• Author / Uploaded
• Che' Smit

• Categories
• Braguero
• Pandeo
• Columna
• Viga (Estructura)
• Fuerza

Citation preview

R. Shankar Nair

Simple Solutions to Stability Problems in the Design Office

Author R. Shankar Nair is a principal with Nair/KKBNA, consulting engineers, in Chicago. He is also senior vice president and national technical director of KKBNA Incorporated, consulting engineers, headquartered in Denver. Nair received his doctorate degree in civil engineering from the University of Illinois, Urbana, in 1969. He spent 15 years with a structural/civil engineering design firm in Chicago and two years as an independent structural engineering consultant before joining KKBNA in 1986.

During his career, Nair has been responsible for the structural design of many high-rise buildings of 30 to 70 stories in Chicago. He has also designed several major bridges, including long-span structures over the Mississippi. These designs have received numerous awards for engineering excellence. An active lecturer, researcher and participant in professional activities, Nair has written numerous technical papers on structural analysis and design. He is a member of several technical committees, including the ASCE Committee on Design of Steel Building Structures, and chairman of the ICTBUH Publications Committee.

Summary Today's computational technology permits rigorous solution of almost any stability analysis problem that might arise in the course of structural engineering design. Complete large-deformation analysis can be performed now on structures for which even a linear analysis would have been practically impossible 30 years ago. Nonetheless, stability analysis and design for stability remain among the most intractable of problems in structural design office practice. The sophisticated tools that exist today for stability analysis have not yet been integrated into normal design office procedure. Stability effects are not included routinely in the analyses that are used for structural design. Consequently, stability remains an issue that must be addressed separately in the design, separate from the basic linear analysis on which the design is based. The treatment of stability effects as a separate issue is not necessarily a source of great inefficiency, thanks to the existence of certain simple but effective methods of considering stability in structural design. As explained in this paper, simple solutions sufficiently accurate for use in design are available for many of the stability problems faced in structural engineering practice. These problems include lateral stability of buildings and towers, connection of columns to floor diaphragms, treatment of floors that are bypassed by the overall lateral load-resisting system, truss bracing, and many other situations.

38-1 © 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

SIMPLE SOLUTIONS TO STABILITY PROBLEMS IN THE DESIGN OFFICE

by R. Shankar Nair

INTRODUCTION

Today's computational technology permits rigorous solution of almost any stability analysis problem that might arise in the course of structural engineering design. Complete large-deformation analysis can be performed now on structures for which even a linear analysis would have been practically impossible thirty years ago. However, these sophisticated computational tools have not yet been integrated into normal design office procedure. Stability effects are not included routinely in the analyses that are used for structural design. Consequently, stability remains an issue that must be addressed separately in the design, separate from the basic linear analysis on which the design is based. The treatment of stability effects as a separate issue is not necessarily a source of great inefficiency, thanks to the existence of certain simple but effective methods of considering stability in structural design. Simple solutions sufficiently accurate for use in design are available for many common stability problems including lateral stability of buildings and towers, connection of columns to floor diaphragms, connection of floor diaphragms to the lateral load-resisting system, and truss bracing. STABILITY OF MULTISTORY BUILDINGS

The special characteristics of multistory buildings have been used to develop very simple techniques for including lateral stability or "P-delta" effects in the design of these structures (1,2). Though orginally developed for tall buildings, these simple techniques are applicable to all buildings in which the floors act as diaphragms that are rigid in their own plane. The rational basis of these techniques is explained in Ref. 1. The procedures are outlined below.

Preliminary Assessment of Stability Effects The importance of lateral stability effects in a building can be assessed even before any analysis is performed. The ratio of actual vertical load on the building to the load that would cause

38-2

© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

lateral buckling (of either a single story, by shear racking, or an entire tall building, as a flexural cantilever) can be estimated using the following equation:

In this equation, "Wind Drift/Height" is the maximum drift/height ratio (either single story or for the entire building) anticipated due to the "Wind Pressure" shown in the denominator. "Depth" is the plan dimension of the building in the direction of wind loading and buckling being considered. "Density" is the total weight of the building and its contents divided by its total volume. Typical values of density range from about 10 lb/cu. ft for steel office buildings to 20 lb/cu. ft for concrete apartment buildings. Concrete office buildings lie near the middle of this range. As an example of the application of Equation 1, consider a steel office building that has a depth of 120 ft in the direction of loading and buckling being considered. It is to be designed to have a drift/height ratio no greater than 1/450 for a uniform wind pressure of 30 psf. The ratio of actual to critical load is:

The corresponding magnification factor (applicable to all moments, forces and displacements caused by lateral loading) is:

For design purposes, load/resistance factors should be applied in the computation of the magnification factor, which yields:

These factors provide a preliminary assessment of the importance of lateral stability effects in the building and can be used in the initial proportioning of members, before any lateral load analysis is done. After linear lateral load analyses have been performed, stability effects can be determined with greater accuracy and included in the design of members and connections by means of the following procedure.

38-3

© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

Procedure for Design The suggested procedure for including lateral stability or "P-delta" effects in the design of buildings is based on the magnification factor concept. Separate magnification factors are used for overall moment or overturning effects and for horizontal shear or "racking" effects. [ In a particular building, either of these effects (overturning or racking) might be insignificant, in which case the suggested procedure will yield an unrealistically high magnification factor for that effect. This is not a major drawback since the high factor would be applied to very small moments and forces.] The magnification factors are determined from the results of linear, first-order analysis of the structure. The procedure is as follows:

1.

Perform linear analysis of the structure for the design lateral loadings.

2.

Compute the critical vertical load for overall flexural (or overturning) buckling, per unit height, from the following:

where H is the total height of the building; f is a lateral load per unit height; and is the lateral displacement at the top caused by load f. [If the loadings in the analysis do not include a uniform lateral loading, for f use the value of uniform load that would produce the same base moment as the lateral loading actually used in the analysis. Alternatively, if the design loadings are extremely non-uniform, include an arbitrary uniform lateral loading among the loadings in the analysis and use this uniform loading for computing .] Equation 2 is illustrated in Figure 1. 3.

Compute the magnification factor for overall moment or overturning effects from the following:

where is the magnification factor; is the critical load determined in Step 2; and p is the actual average vertical load on the building per unit height. (Load factors should be included as explained later.)

38-4 © 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

4.

Apply magnification factor to the following effects of lateral loading: Axial forces in columns; moments and axial forces in shearwalls.

5.

For each story of the building (or for each of a few representative stories) compute the critical load for shear racking buckling, from the following:

where h is the story height; V is the total horizontal shear force in the story for a particular loading condition; and is the lateral deformation of the story caused by that loading. The factor of 0.82 may be replaced by 1.00 if most of the vertical force in the story is in columns or walls that remain essentially straight between floors. Intermediate values may be used. Equation 4 is illustrated in Figures 2 and 3. 6.

Compute the magnification factor for shear racking effects in the story from the following:

where is the magnification factor; is the critical load determined in Step 5; and P is the actual total vertical force in the story. (Load factors should be included as explained later.) 7.

Apply magnification factor to the following effects of lateral loading: Moments ana shear forces in columns; moments and shear forces in beams; shear forces in shearwalls; axial forces, moments and shears in diagonal bracing members.

8.

After all member forces and moments due to lateral loading have been multiplied by or , as appropriate, the members should be designed in accordance with the "braced against sideway" provisions in the design specifications. Column effective lengths may be taken (conservatively) as the actual length between floors. All connections must be designed for the magnified forces and moments.

Load Factors.- When and are to be used to design members and connections for strength and safety, load factors should be applied to p and P and strength reduction factors should be applied to and in Equations 3 and 5. If load factor design is being used, the load and strength factors should be as specified explicitly in the design specifications. If working stress design is being used, a combined load and strength factor of 23/12 may be applied to Load factors and

38-5 © 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

strength reduction factors should not be included in the calculation of and values that are to be used only for evaluating serviceability conditions.

Sidesway Effects of Vertical Loading.- In Steps 4 and 7 of the design procedure, magnification factors are applied to the effects of lateral loading. In buildings in which vertical loading causes significant sidesway, and should also be applied to the sidesway effects of vertical loading. CONNECTION OF COLUMN TO FLOOR DIAPHRAGM

In the design of multistory buildings, it is assumed customarily that individual columns cannot buckle laterally at the floors; they are held by the floor diaphragms, which force all columns to move together at every floor. While this assumption is reasonable (and is implicit in the stability design procedure outlined above), there is considerable disagreement among designers as to the required strength of the column to floor diaphragm connection. Connection strengths ranging from 0.004 to 0.020 of the axial force in the column have been used in design. As illustrated in Figure 4, the force required to restrain a column at a floor can be approximated by the algebraic difference between the products of column force and tilt in the stories above and below the floor. (The "tilt" is the mean out-of-plumbness of the column in the story.) The relationship is approximate in that shear force in the column has been neglected; the shear would tend to reduce the required restraining force.

For multistory buildings, changes in column axial force from story to story are relatively small and the restraining lateral force required at a floor can be represented by the column force below the floor times the difference in tilt in the stories above and below the floor. For a column in a multistory building, the sources of out-of-plumbness are erection imperfections and horizontal floor movements due to deformation of the lateral load-resisting system. Deformation of the lateral load-resisting system can be caused by external lateral loading and also by P-delta effects.

The maximum erection out-of-plumbness may be taken as 0.002, which is the tolerance specified in AISC's Code of Standard Practice. Since the column can tilt in opposite directions in adjacent stories, the maximum difference in tilt above and below a floor due to erection imperfections can be taken as 0.004. The maximum relative horizontal movement of adjacent floors due to deformation of the lateral load-resisting system (caused by lateral loading and P-delta effects) can be expected to be in the range of 0.002 to 0.003 of story height, i.e., the maximum column 38-6

© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

tilt due to floor movements is 0.002 to 0.003. Since column out-of-plumbness caused by load-induced floor movement is not likely to be in opposite directions in adjacent stories, the maximum difference in tilt between adjacent stories due to floor movement may be taken as 0.002. It can be concluded that connections between columns and floor diaphragms should be designed to transmit a horizontal force of 0.006 times the axial force in the column below the floor. [This is in addition to any horizontal force applied directly to the column.] Types of Connections Consider a fully stressed W14X730 Grade 50 column. The force in the column is about 5800 kips. The required capacity of the column to floor diaphragm connection is 0.006 X 5800 = 35 kips. If a 3" thickness of concrete is bearing on the column on all sides, the minimum bearing area is about 3 X 18 = 54 sq. in. and the bearing stress is 35/54 = 0.65 ksi, which is well within allowable limits. It is clear from this example that interior columns fully surrounded by concrete slabs generally have ample lateral restraint and do not require special connections to the floor diaphragms. However, special connections might be required for columns adjacent to edges and openings in floors. If a composite beam from the interior of the floor frames into an edge column, the beam to column connection can be designed for the horizontal restraining force in combination with the usual vertical reaction. This might require an increase in the size of the connection for small beams at heavy columns. Two types of simple, direct connection between column and floor diaphragm are sketched in Figure 5. In one case, a U-shaped reinforcing bar wraps around the outside of the column. In the other type, a long U-bolt (or, alternatively, one or two straight or J bolts) is connected directly to the column. In either case, the anchoring element must be developed into the interior of the concrete floor slab and must extend far enough into the interior to avoid a pulling-out failure of the part of the floor engaged by the bar or bolt. CONNECTION OF FLOOR TO LATERAL LOAD-RESISTING SYSTEM

The connection of columns to floor diaphragms has been discussed. The sum of the horizontal forces transmitted to a floor diaphragm through these connections must, eventually, be fed to the lateral load-resisting system. The connection of the floor diaphragm to the lateral system must be of adequate capacity to achieve this load transfer. 38-7

© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

A simple and conservative approach is to check the connection for a horizontal force of 0.006 times the total vertical force in the story below the floor under consideration, excluding forces in components of the lateral load-resisting system. [This horizontal force is in addition to forces that result directly from external loading.]

The horizontal force of 0.006 times vertical force can be applied in any direction. If torsion is a concern, horizontal forces of 0.006 times column force could be applied in different directions at different column locations, in such a way as to create the greatest torsional loading on the connection to the lateral load-resisting system. The forces suggested for designing or checking the floor diaphragm connections are not additive over successive floors. They are conservative estimates of the maximum force that can occur at a particular floor. It is not necessary to check the lateral load-resisting system for the summation of these forces over two or more floors. FLOOR NOT ENGAGED BY LATERAL LOAD-RESISTING SYSTEM

An elevation of a large braced frame is sketched in Figure 6. This can be a very effective and efficient lateral load-resisting system. However, careful inspection of the frame layout will reveal that even-numbered floors are not provided with lateral restraint by the bracing system. [If all members, including the beam segments on each side of the diagonals, are considered to be pin-ended, the even-numbered floors have no lateral stiffness. If the beams at even-numbered floors are flexurally continuous across their connections to diagonals, the floors would derive a small degree of lateral stiffness from the flexural stiffness of the beams.] Situations similar to that sketched in Figure 6 are not unusual in multistory buildings with large braced frames. Certain mezzanines and partial floors are other examples of floors that are not engaged by the overall lateral load-resisting systems of buildings. One approach to these situations is to add members as necessary to create an essentially rigid connection between the floor in question and the overall bracing system. Another approach is to leave the floor unconnected to the lateral load-resisting system and to account for this in the design of columns and other components.

Figure 7 shows the same structure that was shown in Figure 6, but with additional members to provide lateral restraint to the even-numbered floors. All floors can be considered "braced" if the connections to the lateral load-resisting system meet the strength requirements discussed earlier.

38-8 © 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

If the additional members indicated in Figure 7 are not provided, the even-number floors cannot be considered to be laterally restrained by the overall lateral load-resisting system. The simplest approach, in this case, is to design all columns as though they were laterally unsupported at the unrestrained floors. This procedure might be excessively conservative, especially if some of the columns are much more slender than others. A more reasonable technique is to combine the stiffnesses of all columns into a single imaginary column extending between laterally restrained floors (two stories in the present example), and to check this combined column for the sum of the loads in all columns. Individual columns also should be checked; they may be assumed to be supported laterally at every floor.

The general design approach, when some floors in a multistory building are bypassed by the overall lateral load-resisting system, is to treat the structure between laterally restrained floors as an imaginary single column of appropriate stiffness, supported laterally at top and bottom. If there are no beam-column moment connections, the stiffness of the imaginary column is simply the sum of the individual column stiffnesses. If the framing is more complicated, the subassembly between laterally restrained floors can be analyzed separately (under unit lateral loads at appropriate locations) to determine the characteristics of an equivalent single column. TRUSS BRACING

The restraining forces exerted on an out-of-straight truss compression chord by a bracing system are illustrated in Figure 8. [As a simplifying idealization, lateral flexure and shear in the chord are neglected.] As indicated in the figure, the restraining force at a bracing point is the algebraic difference between the products of chord compression and skew angle on the two sides of the bracing point.

The forces exerted by the out-of-straight chord on the bracing system are, of course, the reverse of the restraining forces on the chord. The resultant lateral shear in the bracing in any panel is the product of chord compression and skew angle in that panel (see Figure 8). This result could be arrived at by summation of the forces at the bracing points. It could also be obtained directly as the lateral component of the compression in the truss chord.

The sources of chord out-of-straightness are construction imperfections and deformation of the lateral bracing system. Deformation of the bracing can be caused by external lateral loading and also by the forces required to restrain the out-of-straight chord.

38-9

© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

AISC's Code of Standard Practice specifies that compression members must be straight to within 1/1000 between points of lateral support. However, there do not appear to be standards for the skew of the straight line between lateral support points ( in Figure 8). In the absence of a specific standard tolerance, this writer assumes a maximum skew of 0.003 and an algebraic difference in skew between adjacent panels of 0.006 due

to imperfections in fabrication and erection. The maximum likely skew due to deformation of the lateral bracing system depends on the type of bracing. A panel of diagonal bracing with a 45" diagonal angle is shown in Figure 9. If the forces on this bracing system cause a strain of 0.00103 (which corresponds to a stress of 30 ksi) in the diagonal and negligible strain in the chord, the resulting skew of the chord would be 0.00207. This result is not very sensitive to the angle of the diagonal. For angles of 22.5° to 67.5° the maximum chord skew caused by a strain of 0.00103 in the diagonal is 0.00293. These

skew calculations are valid both for bracing systems with diagonals that support tension and compression and for those in which diagonals carry only tension, provided that the strain in the cross strut is small. From these simple calculations, the maximum chord skew due to bracing deformation can be assumed to be 0.003 for trusses with diagonal bracing consisting of bolted or welded steel members. This is in addition to the chord skew caused by construction

imperfections, which is also assumed to be 0.003, as discussed earlier. Thus, the maximum total skew in any panel can be taken to be 0.006 and the maximum difference in skew between adjacent panels can be assumed (very conservatively) to be 0.012. It can be concluded that the lateral bracing system should be designed for a transverse shear at any location of 0.006 times the axial force in the compression chord at that location. The

connection of the chord to the bracing system should be designed for a transverse force of 0.012 times the compression in the chord (average of adjacent panels). These shears and connection loads are the requirements for restraining the compression chord against lateral buckling. The bracing system must be designed for externally-applied lateral loading (if present) in addition to the chord-restraining effects. The shears and connection loads suggested for design of the bracing system are the maximum values at each location. These maximum values do not occur simultaneously at different locations

and are not additive or cumulative. Application of the suggested connection design loads as external forces would result in grossly overconservative bracing design. If two or more parallel trusses are braced by a single bracing system, the total force in all chords should be used to determine the design loads on the bracing. Struts connecting remote trusses to the bracing system should be designed for the sum of

the single-point restraining forces (0.012P) from all the trusses

38-10 © 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

restrained by the strut. If there is a possibility of significant length change in the strut, the bracing design factors of 0.006 and 0.012 should be increased to account for the greater possible skew of the truss chords.

The suggested forces and shears for design of bracing to provide lateral restraint to truss chords (0.012 and 0.006 of chord compression) are for diagonally braced systems with the following characteristics: diagonal angles of 22° to 68°; diagonal member axial stress no greater than 30 ksi due to chord-restraining effects plus externally-applied lateral loading; negligible axial strain in bracing system cross struts; negligible axial strain in truss chord due to lateral effects; lateral skew of unloaded truss chord not greater than 0.003. For trusses and bracing systems having different characteristics, different design factors can be derived by modifying the simple calculations that were used to arrive at the 0.006 and 0.012 factors. SUMMARY

Simple solutions have been suggested for some of the stability problems frequently encountered in structural engineering practice. The solutions are approximate, but they are sufficiently accurate for use in structural design in most practical situations. Enough information about the suggested solutions is provided to permit the designer to judge whether the solutions are applicable in particular instances. REFERENCES

1.

Nair, R.S., "Tall Building Stability—Practical Considerations," Materials and Member Behavior, Proceedings of Structures Congress '87, ASCE, Aug. 1987.

2.

Nair, R.S., "A Simple Method of Overall Stability Analysis for Multistory Buildings," Developments in Tall Buildings—1983, Council on Tall Buildings and Urban Habitat, Hutchinson Ross Publishing Co., 1983.

38-11

© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

Figure 1.- Bending and Buckling of Flexural Cantilever

Figure 2.- Lateral Loading and Buckling of Shear Element

Figure 3.- Alternative Loading and Buckling Configurations in Story of Building

38-12 © 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

Figure 4.- Restraining Force on Out-of-Plumb Column at Floor

Figure 5.- Direct Connection of Column to Floor Diaphragm

38-13 © 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

Figure 6.- Braced Frame; Even-Numbered Floors Laterally Unrestrained

Figure 7.- Braced Frame; With Additional Members to Restrain Even-Numbered Floors

38-14

© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.

Figure 8.- Restraining Forces on Out-of-Straight Truss Compression Chord From Bracing System

Figure 9.- Chord Skew Due to Strain in Bracing System

38-15

© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without permission of the publisher.