The 2000 T.R.
Higgins Lecture:
A Practical Look
at Frame
Analysis,
Stability, and
Leaning Columns
Louis F. Geschwindner
Author
Summary
ouis F. Geschwindner is professor of architectural engineering at Pennsylvania State
University and is a Registered
Professional Engineer. He
received his bachelor's degree in
building science from Rensselaer
Polytechnic Institute and both his
master of science in architectural
engineering and his Ph.D. in civil
engineering from Pennsylvania
State University. He has been a
faculty member at Pennsylvania
State for over 30 years, teaching
and conducting research in building structures and he is also in
charge of the undergraduate program in architectural engineering.
Professor Geschwindner is
chairman of the Committee on
Design of Steel Building
Structures of the American
Society of Civil Engineers
(ASCE). He is also a member of
the Committee on Metals, the
LRFD Committee and vice-chair
of the Tension Membrane
Structures Standards Committee,
all of ASCE. Dr. Geschwindner is
a member of the AISC Committee
on Specifications as well as its
Task Committee 10 - Stability.
His memberships also include the
Masonry Society and the
American Society for Engineering
Education.
Teaching advanced level structural courses in the five-year
architectural engineering program
at Pennsylvania State University,
has been the primary thrust of Dr.
Geschwindner's academic career.
He has participated as a lecturer
in the American Institute of Steel
Construction's lecture series and
has developed a short course on
LRFD for practicing professionals.
He has received numerous
awards for outstanding teaching,
including the University-wide
AMOCO Foundation Outstanding
Teaching Award and the AT&T
Foundation Award for Excellence
from ASEE.
T
L
he analysis and design of
unbraced moment frames is a
fairly regular activity in structural
engineering practice yet it can be
a complex structural engineering
problem. Numerous analysis
methodologies are available and
the many commercial software
packages used in practice provide
a variety of approaches to the
problem. Questions arise as to
whether a first-order or secondorder analysis is appropriate,
should an elastic or inelastic
analysis be carried out, what
moment magnifiers should be
used when axial load and moment
act together, and how should column capacity be determined, with
effective length factors or some
other approach. Stability of a column, although often expressed as
a function of the individual
column, is actually a function of all
of the members in the story.
Thus, column design is a story
problem, not an individual column
problem. When unbraced
moment frames support pinended columns, additional problems arise. These pin-ended
columns do not participate in the
lateral resistance of the structure,
but instead rely on the unbraced
frame for their lateral stability.
Thus, the frame must be designed
to accommodate the loads that
are applied to it and provide sufficient stiffness to support any of
these "leaning" columns.
Frame analysis may be
approached by a variety of paths
but linear elastic analysis is
perhaps the most common, as
well as the least complete. A
second order inelastic analysis,
while perhaps the most comprehensive, is also the most complex. Whichever analysis method
is chosen, the design approach
must be compatible. Numerous
approaches have been presented
in the literature to address the
design of frames both with and
without leaning columns.
Although a direct stability analysis
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© 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.
may be performed, the most common approaches still appear to be
those that utilize some form of
simplification. This paper will
briefly review first- and secondorder elastic and inelastic analysis
approaches and a few of the commonly used computer software
packages for unbraced moment
frames. This will be followed with
a discussion of the impact of lean-
ing columns on these analyses.
Once analysis approaches have
been identified, the design
process, where the inclusion of
column effective length comes
into play, will be addressed. The
use of effective length to predict
column capacity will be discussed
and its use to account for the
inclusion of leaning columns will
be addressed. Effective length
calculations will be reviewed with
particular attention to the
approaches presented by Yura,
Lim & McNamara, LeMessurier,
and the equations found in the
AISC LRFD Commentary. The
results from these approaches will
be compared to those of a stability analysis for simple frames that
have been found in the literature.
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A Practical Look at Frame Analysis, Stability, and Leaning Columns
By Louis F. Geschwindner
Introduction
The analysis and design of unbraced moment frames is a fairly regular activity in
structural engineering practice yet it can be a complex structural engineering problem.
Numerous analysis methodologies are available and the many commercial software
packages used in practice provide a variety of approaches to the problem. Questions arise
as to whether a first-order or second-order analysis is appropriate, should an elastic or
inelastic analysis be carried out, what moment magnifiers should be used when axial load
and moment act together, and how should column capacity be determined, with effective
length factors or some other approach. Stability of a column, although often expressed as
a function of the individual column, is actually a function of all of the members in the
story. Thus, column design is a story problem, not an individual column problem. When
unbraced moment frames support pin-ended columns, additional problems arise. These
pin-ended columns do not participate in the lateral resistance of the structure, but instead,
rely on the unbraced frame for their lateral stability. Thus, the frame must be designed to
accommodate the loads that are applied to it despite the reduction in overall frame
stability that results from these "leaning" columns.
Frame analysis may be approached by a variety of paths. Linear elastic analysis is
perhaps the most common, although the least complete. A second order inelastic
analysis, while perhaps the most comprehensive, is also the most complex. And there are
many approaches between these. Whichever analysis method is chosen, the design
approach must be compatible. Numerous approaches have been presented in the literature
to address the design of frames both with and without leaning columns. Although a direct
stability analysis may be performed, the most common approaches still appear to be those
that utilize some form of simplification. This paper will briefly review first- and secondorder elastic and inelastic analysis approaches and a few of the commonly used computer
software packages for unbraced moment frames. This will be followed with a discussion
of the impact of leaning columns on these analyses. Once analysis approaches have been
identified, the design process, where the inclusion of column effective length comes into
play, will be addressed. The use of effective length to predict column capacity will be
discussed and its use to account for the inclusion of leaning columns will be addressed.
Effective length calculations will be reviewed with particular attention to the approaches
presented by Yura, Lim & McNamara, LeMessurier, and the equations found in the AISC
LRFD Commentary. The results from these approaches will be compared to those of a
stability analysis for simple frames that have been found in the literature.
Analysis
The state of the art of structural analysis encompasses a wide range of possible
approaches to the determination of member response to structural loading. Each new
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approach adds or subtracts some aspect of frame or member behavior in an attempt to
properly model the true behavior of the structure. Before looking at the requirements of
the current LRFD specification, it will be helpful to categorize these analysis approaches
and discuss their characteristics. Figure 1 shows a comparison between the loaddeflection curves of a series of analysis approaches.
First-order elastic analysis (1): The first and most common approach to structural
analysis is the linear elastic analysis. In this case, deformations are assumed to be small
so that the equations of equilibrium may be written about the undeformed configuration.
In this case, superposition is valid, any inelastic behavior of the material is ignored and
the load-deflection curve is linear. This is the approach used in the development of the
common analysis tools of the profession, such as slope-deflection, moment distribution
and the stiffness method found in most commercial computer software. In a first-order
analysis, the deformations are determined and then used in turn to determine forces. Once
the forces are determined, the analysis is complete.
Figure 1. Load-deflection history
Linear buckling analysis (2): An analysis carried out for the determination of the elastic
buckling load will result in the determination of a single critical buckling load for a
frame. The critical buckling load may be determined through an eigenvalue solution or
through a number of iterative schemes based on equilibrium equations written about the
deformed configuration. This is the type of analysis that yields the critical buckling load
of the single column and is the basis upon which the effective length factor is founded. It
can be seen in Figure 1 that the results of this analysis do not provide a load-displacement
curve but rather the single value of load at which the structure fails.
Second-order elastic analysis (2): When the equations of equilibrium are written about
the deformed configuration of the structure and the deflections corresponding to a given
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set of loads are determined, the resulting analysis is a second-order analysis. This is the
analysis generally referred to as the P-delta analysis. Two components of these secondorder effects are included in the analysis. When the influence of axial force and member
curvature is included, it is said that the
effects are included and when the sidesway
effects are included it is said that the
effects are included. It is seen in Figure 1 that
the load-deflection history obtained through this analysis approach the value obtained
from the eigenvalue solution. This analysis usually requires an iterative solution so it is a
bit more complex than the first-order analysis. Because of the problems inherent with
iterative solutions, many researchers have proposed one-step approximations to the
second-order analysis.
First-order plastic-mechanism analysis (3): As load is increased on a structure, it is
assumed that defined locations within the structure will reach their plastic capacity. When
that happens, the particular location continues to resist that plastic moment but undergoes
unrestrained deformation. These sections are called plastic hinges. Once a sufficient
number of plastic hinges have formed so that the structure will collapse, it is said that a
mechanism has formed. No additional load can be placed on the structure. This limit can
be seen in Figure 1.
First-order elastic-plastic analysis (4): If the approach to determination of the collapse
mechanism tracks the development of individual hinges, more information is obtained
from this analysis than from the mechanism analysis. It is clear however, that if the same
assumptions as to hinge length and geometry are maintained, the limit of the elasticplastic analysis will be the mechanism analysis as seen in Figure 1.
Second-order inelastic analysis (5): This approach to analysis combines the same
principles of second-order analysis discussed previously with the plastic hinge analysis
just discussed. It is obvious that this approach to analysis is much more complex. It does,
however, yield a more complete and accurate picture of the behavior of the structure,
depending on the completeness of the model used. This type of analysis is what has come
to be known as "advanced analysis." The load-deformation curve for a second-order
inelastic analysis is shown in Figure 1.
In summary, it can be seen that as more complex behavior is taken into account in the
analysis, the predicted load level is reduced. Thus, designers must exercise extreme
caution when using an analysis approach with which they are unfamiliar.
Design
The approach taken for member design must be consistent with the approach chosen for
analysis. There are currently three approaches to the design of steel structures acceptable
under US building codes as they incorporate AISC Specifications (6,7). Currently, the
best tool for steel design is the Load and Resistance Factor Design Specification (LRFD).
However, the Plastic Design (PD) approach is also permitted and the Allowable Stress
Design Specification (ASD) is still used.
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The LRFD specification stipulates, in Section C1, that "Second order effects shall be
considered in the design of frames." The comparable statement in the ASD specification
states, in Section A5.3, that "Selection of the method of analysis is the prerogative of the
responsible engineer." And in Section C1 that "frames...shall be designed to provide the
needed deformation capacity and to assure overall frame stability." The normal analysis
method is one of the linear elastic approaches. The satisfaction of the deformation
capacity and the assurance of stability are left to the engineer.
In order to use the Plastic Design Specification for structural design, a first-order
mechanism analysis must be carried out. Additionally, other restrictions found within the
specification must be satisfied.
Thus, regardless of the specification used, the engineer is required to address the second
order effects. This may be done using a first-order analysis and a code provided
correction for second order effects or it may be addressed through direct use of a secondorder analysis.
Impact of Second-order Effects
Two different second-order effects will impact on the design of a single column. The first,
illustrated in Fig. 2a for a column in which the ends are prevented from displacing, is the
result of the deflection along the length of the column. It can be seen that the moments
along the column will be increased due to the column deflection, by an amount
This
increase in moment due to member deflection is referred to as the member effect.
Figure 2. Influence of Second-Order Effects
The column in Fig. 2b is part of a structure that is permitted to sway laterally an amount
As a result, the moment required in the end of the column to maintain equilibrium in the
displaced configuration is given as
This member moment is referred to as the structure
effect, since the lateral displacement of the column ends is a function of the properties of all
of the members of the structure participating in sway resistance.
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The deflections, and shown in Fig. 2 are second-order deflections, resulting from the
applied loads plus the second-order forces. Although it requires a second-order analysis to
determine the magnitude of the forces, both of these effects appear to be straightforward for
the individual column of Fig. 2. However, when columns are combined to form frames, the
interaction of all of the members of the frame significantly increases the complexity of the
problem. The addition of columns which do not participate in lateral resistance but which do
carry gravity load brings further complexity to the problem.
Computer Approaches
Although the influence of second order effects was easily observed in the single column
of Figure 2, the analysis of an unbraced frame of any realistic proportions could be
expected to require the use of some computer software. The particular software being
used will dictate the methods of analysis available to the engineer. Based on their
showing in a recent AISC survey (8), five commercial software packages were reviewed
to determine their approach to a second order analysis. They are STAAD-III,
GTSTRUDL, RISA-2D, ETABS, and RAM FRAME. All of these programs indicate, in
their literature that they include some form of second-order analysis.
STAAD-III (9) adopts a simplified method to perform the P-delta analysis. The users
manual provides the following steps as a guide for using its P-delta analysis: 1. Perform a
first order analysis to obtain deflections and member forces based on external loads, 2.
Create an additional load vector by combining the previously calculated member forces
and deflections with the external loads, 3. Use this new load vector and the original
stiffness matrix to calculate a revised set of deflections, 4. Determine member forces and
reactions from these revised deflections. Although this is not an exact approach, the
program authors indicate their belief that their approach has been justified by the
literature.
GTSTRUDL (10) uses a direct stiffness formulation about the displaced geometry of the
structure. A secant stiffness approach is used where the structure stiffness matrix is
modified in each step of the analysis until the resulting change in displacements or forces
from one step to the next is within the bounds specified or the maximum number of
specified cycles is reached.
In RISA-2D (11), the P-delta effects are "accurately approximated." Their approach is to
solve the first-order problem and determine the member forces. From these member
forces and displacements, a member shear is determined as
These shears are
added to the original loads and the first-order problem is resolved. The new
displacements are compared to the previously calculated displacements. If the difference
is small enough, the analysis is complete. If a more accurate solution is needed,
additional cycles are performed until the required level of accuracy is reached.
ETABS uses a geometric stiffness correction in which the effects are "exactly
represented"(12). The approach is to use the load P determined from the story mass to
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modify the stiffness matrix, which is then used throughout the analysis, both static and
dynamic. This method has been presented in the literature by the program authors (13).
RAM FRAME (14) has also adopted the geometric stiffness method. However, they
indicate that certain modifications have been made in the implementation of the method
within the specific formulation of their program.
It appears that commercial software for second-order analysis of steel frames is readily
available to the design engineer. Thus, it can be expected that the use of a second-order
analysis will become a more realistic approach for normal design.
Predicting the Critical Buckling Load
When an analysis tool is available to determine the critical buckling load of a frame, there
is no need to worry about "predicting" that load through some other means. Thus, it
might be said that if all structural analysis were carried out as a second order buckling
analysis, there would be no need to spend time discussing the correct approach to
determining a K-factor to use in design. It seems that ever since the K-factor was
introduced into the 1961 AISC Specification, it has generated extensive discussion and
misunderstanding (15). To understand the debate over the K-factor, one must understand
what the K-factor is intended to accomplish. It will be helpful to consider the critical
buckling load of a particular column, determined by one of the second-order analysis
programs, as
It will also be helpful to remember that the critical buckling load of the
perfect column, as derived by Euler, is given as
(1)
Since the column in a steel frame is not likely to be a perfect column, but rather a real
column with multiple imperfections, its critical buckling capacity can be said to be
somewhat less than the Euler column, thus
(2)
If that reduction factor is defined as
it is seen that
(3)
Thus, the K-factor is simply a mathematical adjustment to the perfect column equation to
try to predict the capacity of an actual column. Every method or equation that is proposed
is simply trying to predict accurately the actual column capacity as a function of the
perfect column.
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Perhaps the most commonly used approach to the determination of K-factors is the
nomograph found in the commentary to the LRFD and ASD Specifications (6,7). The
equation upon which the nomograph is based is given here as Eq. 4 (3).
(4)
with
and the A and B subscripts referring to the ends of the column under consideration.
The assumptions used in the development of the nomograph are detailed in the Commentary
to the Specification. One of these important assumptions is "all columns in a story buckle
simultaneously." Although this assumption was essential in the derivation of this useful
equation, it is also one that is regularly violated in practical structures. This assumption is
critical since it eliminates the possibility that any column in an unbraced frame might
contribute to the lateral sway resistance of any other column. A reasoned analysis of the
behavior of columns in actual structures would seem to indicate that strong columns, ones
whose load is not near their capacity, should be able to help restrain weaker columns. Thus,
other approaches to determining the K-factor should be considered.
Figure 3. Symmetric Portal Frame
Second-Order Analysis and Leaning Columns
Without leaning columns: Two simple frames will be discussed so that the impact of secondorder analysis and leaning columns may be understood. The symmetric frame shown in Fig.
3a is subjected to a symmetrically placed gravity load. A first order analysis yields the
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forces shown. Note that there are no column moments and thus, there will be no axial force
and moment interaction. When a lateral load is added as shown in Fig 3b, forces and
moments as shown result from a first-order analysis. In this case, all moment is due to the
lateral load and these moments must be amplified to account for the second-order effects.
For both cases, axial capacity may be determined using the K-factor from the nomograph.
A similar frame, with two different columns, is shown in Fig. 4a, loaded with a
nonsymmetric gravity load. For an LRFD-based design and a first-order analysis, the
sway and non-sway moments must be separated. To do this, the first step is to restrain
sidesway and perform a first-order analysis. A fictitious restraining force, AJR, results as
shown in Fig. 4b. Next, the fictitious force is removed by applying CJF = -AJR,
removing the restraint, and carrying out another first-order analysis. These results are
shown in Fig. 4c. The capacity of the columns must then be checked according to the
code specified interaction equations. Regardless of method, LRFD or ASD, the column
buckling capacity for the first term of the interaction equation can be determined from the
nomograph and the column moments will be amplified to account for second-order
effects using the in-plane K-factor in the calculation of the amplification factors.
Normally, the sway effects of the non-symmetric load will be quite small. If a lateral load
is added, as shown in Fig. 4d, a similar approach can be used; however, the sway
moments will now be significant.
Figure 4. Nonsymmetric Portal Frame
When the frame of Fig. 3a is subjected to a second-order analysis, there will be no change
from the results for a first-order analysis. Thus, the design approach should be the same;
K-factors from the nomograph and no moments to be amplified. If a second-order
analysis is performed for the frame of Fig. 3b, the results will be different from the first-
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order results. The buckling capacity may still be determined through the nomograph Kfactor; however, the moments will not need to be amplified to account for sway since that
was already accomplished through the second-order analysis.
When the frames of Fig. 4 are subjected to a second-order analysis, different moments
result, which account for sway. The buckling capacity may be determined from the Kfactors of the nomograph and moments may be used directly, with no need for
amplification.
With Leaning Columns: When an unbraced frame is called upon to provide lateral
restraint for leaning columns, the first- and second-order analysis may or may not be
sufficient, depending on the simplifications used. In addition, the nomograph approach
(Eq. 4) to determining buckling load may not be a good indicator of column capacity.
When a leaning column is added to the frame of Fig. 3a, the resulting structure is as
shown in Fig. 5. A first-order analysis will yield the same member forces for the
unbraced frame as had been determined for the structure in Fig. 3a. Thus, it appears that
the leaning column has no impact on the original structure. If the structure is subjected to
a second-order analysis, again, no change will be noted.
Figure 5. Symmetric Frame with
Leaning Column
Figure 6. Nonsymmetric Frame
with Leaning Column
If a leaning column is added to the frame of Fig. 4, as shown in Fig. 6, and both the
gravity and lateral loads shown are applied, a first-order analysis will again repeat the
results from the frame of Fig. 4. If a second-order analysis is performed, the results will
be different from those previously determined. They will account for the amplification of
moment due to sidesway of the structure and both loads P and Q; but, as for the frame of
Fig. 5, no account will be taken of the reduced buckling capacity due to the presence of
the leaning column.
If the buckling load for a frame member is to be determined through an approach other than
a complete buckling analysis, a model that will reasonably predict the capacity of a frame
including these leaning columns is needed. Numerous approaches intended to account for
the effect of leaning columns and the sharing of lateral resistance have been presented in the
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literature. These approaches offer a wide range of mathematical complexity and practical
usefulness. Four approaches that have been presented in the literature for including the
leaning column in the determination of column capacity will be discussed along with some
simplified equations that are included in the Commentary of the LRFD Specification. As
always, the designer is called upon to decide on the appropriate approach to use in a
particular design situation.
Effective Length
Modified Nomograph Equation (16): The derivation of Eq. 4 is available in numerous
references, including (3). Following the same procedures and assumptions, with the addition
of the leaning column, as shown in Figure 7, a new equation may be developed.
Figure 7. Restraining and Leaning Columns
Viewing the structure in its displaced equilibrium configuration, the leaning column and the
restraining column are separated as shown in Fig. 7b and c. The load Q on the leaning
column CD must be balanced by the horizontal force,
at D, for equilibrium of the
leaning column. This force must then be applied as a load at B on the restraining column
AB.
Equations of equilibrium at the joints of column AB and the sway equilibrium
equation can be written for the structure in the displaced configuration. Member end
moment equations are then written using the slope deflection method, incorporating the
stability functions (17) necessary to account for the influence of axial load on column AB.
Combining these equations and setting the determinate of the coefficients equal to zero will
yield the following buckling condition equation.
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If the leaning column load is zero, Eq. 5 reduces to Eq. 4. Since neither of these equations
can be solved explicitly, an iterative approach may be used or, in the case of the frame
without leaning columns, the nomograph already discussed may be used.
The Yura Approach (18): This is perhaps the easiest approach to develop since it relies on a
straightforward interpretation of the physical problem. For the unbraced frame shown in Fig.
8, equilibrium will be established for the structure in the undeflected configuration and again
in the deflected configuration. The first-order, undeflected equilibrium configuration forces
are shown in Fig. 8a. If the frame is permitted to displace an amount A, equilibrium in this
displaced configuration will be as shown in Fig. 8b. In order for column EF to be in
equilibrium, a lateral force,
as shown at F is required. This force must be equilibrated
by an equal and opposite force shown at B. Thus, when column AB buckles, it buckles with
a moment of
at its base. It is observed that this is the same moment that would
result if the individual column AB were to buckle under the axial load of (P + Q). The
assumption that the buckling load is (P + Q) is only slightly conservative for the individual
column AB, since the deflected shape due to an axial load and a lateral load differ only
slightly. In order to insure sufficient lateral resistance to buckling for column EF, column
AB must be designed to carry a fictitious load (P + Q).
Figure 8. Equilibrium forces for Yura derivation
In order to compare this approach to others presented in the literature, it is helpful to convert
it to an effective length approach. If column AB is to be designed to carry the load P but
have the capacity (P + Q), a modified effective length factor will be required. is defined
as the effective length factor that would be determined from the nomograph or Eq. 4, which
does not account for the leaning column. In this case
is defined as the effective
length factor that will account for the leaning column. Thus, based on the buckling load
being (P + Q)
(6)
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If the column is to be designed to carry the actual applied load, P, with the leaning column
accounted for through
then
(7)
Solving equations 6 and 7 for their corresponding K's and taking the ratio
yields
(8)
which may be solved for
as
(9)
If column AB from Fig. 8a were designed to carry the load P using the effective length
factor
it would provide sufficient lateral restraint to permit column EF to be designed to
carry the load Q using K = 1.0.
For frames with more than one leaning column and more than one restraining column,
and
will replace P and Q. It should also be noted that this approach maintains the
assumption that all restraining columns in a story buckle in a sidesway mode
simultaneously.
Lim & McNamara Approach (19): Another approach that will account for the leaning
column was proposed by Lim and McNamara for columns of unbraced tube buildings. Their
development is also based on the assumption that all columns in the restraining frame
buckle in a sidesway mode simultaneously; however, they developed the sway buckling
equation through the use of stability functions and an eigenvalue solution.
The resulting effective length factor, accounting for leaning columns is given in their paper
as
(10)
where and are as defined earlier,
and is the eigenvalue solution for a
frame without leaning columns and is the eigenvalue solution for a frame with leaning
columns. The authors suggest that for normal column end conditions,
should
provide a K-factor on the conservative side by at most 2%. Substituting for n and using
the Lim & McNamara approach gives the same K-factor as the modified Yura
approach where
(11)
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Thus, for the story buckling approach, a single multiplier for each story will be sufficient to
modify the individual nomograph K-factors to account for leaning columns.
LeMessurier Approach (20): In his landmark paper, LeMessurier presented a more complex,
yet still very practical approach for frames with and without leaning columns. The basic
equations were developed for a single cantilever column and then extended to the general
frame. Where the previous approach determined a constant value for a story by which the
nomograph value of was modified, this approach determines a constant value for a story
which then multiplies the individual column moment of inertia divided by the column load,
, for each column, i. Thus, the contribution of each column to the lateral resistance is
accounted for individually. The effective length factor for each column that participates in
resisting sidesway buckling, Eq. 46c from the original paper, expressed in the notation of
this paper, is given by
(12)
where
(13)
(14)
= effective length of column i, accounting for leaning columns.
= 0 for leaning columns.
= load on restraining column, i.
= moment of inertia for column, i,
= load on the restraining columns in a story.
= load on the leaning columns in a story
= sum of
= sum of
for each column in the story.
for each column participating in lateral sway resistance
Commentary Equations (6): Although use of Eq. 12 is not particularly complex, the second
edition of the Commentary to the LRFD Specification presented two modified LeMessurier
equations that were thought to be of value to the practicing engineer. Although the third
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edition of the commentary will be significantly simplified in the relevant section, it is
anticipated that there will still be two simplified LeMessurier equations presented. One is
based on the story buckling model while the other is based on a story stiffness model.
For the story buckling model, it is assumed that there is no reduction in column stiffness due
to the presence of axial load. This is accomplished by taking
for all columns, which
leads to
Substitution of these values into Eq. 12 yields:
(15)
Which reduces to
(16)
Equation 16 can be recast into the form of the commentary equation, which is currently
proposed to be
(17)
For a structure in which only one column can be considered to provide lateral stability, the
summation in Eq. 16 are unnecessary and the equation reduces to
(18)
which is the same as the equation that resulted from the modified Yura and Lim &
McNamara approaches, Eqs. 9 and 11 respectively.
For the story stiffness model, stiffness reduction due to axial load is included as though all
columns were cantilevers with a buckled shape in the form of a half sine curve as shown in
Fig. 2b, thus
Since the leaning columns have no lateral stability of their own,
= 0.0 for all leaning columns. The equation given in this paper as Eq. 12 is just one form of
the effective length factor equations given by LeMessurier. Another form that uses, as a
measure of lateral stiffness, the ratio of lateral displacement of a story to the lateral load, is
also available through the same derivation (19). Equation 46d from the original paper, in the
notation of this paper, is given as
(19)
where
the total lateral load supported by the level under consideration,
is the
corresponding lateral displacement of the level and
is the total load on the
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1-16
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given story. In order to account for
on the leaning columns, the load on these leaning
columns must be subtracted from the total load on the story so that
Making this substitution and factoring out
yields
(20)
This equation is somewhat simplified in the second edition commentary as
(21)
If the stiffness reduction due to axial load is applied to all columns, that is the leaning
columns are not excluded, then
would be applied to the total load on the story
and the separation taken to arrive at Eq. 20 would not be necessary. Thus, Eq. 19 would
become
(22)
This equation, recast in the form of the third edition commentary equation is
(23)
These simplifications may not really be necessary since, in the original form, the equations
presented by LeMessurier are not much more complex.
Examples
The following examples will show how these approaches may be used to evaluate
columns in unbraced frames.
Example 1: The unbraced frame with leaning columns as shown in Fig. 9 is to be checked
for strength and stability.
This is the frame introduced by Higgins (21) and
modified with a load factor of 1.43 to permit a check by LRFD. The frame is braced out of
the plane.
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Yura Approach: Since sway will likely control the design of column AB, a W10x39 will be
investigated. Due to symmetry, one half of the load on the leaning columns will be assigned
to a single column, AB.
Figure 9. Symmetric Frame for Example 1
Note that to account for the pin ended beam, the length is modified by the factor 2.
For strength in the y-axis,
thus, use LRFD E2-2.
kips > 57 kips thus, the strength is adequate.
For stability about the x-axis,
therefore from LRFD E2-2,
= 228.0 kips > 207 kips, thus the column will be sufficient to provide stability for the
remaining leaning columns.
Lim & McNamara Approach: Again, a W10x39 will be considered for column AB. Using
the ratio of
As before,
so
As already shown, the column will be adequate for strength.
Now, checking for stability,
and
Thus,
applied load which shows that the column is also adequate
for stability.
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LeMessurier Approach: Since the W10x39 column was shown to be adequate, it will again
be checked. Using the values obtained above, for column AB,
Thus, with
restraint.
from above, the W10x39 will be sufficient to provide lateral
Modified Nomograph Equation: An iterative solution of Eq. 5 with
Q and P as
given above yields
As can be seen, this value compares quite well with the values
already obtained and the column will be adequate.
Commentary Equations: For the first simplified equation, Eq. 16,
as would be
expected from the derivation shown above, since there is only one restraining column.
The use of the second simplified equation, which is based on Eq. 19, requires an analysis of
the structure to determine the ratio of lateral displacement to load. Since the analysis carried
out for the previous approaches assumed that the pin connection at the base was not a true
pin but one which resulted in
this must be included in the analysis. With this
provision accounted for, an arbitrary lateral load of 5 kips results in a deflection
in. Using Eq. 20 yields
Using the equation from the second edition commentary, Eq 21, results in
Geschwindner
1-19
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The proposed equation for the third edition, in the form of Eq. 22, yields
Using any of these K values, the column will prove to be adequate to provide the required
lateral resistance. The results obtained from Eqs. 20 and 21 appear to be quite similar to
those determined from the other approaches presented. The result from Eq. 22 appears to be
somewhat distant from the other results, although the column will still prove to be adequate.
Example 2: The frame shown in Fig. 10 was used by Cheong-Siat-Moy (22) to show that
Yura's approach would not work and by de Buen (23) to present his new approach. The truss
is assumed to provide sufficient rotational restraint at the top to permit that end of column
AB to be treated as a fixed end while the bottom of the column is pinned, thus from the
nomograph,
Recognizing that Yura's approach will yield a larger column than
required and that LeMessurier's approach will give the correct results, as shown by the two
previously cited references, the Lim & McNamara approach will be compared to the
LeMessurier approach. For this example, the column will be taken as a W12x136 and
50 ksi.
Figure 10. Frame for Example 2
Lim & McNamara: With one half of the leaning column load resisted by the single
restraining column, n = 990/330 = 3.0. Thus,
Using
thus from LRFD E2-3,
= 8.48 ksi, thus
Since this approach is essentially the same as the
Yura approach, this result is not entirely unexpected.
LeMessurier: For a fixed-pinned column,
and
so that Eq. 12 yields:
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1-20
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as shown in Ref. 22 and 23.
Using
thus from LRFD E2-3,
thus
Modified Nomograph Equation: Using G values consistent with a true fixed end and a true
This
pinned end, along with
an iterative solution of Eq. 5 results in
result is similar to that previously obtained from the LeMessurier approach.
Commentary Equations: As was shown for example 1, where only one column provides
lateral support, the results from Lim & McNamara and the simplified equation, Eq. 16, are
identical,
The LeMessurier analysis, Eq. 12, assumed that the columns were fixed
at the upper end and had true pins at the lower end. This results in
which is the
assumption used to develop the second simplified equation, Eq. 20. With the lateral
displacement calculated for a cantilever beam with a 5 kip load,
in. This yields
from Eq. 20 and
from Eq. 22. The results from Eq. 20 are identical to
the previously calculated value using the complete LeMessurier approach and the results
from Eq. 22 are the same as that from the Lim & McNamara approach, as would be
expected from the derivations of the simplified equations.
Example 3: An interesting problem originally proposed by Zweig (24) is shown in Fig. 11.
A portion of a large, unbraced one story industrial building with deep roof trusses is shown.
The trusses, which frame in each direction, exhibit an infinitely large stiffness when
compared to the columns. In order to equalize sway restraint in each direction, alternate
columns have their strong axes turned 90°. The nomograph approach with
and
= 10 yields
Without considering leaning columns and using
the
W12x65 proposed by Zweig would be slightly undersized for the required
By
this approach, each column in the building would be the same, designed to carry 234 kips
about its weak axis. However, if the strong axis column is used to brace the weak axis
column, there should be some savings available. Each of the approaches previously
discussed will be used to check a W12x53 column to determine whether this smaller column
would be adequate. Throughout this example, the columns will be taken as pairs, one strong
axis and one weak axis column.
Yura Approach: Using
the capacity of the W12x53 for sway buckling about the
x-axis, column 1, and the y-axis, column 2, will be added. The combined capacity must be
equal to or greater than the total load 2(234) = 468 kips. For
The
Geschwindner
1-21
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Figure 11. Frame for Example 3
combined capacity is 432 + 127 = 559 kips which is greater than the 468 kips combined
load. Thus, the W12x53 will be acceptable for all of the columns.
Lim & McNamara Approach: If column 1 is taken as the restraining column and column 2
as the leaning column, n = 234/234 = 1.0. Thus, for column 2, K = 1.0 and KL = 1.0(20) =
20 ft. For the W12x53 buckling about the y-axis,
For column 1,
and
For buckling about the x-axis,
280 kips > 234 kips. Thus, the W12x53 is adequate for both columns in each direction.
LeMessurier Approach: With
and
for both columns 1 and 2,
and
Thus, from Eq. 12
which yields
and
Thus, for column 2,
thus
and
For column 1,
which is the same as for column 2 so
This
shows that the W12x53 column is adequate for both carrying the load and providing lateral
restraint.
Modified Nomograph Equation: In this case, with end stiffness and load ratio as for the
other approaches, Eq. 5 yields
for the restraining column. The leaning column
would use K=1.0 as with the Lim & McNamara approach. Again, the W12x53 will be
acceptable.
Commentary Equations: For the first simplified equation, Eq. 16,
columns. Thus,
for both
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1-22
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As was the case in example 1, the application of second simplified equation requires the
analysis of the structure with a connection stiffness at the base that will result in
With a 5 kip lateral load, the resulting lateral deflection,
Thus, from Eq. 20,
while Eq. 21 yields
and Eq. 22 yields
As can be seen from these K values, the W12x53 will work satisfactorily for all columns in
the structure when checked through these simplified equations. The significance of this
example, as it was when originally published, is to show that the sharing of lateral stiffness
can be used to the advantage of the structure.
Example 4: The frame shown in Fig. 12, introduced by Geschwindner (25), will be used
to compare the simplified methods for determination of effective length factors with a
true buckling analysis. The frame is supported in such a way that in plane behavior will
be critical. The columns AB and CD as well as the beam BC are W12x136. The other
members are of such a size that their individual characteristics will not control. The
results of a GTSTRUDL buckling analysis for equal loads on columns AB and CD yields
Using Eq. 3 this is equivalent to K=2.232. The nomograph equation
yields K=2.166 or
Figure 12. Frame for Example 4 with Leaning Columns
When equal loads are also applied to columns EF, GH, and JK, GTSTRUDL yields
528.7 kips or K=3.414. The loading on the structure shows two equal loads on the
restraining columns and three equal loads on the leaning columns. This gives P = 2 and Q
= 3. From the modified nomograph equation, K=3.289 and
Eq. 11 yields
K = 3.425 and
The LeMessurier equation, using G = 100,000 to represent the
pin end, yields K = 3.295 and
Geschwindner
1-23
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No matter what approach is taken to account for the leaning columns, it is clear that
they have a significant impact on the stability of the structure. It is also evident that a
second-order elastic analysis will yield the same forces for members AB, CD, and BC,
whether there are loads on the leaning columns or not. Thus, more than a second-order
elastic analysis is needed for the complete design of the structure.
Example 5: Factored loads are now applied to the frame of example 4, as shown in Fig.
13. First- and second-order elastic analyses are performed and, along with the results
from example 4, a check on column CD, with
is carried out.
Using the results from the first order analysis,
and
and
from the LeMessurier analysis including the leaning columns, K=3.295 and
From this,
so LRFD Eq. H1-1a is used. Since the
column moment is from a first order analysis, it must be amplified. This will be
accomplished using the second suggested equation for
Since there are no moments in
the non-sway analysis,
The results of the first-order analysis give a
lateral deflection due to the 20 kip load of 1.738 in. Using these values,
so that
The interaction equation becomes
Since this is less than 1.0, the column will be adequate.
If the results of the second-order analysis are used,
and
ft. Again, the effect of the leaning columns will be included from the LeMessurier
analysis so that,
and LRFD Eq. H1-1a is used
again. Since the column moment results from a second-order analysis, there is no need to
amplify it prior to using the interaction equation, thus
Geschwindner
1-24
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Again, the column is seen to be adequate. It is interesting to note that there is an increase
in the column axial load due to the second order effects that is not included in the
simplified code approach to second-order analysis and that the second-order moments
obtained from the two approaches are quite similar.
Example 6: An interesting structure was presented by Baker (26) to demonstrate the
problems associated with effective length determination when the assumptions of the
nomograph are violated. The frame shown in Figure 14 represents one of two frames
Figure 14. Example 6 Frame with lateral service load
participating in providing lateral resistance for the building. This frame carries a gravity
load of
and provides lateral stability to columns carrying an additional
= 1875 kips, both being one half of the total load for the building. The lateral deflection
of the frame due to
is
with
The total gravity load is
and
The results presented by Baker were in the form of column capacities,
However, a
review of his equations shows that his solution actually uses Eq. 21. Table 1 shows the Kfactors for each of the five columns in this frame as determined through nomograph, the
four simplified equations, and an elastic buckling analysis. It can be seen from Table 1
that the use of the nomograph, Eq. 4, does not predict effective length values that would
subsequently produce accurate elastic buckling values for the columns of this frame. This
is due to the fact that this frame significantly violates the assumptions used to develop the
nomograph. If this structure had been designed with those values, the columns would
have had an expected capacity significantly larger than their true capacity.
The goal in determining an appropriate K-factor is to predict the results of a linear
buckling analysis. The results of a linear buckling analysis from GTSTRUDL, converted
to effective length, are also given in Table 1. The values determined from Eqs. 16, 20, and
21 appear to be fairly consistent although slightly below the elastic buckling values. The
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1-25
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results of Eq. 22 are significantly higher than the elastic buckling values, although they
will yield a conservative solution.
Table 1. Summary of Effective Length Calculations for Example 6
Conclusions
Structural systems which combine unbraced frames with simple gravity columns provide the
structural engineer with a system requiring special attention. In addition, unbraced frames
that do not meet the restrictive assumptions permitting use of the nomograph also pose
interesting problems. It is clear that leaning columns result from the structural framing
arrangement, not from the use of a particular design philosophy. It is also clear that a
specific accounting must be made for the leaning column, even when a second-order
analysis has been carried out. In addition, the design philosophy and the analysis approach
must be compatible.
This paper presented a brief discussion of the full range of approaches that might be used to
carry out a structural analysis. It is well understood that the critical buckling load of a frame
can be determined through an eigenvalue analysis. It is this load that the K-factor is
attempting to predict. Although the K-factor has been a controversial topic from its initial
introduction, it remains a useful tool to measure column capacity. Perhaps the most
troubling aspect associated with the use of the K-factor has been the assumptions included in
the most common predictor equations.
Four approaches from the literature for determination of the K-factor were presented, along
with several simplified equations derived from those procedures. It was shown that through
an iterative solution of Eq. 5, a more accurate value of could be obtained than that from
the nomograph, Eq. 4, when leaning columns are present. In this case, the leaning column
loads are accounted for; however, the other limitations of the nomograph solution are still
present.
The equations proposed by LeMessurier are generally recognized as the most accurate of
those presented. There are two approaches to the use of the LeMessurier equations. One
requires the determination of which may be accomplished through the nomograph, as is
Geschwindner
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normally done, or by an iterative solution of equation, Eq. 4. The other approach uses the
lateral stiffness of the frame, as measured by its lateral deflection due to a lateral load. Either
of these approaches will provide a practical solution to determining column capacity. Thus,
it is not unrealistic to use the LeMessurier equations for effective length factors in normal
engineering practice.
The commentary to the LRFD Specification provides simplified equations, based on the
LeMessurier equations, which are felt to be useful to the designer. The examples presented
here allow for a comparison of results between several of these equations. It appears that
there is some wide variation in results, depending on the choice of approximation. The
assumptions used to develop these simplified equations are presented so the engineer will be
in a better position to decide which should be used in a particular situation.
It was also shown that the use of a second-order analysis does not automatically account for
the presence of leaning columns, but does meet the need for determination of moments for
use in the interaction equations. Thus, both the second-order effects and buckling capacity
must be determined for the analysis and design to be complete. It is simply a matter of
deciding which approach is to be used.
Although the LeMessurier approach is not overly complicated to use, designers wishing to
use an even simpler approach may find that the Lim & McNamara equation for provides
a sufficiently accurate way to account for leaning columns, particularly in preliminary stages
of design. In addition, the LeMessurier equation, which is based on the lateral deflection of
the frame, provides a straightforward approach to the actual calculations, as suggested by
Baker. Although simplified equations are presented in the Commentary to the 1993 LRFD
Specification, and will be included in the 1999 version, there is really no need to use them.
Once the elastic-buckling load of the frame has been determined and the appropriate amount
attributed to the individual columns, design by any approved method may proceed.
References
1. West, Harry H., Analysis of Structures. John Wiley and Sons, New York, 1989.
2. Galambos, T. V., Structural Members and Frames. Prentice-Hall, Inc., Englewood
Cliffs, NJ, 1968.
3. Disque, R. O., Applied Plastic Design in Steel. Van Nostrand Reinhold Co. New York,
1971.
4. Chen, W. F., Goto, Y., and Liew, J. Y. R., Stability Design of Semi-Rigid Frames. John
Wiley and Sons, New York, 1996.
5. Chen, W. F. and Toma, S., Advanced Analysis of Steel Frames, CRC Press, Boca
Raton, 1994.
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This publication or any part thereof must not be reproduced in any form without permission of the publisher.
6. Load and Resistance Factor Design Specification for Steel Buildings. 2nd edition, AISC,
Chicago, IL, 1994.
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Journal, AISC Vol. 31, No. 4, 1994, pp. 141-149.
17. Chen, W.F. and Lui, E.M.. Stability Design of Steel Frames, CRC Press, Boca Raton,
FL., 1991.
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Journal, AISC, Vol. 8, No. 2, 1971, pp. 37-42.
19. Lim, L. C. and McNamara, R. J., "Stability of Novel Building System," Structural
Design of Tall Steel Buildings. Vol. II-16, Proceedings, ASCE-IABSE International
Conference on the Planning and Design of Tall Buildings, Bethlehem, Pa. 1972, pp.
499-524.
20. LeMessurier, Wm. J., "A Practical Method of Second Order Analysis," Engineering
Journal, AISC, Vol. 14, No. 2, 1977, pp. 49-67.
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21. Higgins, T. R., "Column Stability under Elastic Support", Engineering Journal, AISC,
Vol. 2, No. 2, 1965, pp. 46-49.
22. Cheong-Siat-Moy, F., "Column Design in Gravity-Loaded Frame," Journal of Structural
Engineering, ASCE, Vol. 117, No. 5, May 1991, pp.1448-1461.
23. de Buen, Oscar, "Column Design in Steel Frames Under Gravity Loads," Journal of
Structural Engineering, ASCE, Vol. 118, No. 10, October, 1992, pp. 2928-2935.
24. Zweig, A., Discussion of "Column Stability Under Elastic Support," by T. R. Higgins,
Engineering Journal, AISC, Vol. 2, No. 3, 1965, pp. 105-106.
25. Geschwindner, L.F., "Practical Design of Unbraced Moment Frames with Leaning
Columns," Proceedings of Structures Congress 13, Boston, Mass., April 2-5, 1995,
ASCE, pp. 527-542.
26. Baker, W. F., "Practical Problems in Stability of Steel Structures," Proceedings of the
National Steel Construction Conference, Chicago, IL, AISC 1997, pp. 2.1-2.24.
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