- Email: [email protected]
On creep buckling analysis of structures
004~7949181/05lm8~.@/0 Permmon Press Ltd.
ON CREEP BUCKLING ANALYSIS OF STRUCTURES
Department
of Mechanical
JAN WALCZAKt Engineering, Rzeszow Technical University,
Rzeszow, Poland
Abstract-Analysis of creep buckling using ADINA is described and evaluated. The main emphasis is on the selection of solution time steps. An energy method is presented to find an approximate time to buckle. The evaluation of ADINA results is based on another numerical solution of creep buckling columns. We also give some practical guidelines for creep buckling analysis in general.
1. INTRODUCTION
The criterion states, that the dissipative deformation process cannot be unlimited as regards its duration. We assume therefore an energy barrier, the attainment of which leads to a critical state. This is due to the fact that, at a certain instant of time, the material loses its ability to dissipate energy. After series of modifications the criterion may be written as follows:
Creep buckling analysis of columns has been studied by numerous authors. In those investigations it is generally defined in terms of unbounded deflections or deflection rates. Only a few of the authors included the geometrical nonlinearity in the way of large deflection analysis. They presented however different conclusions. Zyczkowski [ I] found that due to the geometrical nonlinearity, the finite critical time for creep buckling would not exist. Huang’s conclusion[2] is that the creep buckling may occur but only if the effect of plasticity is included in the creep law. Both authors have been using in their papers Norton’s nonlinear creep law and the collocation method. Samuelson[3] exposed, using Runge-Kutta methods, that creep buckling occurs with no additional conditions such as Huang’s. This conclusion is in a good agreement with experimental observations and. as it is going to be seen, with ADINA results and those presented in the paper. Nonlinear creep buckling analysis using ADINA requires an unfailing solution strategyI4-41. One of the most important problems is an appropriate selection of time steps irrespective of solution methods (with or without iterations). Observations of creep tests lead to a conclusion, that the critical time (time to buckle) may be depended on that part of energy which is dissipated by the body. Thus, assuming an energy criterion we might be able to estimate the approximate critical time tzl. Having obtained trr and assuming a condition for a single time step AI, we may choose adequately the nbmber of time steps required for an ADINA solution. Another consideration when performing an ADINA creep analysis is that the finite element mesh must also be adequate. For an evaluation of the ADINA creep material model and those above parameters, a numerical solution is employed for creep buckling analysis of columns. The method is based on the exact bending line (large deflection analysis) for which a differential equation is established and solved using the Newton-Raphson iteration method.
f(&, Ed) = const. = K,
where 6, is the effective stress, Ed the specific dissipated energy and K is a material constant. The dissipated energy may be obtained from the equation: Ed =
.&
*cr
dt.
(2.2)
The Norton-Odqvist creep law for a plane stress analysis and incompressible materials may be written as follows:
i,=B*S:-‘. i,=B.S:-'.
(a-*+1 (&+, ),
(2.3)
where S,, S2are principal stresses, and i,, i2 are principal strain rates. Taking into account eqn (2.3) we have:
ri=Slj(X,Yst) &[6j(X,YTt)l ’
=,.,:-‘.[(,,-t,,),,+(~~-~6 & = B ’ ,:+I.
2. THEENERGY CRRERION OF CREEP BUCKLING The idea has been forwarded by Bychawski and Olszak in 1%8[71. They assumed that for visco-elastic bodies. accumulated and dissipated energies may be equally responsible for a transition into a critical state. The idea has been further employed for creep rupture problems by Kopecki and Walczak[8] and [9], where the form of a dissipated barrier has been established.
+Assistant Professor of Mechanical
(2.1)
(2.5)
The form of the function f(Ed) has been established on the grounds of the rheological characteristics of materials, for which the creep curves i = i(t) for constant 6 are known in a broad variability range of the stress. If we know the creep curves, we can evaluate the amount of the energy dissipated up to the critical instant of time for constant S. Thus we are able to evaluate the critical time tz, after the material is no more able to dissipate energy. The analysis of creep data for a number of materials
Engineering. 683
684
J. WALCWK
proves that the amount of energy up to creep failure satisfiesthe followingrelation: jr&, Ed) = $
=
= I(.
(2.6)
This means that the dissipationbarrier depends on two physical parameters, namely the dissipated energy and the effective stress. Taking into account eqns (2.5) and (2.6)we find
where K and fl are material constants, which, for a chromium-molybdenum-siliconsteel at S4ooC,are found to be: ,9 = 0.665,K = 1.K~127 r 10-4. The above formula gives good results in creep rupture analyses, under constant stresses@,91.In creep buckling problems,stresses are not constant and the response of a structure is unknown at the beginningof a analysis. But we need to have this criterion as an additional approach to the problem, so we can use a reduced form of eqn (2.7)to estimate the range of the critical time for a certain value of the effective stress, which we mightcall a representative stress. 3. CREEP BUCKLING ANALYSIS
elements respectively, for a plane stress analysis. In all cases we have been using4 x 4 Gauss integrationpoints. All numerical results are presented in graphic forms. Figure I represents a solution for column No. 1. The finite element mesh shown in the figure, has been estabhshed after a number of tests. As it turned out, it was good enough to model the transversal dimensionof the column using one eight-node element with 4x4 Gauss points, and the longitudinal dimension by six eiements.The time step Al has been chosen on the basis of Samue~sonsolutionsB]. The inte~at~n parameter a has been set to 0.0 by solving without iterations and OS-using iterations. Employing the BFGS iteration method[6]allowed us to use a five times larger time step than in the previous method. This did not cut down however the CPU time. Usingmuch larger time steps, as is shown in Fig. 2, reduced the CPU time and gave inaccurate results (similarly with the modified Newton iteration method),but the critical time was not exceeded. More subdivisions per time steps did not improve significantlyresults in this problem, so only one subdivisionper At has been used. Solutionsfor the column No. 2 are presented in Fig. 3. The best one obtained is that without iterations, for a relatively small time step. The BFGS iteration method gave quite good results for ten times larger time steps,
USING ARINA
As is pointed out by Bathe[5], the nonlinear finite element analysis is still faced with difficulties, particularly in creep buckling problems. A major question relates to the finite element mesh to be used because of expenses and accuracy. Another problem lies in the selectionof an appropriatetime step and this is the most difficult task. The last one we want to underline is a solution method. We should also say there are no simple answers for these questions and they are di@cult to generalize. To show some of the problems we wiIi study two kinds of columns for two different material sets. One of them (columnNo. 1) is relatively short and thick, with a slendernessratio A= 36.9.The other one (columnNo. 2), is long and slender with A= 554.2.Since the columnscan undergo the large deflection analysis, we use the total Lagrangian formuiation[4,5], and the thermo~iasticplastic and creep material mode! for 2/TIsolid elements. The columns are modeled with six and eight, eight-node
Fip.1.
GmI-
If-
I-dr-01. ~-f~~f~ot~on IO---
2-4t-40,y-c0&,8~6s
3-dt-4o,_x-cotut.N-R of_ ._ 4- dt-80, 5%const. BFGS
J- At-c4R K-const , N-R
Fig. 2.
1
685
On creep buckling analysis of structures
Fig. 3. but the modified Newton iteration method failed to converge at an early stage (see Fig. 3). The solution time step Ar is established on the basis of the energy criterion. For this we can use the representative stress S, which might be defined as a mean stress 6, = (60+ 6,.,)/2 = (z&t g/2
(3.1)
where & denotes the initial compressive stress, P is the applied load, W is the section modulus, and f is the assumed maximum deflection of the column. One way to find f may be as follows
f+;
0.2)
where A0 is a limiting slenderness ratio, h-transversal dimension of the column. In our example A = 554.2, A0= 88.0, h = 2.0, thus f = 6.3. Hence 6, = 627.0 kG/cm’
(3.3)
Taking into account eqn (2.7) one can find an approximate time to buckle t:,= 22748.0h.
The form for the critical time step At (eqn (3.7)) gives also good results for column No. 1 and it may be useful in other problems involving creep buckling behaviors. It is also seen from Fig. 3, that the critical time obtained from the ADINA solution is close to 16,Oo hr, which is comparable with the approximate result. As has been mentioned above, for the BFGS and modified Newton iteration methods, the time step At has been selected ten times larger than the above. It would be interested to see CPU times on a IBM 370 machine. Let us make a comparison for solutions presented in Fig. 3:
1 BFGS, 1.06, +%0.58. tno,,cr
noIt.%
Using smaller time steps for the BFGS and modified Newton methods, would increase considerably the CPU times. In the above solutions only one subdivision size per time step is used again. More subdivisions are time consuming with not enough evidence that the results in this problem are improved. The ADINA solutions are next evaluated by a numerical procedure applied to the same columns under the same conditions.
(3.4)
A criterion for a single time step At may be written as follows
4. A NUMEIUCALPROCRDURE FOR CREEP BUCKLING ANALYSIS OF COLUMNS
The creep buckling behavior of a column may be described in the natural coordinates (s, 0) (Fig. 4): which are related to the Cartesian coordinates (x, y) m well known forms which leads to a form x=
,,
E
. B
.
&(n-l)
(3.6) Y=
where E,,~is an inelastic strain tolerance. Taking S, = S, and E101= 0.1, one can find that the time step for this problem should be smaller than 69.2hr, thus it has been assumed as much as 50 hr. Having obtained the approximate critical time tz, and the critical time step, we can perform more adequately the finite element analysis.
I
I
cl I
cos ~9ds (4.1) sin 8 ds,
where B = 0 (s, t) is our investigated solution. To avoid numerical integrations along the thickness of the column we will make an idealization of the cross section, similar to Samuelson’s assumption[3]. The cross section is divided into a number of flanges and cores
686
J.
WALCZAK
consider a deformed element (Fig. 6). We can find from the figure
P
Y 4
E. -
Ev _
I
7rR=X
(4.4)
where ez and E, denotes strains of extreme fibres, H is the distance between the fibres and X is the curvature of the deformed column, and (4.5) Taking the first derivative of eqn (4.4) with respect to time we get t9
1,
Equation (4.6) holds for any par of flanges, thus it may be written in a system of equations
X
‘l=&P~H’, E,’- E,
Fig. 4. (Fig. 5), where the cores do not take normal forces. All dimensions shown in Fig. 5 can be obtained from a comparison of inertia moduli
i=l2 , ,...1 fn.
Hence we obtain .I
Q -
.I eu - 2 i,’ + 2 E,I = %‘(HI - 2 Hi). i-Z
&= [12z
(:Y;“)‘]‘”
(4.7)
(4.8)
For any flange we can evaluate stresses as follows P Hi
6,=6 .-m-2itl I 0 2
61.,=-socose~~*_z~y
(4.2)
h=h/m
(4.9)
where
where m is the number of flanges. The constitutive equation is accepted in the form of Norton creep law
I =g
(rectangular cross section),
and I
Y=
i=;tB.S", where the creep term has the following meaning: B. S” = B . l8r-l.
S.
sin 6 ds.
Differentiating eqn (4.9) with respect to time we have ;I b z.v
= a0 sin 0 . e’ + $$[(,‘sint?ds]. (4.10)
To find a bending line of the column we have to
Fig. 5.
Fig. 6.
687
On creep buckling analysis of structures
Substitution of eqns (4.3) and (4.10) into eqn (4.8) leads to a differential equation which may be written in the form s.$[lsinBds]
n 1
[ . I*
+S,
u
sineds-&cosB
+~*[,Il~~,~(~i:sin,d~+~~cos~~n ” -e*&
(I,‘sin
(4.11)
Bds)]
-(-l)“.B[3lsinBds +60~6~
e
Equation (4.11) can be solved by the Newton-Kaphson iteration method. To achieve the goal it would be convenient to rewrite it as follows ,+a
2 . K=E*
+ 2 1-Z
--
2 E
,+A$,
(-
1)”
+ B(“A’F,
. B(‘+A’F,
(4.16)
_ c+Al~)n
+ ‘+A’s)”
-
B
. (I+A)Fi
-
‘+A’s)”
[
. ‘+Atfii
1
+ (_
1)B.
the curvature at the end of each iteration be within a tolerance of the true solution. Hence the convergence criterion is
(‘+A’,,
(4.12)
+‘+A~s)
where 1]Ki<112 denotes the Euclidean norm of K[lO], and lk is a curvature tolerance. Unfortunately the vector ‘+“K is unknown, so we
have been using a modifiedcriterion
where <
f+Af
* K= f+Ar%‘( Hi - 2 &)
‘+A3 = 60cos (‘+“‘e), (4.13)
4,
(4.17)
which was effective in the analysis. Figure 7 shows the predicted displacement response for column No. 1. As expected, it can be seen that increasingthe number of flangeswe get a better solution, and the approximation by six flanges is an optimum. Figure 8 presents solutions for column No. 2. Finally Figs. 3 and 10 present comparisons of these results and those obtained from the ADINA analysis. Additionally
All above integrals have been carried out numerically by the trapezoid rule. The columns have been divided into twenty elements, and more subdivisions did not change results. To solve eqn (4.2) we have to replace all differentials by finite differences I+AfKW=
[f+At@k,
_
r+Ar&k-1) t+Ar~/k-II
‘K]/&
=
[I+AtK(k-1)
=
[t+Are(k-1)
=
ft+At@k-I)_
_
+
AK
_
SK]/&
to],& t~~]/~t,
where the superscript k denotes iterations. Substituting the above formulas into eqn (4.12) we have AK=
r+Ar@k-l,At
+
‘K_
t+AlK(k--I),
where ‘+“RCk-‘)is the r.h.s. of eqn (4.12).
Since we are seeking the curvature of the column correspondingto time t + At, it is natural to require that
there is presented one of Samuelsonsolutions[3],based on a different method. As can be seen, there are some minor differencesdue to the different assumptions.In the case of column No. 1 only in the ADINA analysis plastic
688
J. WALCZAK
7- ADNA
20 -
2 - gfesented SulutLon 3-~amefsun
-
1f -
Fig. 9.
-
7- ALVNA .?- presented SUhhUff
.
I#
3---2
/ p
2
4
6
8
I
72
f MJf!‘3
Fig. 10. strains have been included, and this accelerated the process. Small deviations of the initial column imperfections give also significant changes in the solutions. It should be underlined that it is difficult to compare any creep solutions based on dislerent methods. From that point of view, it is evident that the agreement is good. 5. CONCLUSIONS (a) In the initial analysis of creep buckling problems, where the response of the structure is unknown, the energy criterion may be useful. It gives the range of the period of time for the analysis with no computational effort. Also the use of the dissipative barrier gives good
results, but has one disadvantage-two more material constants. This is not a problem however, when we deal with a known material (i.e. for which the creep curves are known). It should be recognized that in reality, the criterion is a creep failure criterion determined by the local stress condition. The creep failure may occur prior to creep. buckling, but this is beyond the scope of the paper. (b) The ADINA thermo-elastic-plastic and creep material model gives good results and is relatively inexpensive. As has been shown, there are some minor differences in our comparative analysis, which are mainly due to necessary assumptions in the applied procedure but they testify in favor of ADINA. (c) To obtain a good ADINA solution we have to set up some vital parameters. We can summarize our experiences in the problems analyzed as follows: -The analysis is most economically performed without iterations with the stiffness matrix reformed every time step. -In using the modified Newton or BFGS iteration method it should be recognized that the cost of the analysis can be high. For slender columns the BFGS method is preferable. Using the BFGS iteration method we usually get the same results for five times larger time steps than without iterations. For larger time steps the columns “look stiffer” but the results never exceeded the critical time in either equilibrium iteration method. -It is usually enough to select one subdivision per time step. OnIy for large stress gradients, close to the buckling time, more subdivisions improve the results. (d) The presented numerical solution for creep buckling columns is simple and convenient to use on minicomputers, gives good results and might be applied for any cross section area with one symmetry axis. (e) As is shown in the paper, the ADINA results and the numerical solutions prove that the critical time for creep buckling, defined at the beginning of the paper, exists and this does not depend on the plastic strains in the constitutive equations. Only in column No. 1 the plastic strains have been included in the ADINA analysis. In the numerical procedure elastic and creep strains have been generated in either column. For column No. 2
Fig. II.
On creep buckling analysis of structures
with no plasticity, on the ground of the ADINAanalysis, the finite critical time is likely to exist with no additional conditions for the load level, except that the load must be smaller than the instantaneous buckling load. This conclusion highly disagrees with Huang’s~Z]. (f) The idea of evaluationsof the approximatetime to buckle and the critical time step may be also appliedto a creep bucklinganalysis of spherical axisymmetricshells by using 2/D solid elements for the axisymmetricanalysis. The representative stress S, may be taken from the elastic response (see Fig. 11). (g) It appears that an adaptive time step for the ADINA creep analysis might be very useful and it ap pears that the dissipated energy may be helpful in the selection of the time steps. Acknowkdgemenfs--I wish to take this opportunity to express my appreciation to Prof. Klaus-J~rgen Bathe for valuable discussions and su~estions while I was working with him at M.I.T. REFERENCES 1. M. Zycxkowski, Geometrically nonlinear creep buckling of bars. Creep in Structures, Colloquium held at Stanford University, California, II-15 July, 2960(Edited by N. J. Hoff}. Academic Press, New York ft%2).
689
2. N. C. Huang, Creep buckling of imperfect columns. J. Appl. Mech.. Trans. ASME, March 1916. 3. A. Samuelson, Creep deformation and buckling of a column with an arbitrarv cross section. The Aeronautical Research Institute of Sweden, Rep. 107(1966). 4. K. J. Bathe, ADINA: a finite element program for automatic dynamic incremental nonlinear analysis. Rep. 824481, Acoustics and Vibration Lab., M.I.T., Sept. 1975 (revised Dec. 1978). 5. K. J. Bathe, Static and dynamic geometric and material nonlinear analysis using ADINA. Rep. 82448-2, Acoustics and Vibrat~n Laboratory, Mechanic E~neering Depart” ment, M.I.T., May 1976(revised May 1977). 6. K. J. Bathe and A. Cimento, Some practical procedures for the soiution of nonlinear finite element equations. L Comput. Meth. ADDS.Mech. Enene 22. SMS 119801. 7. Z. By&a&i and W: GI&k, Creep failureof nonlinear rotationalshells.8th Gong.ht. Association for Bridge and Struck-at Engineeting,Zurich, New York, 1968. 8. H. Kopecki and J. Walcxak, The energy dissi~tion barrier as a criterion of creep failure of rotating discs. Archiwum Budowy Maszyn 4 (19761. 9. J. Walczak, Critical states of rotating discs in nonlinear creep conditions (in Polish). Engng Trans. 27, 2 (1979). 10. K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Ekment Analysis. Prentice-Hall, Engiewood Cliffs, New Jersey (1976).
ON CREEP BUCKLING ANALYSIS OF STRUCTURES
Department
of Mechanical
JAN WALCZAKt Engineering, Rzeszow Technical University,
Rzeszow, Poland
Abstract-Analysis of creep buckling using ADINA is described and evaluated. The main emphasis is on the selection of solution time steps. An energy method is presented to find an approximate time to buckle. The evaluation of ADINA results is based on another numerical solution of creep buckling columns. We also give some practical guidelines for creep buckling analysis in general.
1. INTRODUCTION
The criterion states, that the dissipative deformation process cannot be unlimited as regards its duration. We assume therefore an energy barrier, the attainment of which leads to a critical state. This is due to the fact that, at a certain instant of time, the material loses its ability to dissipate energy. After series of modifications the criterion may be written as follows:
Creep buckling analysis of columns has been studied by numerous authors. In those investigations it is generally defined in terms of unbounded deflections or deflection rates. Only a few of the authors included the geometrical nonlinearity in the way of large deflection analysis. They presented however different conclusions. Zyczkowski [ I] found that due to the geometrical nonlinearity, the finite critical time for creep buckling would not exist. Huang’s conclusion[2] is that the creep buckling may occur but only if the effect of plasticity is included in the creep law. Both authors have been using in their papers Norton’s nonlinear creep law and the collocation method. Samuelson[3] exposed, using Runge-Kutta methods, that creep buckling occurs with no additional conditions such as Huang’s. This conclusion is in a good agreement with experimental observations and. as it is going to be seen, with ADINA results and those presented in the paper. Nonlinear creep buckling analysis using ADINA requires an unfailing solution strategyI4-41. One of the most important problems is an appropriate selection of time steps irrespective of solution methods (with or without iterations). Observations of creep tests lead to a conclusion, that the critical time (time to buckle) may be depended on that part of energy which is dissipated by the body. Thus, assuming an energy criterion we might be able to estimate the approximate critical time tzl. Having obtained trr and assuming a condition for a single time step AI, we may choose adequately the nbmber of time steps required for an ADINA solution. Another consideration when performing an ADINA creep analysis is that the finite element mesh must also be adequate. For an evaluation of the ADINA creep material model and those above parameters, a numerical solution is employed for creep buckling analysis of columns. The method is based on the exact bending line (large deflection analysis) for which a differential equation is established and solved using the Newton-Raphson iteration method.
f(&, Ed) = const. = K,
where 6, is the effective stress, Ed the specific dissipated energy and K is a material constant. The dissipated energy may be obtained from the equation: Ed =
.&
*cr
dt.
(2.2)
The Norton-Odqvist creep law for a plane stress analysis and incompressible materials may be written as follows:
i,=B*S:-‘. i,=B.S:-'.
(a-*+1 (&+, ),
(2.3)
where S,, S2are principal stresses, and i,, i2 are principal strain rates. Taking into account eqn (2.3) we have:
ri=Slj(X,Yst) &[6j(X,YTt)l ’
=,.,:-‘.[(,,-t,,),,+(~~-~6 & = B ’ ,:+I.
2. THEENERGY CRRERION OF CREEP BUCKLING The idea has been forwarded by Bychawski and Olszak in 1%8[71. They assumed that for visco-elastic bodies. accumulated and dissipated energies may be equally responsible for a transition into a critical state. The idea has been further employed for creep rupture problems by Kopecki and Walczak[8] and [9], where the form of a dissipated barrier has been established.
+Assistant Professor of Mechanical
(2.1)
(2.5)
The form of the function f(Ed) has been established on the grounds of the rheological characteristics of materials, for which the creep curves i = i(t) for constant 6 are known in a broad variability range of the stress. If we know the creep curves, we can evaluate the amount of the energy dissipated up to the critical instant of time for constant S. Thus we are able to evaluate the critical time tz, after the material is no more able to dissipate energy. The analysis of creep data for a number of materials
Engineering. 683
684
J. WALCWK
proves that the amount of energy up to creep failure satisfiesthe followingrelation: jr&, Ed) = $
=
= I(.
(2.6)
This means that the dissipationbarrier depends on two physical parameters, namely the dissipated energy and the effective stress. Taking into account eqns (2.5) and (2.6)we find
where K and fl are material constants, which, for a chromium-molybdenum-siliconsteel at S4ooC,are found to be: ,9 = 0.665,K = 1.K~127 r 10-4. The above formula gives good results in creep rupture analyses, under constant stresses@,91.In creep buckling problems,stresses are not constant and the response of a structure is unknown at the beginningof a analysis. But we need to have this criterion as an additional approach to the problem, so we can use a reduced form of eqn (2.7)to estimate the range of the critical time for a certain value of the effective stress, which we mightcall a representative stress. 3. CREEP BUCKLING ANALYSIS
elements respectively, for a plane stress analysis. In all cases we have been using4 x 4 Gauss integrationpoints. All numerical results are presented in graphic forms. Figure I represents a solution for column No. 1. The finite element mesh shown in the figure, has been estabhshed after a number of tests. As it turned out, it was good enough to model the transversal dimensionof the column using one eight-node element with 4x4 Gauss points, and the longitudinal dimension by six eiements.The time step Al has been chosen on the basis of Samue~sonsolutionsB]. The inte~at~n parameter a has been set to 0.0 by solving without iterations and OS-using iterations. Employing the BFGS iteration method[6]allowed us to use a five times larger time step than in the previous method. This did not cut down however the CPU time. Usingmuch larger time steps, as is shown in Fig. 2, reduced the CPU time and gave inaccurate results (similarly with the modified Newton iteration method),but the critical time was not exceeded. More subdivisions per time steps did not improve significantlyresults in this problem, so only one subdivisionper At has been used. Solutionsfor the column No. 2 are presented in Fig. 3. The best one obtained is that without iterations, for a relatively small time step. The BFGS iteration method gave quite good results for ten times larger time steps,
USING ARINA
As is pointed out by Bathe[5], the nonlinear finite element analysis is still faced with difficulties, particularly in creep buckling problems. A major question relates to the finite element mesh to be used because of expenses and accuracy. Another problem lies in the selectionof an appropriatetime step and this is the most difficult task. The last one we want to underline is a solution method. We should also say there are no simple answers for these questions and they are di@cult to generalize. To show some of the problems we wiIi study two kinds of columns for two different material sets. One of them (columnNo. 1) is relatively short and thick, with a slendernessratio A= 36.9.The other one (columnNo. 2), is long and slender with A= 554.2.Since the columnscan undergo the large deflection analysis, we use the total Lagrangian formuiation[4,5], and the thermo~iasticplastic and creep material mode! for 2/TIsolid elements. The columns are modeled with six and eight, eight-node
Fip.1.
GmI-
If-
I-dr-01. ~-f~~f~ot~on IO---
2-4t-40,y-c0&,8~6s
3-dt-4o,_x-cotut.N-R of_ ._ 4- dt-80, 5%const. BFGS
J- At-c4R K-const , N-R
Fig. 2.
1
685
On creep buckling analysis of structures
Fig. 3. but the modified Newton iteration method failed to converge at an early stage (see Fig. 3). The solution time step Ar is established on the basis of the energy criterion. For this we can use the representative stress S, which might be defined as a mean stress 6, = (60+ 6,.,)/2 = (z&t g/2
(3.1)
where & denotes the initial compressive stress, P is the applied load, W is the section modulus, and f is the assumed maximum deflection of the column. One way to find f may be as follows
f+;
0.2)
where A0 is a limiting slenderness ratio, h-transversal dimension of the column. In our example A = 554.2, A0= 88.0, h = 2.0, thus f = 6.3. Hence 6, = 627.0 kG/cm’
(3.3)
Taking into account eqn (2.7) one can find an approximate time to buckle t:,= 22748.0h.
The form for the critical time step At (eqn (3.7)) gives also good results for column No. 1 and it may be useful in other problems involving creep buckling behaviors. It is also seen from Fig. 3, that the critical time obtained from the ADINA solution is close to 16,Oo hr, which is comparable with the approximate result. As has been mentioned above, for the BFGS and modified Newton iteration methods, the time step At has been selected ten times larger than the above. It would be interested to see CPU times on a IBM 370 machine. Let us make a comparison for solutions presented in Fig. 3:
1 BFGS, 1.06, +%0.58. tno,,cr
noIt.%
Using smaller time steps for the BFGS and modified Newton methods, would increase considerably the CPU times. In the above solutions only one subdivision size per time step is used again. More subdivisions are time consuming with not enough evidence that the results in this problem are improved. The ADINA solutions are next evaluated by a numerical procedure applied to the same columns under the same conditions.
(3.4)
A criterion for a single time step At may be written as follows
4. A NUMEIUCALPROCRDURE FOR CREEP BUCKLING ANALYSIS OF COLUMNS
The creep buckling behavior of a column may be described in the natural coordinates (s, 0) (Fig. 4): which are related to the Cartesian coordinates (x, y) m well known forms which leads to a form x=
,,
E
. B
.
&(n-l)
(3.6) Y=
where E,,~is an inelastic strain tolerance. Taking S, = S, and E101= 0.1, one can find that the time step for this problem should be smaller than 69.2hr, thus it has been assumed as much as 50 hr. Having obtained the approximate critical time tz, and the critical time step, we can perform more adequately the finite element analysis.
I
I
cl I
cos ~9ds (4.1) sin 8 ds,
where B = 0 (s, t) is our investigated solution. To avoid numerical integrations along the thickness of the column we will make an idealization of the cross section, similar to Samuelson’s assumption[3]. The cross section is divided into a number of flanges and cores
686
J.
WALCZAK
consider a deformed element (Fig. 6). We can find from the figure
P
Y 4
E. -
Ev _
I
7rR=X
(4.4)
where ez and E, denotes strains of extreme fibres, H is the distance between the fibres and X is the curvature of the deformed column, and (4.5) Taking the first derivative of eqn (4.4) with respect to time we get t9
1,
Equation (4.6) holds for any par of flanges, thus it may be written in a system of equations
X
‘l=&P~H’, E,’- E,
Fig. 4. (Fig. 5), where the cores do not take normal forces. All dimensions shown in Fig. 5 can be obtained from a comparison of inertia moduli
i=l2 , ,...1 fn.
Hence we obtain .I
Q -
.I eu - 2 i,’ + 2 E,I = %‘(HI - 2 Hi). i-Z
&= [12z
(:Y;“)‘]‘”
(4.7)
(4.8)
For any flange we can evaluate stresses as follows P Hi
6,=6 .-m-2itl I 0 2
61.,=-socose~~*_z~y
(4.2)
h=h/m
(4.9)
where
where m is the number of flanges. The constitutive equation is accepted in the form of Norton creep law
I =g
(rectangular cross section),
and I
Y=
i=;tB.S", where the creep term has the following meaning: B. S” = B . l8r-l.
S.
sin 6 ds.
Differentiating eqn (4.9) with respect to time we have ;I b z.v
= a0 sin 0 . e’ + $$[(,‘sint?ds]. (4.10)
To find a bending line of the column we have to
Fig. 5.
Fig. 6.
687
On creep buckling analysis of structures
Substitution of eqns (4.3) and (4.10) into eqn (4.8) leads to a differential equation which may be written in the form s.$[lsinBds]
n 1
[ . I*
+S,
u
sineds-&cosB
+~*[,Il~~,~(~i:sin,d~+~~cos~~n ” -e*&
(I,‘sin
(4.11)
Bds)]
-(-l)“.B[3lsinBds +60~6~
e
Equation (4.11) can be solved by the Newton-Kaphson iteration method. To achieve the goal it would be convenient to rewrite it as follows ,+a
2 . K=E*
+ 2 1-Z
--
2 E
,+A$,
(-
1)”
+ B(“A’F,
. B(‘+A’F,
(4.16)
_ c+Al~)n
+ ‘+A’s)”
-
B
. (I+A)Fi
-
‘+A’s)”
[
. ‘+Atfii
1
+ (_
1)B.
the curvature at the end of each iteration be within a tolerance of the true solution. Hence the convergence criterion is
(‘+A’,,
(4.12)
+‘+A~s)
where 1]Ki<112 denotes the Euclidean norm of K[lO], and lk is a curvature tolerance. Unfortunately the vector ‘+“K is unknown, so we
have been using a modifiedcriterion
where <
f+Af
* K= f+Ar%‘( Hi - 2 &)
‘+A3 = 60cos (‘+“‘e), (4.13)
4,
(4.17)
which was effective in the analysis. Figure 7 shows the predicted displacement response for column No. 1. As expected, it can be seen that increasingthe number of flangeswe get a better solution, and the approximation by six flanges is an optimum. Figure 8 presents solutions for column No. 2. Finally Figs. 3 and 10 present comparisons of these results and those obtained from the ADINA analysis. Additionally
All above integrals have been carried out numerically by the trapezoid rule. The columns have been divided into twenty elements, and more subdivisions did not change results. To solve eqn (4.2) we have to replace all differentials by finite differences I+AfKW=
[f+At@k,
_
r+Ar&k-1) t+Ar~/k-II
‘K]/&
=
[I+AtK(k-1)
=
[t+Are(k-1)
=
ft+At@k-I)_
_
+
AK
_
SK]/&
to],& t~~]/~t,
where the superscript k denotes iterations. Substituting the above formulas into eqn (4.12) we have AK=
r+Ar@k-l,At
+
‘K_
t+AlK(k--I),
where ‘+“RCk-‘)is the r.h.s. of eqn (4.12).
Since we are seeking the curvature of the column correspondingto time t + At, it is natural to require that
there is presented one of Samuelsonsolutions[3],based on a different method. As can be seen, there are some minor differencesdue to the different assumptions.In the case of column No. 1 only in the ADINA analysis plastic
688
J. WALCZAK
7- ADNA
20 -
2 - gfesented SulutLon 3-~amefsun
-
1f -
Fig. 9.
-
7- ALVNA .?- presented SUhhUff
.
I#
3---2
/ p
2
4
6
8
I
72
f MJf!‘3
Fig. 10. strains have been included, and this accelerated the process. Small deviations of the initial column imperfections give also significant changes in the solutions. It should be underlined that it is difficult to compare any creep solutions based on dislerent methods. From that point of view, it is evident that the agreement is good. 5. CONCLUSIONS (a) In the initial analysis of creep buckling problems, where the response of the structure is unknown, the energy criterion may be useful. It gives the range of the period of time for the analysis with no computational effort. Also the use of the dissipative barrier gives good
results, but has one disadvantage-two more material constants. This is not a problem however, when we deal with a known material (i.e. for which the creep curves are known). It should be recognized that in reality, the criterion is a creep failure criterion determined by the local stress condition. The creep failure may occur prior to creep. buckling, but this is beyond the scope of the paper. (b) The ADINA thermo-elastic-plastic and creep material model gives good results and is relatively inexpensive. As has been shown, there are some minor differences in our comparative analysis, which are mainly due to necessary assumptions in the applied procedure but they testify in favor of ADINA. (c) To obtain a good ADINA solution we have to set up some vital parameters. We can summarize our experiences in the problems analyzed as follows: -The analysis is most economically performed without iterations with the stiffness matrix reformed every time step. -In using the modified Newton or BFGS iteration method it should be recognized that the cost of the analysis can be high. For slender columns the BFGS method is preferable. Using the BFGS iteration method we usually get the same results for five times larger time steps than without iterations. For larger time steps the columns “look stiffer” but the results never exceeded the critical time in either equilibrium iteration method. -It is usually enough to select one subdivision per time step. OnIy for large stress gradients, close to the buckling time, more subdivisions improve the results. (d) The presented numerical solution for creep buckling columns is simple and convenient to use on minicomputers, gives good results and might be applied for any cross section area with one symmetry axis. (e) As is shown in the paper, the ADINA results and the numerical solutions prove that the critical time for creep buckling, defined at the beginning of the paper, exists and this does not depend on the plastic strains in the constitutive equations. Only in column No. 1 the plastic strains have been included in the ADINA analysis. In the numerical procedure elastic and creep strains have been generated in either column. For column No. 2
Fig. II.
On creep buckling analysis of structures
with no plasticity, on the ground of the ADINAanalysis, the finite critical time is likely to exist with no additional conditions for the load level, except that the load must be smaller than the instantaneous buckling load. This conclusion highly disagrees with Huang’s~Z]. (f) The idea of evaluationsof the approximatetime to buckle and the critical time step may be also appliedto a creep bucklinganalysis of spherical axisymmetricshells by using 2/D solid elements for the axisymmetricanalysis. The representative stress S, may be taken from the elastic response (see Fig. 11). (g) It appears that an adaptive time step for the ADINA creep analysis might be very useful and it ap pears that the dissipated energy may be helpful in the selection of the time steps. Acknowkdgemenfs--I wish to take this opportunity to express my appreciation to Prof. Klaus-J~rgen Bathe for valuable discussions and su~estions while I was working with him at M.I.T. REFERENCES 1. M. Zycxkowski, Geometrically nonlinear creep buckling of bars. Creep in Structures, Colloquium held at Stanford University, California, II-15 July, 2960(Edited by N. J. Hoff}. Academic Press, New York ft%2).
689
2. N. C. Huang, Creep buckling of imperfect columns. J. Appl. Mech.. Trans. ASME, March 1916. 3. A. Samuelson, Creep deformation and buckling of a column with an arbitrarv cross section. The Aeronautical Research Institute of Sweden, Rep. 107(1966). 4. K. J. Bathe, ADINA: a finite element program for automatic dynamic incremental nonlinear analysis. Rep. 824481, Acoustics and Vibration Lab., M.I.T., Sept. 1975 (revised Dec. 1978). 5. K. J. Bathe, Static and dynamic geometric and material nonlinear analysis using ADINA. Rep. 82448-2, Acoustics and Vibrat~n Laboratory, Mechanic E~neering Depart” ment, M.I.T., May 1976(revised May 1977). 6. K. J. Bathe and A. Cimento, Some practical procedures for the soiution of nonlinear finite element equations. L Comput. Meth. ADDS.Mech. Enene 22. SMS 119801. 7. Z. By&a&i and W: GI&k, Creep failureof nonlinear rotationalshells.8th Gong.ht. Association for Bridge and Struck-at Engineeting,Zurich, New York, 1968. 8. H. Kopecki and J. Walcxak, The energy dissi~tion barrier as a criterion of creep failure of rotating discs. Archiwum Budowy Maszyn 4 (19761. 9. J. Walczak, Critical states of rotating discs in nonlinear creep conditions (in Polish). Engng Trans. 27, 2 (1979). 10. K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Ekment Analysis. Prentice-Hall, Engiewood Cliffs, New Jersey (1976).