ashraf habibullah, g. robert morris - makalah

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Seminar dan Pameran HAKI 2009 - “Advanced Nonlinear Analysis for Modern Structures”

ADVANCED NONLINEAR ANALYSIS

FOR MODERN STRUCTURES

Ashraf Habibullah & G. Robert Morris

ABSTRACT Today’s modern structures have created new demands for structural analysis and design. Largerstructures with more exotic shapes are challenging traditional approaches and design codes.Greater performance expectations have become more commonplace, including for the responseto wind, vibration, earthquake, and blast hazard. Continued economic pressure requires moreefficient use of labor and materials, and compressed timetables with rapid design-build projects.This paper discusses the new capacities for analysis and design that are becoming available tomeet these demands. Improved nonlinear analysis techniques provide the designer with more

information on the behavior of the structure. Performance-based design procedures provide amore nuanced approach for designing structures for capacity against failure. Integratedmodeling/analysis/design software provides the framework needed to manage the informationneeded for sophisticated design. All of these require significant computer power, which fortunatelycontinues to grow. Engineers must learn how to harness this power to produce practical designinformation without being overwhelmed by numerical detail.

ABSTRAK Struktur-struktur modern saat ini telah menyebabkan timbulnya kebutuhan baru dalam analisadan desain struktur. Struktur-struktur berskala besar dengan ragam bentuk yang lebih tidak lazimmenantang pendekatan tradisionil dan peraturan-peraturan desain. Performance lebih besar yangdiharapkan telah menjadi sesuatu yang biasa, termasuk respons terhadap angin, getaran, gempa

bumi dan ledakan. Tekanan ekonomi yang berkelanjutan mengharuskan penggunaan tenagakerja dan bahan-bahan yang lebih efisien, dan mempersingkat waktu pengerjaan dengan proyek-proyek design-build yang cepat.Paper ini membahas kemampuan baru untuk analisa dan desain yang tersedia untuk memenuhikebutuhan-kebutuhan tersebut. Peningkatan teknik-teknik analisa nonlinear memberikankonstruktor lebih banyak informasi tentang prilaku struktur. Prosedur performance-based design memberikan suatu pendekatan yang lebih untuk mendesain struktur dengan kapasitas mendekatiruntuh. Software modeling/analisis/desain yang terintegrasi menyediakan kerangka kerja yangdibutuhkan untuk mengelola informasi yang dibutuhkan untuk desain yang rumit. Semua inimembutuhkan kemampuan komputer yang besar, suatu hal yang menguntungkan kemampuankomputer memang terus berkembang. Engineers harus belajar bagaimana memanfaatkan

kemampuan tersebut untuk menghasilkan informasi desain praktis tanpa terkecoh oleh detailnumerik.



Seminar dan Pameran HAKI 2009 - “Advanced Nonlinear Analysis for Modern Structures” 1

ADVANCED NONLINEAR ANALYSIS

FOR MODERN STRUCTURES

Ashraf Habibullah & G. Robert Morris

1. INTRODUCTION

Today’s modern structures have created new demands for structural analysis anddesign. Larger structures with more exotic shapes are challenging traditional approachesand design codes. Greater performance expectations have become more commonplace,

including for the response to wind, vibration, earthquake, and blast hazard. Continuedeconomic pressure requires more efficient use of labor and materials, and compressedtimetables with rapid design-build projects.

Fortunately, new capacities for analysis and design are becoming available to meetthese demands. Improved nonlinear analysis techniques provide the designer with moreinformation on the behavior of the structure. Performance-based design proceduresprovide a more nuanced approach for designing structures for capacity against failure.Integrated modeling/analysis/design software provides the framework needed to managethe information needed for sophisticated design, particularly for rapidly changing design. All of these require significant computer power, which fortunately continues to grow.

2. MODERN STRUCTURES

Buildings continue to get taller and more slender. The structural systems also areevolving, with shear wall becoming more common since the attacks of 9/11. This makesthe consideration of construction sequencing, including age effects (strength, creep, andshrinkage), more important than ever. The seismic performance of such structures is notfully covered by current codes. Comfort and safety also demand the use of specialdamping or active control systems for wind loading.

The desire for signature structures also is leading to more exotic shapes, including theuse of curves, asymmetry, and cantilevered construction. Multiple towers on a commonbase, sometimes connected by bridges at higher levels, are another example ofnontraditional structures. The structural behavior of such systems often requires specialconsiderations that are not well covered by current codes. Construction sequencing, withage effects, can be quite important for determining stress distribution as well as thedeflected shape, which may be significantly unsymmetrical.

Similar considerations affect modern bridges, with increased use of curved girders,prestressing, and segmental construction. Construction sequencing is almost alwaysimportant for composite superstructure design. Increased use of cable-stayed bridgesrequires techniques for determining and controlling cable tension, and controllingvibration due to wind and vehicle loading. Determination of camber is affected byconstruction sequence, age effects, and cable tensioning, depending upon the type of

bridge construction.




3. MANY TYPES OF NONLINEARITY

There are many different types of nonlinearity that may be important in modernstructures. These may typically include: Material stress-strain, such as yielding, cracking,strength loss, and stiffening; Geometric nonlinearity, such as P-delta effects and largedeflections and deformations; Cables and fabrics; Contact and friction; Special devices,such as isolators, nonlinear dampers, and buckling restrained braces (BRB); andNonlinear soil and foundation behavior.

Staged construction also can be considered a type of nonlinearity, whether or not thebehavior is linear within a given stage, since the overall stiffness of the structure ischanging. Changes include the addition and removal of members, modification ofsupports and connections, and changing material behavior due to curing, creep, andshrinkage.

Only some of these types of nonlinear behavior will be present in any given structure. Inaddition, different types of nonlinearity may be important for different aspects of theanalysis and design of a given structure. It is important to have a clear sense of when toconsider nonlinear behavior and to have analysis and design tools that canaccommodate these different behaviors, even for different analyses of the samestructure.

We will now examine nonlinear behavior in more detail.

4. STAGED CONSTRUCTION

The sequence of construction can have a significant effect on the final shape of thestructure, and more importantly, the distribution of stresses, particularly for redundantsystems. This is especially true for concrete structures, which can exhibit creep andshrinkage, as well as increasing stiffness due to continued curing during construction.

Consideration of creep and shrinkage is common for segmentally constructed bridges,where it is necessary to determine the deformed geometry in order that the cantileveredsegments from two piers meet correctly at mid-span. However, it has becomeincreasingly important to consider these effects for modern buildings, where shear wallsand columns may deflect differentially. Curved, unsymmetrical, and other exoticallyshaped buildings also may experience significant lateral and torsional deflection undergravity load, which can be exacerbated by creep and shrinkage.

Staged-construction analysis, properly utilized, allows the accurate determination of thedeformed shape and stress distribution at any time during the construction process andfor years thereafter. Operations that could have a significant effect on the behavior andshould be modeled include: Addition of structural members, with proper account of theirages; Addition and removal of temporary support structures; Changing of supportconditions and member fixity; Jacking, fitting, and tensioning operations that may inducesignificant stresses.

When considering time-dependent effects, the results can be very sensitive to theassumed age of young concrete. Most creep models will give excessive deflections ifstresses are applied to concrete unrealistically early [CEB]. Load should not besupported by the concrete until the age at which the formwork is removed. For staticallydeterminate structures, the amount of creep can be computed easily at any time from the




creep model. For indeterminate structures, creep causes stress redistribution, which inturn modifies the creep behavior. Proper time-integration schemes must be used in thiscase to achieve accurate results [Cook et al], [Zienkiewicz & Taylor]. Traditional creep

models consider the history of every stress increment applied to the material. This canlead to growth in the storage and calculation time required for each time step, especiallywith stress redistribution. The use of Dirichlet series to represent creep behavior cancontrol this data growth and also prevent excessive deflections for young concrete[Ketchum].

Figure 1. Examples of Segmental Construction of Bridges

In addition to being able to change support and fixity conditions, jacking, fitting, andtensioning operations may need to be modeled. Elements that support strain ordeformation loading can be used to jack to a specified displacement, or to pull partstogether. Jacking or tensioning to a specified force requires an iterative procedure thatdetermines the amount of strain/deformation load required to achieve that force. This canbe highly dependent on the flexibility of the surrounding parts of the structure, and insome cases, may not be achievable if equilibrium would be violated.

For structures undergoing significant elastic and/or time-dependent displacement, it isoften desirable to specify the desired final geometry and calculate the initial geometryusing analysis. This provides the size and shape (camber) of the member needed, aswell as where it will be located when added to the structure. Two approaches arecommonly used. In the first case, we start with the entire final structure, then load anddeconstruct it in reverse. The geometry just before the element is removed shows theinitial shape and location of the member when it is added. The second approach is

iterative, building and loading in normal order, then modifying the initial geometry basedon the final deflected shape and re-analyzing. This requires more computational effort,but is required for proper consideration of time-dependent effects and nonlinearity. Built-in tools for using this method simplify the process for the engineer [CSI 2009].

Nonlinear behavior may be important in staged-construction analysis, such as: Gapopening and closing, especially the contact with temporary support structures; Tension-only members; Cable behavior; Concrete cracking; and P-delta effects. Largedeflectionanalysis is not usually required, even when there are significant creep and shrinkagedeflections, since the rotations involved are not usually large enough to change theequilibrium equations. However, curved girders in bridge structures may requireconsideration of large deflections, as well as cases where the structure is assembled by

rotating members into place when they are already partially attached.




To be truly useful, the analysis program should be able to consider linear analyses atany stage in the construction, such as calculating vibration modes or buckling modes, aswell as design load combinations. This can be important to assure the stability and safety

of the structure throughout its construction. The flexibility to consider multipleconstruction scenarios, either from the start or branching at intermediate stages, can bevery powerful [CSI 2009].

5. GEOMETRIC NONLINEARITY AND STABILITY ANALYSIS

For most practical structures, the P-delta effect is more significant than the considerationof large deflections. The exception to this is cables, which should always considerequilibrium in their deflected shape as part of the analysis. However, cables can betreated specially without considering large deflections for the rest of the structure.

The P-delta effect considers the softening and destabilizing effect of compressive forcesand stresses. For most structures, P-delta effects should be taken into consideration,even if only for the initial determination of the stiffness of the structure for an otherwiselinear analysis and design. Taking this a step further, the Direct Analysis Method that isnow part of the AISC 360-05/IBC2006 steel design code [AISC] can directly use the P-delta effects within members, as calculated by analysis, to perform a more accuratedesign than is often achieved using K-factors, which may result in savings of materialcost [Deierlein et al]. This requires a program that can consider internal (so-called“small”) P-delta effects.

Stability analysis can be performed by calculating linear buckling modes or by performingfull geometrically nonlinear analysis. It is important to know when each is useful and how

they may be used together.

Figure 2. Lateral Buckling of a Single Member




Linear buckling analysis determines the safety factor for a given loading before thePdelta effect would destabilize the structure and cause buckling failure. This is alinearized estimate that assumes that the distribution of direct stresses and the deflected

geometry does not change significantly due to the P-delta effect. In the absence ofsignificant material nonlinearity (including gaps, cables, and so on), this assumption isusually reasonable for traditional straight buildings when considering buckling due togravity loads. In the presence of significant asymmetry, where gravity loading may causelateral deflections, the P-delta effect may increase the lateral deflection due to gravity,causing redistribution of axial loads. In such a case, a linearized buckling analysis fromzero, and one performed after a nonlinear P-delta gravity analysis, may produce twodifferent estimates of the buckling factor of safety. The estimate calculated near thedesign load of the building would be the most accurate.

An alternative to linear buckling analysis is to perform a nonlinear P-delta analysis for the

same loading. This will not automatically produce a factor of safety. However, applying asmall simultaneous lateral load will enable the engineer to determine when the lateraldeflections begin to grow rapidly, signaling instability. As the buckling load isapproached, the P-delta analysis loses its validity, since the rotations begin to becomesignificant.

Large deflection analysis is mostly concerned with the effect of rotation, which changesthe direction of axial force and shear, and hence affects the equilibrium equations. In theoccasional situation where post-buckling behavior is of importance, the nonlinearanalysis must consider large deflections rather than just P-delta effects. Nonlinearmaterial behavior also may be critical. Such analyses usually are restricted to specialstructures or subsystems of larger structures, rather than entire buildings or bridges

[Cook et al].

In the analysis of post-buckling behavior, special algorithms usually are required tocontrol the application of the load, which is discussed in text that follows.

6. NONLINEAR STATIC ANALYSIS

Almost all interesting structures require some kind of nonlinear static analysis, even if itis simply to compute the P-delta effect to be included for subsequent linear analyses.Other basic types of nonlinearity that are commonly considered include: Cables;Tension-only bracing; Support gaps and foundation uplift; and Cracked concrete slabs.Such analyses can be carried out for the entire structure at once, or using staged

construction when required.

The question that must be asked is if after applying dead load, prestressing, and otherfixed loading, the structure can be considered essentially linear for all subsequent designloads, such as live load, wind load, and thermal load. If so, the stiffness at the end ofnonlinear analysis can be solved once, and any number of load cases can be appliedand superposed to create design load combinations. The initial nonlinear static caseusually can be solved with little more effort than a linear analysis for well behavedstructures, so that the overall analysis is quiet efficient.

When nonlinearities affect all applied loading, superposition no longer applies, and everyload combination must be separately analyzed using nonlinear analysis. An example ofthis is for lateral loads with tension-only bracing, where the members that participatediffer for loading in the positive and negative directions. When using the Direct Analysis




Method, nonlinear load combinations also are required to correctly determine the internal(small) P-delta effect in each member; linear superposition is not adequate [Deierlein etal].

For many structures, a hybrid approach is possible. A primary nonlinear static analysisfollowed by linear load combinations would cover most of the design needs, butadditional nonlinear static or dynamic analyses would be used for performance-baseddesign or other special cases. Having the ability to flexibly consider these multipleanalysis and design scenarios using the same model is essential for today’s modernstructures.

7. PUSHOVER ANALYSIS

Pushover analysis is a specialized type of nonlinear static analysis that can be used in

performance-based seismic design. Critical components of the structure are modeledwith nonlinear material behavior, and one or more deformation levels identified for eachcomponent corresponding to various performance levels. Examples of performancelevels include immediate occupancy, life safety, and collapse prevention, although anynumber can be considered [ACSE], [AASHTO].

Lateral loading that is distributed similar to inertial forces is applied with incrementallyincreasing magnitude, and each component is monitored for its performance level. For agiven level of loading, corresponding to a particular performance level earthquake,components not having adequate capacity can be identified, and the structural designappropriately modified. Pushover analysis may need to be conducted for a variety of loadpatterns corresponding to different directions of earthquake in an attempt to find all the

weaknesses of the design.

Figure 3. Pushover Analysis Showing Demands on Individual Members

More than most other types of nonlinear analysis, pushover analysis is strictly a designtool and not intended to represent realistic behavior. It combines linear demands withnonlinear capacities, and represents dynamic behavior with static analysis. Because ofthe assumed load patterns, its usefulness is limited to structures where higher modeeffects are not significant, generally structures of less than about ten stories. For taller




structures, full dynamic analysis may be needed for performance-based design. Whenproperly used, pushover analysis can be a powerful design tool, providing great insightinto structural behavior.

8. DYNAMIC ANALYSIS

For traditional structures, design for wind and seismic lateral loads using static loadcases or response-spectrum analysis may be adequate. For larger or more unusualstructures, the traditional assumptions no longer apply, and full time-history analysismust be considered.

Two distinct types of time-history analysis are available – modal superposition and directintegration. Modal superposition is the more efficient of the two, but it is not able toconsider the full range of nonlinear behavior as can direct integration. However, the

unique Fast Nonlinear Analysis (FNA) method of modal superposition can considerlocalized nonlinearity and coupled damping behavior accurately and efficiently, making itan ideal choice for analysis of isolation and energy dissipation systems for seismic, wind,and vibration control [Wilson]. This method makes use of Ritz-type modes that can becomputed using the stiffness at the end of any nonlinear analysis, thus including staged-construction, cable effects, P-delta, and other built-in nonlinear effects [Wilson et al].

For performance-based design of structures for which pushover analysis is not sufficient,direct integration can be used to perform complete nonlinear analysis of the structuresubjected to a suite of representative seismic motions. Unlike pushover analysis, highermode effects are automatically included, which can identify capacity limitations in regionsthat may not be predicted from assumed first-mode loading [ASCE], [CSI 2006]. Direct-

integration analysis also can be used for stability and other types of nonlinear analysisfor which static nonlinear analysis is not adequate. This type of analysis iscomputationally intensive, but modern computational methods and machines, includingparallel processing, have made it a practical possibility.

When needed, both modal superposition and direct integration can be used for the samestructure, each for the appropriate aspect of design. Using a well structured analysis tool,all of the necessary analysis and design can be performed using the same model.

9. COLLAPSE PREVENTION

A practical way to design for collapse prevention is to analyze the structure for a

multitude of scenarios with critical members and components removed [US DOD], [USGSA]. Staged construction can be used for this purpose, removing the membersstatically or dynamically from the loaded state.

Careful attention must be paid to consider the type of nonlinear behavior that needs tobe considered in the vicinity of the removed members. This may include materialnonlinearity and P-delta effects, but in some cases large deflections may be warranted ifcatenary action from beams and slabs is expected to prevent collapse.

While it is sometimes tempting to want to consider the full progressive collapse scenariofrom the failure of one component, this is extremely difficult and time consuming, andcannot consider all the possible paths of collapse. It is more practical to determine thedemands from the removal of one or more critical components and design accordingly.




10. STABILITY PROBLEMS

Mathematically unique solutions are guaranteed for linear analysis and are often assuredwhen the stiffness of the structure is positive, even if it is changing due to nonlineareffects. However, this is no longer true when there is a loss of strength, whether due to asoftening material, fracture, or buckling. The load carried by the failing member must beredistributed to other members, or the overall load-carrying capacity of the structure maybe reduced.

There are two significant issues here: How to control the application of the load when itmust be reduced, and how to internally redistribute the load. For static analysis,controlling the load by monitoring a monotonically increasing displacement (or multipledisplacements) works well for most problems, with arc-length control being useful forcertain special cases. Additional monitoring of the yielding materials also can be used to

control load reversals, when physically impossible negative plastic work is detected.These methods allow for the capturing of snap-back and snap-through behavior, whichoccur in large flexible structures when a local region unloads, causing significant elasticunloading elsewhere [CSI 2006], [CSI 2009].

No unique solution exists for load redistribution in a static analysis. Typical approachesinclude proportionately unloading the entire structure, then reloading; locally unloadingthe failing element and applying the load to the surrounding elements; and reloadingfrom zero with the failing element having reduced secant stiffness. These and othermethods may give different results in many structures.

For more difficult problems, dynamic analysis is often the best approach. The load can

be applied quasi-statically (very slowly), but load redistribution may happen quicklywhere local failures occur. The path of load redistribution is uniquely determined byinertia and represents the most “natural” solution. However, it is unrealistic to expect the“true” solution in these cases, since in the case of rapid strength loss, high speed wavepropagation occurs, which would require detailed modeling not warranted for typicalstructures [Zienkiewicz & Taylor].

11. LOCALIZATION

Another important issue that must be considered is the tendency of material failure tolocalize. A common example of this is the necking that appears in a steel specimen in atensile test. Even though the stress is essentially uniform over the whole central length of

the specimen, once stress loss occurs, failure localizes to a shorter length that is on theorder of the cross sectional dimensions. The remainder of the specimen unloadselastically. In a similar fashion, moment hinges tend to localize, upon strength loss, to alength related to the length of the member and its cross sectional dimensions.

When analyzed using finite elements, numerical localization tends to occur over lengthson the order of the element size, hence the results will continue to change with increasedmesh refinement. To get “realistic” results, the mesh size should be set to the expectedlocalization (or hinge) length, or specialized nonlocal material models should beemployed. For practical engineering purposes, including performance-based design,mesh refinement beyond this level may not be useful for nonlinear analysis.




Figure 4. Reasonable Mesh Size to Avoid Localization

In view of the previous discussion of nonlinear analysis and design needs for modernstructures, we can describe practical analysis and design software to accomplish thesegoals. First the program must provide the tools to easily create and manipulate theessential structural model. This includes drafting capability, import from other software,and powerful and flexible methods for managing the large amount of data required.

Figure 5. Integrated Graphics Provide Needed Visualization

Secondly, the program must be capable of performing a variety of different types of

analysis and design using a single, integrated model. This assures that every member isconsidered in all analyses and is comprehensively designed. Even more importantly,




having a single model means that all changes are made once, minimizing the chance oferror that can easily occur when trying to keep several parallel models up to datesimultaneously.

Flexible load case scheduling is another important requirement for sophisticatednonlinear analysis. This includes the ability to perform complex staged-constructionanalysis and to be able to sequence nonlinear static and dynamic analysis with eachother and with staged construction cases, since path dependence can significantly affectbehavior and design. To be truly useful, the program should be able to consider multipleparallel analysis sequences and to be able to branch from one case into many paths.The ability to selectively run some analyses, to add new load cases without losingprevious results, and to selectively re-run certain cases are all indispensible.

At any step along a given nonlinear analysis sequence, linear analyses may be needed

for computing linear design combinations, modal and buckling analyses, influence-linebase moving-load analysis, and many other purposes. These linear analyses use thestiffness from the nonlinear state under consideration and represent a linear perturbationfrom the nonlinear state.

Since every nonlinear structure is unique, a toolbox of nonlinear analysis methods isrequired. These include event-to-event methods, constant-stiffness and NewtonRaphson iteration, line-search techniques, and multiple time-history methods, such asmodal superposition and implicit/explicit direct-integration. These need to be coupledwith efficient algorithms that take full advantage of the available computer technology,including vector and parallel processing.

Code-based design checking must be integrated so that it can work directly with theresults of any set of linear or nonlinear analysis results. This should not be restricted to asingle set of analysis results, but be flexible enough to envelope the design over any setof appropriate load cases and combinations.

Powerful data manipulation capabilities also are needed to be able to extract the dataneeded for external processing in cases where code-based methods are not adequatedue to the unique or unusual characteristics of some structures.

Figure 6. Integrated and Enveloped Design Results




13. GENERAL GUIDELINES

No matter how sophisticated is the engineering software and computer hardware, themost important ingredient for effective nonlinear analysis and design is the experiencedstructural engineer. Several guidelines are helpful to make effective use of computerizedstructural analysis and design [CSI 2009]:

Remember the goal – structural design, not analysis. The purpose is not topredict the “realistic” behavior of the structure, but to produce a design thatcontrols the behavior.

All nonlinear models have a learning curve. Each structure is unique, and thenonlinear behavior may not always be as expected. Time must be allowed tounderstand the structure.

Start as simply as possible, checking the linear behavior under dead load and

examining the vibration modes. This is a good way to verify structural properties,connectivity, and support.

Add complexity and nonlinearity to the model gradually as your understanding ofthe behavior improves. Perform many quicker analyses first before running longeranalyses.

Use realistic properties. Avoid excessively large “rigid” stiffnesses, and always include realistic inertia.

Use appropriate mesh size, no more refined than necessary. This is necessaryfor efficiency and to avoid unrealistic localization. When detailed stress resultsare needed, consider limited local meshes.

Ignore strength loss whenever possible, or delay its consideration until later in thedevelopment of the model after many of the design decisions have been made

already. Study the sensitivity of the model to different assumptions, including material

properties and the flexibility of the foundation.

Nonlinear analysis is a powerful tool that must be used wisely or the results can beconfusing or even misleading.

14. CONCLUSION

Modern structures provide a challenge to traditional methods of design and analysis,particularly given the higher expectations placed on them for appearance, performance,

and cost-effectiveness. New methods of nonlinear analysis and performance-baseddesign provide the tools to meet these challenges. No single approach is suitable for thedesign of all structures, or even for a given structure. Rather, comprehensive and well-integrated structural engineering software, running on powerful hardware, can providethe experienced engineer with the multiple capabilities needed to produce safe andeffective designs.

REFERENCES

AASHTO (2009). “ AASHTO Guide Specifications for LRFD Seismic Bridge Design”, American Association of Highway and Transportation Officials, Washington, D.C.

AISC (2005). “ ANSI/AISC 360-05: An American National Standard Specification for

Structural Steel Buildings” , American Institute of Steel Construction, Chicago,Illinois.




ASCE (2006). ASCE Standard ASCE/SEI 41-06 “Seismic Rehabilitation of ExistingBuildings”, American Society of Civil Engineers, Reston, Virginia.

CEB (1993). “CEB-FIP Modal Code” , Comite Euro-International Du Beton, Thomas

Telford, LondonCSI (2009). “CSI Analysis Reference Manual” , Computers and Structures, Inc., Berkeley,

California.CSI (2006). “Perform-3D User Guide, V4, Nonlinear Analysis and Performance

Assessment for 3D Structures” , Computers and Structures, Inc., Berkeley,California.

R. D. Cook, D. S. Malkus, and M. E. Plesha (1989). “Concepts and Applications of FiniteElement Analysis” , 3rd Edition, John Wiley & Sons, New York, New York.

G. G. Deierlein, J. F. Hajjar, J. A. Yura, D. W. White, W. F. and Baker, W.F. (2002),“Proposed New Requirements for Frame Stability Using Second-Order Analysis”,Proceedings, Annual Technical Session, Structural Stability Research Council , pp.

1-20.M. A. Ketchum (1986), “Redistribution of Stresses in Segmentally Erected PrestressedConcrete Bridges” , Report No. UCB/SESM-86/07, Department of Civil Engineering,University of California, Berkeley, California.

US DOD (2003). “DOD Minimum Antiterrorism Standards for Buildings”, Unified FacilitiesCriteria, UFC 4-010-01, US Department of Defense, Updated 2003.

US GSA (2003), “Progressive Collapse Analysis and Design Guidelines – for NewFederal Office Buildings and Major Modernization Projects”, US General Services Administration.

E. L. Wilson (2004). “Static and Dynamic Analysis of Structures – A Physical Approachwith Emphasis on Earthquake Engineering ” , Computers and Structures, Inc.,Berkeley, California.

E. L. Wilson, M. W. Yuan, and J. M. Dickens (1982) “Dynamic Analysis by DirectSuperposition of Ritz Vectors”, Earthquake Engineering and Structural Dynamics,Vol. 10, pp. 813 –823.

O. C. Zienkiewicz and R. L. Taylor (1991). “The Finite Element Method ” , 4th Edition, Vol.2, McGraw-Hill, London.

ashraf habibullah, g. robert morris - makalah

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