Gusset plate connections under monotonic and cyclic loading

Introduction
Gusset plate connections under monotonic and
cyclic loading
S.S. Walbridge, G.Y. Grondin, and J.J.R. Cheng
Abstract: A numerical investigation of the monotonic and cyclic behaviour of steel gusset plate connections is conducted
using a nonlinear finite element model. Successive versions of the model, which include the effects of framing
member stiffness, nonlinear material behaviour, initial imperfections, and bolt slip, are formulated and validated by
comparison with test results. A parametric study is then conducted to examine the effects of the load sequence and the
interaction between the gusset plate and the brace member under cyclic loading. This investigation demonstrates that
the cyclic behaviour of gusset plate connections can be modelled accurately using a simplified finite element model.
Gusset plate – brace member subassemblies, wherein the gusset plate is designed as the weak element in compression
rather than the brace member, are shown to have stable behaviour under cyclic loading and better energy absorption
characteristics than similar subassemblies with the brace member designed as the weak element in compression.
Key words: steel, connections, gusset plates, cyclic loading, concentric bracing, buckling.
Résumé : Une étude numérique du comportement monotone et cyclique des raccordements par plaques-goussets en
acier est effectuée en utilisant un modèle non linéaire par éléments finis. Des versions successives du modèle, comprenant
les effets de la rigidité des éléments de structure, le comportement non linéaire des matériaux, les imperfections
initiales et le glissement des boulons, sont formulées et validées en les comparant aux résultats des essais. Une étude
paramétrique est ensuite effectuée afin d’examiner les effets de la séquence de chargement et l’interaction entre la
plaque-gousset et l’élément de structure soumis à des charges cycliques. L’étude démontre que le comportement cyclique
des raccordements par plaques-goussets peut être modélisé avec précision en utilisant un modèle simplifié par
éléments finis. Les sous-ensembles éléments de structure – plaques-goussets, dans lesquels les plaques-goussets, plutôt
que les éléments de contreventement, sont conçues comme étant des éléments faibles en compression, présentent un
comportement stable sous des charges cycliques et possèdent de meilleures caractéristiques d’absorption d’énergie que
des sous-ensembles similaires dans lesquels les éléments de contreventement étaient conçus comme étant les éléments
faibles en compression.
Mots clés : acier, raccordements, plaques-goussets, chargement cyclique, contreventement concentrique, flambage.
[Traduit par la Rédaction] Walbridge et al. 995
Because of the complex behaviour of gusset plate connections
in concentrically braced frames (CBFs), the design of
these structural elements has traditionally involved highly
simplified methods (Whitmore 1952; Hardash and Bjorhovde
1985; Thornton 1984). Although these methods have proven
to be adequate, it is believed that the factor of safety associated
with their usage is highly variable (Kulak et al. 1987).
Received 9 March 2004. Revision accepted 12 May 2005.
Published on the NRC Research Press Web site at
http://cjce.nrc.ca on 7 October 2005.
S.S. Walbridge, 1,2 G.Y. Grondin, and J.J.R. Cheng.
Department of Civil and Environmental Engineering,
University of Alberta, Edmonton, AB T6G 2W2, Canada.
Written discussion of this article is welcomed and will be
received by the Editor until 28 February 2006.
1 Corresponding author (e-mail: scott.walbridge@epfl.ch).
2 Present address: ICOM – Steel Structures Laboratory, École
Polytechnique Fédéral de Lausanne, CH – 1015 Lausanne,
Switzerland.
Until recently, the majority of the research on gusset plate
connections has focused on elastic stress distributions or the
inelastic behaviour of gusset plates loaded monotonically in
tension. Relatively little attention has been given to their behaviour
under compressive or cyclic loading. Typically, concentrically
braced frames are designed to dissipate energy
through yielding or buckling of the brace members under
seismic loading. The remaining members and connections
are designed to carry the forces that are present in the structure
at the load level that causes the brace members to yield
or buckle. This approach embodies the philosophy of capacity
design (Redwood and Jain 1992) now implemented in the
latest edition of the steel design standard CAN/CSA S16-01
(CSA 2001). An experimental study of the behaviour of
steel gusset plate connections carried out at the University of
Alberta (Rabinovitch and Cheng 1993) showed that, under
cyclic loading, the tensile capacity of these structural elements
remained stable over the displacement range studied
(up to approximately 10 mm). The same study also showed
that gusset plate post-buckling compressive resistance, although
less than the initial buckling load, also tends to stabilize
after a few cycles. Based on these observations, a design
approach that would take advantage of the energy dissipa-
Can. J. Civ. Eng. 32: 981–995 (2005) doi: 10.1139/L05-045 © 2005 NRC Canada
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982 Can. J. Civ. Eng. Vol. 32, 2005
tion potential of the gusset plate was proposed. This approach
consists of designing the gusset plate as the weak
structural element in the seismic force resisting system
rather than the brace member.
The following presents an analytical investigation of the
behaviour of gusset plates under monotonic tension, monotonic
compression, and cyclic loading. The first part looks at
the effects of gusset plate support conditions (rigid versus
flexible), initial imperfections in the gusset plate, material
yielding, and bolt slip. Gusset plate behaviour and energy
absorption characteristics are then examined under cyclic
loading. In the second part, a brace member is introduced
into the model. With the modified model, a parametric study
is conducted to examine the interaction between the gusset
plate and the brace member and to determine the effect of
load sequence on the behaviour of gusset plate – brace member
subassemblies.
Background
Behaviour under monotonic loading
Early investigations of the behaviour of gusset plates under
monotonic loading in the elastic range were carried out
by Whitmore (1952) on a gusset-plate connection detail
common to Warren truss-type highway bridges, by Irvan
(1957) on a double-gusset-plate Pratt truss connection detail
and by Hardin (1958), Davis (1967), and Varsarelyi (1971).
These investigations are reviewed in detail in (Walbridge et
al. 1998).
Chakrabarti and Bjorhovde (1983) and Hardash and
Bjorhovde (1985) looked at the inelastic behaviour of gusset
plate connections under monotonic loading. From their tests
and those of other investigators, a block shear model was
proposed to predict the ultimate capacity of gusset plate connections
under tensile loading conditions.
Thornton (1984), proposed a lower bound approach for
determining the compressive strength of steel gusset plate
connections, whereby it is assumed that the compressive
force in the gusset plate is carried by an equivalent column
between the end of the brace member and the edges of the
intersecting beam and column members.
The method proposed by Thornton for calculating the elastic
buckling load was subsequently expanded to include inelastic
effects (Williams and Richard 1986). This was
achieved by using Thornton’s effective column approach in
conjunction with column design equations. The buckling load
thus obtained was then limited by the yield load, calculated
using the effective width proposed by Whitmore (1952).
Hu and Cheng (1987) conducted an experimental and analytical
investigation of the buckling behaviour of gusset plate
connections loaded monotonically in compression. Their test
program focused on the effects of gusset plate thickness, geometry,
boundary conditions, eccentricity, and reinforcement.
The work of Hu and Cheng showed that thin gusset
plates tend to buckle at a load much lower than the yield
load predicted using the effective width proposed by
Whitmore (1952). In general, either sway or local buckling
modes were observed depending on the out-of-plane brace
restraint conditions. Further analytical work showed that an
increase in the stiffness of the gusset-to-brace splice plate
would result in an increase in the buckling strength of the
gusset plate up to a splice plate thickness of two to four
times the gusset plate thickness. It was recommended that
gusset plate connections of this type should be designed so
that the distance between the end of the splice plate (or
splice member) and the edges of the horizontal and vertical
framing members is kept to a minimum.
Yam and Cheng (1993) investigated the effects of gusset
plate thickness and size, brace angle, out-of-plane brace restraint
conditions, bending moments in the framing members,
and out-of-plane eccentricity of the brace load on the
behaviour and strength of gusset plates loaded in compression.
The test specimens used in this investigation were
stockier than those of Hu and Cheng, and as a result displayed
more inelastic behaviour. Yam and Cheng observed
that the compressive capacity of the gusset plate specimens
was almost directly proportional to their thickness. The effect
of beam and column bending moments on the compressive
capacity of the test specimens was found to be small for
the bending moment to brace load ratios studied. The
method proposed by Thornton (1984) for predicting the
compressive capacity of gusset plates was found to be conservative.
Some of the test results presented by Yam and
Cheng are used later in this paper. These results are summarized
in Table 1.
Behaviour under cyclic loading
Jain et al. (1978) studied the effect of gusset plate bending
stiffness and brace member length on the cyclic behaviour of
brace members. Although the brace member was the main
subject of the investigation, three different gusset plates
were tested in conjunction with various brace member
lengths. Of the 18 test specimens comprising the test program,
none were designed with a gusset plate yield strength
lower than that of the brace member. Jain et al. concluded
from their investigation that there is no advantage in making
the flexural stiffness of the gusset plate greater than that of
the brace member. However, they did find that an increase in
flexural stiffness of the gusset plate (up to that of the brace
member) generally results in a decrease in the effective
length of the brace member. This, in turn, tends to improve
the cyclic behaviour of the brace member.
Astaneh-Asl et al. (1981) studied the cyclic behaviour of
brace members composed of back-to-back double angles
connected to gusset plates. The focus of their investigation
was also the brace member behaviour. In-plane and out-ofplane
buckling of the brace members was investigated. Significant
deterioration of the gusset plate capacity was observed
in some of their test specimens. To avoid this severe
strength deterioration, a free buckling length of not more
than two times the gusset plate thickness between the end of
the brace member and the line marking the points of attachment
of the gusset plate to the framing members was proposed.
It should be noted that the cyclic loading tests on
which Astaneh-Asl et al. based this proposition were not
conducted on rectangular-corner gusset plates.
Rabinovitch and Cheng (1993) studied the cyclic behaviour
of steel gusset plate connections. In their investigation,
the effects of gusset plate thickness, geometry, edge stiffeners,
and bolt slip were investigated. A series of five tests on
full-scale gusset plate specimens showed that cyclic loading
causes the compressive strength of the gusset plate to drop
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Walbridge et al. 983
Table 1. Partial summary of Yam and Cheng (1993) gusset plate test specimens.
Specimen Plate size (mm)
to a stable post-buckling level after several load cycles, but
it has little effect on the tensile strength. Although the addition
of edge stiffeners was seen to have little effect on the
initial compressive strength, this addition was shown to significantly
improve the post-buckling compressive strength as
well as the energy dissipation characteristics of the gusset
plates tested. As in the tests by Hu and Cheng (1987), and
Yam and Cheng (1993), the tests by Rabinovitch and Cheng
used test specimens with strong splice and brace members.
The behaviour of the gusset plate was investigated without
considering the interaction of the gusset plate and the brace
member. Test results from Rabinovitch and Cheng will be
used to validate the finite element model developed for the
parametric study presented herein. These results are summarized
in Table 2 and Fig. 1.
Finite element modelling of gusset plates
To predict gusset plate behaviour under monotonic and
cyclic loading, a model was developed using the finite element
program ABAQUS (1995). The model was validated
with data from the experimental investigations of Yam and
Cheng (1993) for gusset plates loaded monotonically in
compression, and Rabinovitch and Cheng (1993) for gusset
plates under cyclic loading. The following procedure was
adopted to develop the model:
(1) A study of inelastic tensile gusset plate behaviour was
performed to investigate the effects of mesh refinement,
strain hardening, and framing member stiffness. The modelled
gusset plates were all loaded beyond their peak tensile
capacities. Since tensile test results were not available, peak
tensile loads from the cyclic tests conducted by Rabinovitch
and Cheng (1993) were used for validation purposes at this
stage.
(2) Initial imperfections were subsequently incorporated
into the model developed in step (1). The modified model
was then used to investigate gusset plate response under
monotonic compressive loading with different imperfection
shapes and magnitudes. The results of this investigation
were compared with some of the test results of Yam and
Cheng (1993).
(3) Finally, the finite element model developed in step
(2) was used to simulate gusset plate behaviour under cyclic
loading. At this stage, a fastener model was developed to
model the bolt slip that was observed for some of the specimens
tested by Rabinovitch and Cheng (1993) (Fig. 1). The
results of this investigation were compared with the test results
from this same reference.
The following presents the details and results of the
above-mentioned process.
Material properties Performance
Young’s
modulus (MPa)
Yield strength
(MPa)
Ultimate
strength (MPa)
Monotonic tension loading
Ultimate tensile
load (kN)
GP1 500 × 400 × 13.3 207 600 295 501 — 1956
GP2 500 × 400 × 9.8 210 200 305 465 — 1356
GP3 500 × 400 × 6.5 196 000 275 467 — 742
Ultimate compressive
load (kN)
Modelling
Four finite element meshes were used to model specimen
A2 from Rabinovitch and Cheng (1993) (Fig. 1 and Table
2), each with an increasing level of refinement.
ABAQUS shell element S4R was used to model the gusset
plate and the T-shaped splice members. Two different material
models were investigated: an elasto-plastic model and an
isotropic strain-hardening model. The adopted material properties
were based on the materials test results reported by
Rabinovitch and Cheng (1993), summarized in Table 2. The
particulars of the two models can be summarized as follows:
The elasto-plastic model assumes linear elastic behaviour
(with a Young’s modulus of 206 000 MPa) until the yield
stress, after which perfect plastic behaviour is assumed.
The isotropic strain hardening model assumes linear elastic
behaviour (again, with a Young’s modulus of
206 000 MPa) until the yield stress. The strain hardening
curve is then defined in terms of true stress versus plastic
strain. The strain hardening curve assumes perfect plastic
behaviour until a plastic strain of 0.025. The true stress
then increases to the ultimate true stress at a plastic strain
of 0.18 (Walbridge et al. 1998).
The bolts were modelled as rigid links between the gusset
plate and the splice members. The displacement and rotation
of the nodes along the connected edges of the gusset plate
were fully restrained, thereby simulating rigid framing members.
The models were loaded by displacing the nodes along
the loaded edge of each splice member (Fig. 2).
The four meshes are shown in Fig. 3. They are numbered
from 1 to 4, and contain 206, 336, 454, and 596 elements,
respectively. The mesh refinement study indicated that with
a mesh consisting of 454 shell elements in the gusset plate,
convergence of the load-displacement behaviour was
achieved (Walbridge et al. 1998). This mesh was thus
adopted for subsequent analyses.
A more realistic model of the actual boundary (support)
conditions was obtained by modelling the beam and column
(framing members), as shown in Fig. 4. The beam and column
were modelled using ABAQUS S4R shell elements
with linear elastic material properties. A more realistic bolt
model was developed using ABAQUS SPRING2 elements.
The SPRING2 element links a global degree of freedom at
one node with a global degree of freedom at another node.
For this model, two springs were required (one for each inplane
displacement degree of freedom) to link each of the
two splice members to the gusset plate at each bolt location.
The stiffness assigned to the SPRING2 elements for this step
was taken from a double shear load test presented by
Wallaert and Fisher (1965). The stiffness value was taken as
the initial slope of the load versus displacement curve for a
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984 Can. J. Civ. Eng. Vol. 32, 2005
Table 2. Summary of Rabinovitch and Cheng (1993) gusset plate test specimens.
Specimen Plate size (mm)
Material properties Performance
Young’s
modulus (MPa)
Yield strength
(MPa)
Ultimate
strength (MPa)
Ultimate tensile
load (kN)
A1 550 × 450 × 9.32 206 000 449 537 1794 1682
A2 550 × 450 × 6.18 206 000 443 530 1340 1128
A3 550 × 450 × 9.32* 206 000 449 537 1884 2004
A4 550 × 450 × 6.18* 206 000 443 530 1265 1149
*Stiffened edges.
Ultimate compressive
load (kN)
Fig. 1. Test frame and axial load versus displacement hysteresis plots for Rabinovitch and Cheng (1993) specimens: (a) A1, (b) A2,
(c) A3, (d) A4.
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Walbridge et al. 985
Fig. 2. Gusset plate connection model for Rabinovitch and
Cheng (1993) specimens.
Fig. 3. Linear elastic mesh refinement study: (a) mesh 1 – 206
elements, (b) mesh 2 – 336 elements, (c) mesh 3 – 454 elements,
and (d) mesh 4 – 596 elements.
typical A490 bolt. This value was determined to be
253 kN/mm.
Results
The effect of strain hardening on the load versus displacement
behaviour in monotonic tension was an increase in the
ultimate tensile capacity (Fig. 5). It was found, however, that
the elasto-plastic material model resulted in a better prediction
of the test specimen behaviour. The reasons that the
models with strain hardening tended to overestimate the ultimate
load are believed to be twofold. Firstly, bolt holes were
not incorporated in the gusset plate model. The resulting excess
material along the yield surface of the gusset plate (assuming
a block shear failure mode) is believed to explain to
a large extent the difference between the test results and the
predictions of the gusset plate model with strain hardening.
Another possible explanation is that no attempt was made to
model local phenomena affecting the gusset plate behaviour
near the bolt holes, such as the tearing observed in some of
the tests (Rabinovitch and Cheng 1993). As the goal at this
stage was to accurately predict the observed load versus displacement
behaviour of the gusset plate with a model suit-
Fig. 4. Finite element model of gusset plate with flexible framing
members.
Fig. 5. Effect of material model and boundary conditions on
monotonic tension behaviour for Rabinovitch and Cheng (1993)
specimen A2.
able for application in large parametric studies such as the
one presented herein, it was decided to retain the simplified
idealization of the gusset plate along with the elasto-plastic
material model for the remainder of the study.
As shown in Fig. 5, the effect of incorporating realistic,
flexible boundary conditions (as opposed to rigid boundary
conditions) was a slight reduction in the stiffness of the gusset
plate in the elastic range and a decrease in the ultimate
tensile capacity. The use of an elastic fastener model did not
significantly affect the predicted load versus displacement
behaviour and had little effect on the ultimate load. The
rigid bolt model and flexible boundary conditions were
therefore used in subsequent analyses.
Table 3 summarizes the predicted and actual tensile capacities
for the test specimens of Rabinovitch and Cheng
(1993), attained with finite element models that incorporated
the elasto-plastic material model, the flexible boundary conditions,
and the rigid bolt model. Good agreement between
the test results and the predicted capacity can be seen in this
table, with test-to-predicted ratios varying from 0.93 to 1.08.
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986 Can. J. Civ. Eng. Vol. 32, 2005
Table 3. Comparison of monotonic analysis with Rabinovitch and Cheng (1993) test results.
Tension loading Compression loading
Specimen
Test capacity
(kN)
Predicted
capacity (kN) Test/predicted Specimen
Monotonic compression loading
Modelling
To analyze the behaviour of gusset plates loaded monotonically
in compression, initial imperfections were introduced
into the model. Since initial imperfections were not
measured for any of the specimens tested by Yam and Cheng
(1993) or by Rabinovitch and Cheng (1993), a number of
initial imperfection shapes and magnitudes were studied.
Three initial imperfection shapes were investigated, namely
a full sine wave, a quarter sine wave, and a shape con-
Test capacity
(kN)
Predicted
capacity (kN) Test/predicted
A1 1794 1923 0.93 GP1 1956 2073 0.94
A2 1340 1245 1.08 GP2 1356 1342 1.01
A3 1884 1928 0.98 GP3 742 711 1.04
A4 1265 1248 1.01
Fig. 6. Initial imperfection shapes: (a) quarter sine wave and (b) buckled configuration.
Fig. 7. Effect of initial imperfection magnitude on gusset plate
behaviour in compression.
Fig. 8. Finite element model of Rabinovitch and Cheng (1993)
stiffened gusset plate specimens A3 and A4.
structed by scaling the buckled configuration of the gusset
plate by an appropriate constant to obtain the desired imperfection
magnitude (Fig. 6). The three initial imperfection
magnitudes used for this comparison were 0.05, 0.5, and
5 mm. In this study the initial imperfection magnitude was
taken as the maximum out-of-plane imperfection assigned to
any of the gusset plate nodes.
The manner in which the clamping of the gusset plate by
the splice members was modelled was found to be important
(Walbridge et al. 1998). The best results were achieved with
a perfect clamping model, that is, with the out-of-plane degrees
of freedom of each splice member node overlapping
the gusset plate linked to that of the corresponding node on
the gusset plate.
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Walbridge et al. 987
Fig. 9. Springs used to model bolt slip: (a) spring 1, (b) spring 2, and (c) spring 3..
Fig. 10. Spring activation sequence for bolt slip model: (a) step1,(b) step2,(c) step3,and(d) step 4. Dashed lines represent springs
that are “active” and solid lines represent the “effective” spring.
Results
On one hand, it was found that initial imperfection magnitude
has a significant effect on the buckling capacity of the
gusset plate; in general, the larger the initial imperfection,
the lower the buckling capacity. On the other hand, it was
found that the shape of the initial imperfection was much
less critical. The quarter sine wave shape was used in subsequent
analyses.
The effect of initial imperfection magnitude was studied
using a model of specimen GP3 from Yam and Cheng
(1993). Figure 7 shows axial load versus displacement
curves for models of the gusset plate connection with three
different initial imperfection magnitudes. The figure
indicates that an initial imperfection magnitude of 0.5 mm
overestimates the capacity of the assembly, whereas a
5.0 mm imperfection results in an underestimation of the test
result.
Based on the results of the analysis described above, new
models of specimens GP1, GP2, and GP3 from Yam and
Cheng (1993) were constructed. A quarter sine wave initial
imperfection with a magnitude of 2.0 mm was used, and
flexible beam and column members along the gusset plate
boundaries were included in the model. Table 3 shows a
summary of the predicted and actual capacity in compres-
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988 Can. J. Civ. Eng. Vol. 32, 2005
Fig. 11. Axial load versus displacement hysteresis for Rabinovitch and Cheng (1993) specimens: (a) A1, (b) A2,(c) A3, and (d) A4.
Fig. 12. Comparison of energy dissipation for Rabinovitch and
Cheng (1993) specimens: (a) A2and(b) A4.
sion for specimens GP1, GP2, and GP3 from Yam and
Cheng (1993). Good agreement between the test results and
the predicted capacity can be seen in this table with test-topredicted
ratios varying from 0.94 to 1.04.
Cyclic loading
Modelling
Finite element models of specimens A1, A2, A3, and A4
from Rabinovitch and Cheng (1993) were constructed based
on the results of the monotonic loading studies, and loaded
in accordance with the actual load histories of the tests
(Rabinovitch and Cheng 1993). The elasto-plastic material
model was adopted, along with flexible beam and column
members along the gusset plate boundaries. Edge stiffeners
were added to the models of specimens A3 and A4 (Fig. 8).
A quarter sine wave initial imperfection with a magnitude of
1.0 mm was used, as this value gave slightly better results
than the 2.0 mm imperfection for the Rabinovitch and
Cheng tests (Walbridge et al. 1998).
To model specimens A1 and A3 from Rabinovitch and
Cheng (1993), bolt slip had to be considered. Bolt slip was
incorporated into the model using ABAQUS SPRING2 elements,
with nonlinear load versus displacement behaviour.
Although this element can be assigned a nonlinear load versus
displacement relationship, it cannot be used directly to
model inelastic behaviour. Thus, to model the cyclic bolt slip
that took place during the testing of these specimens, the superposition
of several spring elements was required. Figure 9
illustrates the load versus displacement relationship for the
three spring elements that were required to model the bolt
slip. Figure 10 illustrates the spring activation and deactivation
sequence required in the analysis to achieve the
desired load versus displacement relationship. Specifically,
in step 1, spring 1 is active (and remains active for all subsequent
steps); in step 2, spring 2 is activated; and in step 3,
spring 2 is deactivated, and spring 3 is activated. Step 4, in
Fig. 10, shows the resulting load versus displacement behav-
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Walbridge et al. 989
Fig. 13. Finite element model of the gusset plate – brace member subassembly.
iour of the bolt slip model for an entire cycle. Some iteration
was necessary in the selection of appropriate values for the
bolt slip load and the assumed amount of slip. Due to the tediousness
of the spring superposition procedure (which required
stopping the analysis and modifying the model after
each half cycle), only a few load cycles were modelled of
the tests on specimens A1 and A3.
Results
Figure 11 shows axial load versus displacement plots for
specimens A1 to A4 from Rabinovitch and Cheng (1993),
along with the hysteresis plots predicted using the corresponding
finite element models. Comparing the hysteresis
plots for specimen A2 and A4, for which several load cycles
were applied in the analysis, it can be seen that the finite element
model predicts the buckling load and subsequent decay
of the post-buckling load resistance under cyclic loading
quite well. The load resistance of the gusset plate in tension
is also matched closely by the numerical model.
A comparison of specimens A3 and A4, which had edge
stiffeners, with A1 and A2 (with no edge stiffener) indicates
that the effect of gusset plate edge stiffeners appears to be a
reduction in the rate of decay of the post-buckling load. This
corresponds well with the test results. Looking at the hysteresis
plots for specimens A1 and A3, it can be seen that using
the bolt slip model shown in Figs. 9 and 10, the finite element
model is also able to predict the behaviour observed in
these tests reasonably well considering the highly random
nature of the bolt-slip phenomenon and the relatively simplistic
nature of the modelling approach employed.
A comparison of cumulative energy dissipation for specimens
A2 and A4 is presented in Fig. 12. The energy dissipated
was determined by calculating the area enclosed by
the hysteresis loop for each cycle. The amount of dissipated
energy predicted by the finite element model is slightly
higher than the test values for both specimens. This is likely
because the elasto-plastic material model was used in the cyclic
loading study. Although a study of the effect of strain
hardening on buckling load showed only a small difference
between material models, the effect of this parameter on the
stiffness of the model once yielding had started was found to
be significant. For specimen A2 especially, it was found that
larger displacements had to be imposed on the compression
side to cause buckling to occur as observed during the test.
This meant that more energy was dissipated in this portion
of each cycle. The cumulative energy dissipation curves for
specimen A2 indicate that most of the difference between
the two curves occurs in cycles 3, 4, and 5. In these cycles,
higher displacements had to be imposed on the compression
side to cause the gusset plate model to buckle during the
same cycle as the test specimen. In subsequent cycles, the
energy dissipated (per cycle) matches quite well.
Parametric study
For the parametric study presented in this section, the gusset
plate model validated in the previous sections was modified
to include a brace member. The model of the gusset
plate and brace member is presented in Fig. 13. The objective
of the parametric study was to expand the experimental
investigation performed by Yam and Cheng (1993) and
Rabinovitch and Cheng (1993) to include the effect of gusset
plate – brace member interaction and load sequence.
Using the modified model, three different load sequences
were investigated. The first, designated LS1, consisted of a
series of cycles of increasing displacement amplitude, starting
with a tension cycle. The increase in the displacement
amplitude with each cycle was taken as the yield displacement,
δy, obtained from a monotonic load analysis of the
gusset plate (based on the recommendations of the Applied
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990 Can. J. Civ. Eng. Vol. 32, 2005
Fig. 14. Summary of load sequences: (a) LS1 – tension first,
(b) LS2 – compression first, and (c) LS3 – tension first with
three cycles at each increment.
Technology Council (Applied Technology Council 1992)
ATC-24 guideline). A preliminary analysis showed that the
yield displacement, δy, was not significantly affected by gusset
plate thickness (Walbridge et al. 1998). Therefore, the
same yield displacement of 2.5 mm was used for every
model. Increments of displacement were added until the total
deflection on the gusset plate was about six times the
yield displacement, that is, 15 mm. This load sequence is illustrated
in Fig. 14a. The second load sequence, LS2, was
similar to LS1 except that it started with a compression cycle
rather than a tension cycle. This load sequence is illustrated
in Fig. 14b. The third load sequence strictly follows
the additional ATC-24 guideline recommendation that three
load cycles be imposed for each load block. This load sequence
is illustrated in Fig. 14c.
Following the load sequence study, gusset plate – brace
member subassemblies were conceived to investigate the following
four types of behaviour:
Brace member yielding in tension before yielding of the
gusset plate (YBT).
Gusset plate yielding in tension before yielding of the
brace member (YGT).
Buckling of the brace member before buckling of the gusset
plate (BBC).
Buckling of the gusset plate before buckling of the brace
member (BGC).
The investigated models covered the above four cases for
three different gusset plate thicknesses: 6, 9, and 12 mm
(corresponding to models GP1, GP2, and GP3, respectively).
For each thickness, two brace sections were selected,
namely, one that would result in failure due to yielding of
the gusset plate in tension (YGT) and one that would result
in failure due to yielding of the brace member (YBT). The
block shear and net section equations in CAN/CSA S16-01
(CSA 2001) were used for predicting the gusset plate and
brace member yield strengths for this step. For each brace
member section, two brace lengths were modeled, namely,
one corresponding to a slenderness, kL/r, of 50 and one corresponding
to a slenderness of 100. These kL/r values were
computed assuming an effective length factor, k, of 1.0 (i.e.,
pin-pin). For most of the gusset plate – brace member combinations
studied, the shorter brace member had a predicted
capacity in compression higher than that of the gusset plate,
whereas the longer brace member had a predicted capacity
lower than that of the gusset plate. The method proposed by
Thornton (1984) and the CAN/CSA S16-01 (CSA 2001) column
curves were used to predict the gusset plate and brace
member buckling loads in this step. The parametric study
was conducted with 450 mm × 550 mm gusset plates, similar
in geometry to the specimens tested by Rabinovitch and
Cheng (1993). Table 4 summarizes the brace member – gusset
plate combinations investigated.
Results of the parametric study
Under cyclic loading, the capacity in the tension cycle
was limited either by yielding of the brace member or the
gusset plate, depending on the brace member – gusset plate
combination under investigation. Yielding of the gusset plate
was determined from a comparison of two plots: applied
force versus gusset plate displacement and applied force versus
total displacement, which includes the gusset plate and
brace member displacements. If yielding of the gusset plate
was taking place, both plots would show nonlinear behaviour.
If only the brace member was yielding, the gusset plate
displacement plot would not show any plastic displacement.
In the compression cycle, the load carrying capacity was
limited by either buckling of the gusset plate, as shown in
Fig. 15, or buckling of the brace member about its weak
axis, i.e., out of the plane of the gusset plate, as shown in
Fig. 16. The last two columns of Table 4 summarize the
modes of failure, that is, the load limiting mechanisms in
tension and compression for each brace member – gusset
plate combination. As detailed in Walbridge et al. (1998),
the predicted and analytically determined failure modes
were identical for all the brace member – gusset plate combinations
investigated.
Effect of load sequence
Results of the load sequence investigation are presented in
Fig. 17 for model GP2B5, which consists of gusset plate
GP2 and brace member B5 as per Table 4. The displacement
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Walbridge et al. 991
Table 4. Gusset plate – brace member subassembly models.
Brace
type
Gusset
plate type
Fig. 15. Buckling of gusset plate (BGC).
Brace
section
plotted in Fig. 17 is the total axial displacement measured at
the location indicated in Fig. 13. Specimen GP2B5 was designed
to have the brace yield in tension before the gusset
plate and the gusset plate buckle in compression before
buckling of the brace member. A comparison of Fig. 17a
and 17b indicates that there is little difference in behaviour
between load sequence LS1 and load sequence LS2. The
gusset plate – brace member assembly behaviour under load
sequence LS3 is similar to that observed under load sequence
LS1 except for the fact that the initial stiffness of the
assembly in tension tends to be smaller for load sequence
LS3. The hysteresis curves for load sequence LS3 indicate
that the capacity of the assembly deteriorates slightly with
Failure mode
Brace
slenderness, kL/r Tension Compression
B1 GP1 W200 × 21 50 YBT BGC
B2 GP1 W200 × 21 100 YBT BBC
B3 GP1 W200 × 27 50 YGT BGC
B4 GP1 W200 × 27 100 YGT BBC
B5 GP2 W200 × 27 50 YBT BBC
B6 GP2 W200 × 27 100 YBT BBC
B7 GP2 W200 × 42 50 YGT BGC
B8 GP2 W200 × 42 100 YGT BBC
B9 GP3 W200 × 31 50 YBT BBC
B10 GP3 W200 × 31 100 YBT BBC
B11 GP3 W200 × 59 50 YGT BGC
B12 GP3 W200 × 59 100 YGT BBC
Note: YGT, yielding of gusset in tension; YBT, yielding of brace in tension; BGC, buckling of gusset in compression;
BBC, buckling of brace in compression.
increasing number of cycles at each loading block. The differences
between load sequence LS3 and load sequence LS1
are believed to be very small. For this reason, the less time
consuming load sequence LS1 was adopted for the remainder
of the study.
Effect of load limiting mechanism
The load limiting mechanism in the tension cycle was either
yielding of the brace member (YBT) or yielding of the
gusset plate (YGT). In the compression cycle, the load limiting
mechanism was either buckling of the brace member
(BBC) or buckling of the gusset plate (BGC). Figure 18
shows the difference in behaviour of the gusset plate – brace
© 2005 NRC Canada

992 Can. J. Civ. Eng. Vol. 32, 2005
Fig. 16. Buckling of brace member (BBT).
member assembly for different load limiting mechanisms for
a 6 mm gusset plate (GP1). Figure 18a presents the behaviour
when the load in tension is limited by yielding of the
gusset plate and the load in compression is limited by buckling
of the gusset plate (YGT/BGC). Figure 18b presents the
behaviour when the limiting condition in tension is yielding
of the gusset plate and the limiting condition in compression
is buckling of the brace member (YGT/BBC). A comparison
of Fig. 18a with Fig. 18b shows that buckling of the brace
member as a limiting condition in the compression range results
in a greater reduction in compression capacity under
cyclic loading and a deterioration of the load carrying capacity
in tension. The same observation is made when Fig. 18b
is compared with Fig. 18c, which represents the behaviour
when the limiting condition in tension is yielding of the
brace member and the limiting condition in compression is
buckling of the gusset plate (YBT/BGC).
The behaviour of the assembly when the limiting mechanism
in tension is yielding of the brace member and the limiting
mechanism in compression is buckling of the brace
member (YBT/BBC) is illustrated in Fig. 17a. As observed
in Figs. 17a and 18b, the reduction in tension stiffness at
zero load can be quite significant when the compression
capacity is limited by buckling of the bracing member (BBC).
This reduction in tension stiffness was not observed when
buckling of the gusset plate limited the compression capacity.
A comparison of the cumulative energy absorbed for the
various gusset plate GP1 subassemblies is presented in
Fig. 19. The figure shows clearly that the specimens for
which the load limiting mechanism in compression is buckling
of the gusset plate (GP1B1 and GP1B3) absorbed significantly
more energy (about twice) than the specimens for
which the load limiting mechanism in compression is buckling
of the bracing member (GP1B2 and GP1B4). Whereas,
the load limiting mechanism in tension does not have a significant
effect on the energy absorption capacity (the tension
capacity of GP1B1 is limited by yielding of the bracing
member, whereas the tension capacity of GP1B3 is limited
by yielding of the gusset plate and both (GP1B1 and
GP1B3) dissipated about the same amount of energy).
Effect of gusset plate thickness
Three different gusset plate thicknesses were investigated,
namely 6, 9, and 12 mm. Figure 20 shows the difference in
behaviour for gusset plate – brace member subassemblies
proportioned to have a tension capacity limited by yielding
of the gusset plate and a compression capacity limited by
buckling of the gusset plate. Although all plate thicknesses
display good behaviour, the thicker gusset plate shows fuller
hysteresis loops. Finally, dashed lines are shown in Fig. 20,
corresponding to the monotonic loading curves for each of
the specimens. In all cases, the monotonic loading curves
were found to represent well an envelope of the hysteresis
under cyclic loading.
Summary and conclusions
Finite element models of gusset plates were developed to
predict results of tests on gusset plates under monotonic
compression, tension, and cyclic loading. The models were
validated for monotonic tension and compression and cyclic
loading. The effects of material inelasticity, large displacements,
and initial imperfections were included in the models.
The load resistance of the gusset plates in tension was
predicted closely by the numerical models. The models were
also seen to predict well the buckling load as well as the
subsequent decay of the compressive post-buckling load resistance
under cyclic loading.
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Walbridge et al. 993
Fig. 17. Effect of load sequence on cyclic response of gusset
plate – brace member assembly: (a) LS1, (b) LS2, and (c) LS3.
A parametric study of the cyclic behaviour of gusset plate
– brace member subassemblies was performed. A series of
parameters were investigated, which included: the effect of
gusset plate – brace member interaction, the effect of loading
sequence and the effect of gusset plate thickness. The
weak gusset plate – strong brace member concept was examined
in the study as an alternative design approach for concentrically
braced frames. The following conclusions can be
drawn from this parametric study:
Load sequence does not have a significant effect on the
cyclic behaviour of the gusset plate – brace member subassemblies.
Limiting the capacity in tension either by brace member
yielding or by gusset plate yielding does not result in a sig-
Fig. 18. Effect of load limiting mechanism on hysteresis curves:
(a) tension and compression – gusset (YGT/BGC), (b) tension –
gusset/compression – brace (YGT/BBC), (c) tension – brace/
compression – gusset (YBT/BGC).
nificant change in behaviour over the displacement range
studied.
Buckling of the gusset plate results in only a small reduction
in capacity and a very stable cyclic behaviour.
Failure in compression by buckling of the gusset plate
rather than buckling of the brace member results in better
energy absorption characteristics.
Although all plate thickness display good behaviour, the
thicker gusset plate shows fuller hysteresis loops
Monotonic load versus displacement plots tended to delineate
the cyclic load versus displacement hysteresis envelope.
© 2005 NRC Canada

994 Can. J. Civ. Eng. Vol. 32, 2005
Fig. 19. Effect of load limiting mechanism on cumulative energy absorption. B1, B3, gusset plate buckling (GBC); B2, B4, brace
buckling (BBC).
Fig. 20. Effect of gusset plate thickness: (a) 6m,(b) 9mm,and(c) 12mm.
In general, hysteresis plots for the weak gusset plate –
strong brace member models exhibited less pinching and
sustained higher post-buckling compressive loads than the
conventionally designed subassemblies.
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© 2005 NRC Canada