Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 2, pp. 515-529, Warsaw 2011

A NONLINEAR MODEL FOR THE ELASTOPLASTIC

ANALYSIS OF 2D FRAMES ACCOUNTING FOR DAMAGE

Antoĺın Lorenzana

ITAP, University of Valladolid, Valladolid, Spain; e-mail: ali@eis.uva.es

Pablo M. López-Reyes

CARTIF Centro Tecnológico, Valladolid, Spain; e-mail: pablop@cartif.es

Edwin Chica

University of Antioquia, Medelĺın, Colombia; e-mail: echica@udea.edu.co

José M.G. Terán, Mariano Cacho

EII, University of Valladolid, Valladolid, Spain; e-mail: teran@uva.es; cacho@eis.uva.es

Asimple and efficient procedure for non-linear analysis of frames is presen-
ted, under the hypothesis that the non-linear effects, if appear, are concen-
trated in thebeam-ends.We consider adamagemodel basedonContinuum
DamageMechanics, but affecting the cross-section as a whole. The elasto-
plastic behaviour is included formulating the tangent elastoplastic stiffness
matrix in such a way that the yield function, in terms of internal forces
(axial, shear and bendingmoment), is affected by the damage in each plas-
tic cross-section.After the verification of themodel, an example of applica-
tion is solved for different assumptions on the yield function (depending on
the internal forces considered)with the damagebeing taken into account or
disregarded. The differences on the collapse load, for each case, are shown
and some conclusions obtained, among them that themethod can evaluate
in a more accurate way the load that causes the collapse of frames under
increasing loading, considering a fully plastic non-linear analysis.

Key words: plastic methods, structural analysis, material nonlinearities,
elastoplastic stiffness matrix, Bonora damagemodel

1. Introduction

In the civil and structural engineering, there are several approaches to deal
with damage. The structural damage can be quantified through a damage
index, which is the value of damage normalized to the failure level of the



516 A. Lorenzana et al.

structure: a value equal to 1 corresponds to the complete structural failure
(Faleiro et al., 2008), so the structure can not withstand further loadings.

In this paper, the damage index is derived fromContinuumMechanics and
Ductile Fracture theories applied tometallic materials. Using standard stress-
strain relationships in elastoplasticity together with thermodynamic laws for
irreversible processes, and assuming that fracture takes place at a certain rate
of plastic deformation, after several mathematical manipulations it is possible
to couple general plasticity theorywith damage theory through the hypothesis
of strain equivalence (Lemaitre, 1985) to relate equivalent plastic deformation
with damage.

The Continuum Damage Mechanics (CDM) approach, initially proposed
by Lemaitre, takes into account the effects associated to a given damage state
through the definition of an internal state variable. The set of constitutive
equations for the damaged material is then derived within a thermodynamic
framework. Many authors have modified Lemaitre’s linear damage accumu-
lation law in order to be able to incorporate experimental damage measure-
ments with different types of materials fitting in it. A nonlinear CDMmodel,
recently proposed by Bonora (Bonora, 1997, 1998; Bonora et al., 2005) is
able to precisely describe the damage evolution for different types of metals
and has been used by other authors (Bobiński and Tejchman, 2006; Cha-
boche, 1984; Chandrakanth and Pandey, 1993, 1995a; Tai and Yang, 1986;
Tai, 1990).

The aim of this paper is to develop a general, accurate, efficient and simple
procedure for solving the fully non-linear problem of framed structures, using
elastoplastic beam finite elements and consideringmaterial nonlinearities and
the loss of rigidity due to the increase of damage in the cross-section andusing
anexplicit formof the tangent stiffnessmatrix, called theelastoplastic damaged
stiffness matrix (Ibijola, 2002; Yingchun, 2004). The basis of this method is
a direct combination of existing formulations (Navier-Bernoulli’s beam theory
and Bonora’s CDM damage model) to determine in a more accurate way the
collapse load of standard frames.

2. Damage model for cross-sections of beams based on CDM

From a general point of view, damage can be defined as a progressive loss
of load carrying capability as a result of some irreversible processes that oc-
cur in the material microstructure during the deformation process (Lemaitre,



A nonlinear model for the elastoplastic analysis... 517

1985). Assuming thatmicro-cracks andmicro-voids have a uniformly distribu-
ted orientation, the scalar D can be defined in terms of the relative reduction
of the cross-section (Lemaitre, 1984)

D=1−
Aeff
A0

(2.1)

where A0 is the initial section and Aeff is the effective area: Aeff =A0(1−D).
For every value of D ∈ [0,1), the effective stress and strain for uniaxial

behaviour can be defined (Simo and Ju, 1987)

Effective stress: σ=
σ

(1−D)
=
F

Aeff
(2.2)

Effective strain: ε=(1−D)ε (2.3)

where ε and σ are the usual strain and stress Cauchy tensors. For a virgin
material, D =D0 ≈ 0 and for a exhausted state D =Dcr < 1, where D0 is
the initial amount of damage and Dcr is the critical damage.
Then, assuming the hypothesis of strain and stress equivalence, themate-

rial behaviour for a damagedmaterial can be written as

ε=
σ

E
=

σ

(1−D)E
(2.4)

and now it is necessary to show the evolution of D from its initial value
D0 (usually 0) to Dcr, value less than or equal to 1 from which the former
expressions are not considered valid.
For the Bonora (1997) model assumed, D depends only, for each material

and temperature T , on the amount of equivalent plastic strain through the
following expressions

φ=Fp(σeq,R,σy)+φ
∗(Y,D,ṗ,T)

(2.5)

Y =−
σ2eq

2E(1−D)2
[2

3
(1+ν)+3(1−2ν)

(σm
σeq

)2]

where φ is the total dissipation potential (in function of the equivalent
stress σeq, material hardening R and yield stress σy), φ

∗ is the damage dissi-
pation potential and Y is the damage energy release rate. Fp is thedissipation
potential associated with plastic deformation, ṗ is the accumulated effective
plastic strain, σm is the hydrostatic stress, σeq is the Von Mises equivalent
stress, ν is thePoisson ratio, E is theYoungmodulusand the relation σm/σeq
is called the triaxiality ratio or stress rigidity parameter (Lebedev et al., 2001).



518 A. Lorenzana et al.

Bonora proposed the following expression for the damage dissipation po-
tential

φ∗ =
[1

2

(

−Y
So

)2 So
1−D

](Dcr−D)α−1/α
p(2+n)/n

(2.6)

where So is amaterial constant, n is theRamberg-Osgoodmaterial exponent,
α is the damage exponent that determines the shape of the damage evolution
and p is the accumulated plastic strain.
Assuming that the rate of the plastic multiplier λ̇ is proportional to the

rate of the effective accumulated plastic strain ṗ

λ̇= ṗ(1−D) (2.7)

and that for proportional loading the kinetic law, according to Lemaitre’s
model, is

Ḋ=−λ̇∂φ
∗

∂Y
(2.8)

the relationship between damage and effective plastic strain is, finally

D=D0+(Dcr−D0)
{

1−
[

1−
ln
p
pth

ln
pcr
pth

]α}

(2.9)

where pth is the plastic threshold value and pcr is the critical plastic value
corresponding to Dcr (Bonora, 1997).

3. Elastoplastic stiffness matrix considering damage

Assumingstandardelastoplastic behaviour (Deierlein et al., 2001) for thebeam
element, with additive decomposition of displacements duep at the ends of the
element into elastic due and plastic dup components

{duep}= {due}+{dup} (3.1)

and that plastic deformation takes place only on the beam-ends (concentrated
plasticity), and hence also damage, the resulting beam model is represented
in Fig.1, where the initial and deformed configurations are shown (note that
the length of small segments at the beam-ends should be infinitesimal).
This concentrated plasticity model does not account for the spreading of

plasticity from outer fibers inwards. This behaviour could be considered using
more advancedmodels like the layered approach (Chandrakanth and Pandey,



A nonlinear model for the elastoplastic analysis... 519

Fig. 1. Beam element with plasticity and damage at its ends

1997) but, in engineering practice, the distributed plasticity models are less
frequently used than frame theories with concentrated plastic hinges (Inglessis
et al., 1999).
The linear elastic response is governed by

{dF}= [K]{due} (3.2)

where [K] is the standard elastic stiffnessmatrix of a beam element and {dF}
is the beam-end force vector which for the 2D case presented in this paper is
{dF}⊤ = {dNx1,dVy1,dMz1, dNx2,dVy2,dMz2}⊤, andthedisplacementvector
is {due}⊤ = {ux1,uy1,θ1,ux2,uy2,θ2}⊤.
In a similar way, it is necessary to determine the relationship between the

increment of force and the increment of elastoplastic displacement {duep}

{dF}= [Kep]{duep} (3.3)

From Eqs. (3.1) and (3.2)

{dF}= [K]
(

{duep}−{dup}
)

(3.4)

so the increment of plastic displacement {dup}, assuming associated flow rule,
can be expressed as

{dup}= {dλ}
{

dZ

dF

}

(3.5)

where Z is the yield function for the beam element and {dλ} is the vector
of so-called plastic multipliers dλ1,dλ2 in each beam-end. Using the plastic



520 A. Lorenzana et al.

consistency condition

dλ

{

=0 if Z < 0 or p<pth

> 0 if Z =0 or p­ pth
(3.6)

together with Eq. (2.8) for the elastoplastic beam element

{dD}= {dλ}
{−∂φ∗
∂Y

}

(3.7)

and the plastic flow rule condition

Ż =

{

∂Z

∂F

}

{dF}+
{

∂Z

∂D

}

{dD}=0 (3.8)

and substituting Eqs. (3.4) and (3.7) into Eq. (3.8), the following expression
for {dλ} can be found

{dλ}=

{

∂Z
∂F

}

[K]{duep}
{

∂Z
∂F

}

[K]
{

∂Z
∂F

}

+
{

∂Z
∂D

}{

∂φ∗

∂Y

} (3.9)

and taking former equations to Eq. (3.3), the final (Chica et al., 2010) rela-
tionship between forces and displacements for the elastoplastic beam element
is

{dF}= [K]
(

1−
[K]

{

∂Z
∂F

}{

∂Z
∂F

}

{

∂Z
∂F

}

[K]
{

∂Z
∂F

}

+
{

∂Z
∂D

}{

∂φ∗

∂Y

}

)

{duep}= [Kep]{duep}

(3.10)
Now it is necessary to relate the term {∂φ∗/∂Y} with known parameters

for the beam element. Using Eq. (2.6)

∂φ∗

∂Y
=
Y

So

(Dcr −D)α−1/α
p(2+n)/n

1

1−D
(3.11)

and substituting the expression for Y given in Eq. (2.5) in Eq. (3.11)

∂φ∗

∂Y
=−

σ2eq
(1−D)2

f
(σm
σeq

) 1

2ESo

(Dcr−D)α−1/α
p(2+n)/n

1

1−D
(3.12)

and using Von Mises plastic criterion for ductile materials, together with the
Ramberg-Osgood (Ramber and Osgood, 1943) power law, the effective equi-
valent stress can be given as a function of the accumulated plastic strain as

σeq
1−D

=κp1/n (3.13)



A nonlinear model for the elastoplastic analysis... 521

where κ is a material constant. Then, substituting Eq. (3.13) into Eq. (3.12),
n vanishes, resulting

∂φ∗

∂Y
=−f

(σm
σeq

) κ2

2ESo

(Dcr−D)α−1/α
p

1

1−D
(3.14)

To determine the term f(σm/σeq)[κ
2/(2ESo)], the following procedure is

used. Equations (2.7) and (2.8), together with Eq. (3.14) leads to

Ḋ=
κ2

2ESo
(Dcr−D)α−1/αf

(σm
σeq

)ṗ

p
(3.15)

and integrating between D0 and Dcr

(Dcr −D0)1/α =
1

α

κ2

2ESo
ln
pcr
pth
f
(σm
σeq

)

(3.16)

where it is possible to identify

κ2

2ESo
f
(σm
σeq

)

=α
(Dcr −D0)1/α
ln pcr
pth

(3.17)

so that finally the referred factor in Eq. (3.10) is now known

{

∂φ∗

∂Y

}

=

[

A1 0
0 A2

]

(3.18)

with

Ai =−α
(Dcr−D0)1/α
ln pcr
pth

(Dcr −Di)α−1/α
pi

1

1−Di
and only the terms involving Z are not yet identified. Z is the so-called yield
function, which includes damage, meaning, for any cross-section, the values of
damage, axial and shear forces and bending moment from which plastic and
damage levels can increase, according to the flow rule.
For simplicity but without loss of generality, we present the derivation

of the yield function for a rectangular b× h cross-section in a 2D beam,
assuming the Von Mises yield criterion, associated flow rule and damage as
defined previously. According to the CDM and neglecting plastic hardening,
the yield criterion is expressed in terms of the effective stress as

Z =
σeq
1−D

−σy ¬ 0 (3.19)



522 A. Lorenzana et al.

where σy is the elastic limit of thematerial, and σeq is given, according to the
von Mises yield criterion, by

σeq =
√

σ2x+3τ
2
xy (3.20)

where σx is the normal stress in the beam due to the axial force and bending
moment and τxy is the shear stress due to the shear force. Although real
materials exhibit some kind of hardening, its effects can be neglected for some
ductile steels as the one used in this paper (S-1015).

As it is common for undamagedmaterials, the values of the plastic bending
moment Mp, plastic axial force Np and plastic shear force Vp that cause the
full yielding of the cross-section of the beam are (Krenk et al., 1999; Neal,
1985; Olsen, 1999)

Mp =
σybh

2

4
Np =σybh Vp =

2σybh

3
√
3

(3.21)

and including the hypothesis of strain equivalence (Lemaitre, 1985), these
expressions change

Mp =
σybh

2

4(1−D)
Np =

σybh

1−D
Vp =

2σybh

3
√
3(1−D)

(3.22)

In the case of a section simultaneously subjected to the bending mo-
ment Mz, axial Nx and shear forces Vy, for the usual case in which plasticity
first appears in the outer part of the cross-section due to Mz, and considering
that the section is fully plastic when the shear stress reaches its maximum
value σy/

√
3 in any internal point of the section, the resulting equation is

Mz =
σybh

2

4
−
N2x
4bσy

−
9

16

V 2y
bσy(1−D)2

(3.23)

Substituting the former expressions into Eq. (3.19), the yield function Z
is obtained and shown in Fig.2 for different values of D

Z =
|Mz|
Mp
+
(Nx
Np

)2 1

1−D
+
1

3

(Vy
Vp

)2 1

(1−D)3
− (1−D)= 0 (3.24)



A nonlinear model for the elastoplastic analysis... 523

Fig. 2. Yield function for different values of D

Finally, once Z is known, the following factors that appear in Eq. (3.10)
can be determined. Substituting Eq. (3.24) into Eq. (3.17) and Eq. (3.18)

{

∂Z

∂F

}

=









A1 B1
1

Mp
0 0 0

0 0 0 A2 B2
1

Mp









⊤

(3.25)
{

∂Z

∂D

}

=

[

C1 0
0 C2

]

where

Ai =
2Nxi

N2p(1−Di)
Bi =

2

3

Vyi
V 2p (1−D1i3

Ci =
(Nxi
Np

)2 1

(1−Di)2
+
(Vyi
Vp

)2 1

(1−Di)4
+1

and so, the elastoplastic damaged stiffness matrix is completely defined. All
the former expressions are put together in a standard incremental algorithm
and implemented in a computer code. In each increment, iterations are needed
to ensure that in any plastic (and damaged) cross-section all the conditions
are fulfilled. The code is checked using a test problem (Fig.3) and applied to
a standard building frame as the one shown in Fig.5.



524 A. Lorenzana et al.

4. Validation

Themethodwas used to solve the problem shown in Fig.3, for which the data
was available in the literature (Inglessis et al., 1999) or could be obtained by
experimental or statistical methods (Rucka and Wilde, 2010; Rinaldi et al.,
2006).

Fig. 3. Test on steel member: specimen and loading

The load is applied by increasing the value of δ in the free end. The
reaction F in this point is plotted vs. δ in Fig.4 where a comparison between
the experimental and numerical results obtained by Inglessis (Inglessis et al.,
1999) and the results using the proposedmethod is shown.
The parameters for the simulation were L = 665mm, E · I = 1.906 ·

107Nmm2, pcr =1.4, pth =0.259, α=0.2175, D0 =0 and Dcr =1. In spite
of the simplicity of the proposedmethod, the results are accurate enough even
for this demanding test, where the damage value in the clamped end reaches
the value of 0.520.

Fig. 4. Experimental vs. numerical results



A nonlinear model for the elastoplastic analysis... 525

5. Example

After validation, we use the method to compare the collapse load of the 2D
frame shown inFig.5 under different yielding assumptions.The frame is clam-
ped on the bottom of its two columns and subjected to a horizontal load in
node 4 of magnitude P = 62.5kN and vertical loads in nodes 2, 3 and 4
of the samemagnitude, which are proportionally increased using the parame-
ter λ. The assumed properties are: L=1m,E =200GPa,A=0.1×0.1m2,
σy = 250MPa (yield stress). The material is a Steel-1015 and its parameters
of evolution of damage are reported in the literature (Le Roy, 1981; La Rosa
et al., 2001) so that pcr =1.4, pth =0.259 and α=0.28.

Fig. 5. Progressive collapse

Three different considerations for the yield function are considered. In the
first one (1), we use the classic plastic method so that plastic hinges can
appear only due to a bending moment. In the second one (2), the axial and
shear forces and bending moment are considered in the yield function, but
damage is not. Finally, in the third case (3), all effects are taken into account.
For all the three cases, the order of appearance of the plastic hinges (1) or the
plastic sections (2) and (3) is 5→ 4→ 3→ 1. The response curves, for λ vs.
horizontal displacement of node 4, are shown in Fig. 6.

For case 1, in which only the bendingmoment is considered, the response
follows a polygonal curve of decreasing slope.When N and V are considered,
together with M (case 2), the response is a continuous curve that is below the
previous polygonal one.When, in addition, damage is considered (case 3), the



526 A. Lorenzana et al.

Fig. 6. Load factor λ vs. horizontal displacement of node 4

response is even lower, showing that the stiffness of the frame decreases when
more sophisticated models are taken into account.

The deformed shape, amplified ×25, is shown in Fig.5 for the loads 1.348,
1.433, 1.486 and 1.633, corresponding to the formation of plastic sections at
5, 4, 3 and 1, respectively, for case 3 (1.348, 1.436, 1.541 and 1.672 for case 2,
and 1.348, 1.531, 1.765 and 1.833 for case 1).

Fig. 7. Evolution of the damage with the load



A nonlinear model for the elastoplastic analysis... 527

For case 3, the evolution of the damage with the load is shown in Fig.7.
Note that the analysis fails to converge once the plastic state is reached in
section 1, so damage can not be evaluated in this section.

6. Conclusions

UsingContinuumDamageMechanics assumptions, a simple and efficient pro-
cedure for the analysis of frames has been developed. One-dimensional finite
elements (elastoplastic beams)are formulated andnon-linear effects (plasticity
anddamage) are supposed to be concentrated in the beam-ends.The resultant
numerical method is incremental and iterations are needed in each increment
to ensure that all the beam-ends would be balanced and comply with plastic
conditions for each level of damage. The stiffnessmatrix depends on geometry
and onmaterial properties, as usual, but also on the yield function Z, plastic
deformation and damage in the beam-ends.
Under increasing loading, once plastic deformation appears in any cross-

-section, damage increases and the stiffness of the beam decreases, and hence
the framebecomesmore flexible.More plastic and damaged cross-sections can
appear and, eventually, for some loading factor, convergence would not be
achieved: it has reached the collapse state. Themore effects are included in Z
(internal forces, damage), the less the collapse load is. For the simplest case
(Z depending only on the bendingmoment) the results obtained coincidewith
the standard plastic analysis based on plastic hinges.

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Nieliniowy model do sprężysto-plastycznej analizy problemu uszkodzeń

dwuwymiarowych ram

Streszczenie

W pracy zaprezentowano prostą i skuteczną metodę nieliniowej analizy dwuwy-
miarowych ram przy założeniu hipotezy, że efekty nieliniowe, jeśli występują, są
skoncentrowane na końcach belek tworzących układ ramy. Rozważono kontynual-
ny model procesu zniszczenia obejmujący przekrój belki jako całość. Właściwości
elasto-sprężyste materiału ujęto poprzez zdefiniowanie macierzy stycznej sztywności
sprężysto-plastycznej w taki sposób, że funkcja uplastycznienia wyrażona w katego-
riach obciążeń wewnętrznych (sił osiowych, tnących oraz momentu gnącego) zależy
od stanu zniszczenia w każdym uplastycznionym przekroju. Po zweryfikowaniu mo-
delu, rozwiązano przykład zastosowania analizy dla różnych założeń narzuconych na
funkcję uplastycznienia (w zależności od wziętych pod uwagę obciążeń wewnętrz-
nych) z uwzględnieniem zniszczenia lub bez. Dla każdego przypadku pokazano róż-
nice w wartościach obciążenia zewnętrznego prowadzącego do wyboczenia ramy oraz
sformułowanownioski.Wykazano, że przedstawionametoda nieliniowej analizy upla-
stycznienia pozwala na bardziej precyzyjne określenie krytycznych obciążeń prowa-
dzących do zniszczenia konstrukcji.

Manuscript received September 30, 2010; accepted for print December 2, 2010