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Plastic Buckling of Columns:
Development of a Simplified Model of Analysis
Filipe Pereira
Instituto Superior Técnico, October, 2016
Abstract
Column buckling is a phenomenon usually associated with slender columns, that buckle in the elastic range. However,
columns made of elasto‑plastic materials of lower slenderness may also buckle in the plastic range. For these columns, the
maximum load bearing capacity is greater than the buckling load.
The difficulty behind the analysis of the column buckling in the plastic range lies in the fact that, right after buckling occurs,
fibres in the convex side of the column start to elastically unload, so, for every step along the analysis of the post‑buckling
behaviour of the columns, each cross‑section point may be in one of three states: plastic loading in compression, elastic
unloading or plastic loading in tension.
This dissertation aims to study the plastic buckling of columns. A simplified model is proposed based on the assumption
that exists a direct relation between the curvature and the transverse displacement of the mid-span section, making it possible
to obtain the complete post-buckling equilibrium path. This simplified model is then generalized, in order to allow for a
more accurate description of the columns, by analysing the equilibrium of multiple cross‑sections. These models are then
evaluated by comparing their results to those of finite element models. At last, the proposed model is used to study a set of
columns of different geometrical and material characteristics.
1 Introduction In structures, columns are elements which are designed to
transmit loads, both vertical and horizontal, to the
foundations. In most structures, during its life cycle, the
vertical loads are predominant and, consequently, the
study of columns under compressive loads has been a
subject of interest for centuries.
The simplicity associated with the geometrically linear
analysis of these elements is gone in the problem when the
column’s buckling has to be studied. Having to combine
the geometric and material non-linearities when studying
the buckling behaviour of columns with elasto-plastic
materials makes this problem much harder to analyse.
The first approach to describe the phenomenon of buckling
was made by Euler, when he determined the critical load
which causes an elastic column to buckle under a
compressive force
𝑃𝑐𝑟 = 𝑃𝐸 =
𝜋2𝐸𝐼
𝐿𝑒2
( 1 )
where 𝐸 is the elastic modulus of the material, 𝐼 is the
moment of inertia of the cross-section about the flexural
axis and 𝐿𝑒 is the effective length of the columns, which is
simply the length of the column for simply supported
columns.
Studies made since the 19th century, and throughout the
last century have concluded that the buckling load of
columns in the plastic range may be determined by the
same expression, but in which the elastic modulus is
replaced by a tangent modulus 𝐸𝑡, which is the modulus at
which the stress of points on the yield surface evolve in the
plastic range.
𝑃𝑐𝑟 = 𝑃𝑡 =
𝜋2𝐸𝑡𝐼
𝐿𝑒2
( 2 )
This expression was first proposed by Engesser in 1889
[1], but when he first did it, he made so under wrong
assumptions, by not taking into account the elastic
unloading that happens in the convex side of the column.
As a result, in 1895 [2] he corrected his original theory by
creating the reduced modulus theory
𝑃𝑐𝑟 = 𝑃𝑅 =
𝜋2𝐸𝑅𝐼
𝐿𝑒2
( 3 )
The reduced modulus, 𝐸𝑅 takes into account the existence
of both plastic and elastic domains in the cross-section,
therefore the condition 𝐸𝑡 ≤ 𝐸𝑅 ≤ 𝐸 is always true. Von
Karman supported this new perspective and determined
expressions for the reduced modulus [3].
However, later studies established that 𝑃𝑡 does, indeed,
predict the buckling load for the situation of plastic
buckling, because at the instant when buckling occurs,
every point in the cross-section is under plastic
compression, and thus have their behaviour controlled by
2
𝐸𝑡. One of the authors who contributed to this conclusion,
Shanley [4], also determined 𝑃𝑡 to be the lowest possible
buckling load, since loads ranging from 𝑃𝑡 to 𝑃𝑐𝑟 are all
possible buckling loads. This load, 𝑃𝑡, is the one which
conditions the behaviour of the imperfect columns
The post buckling behaviour of the columns was, however,
hard to determine. In the decade of 1970, Hutchinson [5],
with the study of his continuous model, explored the
importance of the elastic unloading in the post-buckling
behaviour of the columns. Nevertheless, the post-buckling
equilibrium path wasn’t accurately reproduced, often
because of incorrect modelling of tension yielding at the
convex side.
This dissertation aims to develop a simplified numerical
model of analysis of the plastic buckling, adopting a
simplified hypothesis which makes it possible to establish
the equilibrium of the column mid-span cross-section and
get the complete post-buckling behaviour of the columns,
correctly taking into account the effects of plasticity,
elastic unloading and tension yielding. It also presents a
more sophisticated approach of the model, by generalizing
it in order to model the behaviour of multiple sections
throughout the columns length. These models are then
validated and used in parametric studies.
The models study simply supported columns, and elasto-
plastic materials with plastic hardening.
The model is implemented using MATLAB.
2 Plastic Buckling
2.1 Governing Equations
A deformed column of length equal to 𝐿, under a
compressive load 𝑃, and in a generic post-buckled
position, is represented in Figure 1.
At any time of the analysis, the equilibrium between the
internal and external forces must be satisfied,
𝐹𝑖𝑛𝑡 − 𝐹𝑒𝑥𝑡
= 0 ( 4 )
Figure 1: Deformed column
For situations like the one shown in the previous figure,
the external forces can be represented as
�� = −𝑃
�� = 𝑃(𝑤 + 𝑤0) ( 5 )
Where 𝑤 is the transverse displacement and 𝑤0 represents
the geometrical imperfections which may be present.
On the other hand, the internal forces are entirely
dependent on the stresses, 𝜎33, which develop in the
cross-section as a result of its deformation profile,
𝜀33(𝑥2) = 𝜀(𝑥2) = 𝜀𝑔 + 𝜒 𝑥2 ( 6 )
where 𝜀𝑔 is the axial strain of the cross-section and 𝜒 is its
curvature.
So, for the elastic case, the stresses in a cross-section are
written as
𝜎33 = 𝐸 𝜀33 ( 7 )
For the plastic case, this expression becomes, for each
point in the cross-section
𝜎 = 𝐸 (𝜀 − 𝜀𝑝) ( 8 )
where 𝜀𝑝 the plastic strain. The relation between this
variable and 𝜀 depends on the loading history of the
cross-section points.
In any case, the internal forces of a cross-section may be
written as
𝑁 = ∫ 𝜎33 𝑑𝐴
𝐴
𝑀 = ∫ 𝜎33 𝑥2 𝑑𝐴𝐴
( 9 )
This allows for the equilibrium to be written as
𝑁 − �� = ∫ 𝜎33 𝑑𝐴
𝐴
+ 𝑃 = 0
𝑀 − �� = ∫ 𝜎33 𝑥2 𝑑𝐴𝐴
− 𝑃(𝑤+ 𝑤0) = 0
( 10 )
2.2 Elastic Behaviour
Taking in consideration the equation of the equilibrium of
moments, it is possible to determine the expression which
gives the post-buckling displacements of a column with
elastic behaviour.
From 𝑀 − �� = 0 we can get, for a perfect column
𝐸𝐼 𝜒(𝑥3) − 𝑃 𝑤(𝑥3) = 0 ( 11 )
From the hypothesis of small displacements and Euler-
Bernoulli’s theory of thin beams, the curvature is equal to
𝜒 = −
𝑑2𝑤(𝑥3)
𝑑𝑥32 ( 12 )
so ( 11 ) becomes
−𝐸𝐼
𝑑2𝑤(𝑥3)
𝑑𝑥32 − 𝑃 𝑤(𝑥3) = 0 ( 13 )
The solution of this equation [6] is the one which allows
the description of the column buckling displacements. It
results in
3
𝑤(𝑥3) = B sin(𝜋𝑥3
𝐿) ( 14 )
2.3 Plastic Behaviour
The expression ( 14 ) is no longer valid when the column
starts to experience plasticity. The column is no longer
responding according to the elastic modulus, 𝐸, in every
point of its cross-sections and as a result, the way it
deforms will not be simply described by a sine wave.
Besides the correct formulation of the transversal
displacements, perhaps the biggest challenge, as it was
previously stated, is the correct evaluation of the stresses
along points of the cross-section.
The evaluation of ( 8 ) requires the definition of 𝜀𝑝 at each
step of the post-buckling analysis. The plastic strain
evolves for points being loaded and in yield. For these
points, it can be written
𝑓 = |𝜎| − 𝜎𝑦(𝜀��) = 0 ( 15 )
where 𝜎𝑦(𝜀��) denotes the strain hardening law, which is the
size of the yielding surface and is dependent of 𝜀��, the
accumulated equivalent plastic strain and the constitutive
law being applied. Note that the variation of 𝜀�� and 𝜀𝑝 relate
by 𝜀�� = |𝜀��|.
An incremental evaluation allows to determine the change
of all these properties. Assume 𝜀𝑝0 and 𝜀𝑝
0 at the beginning
of an increment. If
𝑓𝑡𝑟𝑖𝑎𝑙 = 𝐸(𝜀 − 𝜀𝑝0)×𝑛 − 𝜎𝑦(𝜀 𝑝
0 ) > 0 ( 16 )
the values of 𝜀𝑝0 and 𝜀𝑝
0 must be updated according to
𝜀𝑝𝑛 = 𝜀𝑝
0 + Δ𝜀��×𝑛
𝜀𝑝𝑛 = 𝜀𝑝
0 + Δ𝜀�� ( 17 )
For the point to be within the yielding surface this leads to
Δ𝜀�� =
𝑓𝑡𝑟𝑖𝑎𝑙
𝐸 + 𝐻 ( 18 )
and
𝑑𝜎
𝑑𝜀= 𝐸 −
𝐸2
𝐸 + 𝐻 ( 19 )
where 𝐻 is the plastic modulus which depends of the
chosen constitutive law. Three such laws were considered
in this dissertation: the bilinear, the trilinear and a non-
linear law.
Figure 2: Bilinear and trilinear constitutive laws
Figure 3: Non-linear constitutive law
By evaluating ( 10 ) it can be noted that the internal forces
are solely determined by the deformation profile, and,
therefore, 𝜀𝑔 and 𝜒. So these expressions can be written as
𝑁(𝜀𝑔, 𝜒) + 𝑃 = 0
𝑀(𝜀𝑔, 𝜒) − 𝑃(𝑤 + 𝑤0) = 0 ( 20 )
The relation between 𝜒 and 𝑤 is the basis behind the
developed simplified model. Assuming the buckling mode
( 14 ) remains valid in the plastic case, we have
𝜒(𝑥3) = −𝑑2𝑤(𝑥3)
𝑑𝑤2= 𝐵
𝜋2
𝐿2𝑠𝑖𝑛 (
𝜋𝑥3
𝐿) ( 21 )
So
𝜒(𝑥3)
𝑤(𝑥3)=
𝜋2
𝐿2⇒ 𝜒(𝑥3) =
𝜋2
𝐿2 𝑤(𝑥3) ( 22 )
This relation turns the system of equations defined by ( 20
) into a solvable incremental succession of systems of two
non-linear equations.
3 Development of the Simplified
Model
3.1 Simplified model
The simplified model is about extending the validity of the
relation presented in ( 22 ) to the plastic buckling analysis.
So, at the mid-span section, starting increments are made
to 𝑤, starting from 0, changing the value of 𝜒 at the same
time. This results in a situation where the equilibrium, ( 14
), is not satisfied.
𝑅𝑁
0 = 𝑁(𝜀𝑔, 𝜒) + 𝑃 ≠ 0
𝑅𝑀0 = 𝑀(𝜀𝑔, 𝜒) − 𝑃(𝑤 + 𝑤0) ≠ 0
( 23 )
Therefore, an iterative process must be made to find the
values of 𝜀𝑔 and 𝑃 which re-establish the equilibrium
For the re-establishment of the equilibrium, the Newton-
Raphson method is used, and may be described as
𝑅𝑛 = 𝑅0 +
𝑑��
𝑑Δ𝑋 Δ𝑋 + (… ) = 0 ( 24 )
in which 𝑅0 groups the equilibrium errors at the beginning
of the increment ( 23 ), Δ𝑋 groups the variations of the
dependent variables of the problem, 𝛥𝜀𝑔 and 𝛥𝑃, and 𝑑��
𝑑Δ𝑋
4
is the tangent matrix, which dictates the way �� will change
in each iteration, until the situation of 𝑅𝑛 = 0 is reached.
The definition of 𝑑��
𝑑Δ𝑋 is as follows
𝑑��
𝑑Δ𝑋 =
[ 𝜕𝑅𝑁
𝜕𝜀𝑔
𝜕𝑅𝑁
𝜕𝑃
𝜕𝑅𝑀
𝜕𝜀𝑔
𝜕𝑅𝑀
𝜕𝑃 ]
( 25 )
After the definition of this matrix. the dependent variables
in the next iteration is
𝑋𝑛 = 𝑋0 −
𝑑��
𝑑Δ𝑋
−1
R0 ( 26 )
The updated value for 𝑃 is used to define the new external
forces
𝑁𝑛 = −𝑃𝑛
𝑀𝑛 = 𝑃𝑛(𝑤 + 𝑤0) ( 27 )
whereas the updated value for 𝜀𝑔 is used to define the new
deformation profile
𝜀33
𝑛 = 𝜀𝑔𝑛 + 𝜒 𝑥2 ( 28 )
The stress 𝜎33 is re-calculated, which results in the updated
values of the internal axial force and bending moment, 𝑁𝑛
and 𝑀𝑛. It is also in this step that the tangent matrix is
updated.
After these steps are taken, the values of 𝑅𝑁𝑛 and 𝑅𝑀
𝑛 are
evaluated and if they are not reasonably close to zero, the
iterative process is repeated. Otherwise, a new increment
of 𝑤 takes place, and the calculation of the post-buckling
trajectory continues.
3.2 Generalized Model
The generalization of the simplified model consists in
expanding the analysis by considering multiple cross-
sections throughout the column length.
For 𝑛 sections analysed, 𝑛 sine functions must be
considered, each corresponding to higher levels of
buckling modes. They are defined as
𝑤𝑗 = 𝐵𝑗 sin (
𝑘𝜋𝑥3
𝐿) ( 29 )
where 𝑗 = 1,… , 𝑛, and 𝑘 = 2𝑗 − 1 correspond to the odd,
symmetrical, buckling modes (see Figure 4).
Because of the symmetry of the problem, the sections
analysed are all in one half of the column, and they divide
the column in equal lengths.
Figure 4:Representation of 3 of the n buckling modes which can be
considered
For this model, the incremental variable is 𝐵1. It’s the
change in this variable which leads to the non-satisfaction
of the equilibrium equations which are now
𝑁𝑖(𝜀𝑔𝑖
, 𝜒𝑖) + 𝑃 = 0
𝑀𝑖(𝜀𝑔𝑖, 𝜒𝑖) − 𝑃 (𝑤𝑖 + 𝑤0𝑖) = 0
( 30 )
In which 𝑖 denotes the analysed cross-section.
By having 𝐵1 as an independent variable, this problem
becomes dependent of 2𝑛 variables: the axial strain at
each section analysed, 𝜀𝑔𝑖, the remaining factors for the
sine functions 𝐵2, … , 𝐵𝑛 and the load 𝑃.
The variables 𝜒𝑖 = 𝜒(𝑥3 = 𝑥𝑖) and 𝑤𝑖 = 𝑤(𝑥3 = 𝑥𝑖) aren’t
direct variables of the problem because they are solely
dependent on the 𝐵𝑗 variables. They are obtained as
𝑤𝑖 = ∑𝐵𝑗 sin (
𝑘𝜋𝑥𝑖
𝐿)
𝑛
𝑗=1
( 31 )
𝜒𝑖 = ∑
𝑘2𝜋2
𝐿2sin (
𝑘𝜋𝑥𝑖
𝐿) 𝐵𝑗
𝑛
𝑗=1
( 32 )
where 𝑤0𝑖 defines the value of the initial geometrical
imperfection at section 𝑖
𝑤0𝑖 = 𝑤0(𝑥𝑖) = 𝐵0 sin (
𝜋𝑥𝑖
𝐿) ( 33 )
Finally, we can write similarly to what was previously
done
𝑅𝑁𝑖
0 = 𝑁𝑖 (𝜀𝑔𝑖, 𝜒𝑖) + 𝑃 ≠ 0
𝑅𝑀𝑖
0 = 𝑀 (𝜀𝑔𝑖, 𝜒𝑖) − 𝑃(𝑤𝑖 + 𝑤0) ≠ 0
( 34 )
In this problem, then, we have a system of 2𝑛 equations to
be solved by changing the 2𝑛 variables mentioned before.
Once more, the re-establishment of the equilibrium lies in
the application of the Newton-Raphson method ( 24 ), the
difference being that the number of equations and
variables is different. 𝑋 groups 𝜀𝑔𝑖 (for 𝑖 = 1,… , 𝑛), 𝐵𝑗 (for
𝑗 = 2,… , 𝑛) and 𝑃, so Δ𝑋 groups their variations.
The tangent matrix is for this case
𝑑��
𝑑Δ𝑋 =
[ 𝜕𝑅𝑁𝑖
𝜕𝜀𝑔𝑖
𝜕𝑅𝑁𝑖
𝜕𝐵𝑗
𝜕𝑅𝑁𝑖
𝜕𝑃
𝜕𝑅𝑀𝑖
𝜕𝜀𝑔𝑖
𝜕𝑅𝑀𝑖
𝜕𝐵𝑗
𝜕𝑅𝑀𝑖
𝜕𝑃 ]
( 35 )
5
with 𝑖 = 1,… , 𝑛 and 𝑗 = 2,… , 𝑛.
The process that follows is the same which has already
been explained for the simplified model. The variables are
updated according to ( 26 ), however, since the values of
𝐵2, … , 𝐵𝑛 changed, the transverse displacement and the
curvature in each section also has to be updated, according
to ( 31 ) and ( 32 ), resulting in new values at each iteration,
which we’ll call 𝑤𝑖𝑛 and 𝜒𝑖
𝑛.
The external forces are updated as in ( 27 ), with 𝑁𝑛 = −𝑃𝑛
in every section analysed, and 𝑀𝑛 = 𝑃𝑛 (𝑤𝑖𝑛 + 𝑤0𝑖).
The deformation profile is also updated through
𝜀33𝑖
𝑛 = 𝜀𝑔𝑖
𝑛 + 𝜒𝑖𝑛 𝑥2 ( 36 )
The procedure for the calculation of the internal forces for
each section is the same as it was previously explained.
Once more, after these steps are taken, the values of 𝑅𝑁𝑛
and 𝑅𝑀𝑛 are evaluated and if they are not reasonably close
to zero, the iterative process is repeated. Otherwise, a new
increment of 𝐵1 takes place, and the calculation of the
post-buckling trajectory continues.
3.3 Numerical Implementation
The models described were implemented in MATLAB,
where functions were developed for the analysis of simply
supported columns with rectangular and I cross-sections
(Figure 5 and Figure 6).
Figure 5: Simply supported column
Figure 6: Considered cross-sections
The functions allow for the choice of 3 different
constitutive laws: bilinear, trilinear and non-linear.
Geometric and material imperfections may be taken into
account.
The program’s inputs are the geometrical (the dimensions
of the column and the cross-sections as shown above) and
the material properties (the parameters which define the
constitutive laws. The discretization of the cross-section,
i.e., the dimension of the areas of integration, may also be
chosen, by choosing the values of 𝑑𝑥 and 𝑑𝑦.
Figure 7: Discretization of the cross-sections
Its most interesting outputs are the complete load-
displacement trajectory of the columns, and the possibility
of saving for each increment of the function, the evolution
of the stresses in each of the integration points.
3.4 Comparing the Simplified and the
Generalized Models
In order to evaluate how the simplified model compares to
the more sophisticated generalized model, a comparison
was made, analysing the resulting trajectories.
For this comparison, it was chosen a column with the
following properties: 𝑏 = 0,200 𝑚 (the dimension in the
direction of buckling), ℎ = 0,200 𝑚, 𝐿 = 2,5 𝑚. It was
chosen a bilinear law, with 𝐸 = 210 𝐺𝑃𝑎 and 𝐸𝑡 =
63 𝐺𝑃𝑎 (𝐸𝑡 𝐸⁄ = 0,30), and an initial yielding stress 𝑓𝑦 =
235 𝑀𝑃𝑎, with a discretization of the cross-section in 200
areas of integration along 𝑏 (𝑑𝑥 = 1 𝑚𝑚)
The results were the following (see Figure 8):
Figure 8: Load-displacement trajectories for the column, for several
cross-sections. The dashed line is the simplified model.
The dashed curve, which stands out, is the result of the
application of the simplified models. The other is a
superposition of curves resulting from the use of the
generalized model, for 2, 3, 4, 5, 7 and 10 sections. At the
scale presented they are all indistinguishable. The analysis
made of the results has shown that the difference between
the results of the 5 and 10 curves are minimal, so the model
with 5 sections was taken as reference.
Regarding the simplified model, the resulting curve is
clearly different, but still a very good result, considering
the simplicity of the model. Also, evaluating the relative
error, it results that, although the error in the displacement
8000
10000
12000
14000
16000
18000
20000
0 0,02 0,04 0,06 0,08 0,1 0,12
P(k
N)
w(m)
6
at the point of maximum load is around 16%, the error in
the estimation of the load itself was only 0,93%, relative
to the result of the 5 sections’ curve.
3.5 Influence of the Number of
Integration Areas
In the last section, it was revealed that, although the
number of cross-sections considered along the span of the
column influence the results, they aren’t largely different.
In this section, for the same column and for the simplified
model, it was studied the influence, in the results, of using
different numbers of integration areas. The number of
areas considered were 2, 4, 8, 16, 25, 50, 100, 200 and
2000.
The results led to the conclusion that, for this column, the
examples with less than 16 areas don’t come close to
reproducing the post-buckling trajectory of the column as
seen in Figure 9: Load-displacement trajectories for the
column, analysed with different numbers of integration
areas. Bottom curve: 2 areas; top curve: 16 areas., where
the results for the use of 2, 4, 8 and 16 integration areas are
shown.
Figure 9: Load-displacement trajectories for the column, analysed with
different numbers of integration areas. Bottom curve: 2 areas; top curve:
16 areas.
The trajectories are much more influenced by the lack of
discretization along the cross-section than along the
column’s length. This is because the lack of integration
points doesn’t allow for the correct evaluation of the
stresses along the cross-section. The phenomenon of
elastic unloading, which was proven to be of great
importance, even dating back to the studies of Hutchinson,
is only very roughly reproduced by the models with few
integration points. That idea is exemplified in Figure 10,
where 2 integration points were used.
Figure 10: Real stress profile (dashed line) versus the stresses in the 2
integration points, xi(1) and xi(2)
The use of a higher number of integration points gradually
lead to better results. The curve with 2000 areas precisely
reproduces the bifurcation load estimated by Engesser’s
expression, 𝑃𝑡, for this column. For all the subsequent
analysis, 200 integration areas were considered.
4 Model Validation Using a
Finite Element Analysis
4.1 Plane Stress Model
The same column analysed in section Figure 8 was
modelled and analysed using ADINA [7], which is able to
perform physical and geometrical non-linear analyses. The
results given by ADINA in the plane stress analysis are
here compared to the ones of MATLAB where 5 sections
were considered. The plane stress model defined in
ADINA is shown in
Figure 11: Defined plane stress model
The imperfect model was made by offsetting P2 upwards,
so that the middle plane of the column had a triangular
shape, like shown in
Figure 12: implementation of the geometric imperfection
Since the imperfections in MATLAB were being
implemented through a sine wave, in order to get a “perfect
fit” between the results given by ADINA and the ones
given by the MATLAB model, changes were made so that
the implementation of the imperfections in the developed
model also resulted in a triangular shape. To do that, the
imperfections were implemented in the form of a Fourier
series with a triangular shape, using the first 5 terms.
𝑤0𝑖 = 𝐵0
8
𝜋2∑
(−1)(𝑛−1)/2
𝑛2sin (
𝑛𝜋𝑥𝑖
𝐿)
∞
𝑛=1,3,5,…
( 37 )
8000
10000
12000
14000
16000
18000
20000
0 0,02 0,04 0,06 0,08 0,1 0,12
P(k
N)
w(m)
7
4.2 Comparison of Results
The results for the perfect column and for an imperfect
column with 𝑤0 = 0,01 𝑚 are represented in Figure 13
and in Figure 14 (the MATLAB results are the red, dashed,
curves, while ADINA’s are the blue curves).
Perfect Column (𝒘𝟎 = 𝟎)
Figure 13: Comparison of the results for the perfect column.
MATLAB– red; ADINA– blue
Imperfect Column (𝒘𝟎 = 𝟎, 𝟎𝟏 𝒎)
Figure 14: Comparison of the results for the imperfect column.
MATLAB– red; ADINA– blue
The graphics show that the generalized model developed
in MATLAB matches the results given by a sophisticated,
and harder to use, finite elements’ program. Although the
results from the simplified model differ from these, it still
offers a very good estimate of the maximum load.
5 Parametric Studies
Having established that the models created are able to
reproduce column’s behaviour, in this section parametric
studies are performed in order to better understand the
post-buckling behaviour for different situations.
These studies were performed for columns of different
slenderness ratios, 𝜆 = 𝐿 𝑖⁄ , but the columns weren’t
directly picked by their slenderness ratios, but rather for
their buckling/yielding load relations, 𝑃𝑏𝑖𝑓/𝑃𝑦.
The effect of geometric (𝑤0 = 0,001 and 0,01 𝑚) and
material imperfections was also studied for the bilinear
law.
5.1 Rectangular Cross-section with
Bilinear and Trilinear Laws
The cross-section dimensions are 𝑏 = 0,200 𝑚 and ℎ =
0,100 𝑚. The parameters which define the constitutive
laws are 𝜎𝑦0 = 235 𝑀𝑃𝑎, 𝜎𝑦1 = 450 𝑀𝑃𝑎, 𝐸 =
210 𝐺𝑃𝑎, 𝐸𝑡 = 𝐸𝑡1 = 63 𝐺𝑃𝑎 and 𝐸𝑡2 = 0. For this
column, we have 𝑃𝑦 = 4700 𝑘𝑁.
The column here represented are:
a) 𝑃𝑏𝑖𝑓 = 𝑃𝑡 = 2𝑃𝑦 (𝐿 = 2,10 𝑚);
b) 𝑃𝑏𝑖𝑓 = 𝑃𝑦 = 1,5𝑃𝑡 (𝐿 = 3,43 𝑚);
c) 𝑃𝑏𝑖𝑓 = 𝑃𝑦 = 2𝑃𝑡 (𝐿 = 4,20 𝑚);
d) 𝑃𝑏𝑖𝑓 = 𝑃𝐸 = 0,5𝑃𝑦 (𝐿 = 7,67 𝑚).
The resulting post buckling curves for these columns were
as shown in Figure 15. The bilinear law is represented, in
the results, with a black line while the trilinear is the red
line.
Figure 15: Resulting post-buckling curves for different columns with the
two laws. Bilinear-black; trilinear-red
It can be seen that in the d) curve, both laws are
coincidental, in the range of results represented. This
happens because the second yielding level, for the trilinear
law, isn’t reached, which is the reason why, in curves b)
and c), the results diverge. In curve a) the trilinear law
can’t reach the load of 𝑃𝑏𝑖𝑓 = 𝑃𝑡 = 2𝑃𝑦, because that
would correspond to a level of stresses applied in the
8000
10000
12000
14000
16000
18000
20000
0 0,02 0,04 0,06 0,08 0,1 0,12
P(k
N)
w(m)
2000
4000
6000
8000
10000
12000
14000
16000
18000
0,00 0,02 0,04 0,06 0,08 0,10 0,12
P(k
N)
w(m)
0
0,5
1
1,5
2
2,5
3
0 0,05 0,1
P/P
y
w(m)
a)
b)
c)
d)
8
section which doesn’t comply with the limit of 𝜎𝑦1 =
450 𝑀𝑃𝑎. 2𝑃𝑦 leads to an applied load of 𝑃 = 9400 𝑘𝑁,
and the trilinear law can only support a load of 𝑃 =
9000 𝑘𝑁, so as it reaches this value it buckles and
immediately starts losing load bearing capacity.
5.2 Effect of the Imperfections for the
Bilinear Law
Below, in Figure 17, it is shown, for the same columns
from the last section, the effects of geometric and material
imperfections. The material imperfections analysed have
the profile, along 𝑏, as seen in Figure 16.
Figure 16: Material imperfection, in the rectangular cross-section, along 𝑏
where the long dashed lines represent a geometric
imperfection of 𝑤0 = 0,001 𝑚 and the smaller dashed
lines an imperfection of 𝑤0 = 0,01 𝑚.
It can be seen from the figure that:
In a general way, the presence of geometric
imperfections lowers the maximum load;
The columns of lowest slenderness (𝐿 = 2,10 𝑚) are
particularly sensible to the geometrical
imperfections. It’s shown that even for the smallest
geometric imperfection there’s a big deviation from
the trajectory of the perfect column;
These columns aren’t very sensible to the material
imperfections;
For the second set of columns, the presence of
material imperfections results in a bigger bifurcation
load, for both 𝜌 = −0,5. For 𝜌 = +0,5 the
bifurcation load remains the same, but the initial
slope of the curve is bigger;
For the third set of columns the bifurcation load is
raised for 𝜌 = −0,5 and lowers for 𝜌 = +0,5;
For the slenderest columns, the bifurcation is elastic.
and for the situation with no material imperfections,
the bifurcation follows a straight, horizontal line,
until the point yielding is reached. For the version
with material imperfections, we can see that they both
buckle for the same value of 𝑃 𝑃𝑦⁄ = 0,5, but the
version with 𝜌 = −0,5 has an initial “break” right
after buckling, and a second break around 𝑤 =
0,075 𝑚, whereas the version with 𝜌 = +0,5 seems
to follow a steeper, continuous, descending path after
buckling;
In general, it can be said that the results are more
adversely influenced with 𝜌 > 0.
These results seem to be consistent with the ones obtained
by Ritto Corrêa [8], in his study of the continuous model
of Hutchinson.
Figure 17: Effects, in post-buckling behaviour, of geometric and material
imperfections
6 Conclusions In this paper, a simplified model for the study of the
buckling and post-buckling behaviour of elasto-plastic
columns was introduced, based in the application, for this
situation, of the simple relation that there is in the elastic
case between the transverse displacement of the column
and the curvature of its sections. This allowed for the load-
displacement trajectory of the columns to be obtained
through the analysis of the equilibrium, in the mid-span
cross-section, of the internal and external forces, through
an incremental/iterative method, with the equilibrium
equations being checked for every value of transverse
𝝆 = −𝟎, 𝟓 𝝆 = 𝟎 𝝆 = +𝟎, 𝟓
9
displacement, 𝑤, considered, relying on the variation of
only 2 dependent variables, 𝜀𝑔 and 𝑃.
When considered the comparison with more sophisticated
finite elements models, it was shown that the simplified
reproduces with, a good level of accuracy, the post-
buckling behaviour of columns, which is, in itself, a very
interesting and satisfying result to be taken.
7 References
[1] F. Engesser, “Ueber die Knickfestigkeit gerader
Stäbe,” Zeitschrift für Architektur und
Ingenieurwesen, 1889.
[2] F. Engesser, “Uber Knickfragen,” Schweuzerische
Bauzeitung, 1895.
[3] T. Von Karman, “Untersuchungen uber
Knickfestigkeit,” Mitteilungen über
Forschungsarbeiten auf dem Gebiete des
Ingenieurwesens, 1910.
[4] F. Shanley, “Inelastic Column Theory,” Journal of
Aeronautic Science, vol. 14, 1947.
[5] J. W. Hutchinson, “Plastic Buckling,” Advances in
Applied Mechanics, 1974.
[6] F. Virtuoso, “Estabilidade de Estruturas. Colunas e
Vigas-coluna,” em Folhas da Disciplina de
Estruturas Metálicas - IST, 2013.
[7] I. ADINA R&D, Theory and Modelling Guide, 2015.
[8] M. Ritto-Corrêa, Estabilidade Elastoplástica de
Colunas: Estudo do Modelo Contínuo de Hutchinson,
1996.