1

Plastic Buckling of Columns:

Development of a Simplified Model of Analysis

Filipe Pereira

filipe.s.pereira@tecnico.ulisboa.pt

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.