ty’ and ‘tz’ or ‘t2’ and ‘t3’) option. I have tried with various programs this sectional dimension input. In most cases width = 200 and height (or depth) = 400 worked.
  25. 25. Computer Aided Structural Analysis - 25 - Figure 6-2 Sometimes you may need to specify inertia directly especially for irregular shaped sections. Normally the programs offer only ‘Iz’ and ‘Iy’ options. The most often used is the ‘Iz’. For the beam discussed above, Iz = 200x3003 /12 and Iy = 300x2003 /12. For the column, Iz = 200x4003 /12 and Iy = 400x2003 /12. One important thing you must understand is the concept of ‘member local axes’. In most analysis programs, the ‘local axes’ settings are different from ‘global axes’. Normally, the ‘local axes’ are defined as shown in figure 6-3. Y Z Figure 6-3 Some programs can display ‘local axes’ for all members. Please explore your program’s resource files to see how it handles display of local member axes. XZ Y
  26. 26. Computer Aided Structural Analysis - 26 - Y Z X Y X Z Figure 6-4 The above figure shows orientation of local axes for the inclined member. Note that in your program, the orientation for local axes may be slightly different; for example, direction of Z axis may be in opposite direction. The orientation of global axes is also shown in blue color. It is clear that, when you are defining section properties in terms of local axes, even an ‘inclined’ member is considered as ‘straight’. We shall come to local and global axes story again when we discuss interpretation of analysis output.
  27. 27. Computer Aided Structural Analysis - 27 - 7. Specifying Loads All programs have the option for specifying concentrated and uniformly distributed loads. Some programs allow you to assign a point load on beam without creating a node at that point (the program itself creates a node there internally) where as most programs require that you can assign concentrated loads only at nodes. So, you may need to ‘split’ the member to create intermediate nodes. If your concentrated load is inclined, you better resolve it into horizontal and vertical components yourself and then apply them. Specifying UDL is easy. However, trouble arises, then the load becomes varying. The most common example of varying load is on the beams coming from slabs as shown in figure 7-1. The lengths of the beams are ‘L’. Unfortunately, very few programs will calculate distributed loads form slabs automatically. More often than not, you’ll have to specify the slab load yourself. Some programs allow specifying trapezoidal loads on beam members, however, some allow only triangular load. In that case, you need to ‘split’ each beam into three segments (not necessarily equal) and apply triangular loads at end segments and UDL on mid segment. αL Total Load = W (N) UDL = w (N/m) Figure 7-1 Figure 7-2 Yes, this is somewhat cumbersome if you have, say 200 beam members! But you can avoid trapezoidal loads all together with slight loss of accuracy as shown in figure 7-2. If we equate fixed end moments in two beams (of figure 7- 1 and 7-2), we get 1 – 2α2 + α3 wL2 (1 – 2α2 + α3 ) x W -------------- WL = ------ or w = ----------------------- … (7.1) 12 x (1-α) 12 (1 – α) x L So, the ratio of mid span moment of trapezoidal/uniform r = [(3-4α2 )WL/(24(1-α)] / [(1-2α2 +α3 )(W/L)(L2 /8) / (1-α)] = (1 - 1.33α2 ) / (1 - 2α2 + α3 ) If we tabulate α vs. r values as shown below
  28. 28. Computer Aided Structural Analysis - 28 - α 0 0.2 0.375 0.5 r 1 1.02 1.05 1.068 It is seen that maximum difference of mid span moment for figure 7-1 and 7-2 is 6.8%, which is quite small. So, we can safely replace trapezoidal load with UDL whose magnitude is given by w as shown in (7.1). Under some circumstances, you may have non-linearly varying (e.g. parabolic) type of loads. Except in high-end FEA programs, you can’t input the load through equation. The only way out is, split your member into several sections, and specify concentrated loads varying through nodes (or UDL varying through segments). More number of divisions, better the result is. You may wonder whether you can model all the slabs in your building frame using plate elements instead of converting loads to beams as shown in figure 7- 1. Of course you could. But there are several disadvantages! First of all, your analysis program must have ‘plate’ element to do this. Many frame analysis programs don’t have plate element! If you use ‘plates’, then you must ‘mesh’ it before running analysis. If you have, say 100 slabs (i.e. plates) with 10x10 mesh, you’ll have 11x11x100 = 12,100 extra nodes compared to that you’ll have if you transfer the loads on beams. Not only this takes much more time to have your analysis done, but also it will swamp you with tons of output (just count the number of total plate elements – their stress values etc.)! It has been proved that with the conventional slab load distribution as shown in figure 7-3, you’re quite correct. Figure 7-3 Another type of load, which often creates problem, is due to hydrostatic of earth pressure. If your analysis program has easy method to specify such type of loads, consider yourself really lucky! If you’re applying hydrostatic load on a plate element, apply load before meshing the plate. Sometimes the program allows you to specify separate load at four nodes of the plates (and intermediate values are interpolated) though this is not really necessary in day to day analysis. Hydrostatic load normally takes the shape as of figure 7-4.
  29. 29. Computer Aided Structural Analysis - 29 - Figure 7-4 To specify you generally supply fluid height, axis and density. Now the density may be tricky. For example, in the above figure, the load acts towards the plate. But what to do if we want make it act in opposite direction (i.e. away from plate)? Surprisingly, changing the density into negative works! (Argh!) (I don’t know whether all programs behave in this way, but I found this trick works in Visual Analysis). Surcharge or earth pressure load can be specified in the same way as that of hydrostatic load. If your load needs to be like figure 7-5, then just place the fluid level at higher level. Water level Figure 7-5 Figure 7-6 Figure 7-6 shows another trick where you need to superpose two types of loads to get the desired resultant load distribution. Uniform pressure on plates can easily be applied. While you analyze water tanks, these tricks come handy. When you’re applying distributed load on inclined member, it may act in two different manners as shown in figures 7-7 and 7-8.
  30. 30. Computer Aided Structural Analysis - 30 - Figure 7-7 Figure 7-8 In figure 7-8, the load is projected on horizontal axis. This is the common case. In figure 7-7, the load is acting perpendicular to the member. Most analysis programs can handle both types of loading conditions shown. But it’s your responsibility to apply correct method.
  31. 31. Computer Aided Structural Analysis - 31 - 8. Column Buckling test Solve the following problem: A steel (E=200 GPa) column of 100x100 mm square cross section (I = 8.33E-6 m4 ) is 5 m long and fixed at bottom end. A load is applied axially to the column. Find the buckling load. By calculation, buckling load is given by Pcr = π2 EI/4L2 = 165 kN Figure 8-1 First draw the column, define the column properties and then apply any load, which you know that is well below Pcr. Now see if the column buckles! No, it won’t. To get the correct result you must activate the “Frame instability” or “P- ∆” analysis option yourself to force the computer to make iterations! Now gradually increase the load and re-analyze. At one instant, the computer will show you a message, which will say that the program has encountered a negative diagonal term in member stiffness matrix and analysis will terminate. Note this load. This is the minimum buckling load. You are likely to see that even when the column buckles, the deflected shape of the column is drawn straight! You may ask why computer can’t account for buckling in normal analysis. Well, most analysis programs, by default, perform first order analysis. That is, it sets stiffness matrix, solves it and then calculates axial forces from it. When you instruct it to perform P-∆ analysis, it performs iteration to find out actual axial forces. Remember if you are using a very cheap program or some non- commercial program, it may not have P-∆ analysis option! Be careful! You may wonder, why the computer itself does not choose P-∆ analysis always. Hmm, it would have been nicer. But think of the time required for performing such analysis. I once analyzed a 20 storied 3D frame in VA, which had 10 bays in both x and y direction. With 233 MHz, 16 MB RAM computer it took me 20 minutes to perform first order analysis. If you want to perform P-∆ analysis for
  32. 32. Computer Aided Structural Analysis - 32 - such structure using a standard PC and inexpensive program, chances are that your system will crash! Check it!
  33. 33. Computer Aided Structural Analysis - 33 - 9. Portal and Cantilever method You may have been taught to use portal and cantilever method for analysis of effect of lateral loads in frames. Both of these methods assume a point of contraflexure at mid point of beams and columns, which is often grossly inaccurate. Just analyze any frame subjected to lateral loading by these methods and then compare the results with exact analysis by computer. You will find as much as 50% to 60% difference of moments and shear forces. If computer is available, you must not use these methods. Even for preliminary analysis, when you do not know the size of the members in the structure, still these methods are not useful. You can do the same easily by using computer.
  34. 34. Computer Aided Structural Analysis - 34 - 10. Deflection of Reinforced Concrete member Consider a simply supported beam made of reinforced concrete. It is loaded by uniformly distributed load. How do you calculate its deflection at midpoint? You may, of course, use the familiar equation ∆ = 5wL4 /384EI. But remember, here you must use effective moment of inertia of the section and not the gross moment of inertia of the section if applied moment (wL2 /8 in this example) exceeds cracking moment capacity. Here w stands for dead + live load. Most computer programs do not take into account the reduced moment of inertia because of cracking. Since sometimes Ie comes equal to 50% of Ig, when you do not calculate Ie, you may just double the deflection as found from computer analysis which takes Ig. Please note that, for all members you may not need to use Ie because for all members calculated moments may not exceed cracking moments. Once you have got Ie, you can use the same analysis program to find out the deflection of desired members. But you must note following things. 1. To find out deflection at middle of a beam, you must have a node there. You can achieve this by splitting the beam into two members. Most analysis programs have the option of doing this. 2. Changing I values of some members does not alter moment and shear values which you have got previously using Ig. 3. Ie can be calculated only when you have designed the member i.e. you have specified number and diameter of reinforcement bars. 4. When you are specifying I value explicitly, ensure that you do not define beam width and depth or radius, otherwise you may get absurd results. The formulas for calculating cracked moment of inertia are given below (Ref. 1). For rectangular beam reinforced for tension only: Icr = b(kd)3 /3 + nAs(d-kd)2 Where k = ((2ρn + (ρn)2 )0.5 – ρn and ρ = As/bd For a beam with both tension and compression reinforcement: Icr = b(kd)3 /3 + (2n-1)As’(kd-d’) + nAs(d-kd)2 Where k = ((2n(ρ+2 ρ’d’/d) + n2 (ρ +2 ρ’)2 )0.5 – n(ρ + 2 ρ’), ρ = As/bd and ρ’ = A’s/bd For a T-beam with k>hf Icr = bw(kd)3 /3 + (b-bw)hf 3 /12 + (b-bw)hf(kd-hf/2)2 + nAs(d-kd)2 Where k = (ρ n + 0.5(hf/d)2 )/( ρ n + hf/d) and ρ = As/bd
  35. 35. Computer Aided Structural Analysis - 35 - For a T-beam with k>hf – use same equation as that for a rectangular beam. In all cases n = Es/Ec. Modulus of elasticity of concrete is given by Ec = 5700√fck MPa if fck (MPa) is measured as cube compressive strength of concrete; and Ec = 4700√fck MPa if fck is cylinder compressive strength. What is said above stands for short-term (immediate) deflection. You must add long term deflection due to creep and shrinkage as well. This additional deflection can be obtained by multiplying the short-term deflection (discussed above) due to dead load (+ live load, if live load remains in place for extended periods of time) by creep factor ξ = ν/(1+50ρ’) Where ν = 0.787(months) 0.229 but not greater than 2.0 and ρ’ = area of compression steel/gross cross sectional area of the member. This simple trick works for 1 dimensional member only i.e. for beams. For 2 way members e.g. slab, things are not as easy. We shall discuss later how to find the deflection of 2-way slab by using finite element analysis. Most codes provide you minimum depth of members if you do not calculate deflection. But these values are always overestimated and thus lead to uneconomical design for multistory buildings. Don’t be lazy. Always calculate deflection, you can save money!
  36. 36. Computer Aided Structural Analysis - 36 - 11. Shear deformation Most programs do not take into account deformation due to shear force. For normal beams where depth of beams are much less than their lengths, neglecting shear deformation does not lead to erroneous result but where length of beam is very close to depth of beam it can lead to large error. In fact if (Length of beam/Depth of beam) < 2 then the beam is termed as deep beam. There, shear deformation must be taken into account. Consider the following problem. A point load of 1 MN is applied at the free end of 1-m long steel cantilever beam. The cross section of the beam is 400x600 mm. The total deflection is ∆ = PL3 /3EI + 6PL/5GA (the equation comes from strain energy theorem) = 0.000234 + 0.00007143 = 0.0003029 m. See what deflection your program shows! Chances are that it will show only 0.000234 m. So, what do you learn? Now make the beam section 600x400 mm and you will find that the total deflection is very near to bending deflection. Some programs offer option for specifying shear area. In that case, they can take into effect of shear deformation. Check whether your program has this option. Figure 11-1
  37. 37. Computer Aided Structural Analysis - 37 - 12. Inclined support If your program supports specifying inclined local axes for a particular member, you are lucky. In this case you just need to mention in what angle you want to rotate the local axes of the selected member; then you will specify the joint restraints in usual manner and it will be considered as an inclined support. But if your program does not have this feature, you need to try out something else. You can achieve this by specifying a “spring” of infinite stiffness. Normally you can specify a spring at any angle. The spring reaction is the resultant of X and Y components of reaction. In case of roller support you will get the reaction automatically from spring reaction. Think what you have learnt… How many of following analysis methods you have learnt in the university? – Moment distribution, Slope deflection, Portal, Cantilever, Kani’s rotation contribution, Conjugate beam, Graphical – Funicular polygon & Maxwell diagram – Williot-Mohr diagram, Three moments theorem, Column analogy, Moment area, Substitute frame, Method of joints & method of section for trusses. Probably you know all or most of the above classical methods of analysis. Now be honest, how many of the above methods you still use to solve structures after you have started using computer analysis programs? Probably none! Academic people will argue that all the said methods are to be mastered for a better understanding of structural response. Do you think so? I don’t. Well, among the methods listed above, the moment distribution is most popular. This is quite logical, because this method is easy, does not involve solution of simultaneous equations and converges rapidly. We shall discuss substitute frame method later (see section 13) while considering maximum bending moment, shear force etc. in building frames. Did you notice that all these methods are used for frame analysis only? You may like to know that 80% to 90% of all real world structures analyzed are frame structures. Although you have learnt flexibility and stiffness approach while studying computer method of analysis, only stiffness method is used in computer programs. Modern world’s most powerful analysis method – finite element method is also a stiffness method in essence.
  38. 38. Computer Aided Structural Analysis - 38 - 13. Maximum bending moment, shear force and reaction in building frame: Substitute (Equivalent) frame A frame member will not experience maximum bending moment, shear force and reaction when it is fully loaded. Here we shall combine classical approximate substitute frame with computer analysis. But before that note the following live load distribution criteria. To get this Do this Maximum positive bending moment at center of span Load that span and then alternates spans Maximum positive bending moment at center of span Load adjacent spans and then alternate spans Maximum negative bending moment at support Load adjacent spans and then alternate spans Maximum column reaction Load adjacent spans and then alternate spans Maximum positive bending moment at support Load all spans except adjacent spans In all cases, dead load must always be applied over all spans. Some codes say that if live load intensity does not exceed 75% of dead load intensity, then you can load all spans together with dead and live load without any combination. But if you have computer, it is always better to perform the actual combination to get maximum values of force and moment. In classical substitute frame (see figure 13-1), we isolate one single floor with the assumption of columns at top and bottom floors are fixed. Then we apply the combinations described above to get maximum member forces. In case of computer analysis, though you still need to apply the live load in same combination as discussed above, yet you need not isolate one particular floor. Rather, you should just apply the required span load combination in any floor. In case of regular shaped building elevation, result obtained from one floor will be same for other floors. For example, in the (figure 13-2) shown, the load combination stands for maximum negative support moment in first interior column (actually both interior columns since this structure is symmetric) in 2nd floor (bottom most floor, i.e. ground floor is normally denoted by “0” in structural analysis convention). The value obtained for support moment under this condition will also be the maximum support moment for 1st , 3rd and 4th
  39. 39. Computer Aided Structural Analysis - 39 - floor. (Though it is customary to use reduced live load in roof level). Similarly, other load combinations can be used in same manner. Figure 13-1 Figure 13-2 4 3 2 1
  40. 40. Computer Aided Structural Analysis - 40 - 14. Support settlement Take any statically determinate structure, for example a simply supported beam or a simple truss. Apply a settlement in one of its supports. Now analyze the structure. You will see an interesting phenomenon. Though the program will correctly say zero member force, still it will draw a bending moment or axial force diagram! In a statically determinate structure, there should not be any member force developed due to support settlement. Hooray, you have discovered a bug in the program! The reason of this awkward shape can be explained. Although the member force is zero, the program calculates it as a very small (say 10-100 ) number. The graphic code picks up this small but finite number and draws the force diagram. Now take a statically indeterminate structure. Say a continuous beam. Make one of its support settle to an amount and perform the analysis. You should find some member forces in the beam. Well, now take a statically indeterminate truss. The truss should be externally indeterminate. For example, you can take a 2-support truss whose both supports are hinged (pinned) as shown in figure 14- 1. Now apply a downward settlement in any one of its supports and analyze the structure. Most likely, you will see zero force in all members after the analysis. This is not correct! Figure 14-1 Most standard analysis package use truss stiffness matrix based on ignoring the support displacement perpendicular to member’s local axis. That’s why you get the wrong answer. But since the members’ length change, there are strains, which would create the axial forces. If you want to know the actual member forces after such support settlement, you need to modify the member stiffness matrix considering the displacement in perpendicular direction as well. Unfortunately, you can’t do it with most available programs.
  41. 41. Computer Aided Structural Analysis - 41 - 15. 2D versus 3D For symmetrical structures, often it is possible to convert the 3D model to 2D for easier input and analysis. For example, consider the following structure as shown in figure 15-1. Figure 15-1 You can easily analyze just one plane frame as shown in fig. 15-2. Whenever possible, try to convert 3D structures into 2D in this way. 2D structures are not only easier to model, but also they can be ‘handled’ and analyzed much more easily compared to 3D structures.
  42. 42. Computer Aided Structural Analysis - 42 - Figure 15-2 Look, in fig. 15-1, the loads are towards X direction. If there were additional loads of same type towards Z direction, you could adopt similar 2D frame (on YZ plane) as shown in fig. 15-2. You can then superpose the result as long as it is a linear structure with material and member section properties are the same. How about dynamic analysis of the frame shown? Is it possible to convert 3D into 2D? I shall discuss this when covering dynamic analysis in detail.
  43. 43. Computer Aided Structural Analysis - 43 - 16. Curved member Most frame analysis programs do not have curve element. You will need to replace the curved member by a number of straight members. Obviously, more the number of straight members used better the accuracy is. While drawing straight members for curve elements, it is a good idea to change grid setting into “polar” form instead of normal rectangular setting. Another way of doing this is to figuring the straight members’ nodal co-ordinates in spreadsheet (for example, Microsoft Excel or Lotus 123). This is useful when the equation of curve is known as y = f(x) e.g. parabolic arch. By using spreadsheet’s built-in commands, you can easily find out y co-ordinates of the curve against each x co-ordinate. Some programs can “copy” and “paste” member and nodal information to and from spreadsheet file. You may like to know that it is theoretically possible to create stiffness matrix of a curved member. Now solve the following two hinged parabolic arch. -50 kN/m (downward) Y 4 m X 20 m Figure 16-1 The answer is: left vertical reaction 375 kN ↑, right vertical reaction 125 kN ↑, horizontal reactions are 312.5 kN inward at both ends. With 20 straight-line segments, you should get exact answer within 1% accuracy. Note that the theoretical answer has been obtained by (H = ∫ My dx / ∫ y2 dx) formula.
  44. 44. Computer Aided Structural Analysis - 44 - 17. Tapered section Many programs have the option of specifying tapered or variable cross sectional members. If so, you’re lucky. If not, still you’re lucky as you are reading this book! To specify a tapered section by yourself, you should ‘break’ the members into a number of parts (more the number, better is the result). Then you should specify various A (areas) and I (inertia) for each segment. This will become clear from the following problem. 1.5 -5000 1.5 2 4 15 15 Figure 17-1 The figure shows a tapered beam. Hinged at left end and fixed at right end. A clockwise moment (hence minus sign) of 5000 has been applied at middle of the beam. It is required to analyze the beam. The cross sections at both ends have been shown. Please note that the beam has been ‘divided’ into 8 sections. Width of the sections is same throughout. But the depth is varied as (from left most section) 2, 2.25, 2.5, 3, 3.25, 3.5, 3.75, 4. The calculated reaction at left end is – 163 (i.e. downward) and at right end is +163 (i.e. upward) compared to theoretical answer of 170. The bending moment at just left of mid-point of beam is – 2443 (theoretically –2549) and that of right is 2557 (theoretically 2451). You will get more accurate answer if you divide the beam into more number of elements. Note another interesting point that, in this problem, I didn’t specify any unit or E value of material! You should get same answer whatever unit you use. Although some programs do allow you to specify “linearly” tapered members; you still need to apply this trick for “non-linearly” (e.g. cubic or parabolic) tapered members.
  45. 45. Computer Aided Structural Analysis - 45 - 18. Nodes connected by a spring Many programs allow you to define a spring support, but none will allow you to connect two nodes by a spring. But you can achieve this! Replace the spring by a member connected between those two nodes where the spring is required. Choose the properties of that member so that stiffness of spring equals AE/L of that connected member. E should be same as that of material of the spring. Choose A and L value properly, keeping L small; because if you choose large L, the member will buckle easily. Also, do not forget to release moment on this member i.e. this spring replacement member should carry axial load only. After analysis, you must check whether axial load in spring replacement member is below its buckling load (π2 EI/L2 ). This can be automatically checked if you activate P-∆ analysis (see section 8) option in your program. This trick works! Figure 18-1
  46. 46. Computer Aided Structural Analysis - 46 - 19. Sub-structuring technique and symmetry (break them into pieces…) In the analysis of large structures, it is often possible to consider only a part of the structure rather than the whole. This approach is useful to reduce the labor (cost and time) of preparing the data, of computing and of interpretation of the results. When an isolated part of a structure is analyzed, it is crucial that the boundary conditions ‘sub-structures’ accurately represent the conditions in the actual structure. As a first simple example, consider the following structure as shown in figure 19-1. You are required to analyze the structure. 10 kN A = 2002 mm² 4 m E = Steel -5 kN/m 20 kN 4 m 4 m 4 m 4 m Figure 19-1 If you separate the upper floor and then analyze only that portion, you will get the result as shown below. 10 kN 11.44 kNm 11.44 kNm 5 kN 5 kN 4.28 kN 4.28 kN Figure 19-2 With the result shown above, the applied loads on the bottom floor of the actual structure will be as shown below.
  47. 47. Computer Aided Structural Analysis - 47 - 11.42 kNm 11.42 kNm -5 kN/m 25 kN 4.28 kN 5 kN 4.28 kN A B C D Figure 19-3 Observe that on leftmost node, 25 kN loads comes from 20 kN applied at that node and 5 kN reaction from upper floor. The reactions you will get in the lower floor should be same as that of obtained if you considered the whole structure as shown in figure 19-1. For your check, the ultimate results are as given below for problem figure 19-1. Node Fx kN Fy kN Mz kNm A -6.223 -14.85 15.58 B -5.99 30.54 15.18 C -11.2 20.64 22.08 D -6.583 3.675 15.81 From the above example, it is clear that; you need to apply opposite of reactions as loads on lower floor frames. The procedure described here seems too meager for this particular structure, but this method is an absolute must for doing a fine meshed finite element analysis. It may happen that, if you run the whole structure once, it may exceed the program’s or your computer’s resource limit. That’s why it’s so important to ‘break them into pieces’. Whenever possible, try to design symmetrical structure as much as possible. They behave better than unsymmetrical ones. For symmetrical structures, this sub-structuring technique is a great time saver. When the structure has one or more planes of symmetry, it is possible to perform the analysis on one-half, one- quarter or an even smaller part of the structure, provided that the appropriate boundary conditions are applied at the nodes of the plane(s) of symmetry. Followings are some examples of exploiting symmetry of structures.
  48. 48. Computer Aided Structural Analysis - 48 - Continuous beams, with even number of spans. Actual beam Figure 19-4 Symmetry utilized beam Fixed Figure 19-5 Continuous beams, with odd number of spans. Actual beam Figure 19-6 Symmetry utilized beam Z direction rotation fixed Figure 19-7 The ‘key’ to utilizing symmetry, is applying proper boundary condition. Remember, in order to take advantage of symmetry, both the structure (geometry and material) and the applied load must be symmetric. Although, you can still take advantage of symmetry even if the loading is ‘anti-symmetric’ (i.e. one half of the loading is similar to other half in magnitude but opposite in direction), the procedure will be somewhat screwy. In all cases, our sign convention is same as described in section 4 earlier. Now consider plane frames with even number of bays as shown in fig. 19-8. This frame can be detached, after applying proper boundary condition, as shown in fig. 19-9. Plane frame with odd number of spans has been shown in figure 19-10. Here you will have to apply boundary condition of X translation and Z rotation prevented in mid points of the middle beams as shown in fig. 19-11. Symmetrical structures are not only easier to analyze but also perform better than unsymmetrical structures in real life!
  49. 49. Computer Aided Structural Analysis - 49 - Figure 19-8 Fixed Fixed Fixed Figure 19-9
  50. 50. Computer Aided Structural Analysis - 50 - Figure 19-10 X translation & Z rotation fixed X translation & Z rotation fixed X translation & Z rotation fixed Figure 19-11 Exercise Solve some problems yourself on the basis of above example models. Unless you analyze the models and visualize the results, things will not be crystal clear to you. If you face any problem, don’t hesitate to send me an e-mail! You will find advanced info on 3D structures’ symmetry, plate’s symmetry etc. in later sections.
  51. 51. Computer Aided Structural Analysis - 51 - 20. Staircase analysis A staircase is actually a folded plate structure. But in our traditional simplified method of analysis, we consider it as a straight beam. How far is this assumption justified? Consider the figure of the staircase shown below. Figure 20-1 The first figure shows exact shape of a flight of a staircase with loads (including self-weight). The second figure is the approximation of the same staircase as simple beam. The section of concrete staircase may be taken as 1-m width x 150-mm depth. The length of simple beam equals 1.25 + 2.75 + 1 = 5 m. Theoretically, loading on landing should be less than that of inclined flight. In approximate calculation, it is assumed same load is acting through out the span for conservative result. The results of both analyses are shown in next figure. In this case we have considered the staircase as simply supported. Depending on casting, it may be fixed-fixed or fixed-pinned as the case may be. In fact staircases are more often analyzed as fixed-fixed support condition. From the analysis it is found that maximum mid span moment is almost same in both analyses. Shear forces (reactions) are also more or less equal. This proves that approximate analysis of staircase is not really inaccurate! In hand calculation, moment was computed using simple M = wL2 /8 formula. We shall venture on folded plate analysis in detail in some later section.
  52. 52. Computer Aided Structural Analysis - 52 - Figure 20-2 This analysis was done in Visual Analysis 3.5. An interesting point is to note that, both structures were analyzed as a single file. This is applicable to most analysis programs. You may analyze as many as separate structures in a single file even they are not connected together. Now think about the following paradox. Following three beams are all simply supported (left end pinned and right end roller). Their projected length on plan is same in all cases (say 10 m). They are all acted by same uniformly distributed load on ‘projected’ length (say 10 kN/m). Find out what will be the bending moment at mid spans. 10 m 10 m 10 m Figure 20-3
  53. 53. Computer Aided Structural Analysis - 53 - What result do you see? The bending moment (and reactions as well) is same in all cases! If you took w = 10 kN/m and L = 10 m, then Mmax = wL2 /8 = 10*102 /8 = 125 kNm. It shows that, for the simple beam, bending moment is same irrespective of beam’s geometry. This happens because all three beams shown are statically determinate structures. Now make all the beams fixed at both ends. Now re-analyze them and you will see different bending moments for all cases. The example problem I presented in this section for staircase, was simply supported in both ends. That’s why you got same bending moment! Had they been fixed at ends, the results would not have matched. However, they still would not differ appreciably from traditional straight beam calculation. Still in doubt why you got same result for statically determinate beams? Well, the reason is simple. As the beams were simply supported, horizontal reactions at supports are zero (since we have only loading acting downward). So, moment due to ‘eccentricity’ of geometry is also equal to zero. This will be from following figure. Internal moment developed Horizontal reaction H ex = eccentricity x Figure 20-4 This internal moment (= Hex at any section of distance x from end) causes the bending moment to differ from the value as in case of straight beams (where ex = 0 at all sections). In case of statically indeterminate beams, both H and ex are non-zero. So, the internal forces differ depending on geometry of the beam. When you analyzed two hinged arches as a student you probably used the equation: Arch moment = Beam moment – He. Didn’t you?
  54. 54. Computer Aided Structural Analysis - 54 - 21. Cables It is possible to analyze cables with a mere frame analysis program. A cable carries ‘tension’ only. So, you should define a cable in the same way as truss member (which carries axial force only) but additionally you will have to specify that it can take tension only (no compression). Some analysis programs may not have the option of defining a tension only member! Once you have specified cables in this way, the analyses are pretty straightforward. While viewing the result, you should check whether cables’ axial force diagram shows tension only (generally positive number) and no bending moments. That’s all. An example of cable structure is shown in fig 21-1. After performing the analysis, check your answer with exact result as given.
  55. 55. Computer Aided Structural Analysis - 55 - Figure 21-1
  56. 56. Computer Aided Structural Analysis - 56 - Vertical reaction at A = 104 kN (down), at B = 250 kN (up). Moments: MAB = 0, MBA = 75, MBE = -117, MEB = 41, MEC = -41, MBD = 41, Mid span of EC = 84 kNm. Figure 21-2 It is also possible to analyze the cable shown in figure 21-2. Use suitable values for span, sags and loads. Then find out the tension in cables. This is given as an exercise to you! If the loads are all unequal, the tensions in the cables will be different. Check if equation of static equilibrium is satisfied at each node.
  57. 57. Computer Aided Structural Analysis - 57 - 22. Pre-stressed cable profile Does your program offer specifying pre-stressed cable profile? If yes, then good. If not then read the following tricks. Observe sign conventions carefully. PyA PyB yA θA θB yB PθA PθB c P P Upward UDL w = 8Pc/L² c θA = (4c + yA – yB)/L L/2 L/2 θB = (4c – yA + yB)/L Actual pre-stressed cable Equivalent load Figure 22-1 PyA PyB yA θA θB yB PθA PθB P P(θA + θB) P Total length L Actual pre-stressed cable Equivalent load Figure 22-2 Observe the figures very carefully. They are really confusing! Try to comprehend the following worked out problem. Please note that the θ values are in radians. Note that the yA and yB indicate eccentricity of the cable at supports in upward direction from center of gravity of concrete (cgc) line. Upward distance is positive at supports and downward distance is positive at mid spans for pre-stressed cable profile (majority of standard analysis programs follow this sign convention). If the cable distances are of opposite sense compared to what shown in above figures, then ‘arrows’ of moments will be reversed.
  58. 58. Computer Aided Structural Analysis - 58 - With reference to the figure 22-3, the calculation is shown below. For left span, L = 15 m, yA = 0.5 m, yB = 0.5 m. So, θA = (0.5+0.5)/10 = 0.1 rad And θB = (0.5+0.5)/5 = 0.2 rad So, moments are 500 x 0.5 = 250 kNm on left end and 500 x 0.5 = 250 kNm on right end of left span. Concentrated force at 10 m from left span is 500 x (0.1 + 0.2) = 150 kN. For right span, L = 15 m, yA = 0.5 m, yB = 0.4 m. So, θA = (4 x 0.6 + 0.5 – 0.4)/15 = 0.17 rad And θB = (4 x 0.6 – 0.5 + 0.4)/15 = 0.15 rad And c = (0.5 + 0.4)/2 + 0.6 = 1.05 m So, equivalent upward load w = 8 x 500 x 1.05/152 = 18.67 kN/m. Also, support moment at left end of right span is 250 kNm and on right end is 500 x 0.4 = 200 kNm. So, the equivalent forces on the beam will be of as shown in figure 22-4 (axial force P is not shown). Figure 22-3 250 kNm 250 kNm 200 kNm 250 kN 150 kN 250 kN 18.67 kN/m 250 kN Figure 22-4 Now the forces shown in blue color will go to support directly. Moments shown in orange color will cancel each other. So, the remaining forces that will act are
  59. 59. Computer Aided Structural Analysis - 59 - shown in green color. The ultimate equivalent load will be that of as shown in figure 22-5. 250 kNm 200 kNm 150 kN 18.67 kN/m Figure 22-5 So, for pre-stressing force, the beam should be analyzed for the loading shown above. Naturally, the beam will also carry dead load and live load as well. Analyze the beam for these loads as separate cases and then combine the results as desired. In actual practice there are always more than one cables. You can analyze effect of each cable separately and then superpose to get the net result. Also remember that, there is a uniform compressive stress ‘P/A’ in the concrete in addition to the bending stress due to pre-stress, dead load and live load. For more information on this subject, please see any standard textbook on pre- stressed concrete. I have shown here only linear and parabolic cable profile. Although parabolic profile is the most common, there are other types of profiles possible. See your textbook for details.
  60. 60. Computer Aided Structural Analysis - 60 - 23. Finite Element Analysis (FEA) Method is approaching… We now come to the most outstanding and most versatile method of structural analysis: the Finite Element Method. It has made possible to analyze virtually all kinds of structures that human brain ever can imagine! If you have studied finite element before, you may skip this section. Those who did not, I present a very very brief introduction of the subject. There exist more than 1001 books in this subject. But I warn you; the theory of finite element analysis is very complex! What is meant by finite element? The answer is any element, which is not infinite. Don’t be exasperated; this is the real definition of finite element. Did you play with mechano when you were a child? Just think how you built a model car or house by “Lego” parts? Now consider each part of mechano as “finite element”. A number of mechano elements were needed to build your model car or house. Now consider a frame. It is made of a number of beam/column members or “bars”. Here the “bars” are “finite elements” of the “frame”. I hope you have probably realized now that the frame analysis, so far what we have discussed in preceding sections, is actually finite element analysis in essence where each finite element is a “bar”. Figure 23-1 This is the longitudinal section of a beam shown. That is, you are viewing a beam from its length side. Observe that here we consider the beam as 2- dimensional “Plane stress” structure. Don’t confuse this with 2D or 3D frame. By 2D or 3D frame we actually mean “Plane” and “Space” frame. In all previous cases, we treat all beams as “bars” like a “stick”. But in the above figure we are treating the beam taking into effect of its length as well as depth. That’s why it is 2D. Had we considered the width of beam in the analysis, it would have been termed as 3D solid. Pretty confusing! Look, there is a “cut” in the beam. The beam is simply supported, left end pinned, right end roller. It is X Y
  61. 61. Computer Aided Structural Analysis - 61 - loaded by a uniformly distributed load. We like to find out the stresses at various points of the beam. Please note that analysis of this problem by classical method is close to impossible. So, first we divide the beam into a number of “triangular finite elements”. Then we shall determine the member stiffness matrix [k] of each individual triangular element and ultimately we shall have to combine the member stiffness matrices into “global stiffness matrix” [K]; pretty much the way we did in case of frame analysis. Then we shall have to apply the boundary condition on [K] matrix. Figure 23-2 After that we need to construct force matrix [P]. For this, distributed load must be converted to appropriate nodal loads by applicable equations. So, our problem can be represented by familiar equation [P] = [K][D]. From this equation we can solve for [D] and then we can find out nodal stresses form equation [σσσσ] = [C][εεεε] where [C] matrix differs in various cases like plane stress, plane strain etc. We are describing this problem as plane stress because we considered only 2 dimensions (X and Y) and stress variation along width (Z direction) has not been taken into account. That means we have taken care of only σx, σy and τxy. In this problem we considered the beam is made of “triangular” finite elements, but we could have also considered it is made of “rectangular” finite elements as shown in figure 23-3. Figure 23-3 If you analyze the beam with both triangular and rectangular elements as shown above, you will see that you get accurate answer when you use rectangular elements. It proves one very fundamental concept of finite element analysis: You must choose proper element for particular problem. You do get correct result with triangular finite element but you must use very fine mesh compared to rectangular element. In general, triangular element is not a good choice. If you are interested to know why triangular element behaves in such way, you should consult any standard finite element analysis textbook. X Y
  62. 62. Computer Aided Structural Analysis - 62 - Figure 23-4 As a crude rule, when you use triangular you will normally need much finer “mesh” than rectangular elements. The assembly of elements in finite element analysis is called “mesh”. Most powerful finite element programs can generate mesh automatically if you specify the boundary surfaces of the models. If you want to analyze the same beam in 3D then your model will look like as shown in figure 23-5. Figure 23-5 In this case finite element will be 3D solid element like shown in figure 23-6. Figure 23-6 This is an 8 noded finite element because it has 8 nodal points. If its each vertex has one additional point in the middle, then it would have been 20 noded finite element. Higher is the number of nodal points in an element better is the accuracy of the solution. But higher noded elements are difficult to calculate even with a computer since total number of nodes increases the size of global stiffness matrix. Whatever element you use, it must be compatible. Compatibility means there must not be any discontinuity or overlapping among the elements when the analysis model deforms under applied load. You can combine more than one kind of element in single structure. You should use more number of elements where you anticipate stress variation is more irregular. There are a lot more other finite elements in addition to basic triangular and rectangular elements discussed above.
  63. 63. Computer Aided Structural Analysis - 63 - One distinguishing feature of finite element method is that it does not provide “closed form” solution. Every problem in finite element analysis is unique. This probably needs little more explanation. Think of a simple beam. In classical method of analysis, you can make a program, which takes L, E, I and w as input and computes deflection at any point by solving the equation of elastic line, which can be easily formulated. But in case of finite element analysis, if you change the length of the beam, it becomes another new problem because the geometry of the model changes. Of course you can change E or w values or boundary condition without remodeling the whole structure. Another aspect of Finite element analysis is that it almost always produces an approximate result. I used the word “almost” because finite element analysis does produce exact result only when the finite element is “bar” that is in “frame structures”. You may be wondering that if finite element method can solve any structure, then what is the justification of studying classical methods of analysis. Aha! A real question indeed! You can realize it yourself. Just think of solving a simple beam in finite element method (this is presented just after this section). After you solve this beam by finite element method, you can easily check whether the result is correct or not by comparing the answer obtained by classical method. But now imagine the analysis of the fuselage of an airplane or the propeller of a ship. How do you check the correctness of these analyses? Therefore you must accept the finite element analysis result as exact result! That’s why it is so important that finite element analysis models must be created to simulate the actual structure as much as possible. You must use proper combination of finite elements, sufficiently accurate mesh, proper load and applicable boundary conditions. It is often a common practice to analyzing the structure first with a particular mesh and then repeating the whole analysis after doubling the mesh to see whether the result converges. But this method has drawbacks! Your program cannot analyze the structure if your number of mesh nodal points exceed the program’s capacity. Moreover, it is very difficult to predict beforehand what particular “finite element” will best simulate the structure. This is especially a demanding task for very complex structures. Finite element method is nowadays widely used in all branches of aerospace engineering, bio-medical engineering, mechanical engineering and structural engineering etc. Some manufacturing companies spend millions of dollars every year in finite element analysis! I am concluding finite element introduction here. But you must realize that it is not so easy as it seems. Researchers are still developing new finite elements.
  64. 64. Computer Aided Structural Analysis - 64 - Sometimes even the most expensive finite element analysis programs produce wrong answer to complex problems. If you feel inclined to know more about this wonderful (?) tool of analysis, I strongly recommend that you to go through some standard finite element method textbooks. One word of advice, many engineers tempt to use finite element analysis everywhere even when it is possible to analyze the particular structure using classical method of analysis. My main aim is to make you realize that finite element analysis is required only when it is absolutely necessary. Remember that finite element analysis programs are very expensive and they also demand great part of contribution from you for preparing input and interpreting output. Typically, a finite element analysis consists of following steps. 1. Defining the model (i.e. drawing it either in the finite element program’s graphical interface or importing it from a CAD program). 2. Creating the mesh (most programs can automatically generate mesh for best result). 3. Defining the boundary conditions. 4. Defining the loads. 5. Performing analysis (may take hours for complicated models!) 6. Interpreting the result (very important). The steps are pretty straightforward. But there are many glitches! In next page you will find an exercise of simply supported beam with uniformly distribute load analyzed by FEA method. This example is for your understanding of the basic concept of FEA only. In practice, this problem should be solved by simple flexure formula of σ = My/I. Remember this!
  65. 65. Computer Aided Structural Analysis - 65 - Exercise A 5-m steel (E=200GPa) beam has width 200 mm and depth 500 mm. It is loaded by 10-kN/m uniformly distributed load. ν = 0.3. Its left end is hinged and right end is roller. Find deflection at mid point and maximum bending stress in the beam by finite element analysis. Try following modeling: 1. Plane stress analysis with 20 rectangular elements, each 0.5x0.25 m size. That means there are 10 elements in X direction and 2 elements in Y direction. You can convert the uniform load into nodal loads by applying 0.25 kN at extreme nodes and 0.5 kN at intermediate nodes. 2. Solid model analysis. Use standard solid brick or tetrahedral element. Most finite element analysis programs offer these elements. Figure 23-7 In case of plane stress model formulation, you should use plate finite element whose thickness will be equal to the depth in Z direction. In this problem, this is equal to width of the beam. After performing the finite element analysis, you should get the answer: mid point deflection 1.95x10-4 m maximum stress 3.75 MPa. Your program may display slight different result due to numerical round off in calculation. The deflected shape should resemble the following figure. Original shape is shown by dotted line. This 2D-beam analysis was performed in Visual Analysis. For 2D analysis, after modeling your structure should look similar to this figure.
  66. 66. Computer Aided Structural Analysis - 66 - Figure 23-8 The figure 23-9 shows one of mid plane stresses, local σx distribution. Your program should have the option to display other stresses e.g. σy , τxy etc. Interpreting the finite element analysis result is very important. It is expected that you spend equal or more time in interpreting analysis result compared to the time previously spent in preparation of the model. Later we shall see how finite element analysis can produce incompatible result. There you will realize why it is essential to learn some theory behind the finite element analysis. Figure 23-9
  67. 67. Computer Aided Structural Analysis - 67 - To analyze the beam as 2D, you should not face any difficulty. However, you should take into account many other things when you analyze the same beam as 3D solid. Firstly, what will be the load? Look, here we’ve applied a total load of 10 kN/m x 5 m = 50 kN acting on the upper face area of 5 m x 0.2 m = 1 m. So, the applied load we have to specify as 50 kN/m² pressure normal to the upper surface. Be careful about the load’s direction. Now, comes the main hurdle, the meshing. If you are using a high-end FEA program, it will mesh the model itself. By default, the program will mesh it by using brick elements or tetrahedral elements. The next figure shows the beam with automatically generated tetrahedral mesh. You may note that, such high density meshing is not really required for this very problem. If you manually mesh with 20 numbers (2 elements along depth and 10 elements along length, similar to shown in fig. 23-7) 8-noded solid elements, (as in SAP2000) you will get exact result for this problem. Left end boundary condition is X, Y, Z translation fixed along bottom edge and Z translation fixed along bottom edge on right end. Figure 23-10 The next figure shows stress (σx) diagram on displaced shape. Figure 23-11 This 3D analysis was performed in Cosmos/Design Star.
  68. 68. Computer Aided Structural Analysis - 68 - Did you see that finite element analysis programs normally give you output in the form of nodal displacements and stresses. It does not show you bending moment or shear force diagram. Why? Well, why do you need bending moment and shear force values? To calculate stresses later, isn’t it? Finite element analysis programs directly give you the stress values!

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