Computers & S/rucrures Vol. 34. No. 2. pp. 215-223.
Printed in Great Britain.
FINITE
ELEMENT
1990
0
ANALYSIS
OF COLUMN
CKMS-7949po $3.00 + 0.00
I990 Pergamon Press plc
BASE PLATES
NATARAJAN KRISHNAMURTHYand DAVID P. THAMBIRATNAM
Department of Civil Engineering, National University of Singapore, Kent Ridge, Singapore
(Received
8 February
1989)
Abstract-Column
base plates have been analyzed and designed traditionally on the assumptions that the
plate is rigid and that the plate thickness can be determined from the cantilever action of the plate
projections beyond the column face. Recent investigations have attempted to investigate the actual
behavior of column base plates and quantify the differences between the real and assumed behavior. This
paper describes the salient features of the two-dimensional finite element analysis of certain base plates
which had been tested in the laboratory, and presents the significant findings. The computer results confinn
the observed behavior of the test specimens to a reasonable extent.
NOTATION
f
”
s
P
M
stress or other quantity (with appropriate subscript)
used in Richardson’s extrapolation formula.
number of nodes in the finite element analysis.
top surface strain at critical section of columnbase plate junction on the side of eccentricity.
load on column-base plate.
gradient in S-P graph.
INTRODUCTION
The classical design of column base plates is based on
the assumption that the base plate is rigid, and hence
the pressure
distribution
under the plate due to
vertical load and bending moment on the column is
linear, as in Fig. 1. Further, to determine the thickness of the plate, the projections of the plate beyond
the column are treated as cantilevers loaded by the
reactive pressure from the support. It has long been
known that this procedure reflects an over-simplification of the real situation, but the method is universally used only because it gives (mostly)_ safe-even
plates no larger than the column itself. Murray [3]
proposed a variation of Stockwell’s localized pressure
distribution,
and also developed a yield line formula
for the thickness of the base plate, also for lightly
loaded base plates.
While analytical approaches
to base plate design
have been very limited, experimental
investigations
into base plate behavior have been more numerous.
De Wolfe and associate [4,5] and La Fraugh and
Magura [6] have published their findings from tests
on specific configurations
of column base plates.
Thambiratnam
and associates
have conducted
a
series of experimental investigations on eccentrically
loaded column base plates, of the configuration
shown in Fig. 2, at the National University of
Singapore (NUS). The tests have been described and
evaluated in [7-91. Significant details and findings
are given in the paper by Thambiratnam and Paramasivam [lo]. Three series of tests were conducted
(including some replications) with the same column
section. but with three different thicknesses of the
obviously excessivedesigns
in the absence of a more
realistic method. In recent years, attempts have been
made to investigate analytically and experimentally
the true behavior of column base plates. Fling [l]
introduced limitations of deflection at the free edge of
the plate as an additional constraint, and also invoked the yield line theory for the determination of
the plate thickness for the required capacity. He
considered only concentric loads and assumed uniform pressure distribution under the base. He mentioned that the bearing pressure was ‘probably
somewhat higher’ in the region where the column
directly bore down on the plate, but noted that this
real situation would result in lower plate deflections
and moments, thus leaving the assumption of uniform pressure distribution a conservative one. Stockwell [2] proposed the concept of localized pressure
distribution, over rectangular areas under the column
flanges and web, with particular reference to ‘lightly
loaded
CAS 34,2-z
base
plates’
meaning
those
which
Anumtd
Bearing Rcssure
Assumed Plate
Bending Moment
Fig. 1. Column and base plate under axial
require
moment.
215
force and
216
NATARAJANKRISHNAMURTHY
and DAVIDP. THAMBIRATNAM
the
Fig. 2. Isometric view of set-up.
base plate, namely 16, 19 and 22mm (S/8, 3/4 and
7/8 in.), and for four different eccentricities, namely
38, 89, 140 and 191 mm (1.5, 3.5, 5.5 and 7.5 in.). The
loads were increased up to complete failure of the test
specimens generally in increments of 20 kN (4.5 kips).
The reason that the design of column base plates
has not advanced beyond the utilization of simplified
design modes based on strength of materials and yield
line theories is that although it could be foreseen that
the reactive pressure under column base plates would
be highly localized, no classical method could predict
the distribution of this variation. The advent of
digital computers and the consequent burgeoning of
the finite element method have presented investigators with a powerful new tool to attack such
complex problems with ease and accuracy.
The first author has utilized the finite element
method extensively for the analysis of steel bolted
end-plate connections as in Fig. 3. An end-plate
connection may be considered as a column base plate
rotated by 90”, and vice versa, as in Figs 1 and 3. The
problems of (and solutions for) the localized bearing
pressure and variable contact between the back of the
plate and the support would be very similar. Most of
Fig. 3. Bolted end-plate connection
work on end-plate connections by the first author
and his assistants has been summarized in [1 11. The
end-plate design procedure he proposed based on the
research findings has been adopted by the American
Institute of Steel Construction and the Metal Building Manufacturers
Association of the U.S.A. [12]. A
special purpose computer program called FEABOC
(for Finite Element Analysis of Bolted Connections)
was developed during his research on bolted connections [13]. Many tests were conducted on prototype
and model specimens of end-plate connections and
their components, to confirm the finite element results, or alternatively (where the discrepancies were
too large) to refine the computer models, and finally
to validate the design procedure.
It was obvious that the knowledge and experience
gained from, as well as the software developed for,
the end-plate research could all be mobilized for the
column base plate investigation. Thus to supplement
and complement the findings of the experimental
investigation at NUS, as well as to generate additional information, the authors initiated finite element analysis of the test specimens by the FEABOC
program [14, 151. Recently, Murray [3] published
results of a finite element analysis of a lightly loaded
base plate.
In the sections to follow, the essential features of
the authors’ computer analysis and findings are
presented, together with comparisons with the test
results, wherever possible.
FINITE ELEMENT ANALYSIS
The theory
application
too
of the finite element
to stress and deformation
method
and its
problems
are
well known to warrant detailed explanation here.
Only the unique features of the finite element modeling of the column base and the computer program
used will be described.
Details of the FEABOC program are given in [13].
In brief, it is a linear elastic displacement formulation
incorporating the bar or spring, the constant strain
triangle, the hybrid rectangle, the sub-parametric
quadrilateral, and the sub-parametric solid block in
its menu of elements. Because the deformed profile of
the plate under any loading is unknown to start with,
it is determined iteratively. Initially, the plate is
assumed to be in complete contact with the support.
After each cycle of analysis, the program checks the
reactions at the support nodes, and releases those
nodes on the underside of the plate that tend to pull
away, setting them free to move in the next cycle, thus
permitting separation of the plate from the support in
that region. It also checks the displacements of the
previously released nodes, and resinstates the support
at those nodes that have moved back too far encroaching into the support region. For any particular
load, the analysis is considered complete when there
is no change in the support codes for the nodes at the
back of the plate. The program will start with zero
(external) load, to analyze the domain for only the
Analysis of column base plates
217
bolt pretension, because even the bolt pretension will
deform the plate. At the end of this zero load run, the
bolt end nodes will be constrained in the stretched
condition, before going on to the analysis for the
specified external loads in sequence. The deformed
profile for one loading will be used as the initial
condition for the next loading.
The domain modeled for the analysis of the
column base plate consisted of the column stub, base
plate and the single anchor bolt as shown in Fig. 4.
The concrete block support for the steel base plate
was assumed rigid for this preliminary study. In the
early runs, only the portion of the bolt shank passing
through the base plate was modeled as part of the
finite element analysis domain. But the agreement
in computed and measured results was rather unsatisfactory. It was subsequently noted from the test
logs that in most of the tests the bolt had broken its
bond from the surrounding concrete during the test.
To simulate this situation, the entire length of bolt
was included in the analysis domain, as shown in
Fig. 4. With this modification, the results agreed
much better.
h
(al
102mm x 102mm
x 12.7mm
Box Section
102mm x 356mm
x 12.7mm Thick Plate
4Y-t
Concrete Block
I
I
TV
c II
;
’
1
I
I
‘-f-t-+
I
(b 1
II
1I
4
1I
1I
Diameter
d9mm
I;
(c)
I
Fig. 4(a)+c).
Fig. 4. Domain for finite element analysis. (a) Stress distribution at top of column stub. (b) Elevation. (c) Plan. (d)
Enlarged view of mesh scheme used and nodal forces
applied.
The eccentric load was applied to the mode1 in its
equivalent combination of concentric load and bending moment, through nodal concentrated loads corresponding to the linear stress distribution acting
over the contributory areas at the top of the column
stub. The domain was analyzed as a two-dimensional
plane stress problem in the plane parallel to the
column web. The widths of the base plate and column
flange, and the combined thickness of the column
webs were input as the thicknesses of the various
elements. The hexagonal bolt heads and the circular
bolt shanks were idealized with equivalent rectangular cross-sections. The bolt shank was enabled to
stretch independently of the plate through which it
passed, by provision of two separate sets of nodes and
elements, one for the bolt and the other for the plate
(with the latter’s width reduced by the width of the
bolt hole), connected only at the common line of
nodes under the bolt head.
A convergence study was conducted on four different meshes, designated ‘coarse’, ‘medium’, ‘fine’ and
‘extra-fine’, with 60, 107, 159 and 203 nodes and 40,
79, 124 and 163 rectangular elements, in the first case.
It was noted that although there was a distinct trend
towards asymptotic convergence, the results for the
fine and extra-fine meshes were not close enough to
warrant their acceptance as sufficiently accurate. As
the available computer resources precluded further
mesh refinement, it was decided to analyze all the
cases with both fine and extra-fine meshes, and
218
NATARAJAN KRBHNAMURTHY and DAVID P. THAMLIRA~NAM
produce any stress or deformation in the analysis
domain, but served only to fix the bolt end in place.
To develop the finite element mesh for each case,
a pre-processor was written which, with the input of
the dimensions of the column, base plate and the bolt,
and the number of divisions in the various components, would generate the node coordinates and
element incidences, and file them in the format
needed by the FEABOC program. Likewise, another
pre-processor program was written to generate and
file the concentrated loads to be applied on the nodes
at the top of the column stub, corresponding to
specified values of applied load and eccentricity for
the particular mesh in each case.
As the quantities measured in the tests were the
strains on the top and bottom surfaces of the base
plate, a post-processor program was written to calculate the strains from the values of the nodal stresses
derived from the finite element analysis. The nodal
stresses themselves were determined as the averages
of the stress values for all the elements meeting at
every node. An exception was made for the critical
location [B in Fig. 4(d)] at the junction of the column
flange and base plate projection, where the contribution of the column elements was omitted from the
averaging process as having an excessively adverse
effect due to the high vertical compression.
not
t
* f,
A=
5
_--__
; ff
‘G
‘;
___- __--
__
--p‘--__
-I
P
n:,
I
I
I
I
I
I
I
I
0
0
lllnf 1
It&l
Fig. 5. Richardson’s
lllnd
extrapolation
-Illnl
II/n,1
scheme.
estimate improved values by the application of
Richardson’s extrapolation
to corresponding
sets
of results from the two meshes, as in Fig. 5. Plots of
critical stress vs various powers of the mesh size, in
the entire analysis domain as well as in the plate
projection on the side of the load eccentricity [BA in
Fig. 4(d)], were examined for suitability of extrapolation. It was determined that plots of critical stress
vs the reciprocal of the number of nodes (n) in the
plate projection gave straight lines. Hence the following formula representing the linearity of these variations was applied to the desired quantities to obtain
the converged value f0 as
FINITE
f =f +
0
e
nr(L -h)
(4 - 4)
in whichfis the stress or other critical quantity to be
estimated, n is the number of nodes in the plate
projection BA, and subscripts f and e refer to the fine
and extra-fine meshes. The bolt had not been pretensioned in the tests, and hence the zero load run did
ELEMENT
RESULTS
All 12 test specimens, corresponding to the three
plate thicknesses, each for four eccentricities, were
analyzed for zero load, half the failure load, and the
failure load. Because the program used in the analysis
was a linear elastic one, ‘failure’ was defined as first
yield, and the failure load was taken as that value in
the test for which the maximum measured strain was
(31.5kipsl
(674 kipsl
(98.9 kips)
Fig. 6. Finite element
and theoretical
bearing pressure distributions
89 mm (3.5 in.) eccentricity.
for 22 mm (7/8 in.) plate loaded
at
Analysis of column base plates
219
.
Fig. 7. Finite element and theoretical bearing pressure distribution for 22 mm (7/8 in.) plate loaded with
100 kN (22.5 kips) at various eccentricities.
closest to the yield strain for the mild steel used,
namely 0.0012 mm/mm (in./in.).
The results from the finite element analysis that
were considered critical for evaluation were:
(a) reactive pressures under the base plate, taken as
equal to the compressive stresses at the nodes on the
bottom surface of the plate;
(b) bending stresses in the plate projection BA on
the side of the eccentricity.
Bearing pressures
Conventionally, it is assumed that the base plate is
rigid and hence the base pressure under the plate
varies linearly. Where the eccentricity of the load is
beyond the ‘kernel’ of the plate section, the base is
treated similar to a concrete cracked section, effective
only in compression. Anchor bolts will provide any
necessary tension to balance overturning moments.
In this approach, the maximum base pressure should
occur at the plate edge on the side of the load
eccentricity.
However, with the thicknesses of base plates used
in practice, their behavior could be far from rigid, and
the base reaction is more like a pressure bulb for a
flexible foundation; the maximum pressure is not at
the free edge of the plate but somewhere between the
edge and the column flange on the side of the
eccentricity. Even at low eccentricities, the plate is not
in contact with its support over its entire length.
Figure 6 shows the assumed (rigid plate) and
computed (flexible plate) pressure distributions for
the typical case of a 22 mm (7/S in.) base plate under
various loads at an eccentricity of 89 mm (3.5 in.). In
the case shown as well as in most of the cases studied,
as the loading is increased, for a particular plate
thickness and load eccentricity, the shape and extent
of the pressure bulb remain the same for all practical
purposes, and only the ordinates increase in proportion, as shown in Fig. 6. At a constant load as the
eccentricity increases, the pressure bulb predictably
shifts towards the plate edge, the shape and extent
changing, as shown in Fig. 7. In this figure, the
maximum base pressure increases with increasing
eccentricity. The maximum value of bearing pressure
computed by the finite element analysis for the cases
studied was found to vary from twice to more than
five times the maxims
value calculated by the
conventional rigid base method.
Bending stresses and strains
The flexible bearing pressure is much higher than
the rigid bearing pressure as previously mentioned.
But the maximum bending moment, as quantified by
the extreme upper and lower fiber bending stresses in
the plate, is about half the theoretical value or smaller
for the flexible plate. Figure 8 depicts the assumed
and computed variations of extreme fiber stresses in
the plate projection for the same typical case shown
earlier at two different loads.
It may be noted from the figure that according to
the finite element analysis while the top surface stress
is maximum at the column face, the bottom surface
stress is maximum a short distance away from the
column face. This happened in all the cases studied
and for all loadings. It is believed that this is the effect
of the downward and outward dispersion of the
column load, particularly at the flange, through the
base plate thickness. In the present analysis the small
value of the plate moment despite the larger localized
pressure may be explained by the fact that the
pressure bulb is much closer to the column flange
NATARAJAN
220
KRISHNAMURTHY
and DAVIDP. THAMMRATNAM
F-l
t
-
300 kN (674
kipsl
110
kips)
kN (964
Fig. 8. Finite element and theoretical bending stress distributions for 22 mm (718in.) plate loaded at 89 mm
(3.5 in.) eccentricity.
than the centroid of the trapezoidal distribution
assumed for the rigid plate.
Figure 9(a,b) shows top and bottom surface strains
for all three plate thicknesses for a ioad of 140 kN
(31.5 kips) at an eccentricity of 89 mm (3.5 in.). These
curves are typical for two other eccentricities studied,
namely 140 mm (5.5 in.) and 191 mm (7.5 in.).
The following general characteristics may be noted.
(1) The column flanges (3 and C) are the locations
of the largest bending strains in the base plate.
(2) The top strains vary relatively little for all three
thicknesses on the side of the eccentricity, but are
quite different on the other side, with the thinner
plates developing much larger maximum strains.
(3) The bottom strains increase with decrease in
plate thickness although not proportionately, but the
increase is much less than by the conventional theory
of a cantilever beam loaded with a linear pressure
diagram. The increase is larger on the anchor bolt
side.
The variations for the smallest eccentricity studied,
namely 38 mm (1.5 in.), were quite different from
those for the other eccentricities. There were little or
no bending strains on the side away from the ec-
1:
0
10
16mm
-
22mm
---
160
60
120
Distance,
mm
200
240
16mm
-
22mm
---
4
I
LO
60
120
160
Distance,
mm
200
Fig. 9. Bending strains on top and bottom surfaces for 140kN (31.5 kips) load at 89mm (3.5 in.)
eccentricity. (a) Top strain. (b) Bottom strain.
c
210
Analysis of column base plates
900
221
(a)
ezl9lmm
e=KOmm
II
100
700
$/
e=lllbT
600
e.38
100
Load,
300
200
Load,
kN
kN
Fig. 10. (a) Variation of strain at B with load for 16mm (5/8in.) plate for various eccentricities. (b)
Variation of strain at B with load for 19mm (3/4 in.) plate for various eccentricities. (c) Variation of strain
at B with load for 22 mm (7/8 in.) plate for various eccentricities.
centricity, and the peak strains at the column flange
on the side of the eccentricity were affected only a
little by the variations in plate thicknesses.
The maximum bending strains S at the plate top
surface (B) plotted against the applied load P were
straight lines for all three thicknesses through the
origin and all four eccentricities, in the form:
S=mP.
(2)
The slopes M themselves varied uniformly and
smoothly with variations in plate thickness and
load eccentricity, as depicted in Fig. lO(a-c). Charts
such as Fig. 10 could perhaps prove useful in checking the maximum bending strain or conversely in
obtaining a preliminary thickness for a desired load
and eccentricity.
EXPERIMENTAL EVALUATION
Although the project was not originally conceived
to include comparison
of finite element and
experimental studies, the finite element model was
designed and modified as necessary, to simulate the
experimental work as far as possible
However, due to limitations of time and resources,
certain aspects of the prototype could not be included
in the finite element model. In particular, the concrete
block was omitted from the model, and the support
for the steel base plate was assumed rigid; hence any
interaction between the column base and the concrete
was ignored in the analysis. In the test, it was noticed
that the concrete block cracked under high loads at
low eccentricities.
Secondly, in the test specimens a single anchor bolt
NATARAJAN KRISHNAMURTHY and DAVID P. THAMBIRATNAM
222
was used on the side away from the load eccentricity
[Fig. 4(b,c)] to hold down the plate. The nut was to
be turned snug tight on this bolt. The bolt, as well as
the lack of (signi~cant) pretension in it, was simulated
in the computer model and FEABOC program. But
the bolt extended all the way from the top of the base
plate to the bottom of the concrete pedestal, a length
of about 305 mm (12 in.); upon dismantling the specimens after the test, the bond between the bolt and
the suKounding concrete was noticed to have been
broken. How far this separation extended into the
block, and how it progressed during the loading
increases, were of course, unknown.
soot
fa 1
p
l&OkN
300kN
UOkN
C~mpuled
-
The finite element model was found to be very
sensitive to the bolt embedment and bond length. At
first when full bond was assumed, leaving only the
length of bolt shank passing through the base plate
to be stretched in the model, the computed results
did not agree with the measured results. The model
was therefore modified in all subsequent analysis to
include the entire length of the bolt, to simulate
complete separation of the bolt from the surrounding
concrete. Figure 11 depicts the strain dist~butions at
top and bottom obtained in the present analysis,
together with the corresponding test results, for the
case of the 22 mm (7/8 in.) plate at eccentricities of
rest
c--e
1200 ( b 1
------a
1000
1100
p
-wd
’
0
20
’
Lo
’
64
’
w
’
’
’
’
100 120 uo 160
*
180
’
’
200 ZM
’
ILOkN
-
c--a
300kN
-
.----
LwkN
-
c--.
I
’
’
L ’
’
’
’
L
80 100 120110 160 1e4200 220z&l
210
Distance , mm
Distance, mm
1100
(d 1
I
I\
#’ \
IO00
t
-6041
’
’
0
20 Lo
Oistance , mm
’
60
’
00
’
100
’
’
120 lb0
’
’
160 110
’
’
*
28) 220 ZUJ
Distance, mm
Fig. 11. (a) Top surface strain in 22 mm (7/8 in.) plate loaded at 89 mm (3.5 in.) eccentricity. (b) Bottom
surface strain in 22 mm (7/8 in.) plate loaded at 89 mm (3.5 in.) eccentricity. (c) Top surface strain in
22 mm (7/S in.) plate loaded at I91 mm (7.5 in.) eccentricity. (d) Bottom surface strain in 22 mm (7/S in.)
plate loaded at 191 mm (75 in.) eccentricity.
Analysis of column base plates
89 mm (3.5 in.) and 191 mm (7.5 in.). It can be observed that the results for maximum strains (at the
critical section B) on the side of the load eccentricity
compare reasonably well. Comparison
of strain
distributions in all other cases were analogous. Figure
11 also displays another characteristic difference
between the computed and observed strain distributions. At the column face away from the load
eccentricity, marked C in the figure, the test results
show very small strains, but the finite element results
show peaks as high as or higher than in the rest of
the plate. This phenomenon is more emphasized in
the case of the higher eccentricity [Fig. 1l(c, d)].
The explanation for this drastic discrepancy lies in
the fact that the finite element analysis was a plane
stress formulation parallel to the column web and the
plane of the loading; all effects in the transverse
direction were assumed constant, in a way averaged
for the whole width. Thus, the bolt would certainly
have provided a highly localized constraint to the
plate against being lifted up, but these effects would
have died off toward the edges where the strain gages
were located. The computer model, on the other
hand, ‘spread’ the support reaction across the entire
width, and caused the spurious high bending strains
at the column face. This spreading effect would not
cause as much discrepency elsewhere because the
column itself would tend to distribute the effects
laterally over the entire width; the cantilever projection on the side of the eccentricity would also tend to
bend uniformly as a wide beam in that region.
CONCLUSION
Eccentrically loaded steel base plates were analyzed
by the two-dimensional finite element method for
different plate thicknesses and load eccentricities.
Computed results were evaluated to determine the
influence of various parameters, and also compared
with the limited measured data available.
The agreement between the computed and measured results was not sufficiently good to provide
conclusive proof of the precision of the finite element
model (or the tests). However, the behavior patterns
predicted by computer and observed in the tests were
qualitatively similar, and quantitatively of the same
order of magnitude especially at the critical section B.
The finite element results were consistent in their
predictions, displaying uniform and smooth varichanges, and predictable
ations, proportionate
trends. This further confirmed the validity of the
finite element model and analysis.
General conclusions from the analysis may be
summarized as follows.
(2) Design on the basis of rigid plate behavior
could be over-conservative.
(3) Finite element analysis provides a powerful tool
for the investigation of column base plates to improve
understanding of their behavior and to develop a
better design method.
Acknowledgemenfs-The
authors are grateful to the
National University of Singapore Computer Centre for the
computer facilities and services provided, and to final year
civil engineering students Messrs C. B. Oh, P. Piyush
Kumar, G. Jagathisan and M. Y. Batcha who did the
computer analyses as part of their final year project.
REFERENCES
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
(1) The actual behavior of steel base plates is
considerably different from their assumed behavior as
rigid plates.
223
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781-794 (1978).
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bolted connection behavior. Comput. Struct. 11, 7582
(1980).
N. Krishnamurthy, A fresh look at bolted end-plate
behavior and design. Engng J. AISC 15, 39-49 (1978).
N. Krishnamurthy, FEABOC: finite element analysis
of bolt connection. Proc. Eighth Conf. on Electronic
Computation, Am. Sot. civ. Eigrs, pp. 312-325 (1983).
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