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Development of a simplified plate element for large deflection elasto-plastic finite element analysis
Compurrrs
Pergamon PII: SOO45-7949(%)00022-3
& Strucrures Vol. 61. No. I. pp. 183-188, 1996 Copyright SC 1996 Elsevier Science Ltd Prmted in Great Britain All rights reserved 004%7949196 $15.00 + 0.00
DEVELOPMENT OF A SIMPLIFIED PLATE ELEMENT LARGE DEFLECTION ELASTO-PLASTIC FINITE ELEMENT ANALYSIS
FOR
Ai-Kah Soh, Chee-Kiong Soh and Kay-Hiang Hoon Nanyang Technological University. Nanyang Avenue, Singapore 639798 (Receiaed 6 October 1994)
Abstract-Numerous researchers have performed large deflection elasto-plastic finite element analysis of stiffened plate structures, using plate elements which employed various iterative/incremental procedures for nonlinear analysis. However, such analysis always requires a large number of plate elements, which become a heavy burden on the computer in terms of computation time and disk storage. Fortunately, the problem can be reduced by employing the one-dimensional version of plate elements to simulate the stiffeners of stiffened plate structures. This type of element has been developed in the present study, by employing the multi-layer approach which allows yielding at different levels of the plate element. Special care has been taken to avoid the “self-straining” problem of plate elements. Moreover, an improved iterative/incremental procedure has been proposed to establish a more reliable and accurate solution. The results obtained by employing this element type for a test example are found to be in good agreement with those obtained by other authors using traditional large deflection elasto-plastic finite element analysis. Copyright 0 1996 Elsevier Science Ltd.
INTRODUCTION
NOTATION
t,i
A N M A I E I\, d (30 4 u H P d ‘
WI
PI
Cartesian coordinates displacements in the x-, y- and z-directions, respectively rotation normalized local x coordinate curvature direct strain total, in-plane, large deflection and bending strains, respectively, in the x-direction total strain at the datum in the x-direction increment axial load in the plate element bending moment about the datum line cross-sectional area of plate element second moment of area of the plate element about the neutral axis Young’s modulus width of plate element direct stress yield stress potential energy strain energy virtual work due to applied loads external force generalized displacement tangent elasto-plastic modular in-plane shape function out-of-plane shape function in-plane nodal displacements of an element out-of-plane nodal displacements and rotations of an element in-plane strain matrix bending curvature matrix slope matrix
Subscripts e
P C
elastic plastic with respect to neutral axis
Many researchers have studied the structural response of stiffened plate structures for which failure often involves both instability and material yielding
in some of their plate components [l]. A significant amount of such study was performed by Soreide and Moan [2], Crisfield [3] and Frieze et al. [4], using the finite element technique. Several iterative/incremental procedures [3] had been proposed for the large deflection elasto-plastic finite element analysis of thin plated structures. However, these procedures often encountered convergence problems in the analysis of stiffened plate structures in which complicated buckling modes and welding residual stresses occurred in both the plates and stiffeners. A solution procedure for overcoming the convergence problem was proposed by Riks [5] and later modified by Crisfield [6] to make it suitable for adoption by the finite element technique. The full section yield criterion, in which the behaviour of the material is assumed to be elastic until yield occurs, at which time the shell element suddenly becomes fully plastic, was employed by Crisfield [7] to perform the elasto-plastic finite element analysis. This full section assumption is not as accurate as considering plastic flow through the depth of the plate thickness, but it has the advantage of reducing the computation time and disk storage required considerably. Moreover Crisfield [7] has shown that both approaches produced very similar results in a large deflection elasto-plastic analysis of plates. Thus, Crisfield’s formulation based on full section yield criterion is commonly employed to perform elasto-plastic finite 183
184
Ai-Kah Soh et al.
element analysis. However, collapse analysis of complex stiffened plate structures will require numerous plate elements, which will be a heavy burden on the computer in terms of computation time and disk storage. Thus, it would be useful to develop the one-dimensional version of plate elements for simulating the stiffeners of complex stiffened plate strucutres. The development of this element type will be discussed in this paper. Moreover, some improvements made to the traditional approach of formulating plate elements for large deflection elasto-plastic analysis will also be discussed.
By differentiating
6, = g =
eqn (1) we obtain
PI{+ = & b5~zl{u}‘,
where [B] is the in-plane strain matrix. By differentiating eqn (2) we obtain
1
for i = 1,2,
x (a(35’ - 1 + 25&))T ANALYSIS
The in-plane displacement function of the onedimensional version of a plate element is given by u(5) =
[N’lT{uJe,
where [G] is the slope matrix of the plate element. By differentiating eqn (2) twice, we obtain x = -g
= [H](a)’
=
[(=&<,r)i(-&(3C+C))r]
for i = 1,2; &= 1
{u}’ = column vector consisting of in-plane nodal displacements. The out-of-plane displacement for describing the bending behaviour of a one-dimensional plate element of length 2a is given by w(5) = tll + a25 + a#
+ ad3 = [N”lT{a}‘,
where aE = arbitrary constants, shape function of element
=
(2)
NI [I
[No] = out-of-plane
N
{$={a;ax)
= column vector consisting displacements and rotations.
of out-of-plane
1,2,
(5)
[G]=[(2)T
-a
-i]
fori=l,2.
(6)
Geometric nonlinearity is obtained by working with the standard linear forms in an iterative manner. The displacement at any depth of a plate element can be expressed in terms of that at the datum as follows: ll: = u - z(aw/ax),
(7)
where u and u, are the displacements in the x-direction at the datum and at distance z from the datum, respectively. The strains of a plate element are defined in terms of the displacements at the datum as follows:
?
and
{ale=
fori=
where [H] is the bending curvature matrix. Soh and Soh [8] have discussed the problem of “self-straining” in plate elements, and the one point integration procedure was used to overcome it. The same procedure is adopted here and, consequently, the [G] matrix is modified as
and
5, = - 1 and
(4)
(1)
. . where [N’] = in-plane shape function of element
N, = (1 + 5,012
(3)
= at4iax +
;(awjaxy - z(a2w/ax2), (8)
nodal where t, , E,,, t,~ and txb are the total, in-plane, large
185
plate, i.e.
deflection and bending strains, respectively.
hl
+qg$g)
ZCT,~ dz
M = I,
shl
= I,
h2 (z, + Z&L dz s h,
=M,+zsN,
(13)
Therefore, the total strain increment is given by
where 1, = width of the plate element, zE= distance from the neutral axis of the plate element and
= Ata + ZAX,
(9)
where At,,, A2c,,= first- and second-order of large deflection strain increments, respectively, At, = total strain increment at the datum, Aei, Atlb = in-plane and bending strain increments, respectively. Plasticity
of the plate element
The elasto-plastic theory allows the estimation of stresses and strains in the plastic region, and subsequently establishes the constitutive relations which define the relation between incremental stress and incremental strain [3]. A multi-layer approach, which allows yielding at different levels, is employed to handle plasticity problems in the plate element. The stress at any level z from the datum line is given by
o,=N+ A
Mz
(I+ AZ:)’
the plate element In the absence of yielding, the elastic stiffnesses are given by E,A and EJ, where E, is the modulus of elasticity of the plate element. With the inclusion of plasticity, the total stiffnesses are given by EA = E,A, + EpAp EI = EJ, + E,Z,, 1
(14)
where A,, I, and A,, 1, are properties in the elastic and plastic regions, respectively; and Ep is the modulus of plasticity of the plate element. The axial load in the plate element can also be expressed as
(10) N = EAc, = EA(c,
where zI = distance between the neutral axis of the plate element and the datum line, N = axial load in the plate element, M = bending moment about the datum line, A = cross-sectional area of the plate element and I = second moment of area of the plate element about the neutral axis. Since each layer of the plate element can be treated as a one-dimensional line segment, the uniaxial yield criterion can be employed. The yield criterion is given by
F=$l,
= moment about the neutral axis of
(15)
Thus, M=M,+z,N = EIx + zs EA(e,
+ zsx)
= z, EA6.w + (EZ + z,‘EA)x.
In the case of elastic-perfectly behaviour, Ep = 0. Therefore,
(11)
where a0 = yield stress in the plate element. The axial load and bending moment can be obtained by integrating through the depth of the
+ ZJ).
N = EA&
(16)
plastic stress-strain
+ z&
(17)
and M = zPECAFG + (EL
+ z,zEeAc)x.
(18)
186
Ai-Kah Soh et al.
St@ess
formulation
-
The variational principle of minimum potential energy is employed in the formulation of the element stiffness matrix. At the state of equilibrium, the potential energy of the system becomes stationary with regard to all kinematically admissible variations in displacements. In the absence of body forces, the potential energy of a body may be written as +=U+H,
(P+AP)AddS ss
=
(N AC,,+ M Ax) dx s
(19) + z2 Ax’) dvol. -
where strain energy,
(P + AP) Ad dS ss z
CJ= jO, ([odr)dvol.
A4=
(NAc,,+MAy)dx+; s
and the virtual work due to applied loads,
+2(%)(g)(%) H= -
ss
P(d, - do) dS.
An increment of the total potential energy is given
+(g>‘(g>‘]dx+c*[[T
by
( >I
aw aAw
Ac$ =
+a~ yy
({c~}~{A\t}+ ;{Ac~}‘{Ac}) dvol. sWI
-
(P + AP) Ad dS -
AP(d - do) dS.
(20)
+e**jAX’dx-1
Axdx
(P+AP)AddS,
sS
sS
(21) The last term of eqn (20) does not involve the increment of deflection, Ad. It will, therefore, vanish when variations with respect to Ad are made on the total potential energy. For this reason the term will be omitted for the following development. Therefore, an increment of the total potential energy for a one-dimensional plate element is given by
AC#J =
where c is the tangential elasto-plastic modular; and c* and c** are related to c. Note that all terms involving third- or fourth-order of aAw/ax and aAu/ax and their products have been ignored. Equations (3j-45) can be expressed in an incremental form as follows:
(0, At, + tA0.y At,) dvol s“0,
. -
(22)
(P+AP)AddS ss e
(N Aest + M Ax) dx
= s
+;
By applying the principle of stationary potential energy, i.e. S(A#J) = 0, we obtain
s
c(Ae,, + z Ax)’ dvol. WI.
{P}’ + {AP}’ - (P}’ = [V{Adp,
total
(23)
187
Development of a simplified plate element where {P>’ is the vector of total external forces prior
to the application of the incremental loads; {BP}’ is the corresponding vector of the incremental loads; {P}’ is the internal load vector; [K]’ is the tangent stiffness matrix of the element, and {Ad}’ is the vector of the incremental nodal displacements. Equation (23) can also be expressed as
details of which can be found in Crisfield [6], is employed in the present analysis, except that a more accurate approach of calculating stresses on the yield surface is adopted. The accuracy of estimating these stresses has significant effects on the rate of convergence. In the present procedure, the concept of strain sub-increment is employed, in addition to the mean average stress approach [3] which has been commonly employed by researchers, by further dividing incremental strains into sub-increments. Sub-incremental stresses are then calculated using the more accurately determined average stresses. The procedure of the strain sub-increment approach is as follows:
where (P,} and {Pb) are the vectors of in-plane and out-of-plane forces, respectively, prior to the application of the incremental loads; (API}, {APL,} (1) Divide the incremental strain and {P, }, {&,} are the corresponding incremental and increments. i.e. internal loads, respectively; and [Ki], [Kb] and [Kc] are the in-plane, bending and coupled stiffness subma{At,,} =A {ACT)’ trices, respectively.
{P,} =
into n sub-
where {ALL}= incremental strain vector for the jth ith increment, and iteration of the {A+} = incremental strain vector for the kth sub-increment. (2) Determine the mean average stress vector, {a), which will then be used to determine the tangential elasto-plastic modular matrix, {ET}, i.e.
N[B] dx s
{I+,}=
and
[K~]= [c[BI[BIT dx
[Kb] = 2
s
c**[H][HITdx + [(N+
x [G][GIT dx +
where {u, _, } = stress vector on completion of the (i - 1)th increment, {o,.~_, } = stress vector on completion of the (k - 1)th sub-increment of the ith increment and F = plastic potential function. (3) Compute the sub-incremental stress vector, {AQ}, and add it to the previous stress vector to produce the current stress vector, i.e.
@>‘)
aw(~[GIT
c* z
{ Au,.~} = [ET]{ kt +
{ 6,~ ) = { ur.i-
[K,I=
}
KWIT) dx dx.
S(
) SOLUTION
I }
+
{
Aus.~. }
(4) Apply a factor, R, to the stress vector to bring J
PROCEDURE
A number of solution procedures are available for the large deflection elasto-plastic analysis of thin plated structures, e.g. the modified Newton-Raphson and Secant iterative methods [3]. The fixed length modified Riks’ procedure, which is the combination of Riks’ procedure [5] and the modified NewtonRaphson method [3], has been successfully employed by Crisfield [6] to perform nonlinear analysis of stiffened plates and shallow shells. This procedure,
L
-600mm
lnllial deflection, WC, = L/l00
E =ZQGPa b.
= 35oYPa
Fig. 1. An axially compressed strut
188
Ai-Kah a
12 equal length Iha element0 of6 layers each
Fig. 2. Finite element model for strut analysis.
it to the yield surface,
i.e.
{a} = R{O,.k}. (5) Repeat sub-increment.
steps
(lt(4)
for
the
next
Soh et al.
handled using five gaussian integration stations through the depth of the section. In the present study, the strut was divided into 24 equal length line elements, as shown in Fig. 2. Each element has two gaussian points which are compatible with the nodal points of the associated plate elements, as shown in Fig. 2. Each line element was sub-divided into six layers so that plastic yielding could be treated in each individual layer, when the average stress at the gaussian point reached a certain value.
DISCUSSION AND CONCLUSIONS
strain
TEST EXAMPLE
The reliability and accuracy of the proposed multi-layer approach for handling the plasticity problems of plate elements can be illustrated by analysing an axially compressed imperfect elastoplastic strut, as shown in Fig. 1. The strut has a slenderness ratio of 91 and a maximum initial deflection of O.OOlL. The dimensions of the strut are chosen such that the results of the present analysis can be compared with those obtained by Crisfield [9], who used the methods to obtain the results. The first method employed a single layer formulation technique in which the modified Ilyushin full section yield criterion was adopted. The second method employed the Ritz’s procedure in which the deflection shape was represented by a sine wave and plasticity was
Figure 3 shows the curves of a normalized compressive load vs normalized central deflection, obtained by the authors and Crisfield [9]. It is obvious that the load, at which the first fibre yield occurred, obtained by the authors is slightly lower than that predicted by Crisfield using the full section yield criterion, and is in excellent agreement with that predicted by Crisfield using Ritz’s procedure. The post-yielding curve obtained by the authors is, in general, slightly lower than those obtained by Crisfield by about 1.5% of the yield load. The above illustration shows that the multi-layer approach for handling plasticity problems of one-dimensional plate elements is reliable and accurate. This type of plate element can be employed to simulate stiffeners in the modelling of stiffened plate structures for large deflection elasto-plastic finite element analysis.
REFERENCES
1. C. Y. Chia, Nonlinear Analysis of Plates. McGraw Hill, London (1980). 2. T. H. Soreide and T. Moan, Nonlinear Maierial and Geometric Behaviour of Stiffened Plates. Division of Ship Structures, The Norwegian Institute of Technoloev. The Universitv of Trondhlim. Norwav (1975). of 3. M. A. Crisfield, Nonlinear Finite Element k,alysk Solids and Structures, Vol. I. Wiley, New York (1991). and P. J. Dowling, 4. P. A. Frieze, R. E. Hobbs Application of dynamic relaxation for large deflection elasto-plastic analysis of plates. Comput. Strucf. 8, 301-310 (1978). approach to the solution of 5. E. Riks, An incremental mapping and buckling problems. Int. J. Solids Struct. 15, 524-551 (1979). non-linear analysis of 6. M. A. Crisfield, The automatic stiffened plates and shallow shells using finite elements. Proc. Institute of Civil Engineers, Part 2, December, pp. 891-909 (1980). Approximation in the non-linear 7. M. A. Crisfield, analysis of rectangular plates using finite elements, TRRL Report SRSIUC, U.K. (1974). 8. A. K. Soh and C. K. Soh, Elimination of self-straining in four-noded plate elements for large deflection analysis. Comput. Struct. 47, 341-342 (1993). buckling 9. M. A. Crisfield, Large deflection elasto-plastic analysis of eccentrically stiffened plates using finite elements, TRRL Report LR725, U.K. (1976). v,
Fig. 3. Relation
between load and deflection compressed strut.
for an axially
Pergamon PII: SOO45-7949(%)00022-3
& Strucrures Vol. 61. No. I. pp. 183-188, 1996 Copyright SC 1996 Elsevier Science Ltd Prmted in Great Britain All rights reserved 004%7949196 $15.00 + 0.00
DEVELOPMENT OF A SIMPLIFIED PLATE ELEMENT LARGE DEFLECTION ELASTO-PLASTIC FINITE ELEMENT ANALYSIS
FOR
Ai-Kah Soh, Chee-Kiong Soh and Kay-Hiang Hoon Nanyang Technological University. Nanyang Avenue, Singapore 639798 (Receiaed 6 October 1994)
Abstract-Numerous researchers have performed large deflection elasto-plastic finite element analysis of stiffened plate structures, using plate elements which employed various iterative/incremental procedures for nonlinear analysis. However, such analysis always requires a large number of plate elements, which become a heavy burden on the computer in terms of computation time and disk storage. Fortunately, the problem can be reduced by employing the one-dimensional version of plate elements to simulate the stiffeners of stiffened plate structures. This type of element has been developed in the present study, by employing the multi-layer approach which allows yielding at different levels of the plate element. Special care has been taken to avoid the “self-straining” problem of plate elements. Moreover, an improved iterative/incremental procedure has been proposed to establish a more reliable and accurate solution. The results obtained by employing this element type for a test example are found to be in good agreement with those obtained by other authors using traditional large deflection elasto-plastic finite element analysis. Copyright 0 1996 Elsevier Science Ltd.
INTRODUCTION
NOTATION
t,i
A N M A I E I\, d (30 4 u H P d ‘
WI
PI
Cartesian coordinates displacements in the x-, y- and z-directions, respectively rotation normalized local x coordinate curvature direct strain total, in-plane, large deflection and bending strains, respectively, in the x-direction total strain at the datum in the x-direction increment axial load in the plate element bending moment about the datum line cross-sectional area of plate element second moment of area of the plate element about the neutral axis Young’s modulus width of plate element direct stress yield stress potential energy strain energy virtual work due to applied loads external force generalized displacement tangent elasto-plastic modular in-plane shape function out-of-plane shape function in-plane nodal displacements of an element out-of-plane nodal displacements and rotations of an element in-plane strain matrix bending curvature matrix slope matrix
Subscripts e
P C
elastic plastic with respect to neutral axis
Many researchers have studied the structural response of stiffened plate structures for which failure often involves both instability and material yielding
in some of their plate components [l]. A significant amount of such study was performed by Soreide and Moan [2], Crisfield [3] and Frieze et al. [4], using the finite element technique. Several iterative/incremental procedures [3] had been proposed for the large deflection elasto-plastic finite element analysis of thin plated structures. However, these procedures often encountered convergence problems in the analysis of stiffened plate structures in which complicated buckling modes and welding residual stresses occurred in both the plates and stiffeners. A solution procedure for overcoming the convergence problem was proposed by Riks [5] and later modified by Crisfield [6] to make it suitable for adoption by the finite element technique. The full section yield criterion, in which the behaviour of the material is assumed to be elastic until yield occurs, at which time the shell element suddenly becomes fully plastic, was employed by Crisfield [7] to perform the elasto-plastic finite element analysis. This full section assumption is not as accurate as considering plastic flow through the depth of the plate thickness, but it has the advantage of reducing the computation time and disk storage required considerably. Moreover Crisfield [7] has shown that both approaches produced very similar results in a large deflection elasto-plastic analysis of plates. Thus, Crisfield’s formulation based on full section yield criterion is commonly employed to perform elasto-plastic finite 183
184
Ai-Kah Soh et al.
element analysis. However, collapse analysis of complex stiffened plate structures will require numerous plate elements, which will be a heavy burden on the computer in terms of computation time and disk storage. Thus, it would be useful to develop the one-dimensional version of plate elements for simulating the stiffeners of complex stiffened plate strucutres. The development of this element type will be discussed in this paper. Moreover, some improvements made to the traditional approach of formulating plate elements for large deflection elasto-plastic analysis will also be discussed.
By differentiating
6, = g =
eqn (1) we obtain
PI{+ = & b5~zl{u}‘,
where [B] is the in-plane strain matrix. By differentiating eqn (2) we obtain
1
for i = 1,2,
x (a(35’ - 1 + 25&))T ANALYSIS
The in-plane displacement function of the onedimensional version of a plate element is given by u(5) =
[N’lT{uJe,
where [G] is the slope matrix of the plate element. By differentiating eqn (2) twice, we obtain x = -g
= [H](a)’
=
[(=&<,r)i(-&(3C+C))r]
for i = 1,2; &= 1
{u}’ = column vector consisting of in-plane nodal displacements. The out-of-plane displacement for describing the bending behaviour of a one-dimensional plate element of length 2a is given by w(5) = tll + a25 + a#
+ ad3 = [N”lT{a}‘,
where aE = arbitrary constants, shape function of element
=
(2)
NI [I
[No] = out-of-plane
N
{$={a;ax)
= column vector consisting displacements and rotations.
of out-of-plane
1,2,
(5)
[G]=[(2)T
-a
-i]
fori=l,2.
(6)
Geometric nonlinearity is obtained by working with the standard linear forms in an iterative manner. The displacement at any depth of a plate element can be expressed in terms of that at the datum as follows: ll: = u - z(aw/ax),
(7)
where u and u, are the displacements in the x-direction at the datum and at distance z from the datum, respectively. The strains of a plate element are defined in terms of the displacements at the datum as follows:
?
and
{ale=
fori=
where [H] is the bending curvature matrix. Soh and Soh [8] have discussed the problem of “self-straining” in plate elements, and the one point integration procedure was used to overcome it. The same procedure is adopted here and, consequently, the [G] matrix is modified as
and
5, = - 1 and
(4)
(1)
. . where [N’] = in-plane shape function of element
N, = (1 + 5,012
(3)
= at4iax +
;(awjaxy - z(a2w/ax2), (8)
nodal where t, , E,,, t,~ and txb are the total, in-plane, large
185
plate, i.e.
deflection and bending strains, respectively.
hl
+qg$g)
ZCT,~ dz
M = I,
shl
= I,
h2 (z, + Z&L dz s h,
=M,+zsN,
(13)
Therefore, the total strain increment is given by
where 1, = width of the plate element, zE= distance from the neutral axis of the plate element and
= Ata + ZAX,
(9)
where At,,, A2c,,= first- and second-order of large deflection strain increments, respectively, At, = total strain increment at the datum, Aei, Atlb = in-plane and bending strain increments, respectively. Plasticity
of the plate element
The elasto-plastic theory allows the estimation of stresses and strains in the plastic region, and subsequently establishes the constitutive relations which define the relation between incremental stress and incremental strain [3]. A multi-layer approach, which allows yielding at different levels, is employed to handle plasticity problems in the plate element. The stress at any level z from the datum line is given by
o,=N+ A
Mz
(I+ AZ:)’
the plate element In the absence of yielding, the elastic stiffnesses are given by E,A and EJ, where E, is the modulus of elasticity of the plate element. With the inclusion of plasticity, the total stiffnesses are given by EA = E,A, + EpAp EI = EJ, + E,Z,, 1
(14)
where A,, I, and A,, 1, are properties in the elastic and plastic regions, respectively; and Ep is the modulus of plasticity of the plate element. The axial load in the plate element can also be expressed as
(10) N = EAc, = EA(c,
where zI = distance between the neutral axis of the plate element and the datum line, N = axial load in the plate element, M = bending moment about the datum line, A = cross-sectional area of the plate element and I = second moment of area of the plate element about the neutral axis. Since each layer of the plate element can be treated as a one-dimensional line segment, the uniaxial yield criterion can be employed. The yield criterion is given by
F=$l,
= moment about the neutral axis of
(15)
Thus, M=M,+z,N = EIx + zs EA(e,
+ zsx)
= z, EA6.w + (EZ + z,‘EA)x.
In the case of elastic-perfectly behaviour, Ep = 0. Therefore,
(11)
where a0 = yield stress in the plate element. The axial load and bending moment can be obtained by integrating through the depth of the
+ ZJ).
N = EA&
(16)
plastic stress-strain
+ z&
(17)
and M = zPECAFG + (EL
+ z,zEeAc)x.
(18)
186
Ai-Kah Soh et al.
St@ess
formulation
-
The variational principle of minimum potential energy is employed in the formulation of the element stiffness matrix. At the state of equilibrium, the potential energy of the system becomes stationary with regard to all kinematically admissible variations in displacements. In the absence of body forces, the potential energy of a body may be written as +=U+H,
(P+AP)AddS ss
=
(N AC,,+ M Ax) dx s
(19) + z2 Ax’) dvol. -
where strain energy,
(P + AP) Ad dS ss z
CJ= jO, ([odr)dvol.
A4=
(NAc,,+MAy)dx+; s
and the virtual work due to applied loads,
+2(%)(g)(%) H= -
ss
P(d, - do) dS.
An increment of the total potential energy is given
+(g>‘(g>‘]dx+c*[[T
by
( >I
aw aAw
Ac$ =
+a~ yy
({c~}~{A\t}+ ;{Ac~}‘{Ac}) dvol. sWI
-
(P + AP) Ad dS -
AP(d - do) dS.
(20)
+e**jAX’dx-1
Axdx
(P+AP)AddS,
sS
sS
(21) The last term of eqn (20) does not involve the increment of deflection, Ad. It will, therefore, vanish when variations with respect to Ad are made on the total potential energy. For this reason the term will be omitted for the following development. Therefore, an increment of the total potential energy for a one-dimensional plate element is given by
AC#J =
where c is the tangential elasto-plastic modular; and c* and c** are related to c. Note that all terms involving third- or fourth-order of aAw/ax and aAu/ax and their products have been ignored. Equations (3j-45) can be expressed in an incremental form as follows:
(0, At, + tA0.y At,) dvol s“0,
. -
(22)
(P+AP)AddS ss e
(N Aest + M Ax) dx
= s
+;
By applying the principle of stationary potential energy, i.e. S(A#J) = 0, we obtain
s
c(Ae,, + z Ax)’ dvol. WI.
{P}’ + {AP}’ - (P}’ = [V{Adp,
total
(23)
187
Development of a simplified plate element where {P>’ is the vector of total external forces prior
to the application of the incremental loads; {BP}’ is the corresponding vector of the incremental loads; {P}’ is the internal load vector; [K]’ is the tangent stiffness matrix of the element, and {Ad}’ is the vector of the incremental nodal displacements. Equation (23) can also be expressed as
details of which can be found in Crisfield [6], is employed in the present analysis, except that a more accurate approach of calculating stresses on the yield surface is adopted. The accuracy of estimating these stresses has significant effects on the rate of convergence. In the present procedure, the concept of strain sub-increment is employed, in addition to the mean average stress approach [3] which has been commonly employed by researchers, by further dividing incremental strains into sub-increments. Sub-incremental stresses are then calculated using the more accurately determined average stresses. The procedure of the strain sub-increment approach is as follows:
where (P,} and {Pb) are the vectors of in-plane and out-of-plane forces, respectively, prior to the application of the incremental loads; (API}, {APL,} (1) Divide the incremental strain and {P, }, {&,} are the corresponding incremental and increments. i.e. internal loads, respectively; and [Ki], [Kb] and [Kc] are the in-plane, bending and coupled stiffness subma{At,,} =A {ACT)’ trices, respectively.
{P,} =
into n sub-
where {ALL}= incremental strain vector for the jth ith increment, and iteration of the {A+} = incremental strain vector for the kth sub-increment. (2) Determine the mean average stress vector, {a), which will then be used to determine the tangential elasto-plastic modular matrix, {ET}, i.e.
N[B] dx s
{I+,}=
and
[K~]= [c[BI[BIT dx
[Kb] = 2
s
c**[H][HITdx + [(N+
x [G][GIT dx +
where {u, _, } = stress vector on completion of the (i - 1)th increment, {o,.~_, } = stress vector on completion of the (k - 1)th sub-increment of the ith increment and F = plastic potential function. (3) Compute the sub-incremental stress vector, {AQ}, and add it to the previous stress vector to produce the current stress vector, i.e.
@>‘)
aw(~[GIT
c* z
{ Au,.~} = [ET]{ kt +
{ 6,~ ) = { ur.i-
[K,I=
}
KWIT) dx dx.
S(
) SOLUTION
I }
+
{
Aus.~. }
(4) Apply a factor, R, to the stress vector to bring J
PROCEDURE
A number of solution procedures are available for the large deflection elasto-plastic analysis of thin plated structures, e.g. the modified Newton-Raphson and Secant iterative methods [3]. The fixed length modified Riks’ procedure, which is the combination of Riks’ procedure [5] and the modified NewtonRaphson method [3], has been successfully employed by Crisfield [6] to perform nonlinear analysis of stiffened plates and shallow shells. This procedure,
L
-600mm
lnllial deflection, WC, = L/l00
E =ZQGPa b.
= 35oYPa
Fig. 1. An axially compressed strut
188
Ai-Kah a
12 equal length Iha element0 of6 layers each
Fig. 2. Finite element model for strut analysis.
it to the yield surface,
i.e.
{a} = R{O,.k}. (5) Repeat sub-increment.
steps
(lt(4)
for
the
next
Soh et al.
handled using five gaussian integration stations through the depth of the section. In the present study, the strut was divided into 24 equal length line elements, as shown in Fig. 2. Each element has two gaussian points which are compatible with the nodal points of the associated plate elements, as shown in Fig. 2. Each line element was sub-divided into six layers so that plastic yielding could be treated in each individual layer, when the average stress at the gaussian point reached a certain value.
DISCUSSION AND CONCLUSIONS
strain
TEST EXAMPLE
The reliability and accuracy of the proposed multi-layer approach for handling the plasticity problems of plate elements can be illustrated by analysing an axially compressed imperfect elastoplastic strut, as shown in Fig. 1. The strut has a slenderness ratio of 91 and a maximum initial deflection of O.OOlL. The dimensions of the strut are chosen such that the results of the present analysis can be compared with those obtained by Crisfield [9], who used the methods to obtain the results. The first method employed a single layer formulation technique in which the modified Ilyushin full section yield criterion was adopted. The second method employed the Ritz’s procedure in which the deflection shape was represented by a sine wave and plasticity was
Figure 3 shows the curves of a normalized compressive load vs normalized central deflection, obtained by the authors and Crisfield [9]. It is obvious that the load, at which the first fibre yield occurred, obtained by the authors is slightly lower than that predicted by Crisfield using the full section yield criterion, and is in excellent agreement with that predicted by Crisfield using Ritz’s procedure. The post-yielding curve obtained by the authors is, in general, slightly lower than those obtained by Crisfield by about 1.5% of the yield load. The above illustration shows that the multi-layer approach for handling plasticity problems of one-dimensional plate elements is reliable and accurate. This type of plate element can be employed to simulate stiffeners in the modelling of stiffened plate structures for large deflection elasto-plastic finite element analysis.
REFERENCES
1. C. Y. Chia, Nonlinear Analysis of Plates. McGraw Hill, London (1980). 2. T. H. Soreide and T. Moan, Nonlinear Maierial and Geometric Behaviour of Stiffened Plates. Division of Ship Structures, The Norwegian Institute of Technoloev. The Universitv of Trondhlim. Norwav (1975). of 3. M. A. Crisfield, Nonlinear Finite Element k,alysk Solids and Structures, Vol. I. Wiley, New York (1991). and P. J. Dowling, 4. P. A. Frieze, R. E. Hobbs Application of dynamic relaxation for large deflection elasto-plastic analysis of plates. Comput. Strucf. 8, 301-310 (1978). approach to the solution of 5. E. Riks, An incremental mapping and buckling problems. Int. J. Solids Struct. 15, 524-551 (1979). non-linear analysis of 6. M. A. Crisfield, The automatic stiffened plates and shallow shells using finite elements. Proc. Institute of Civil Engineers, Part 2, December, pp. 891-909 (1980). Approximation in the non-linear 7. M. A. Crisfield, analysis of rectangular plates using finite elements, TRRL Report SRSIUC, U.K. (1974). 8. A. K. Soh and C. K. Soh, Elimination of self-straining in four-noded plate elements for large deflection analysis. Comput. Struct. 47, 341-342 (1993). buckling 9. M. A. Crisfield, Large deflection elasto-plastic analysis of eccentrically stiffened plates using finite elements, TRRL Report LR725, U.K. (1976). v,
Fig. 3. Relation
between load and deflection compressed strut.
for an axially