Internal variable formulations for the plastic analysis of plane frames

EngineeringStructures, Wol. 17, No.

U T T E R W O R T H I N E M A N N

0141-0296(94)00005-0

3, pp. 214-220, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0141~0296/95 $10.00 + 0.00

Internal variable formulations for the plastic analysis of plane frames C. P. Roth and W. W. Bird Institute for Structural Engineering, University of Stellenbosch, Stellenbosch 7600, South Africa

J. B. Martin Centre for Research in Computational and Applied Mechanics, University of Cape Town, Rondebosch 7700, South Africa (Received October 1992; revised version accepted February 1994)

There are a number of methods available for the plastic analysis of plane frames; of these the most widely used are those which are rooted in mathematical programming techniques. This paper presents a formulation which utilizes the thermodynamically b a s e d internal variable approach where plastic hinges are characterized as internal variables. The algebraic equations resulting from the formulation are versatile and it is illustrated how they can be incorporated in an algorithm to solve incremental problems. A number of examples illustrate the application and effectiveness of the formulation. Keywords: internal variable, plastic analysis, plastic hinges, plane

frames The use of limit state design for frame structures has gained widespread acceptance. It is generally recognized that the dominant ultimate limit state for many steel frames is plastic collapse, and the design of such structures is based on the plastic collapse load. This collapse load, or alternatively, the proportional load at which the onset of plastic flow in the structure takes place, is determined by means of limit analysis. Limit analysis utilizes upper bound (kinematic theorem) and lower bound (static theorem) approaches to determine the load factors at which collapse will take place. These methods, discussed in some detail by Baker et al.l, Neal 2 and Baggett et al. 3 form the basis for the mathematical programming methods used in packages such as STRUPL4. Another approach used in determining limit loads is incremental elastic plastic analysis and Baggett et al. 3 suggested an algorithm which utilizes iterative procedures for the incremental elastic plastic problem. Much of the research to date has concentrated on materials and structural members exhibiting perfectly plastic or strain hardening behaviour. Recently, strain softening behaviour as observed in reinforced concrete sections for example, has been discussed by various authors 5 7. In this paper we propose an approach similar to that of Bagget et al. 3 In our case, however, the rotations of the plastic hinges are characterized as internal variables, resulting in a formulation that can easily accommodate perfectly plastic, softening, or hardening material behaviour. A further advantage of the algorithm is that the iterations

within each load step involve only the nodes where yield is likely to occur. The formulation has been implemented in a computer program capable of analysing plane frame structures, and the effectiveness is illustrated by means of numerical examples taken from the literature.

Formulation of the structural problem The proposed formulation is the thermodynamically based internal variable formulation discussed by Martin 8. The displacements of the structure, which in this case is a plane frame, can be represented by the displacement vector u. Plastic hinges i.e. internal slips are represented by the components of the vector 6. In order to identify the vectors u and 6 , consider the single frame element with six local degrees-of-freedom as depicted in Figure 1. The element is located between two nodes i and j each with three global degrees-of-freedom. The nodes with four elements connected to each are indicated as crosses in Figure 2. In principle, however, any number of elements can be connected to each of the nodes. Figure 3 shows an element in its deformed state with all END 1

Figure 1 Degrees-of-freedom of frame element

214

END 2

Internal variables for plastic analysis of plane frames: C. P. Roth et al.

215

where K e is the standard elastic element stiffness matrix for a frame element. Substituting the expression for u e in the above equation it follows that each element's strain energy can be written as

Figure 2 Typical element in frame

displacements and rotations shown in their positive sense. It is possible for plastic hinges to form at both ends of the element. The plastic rotations of these are denoted by ~b~ and ~b2. We define u ~ as the element displacement vector, where

1 I U" Ke I ui } -PluilTKeNec~e+~eTNeTKeNe~ l e F< = 2 tujJ

uj

tu A

(3) or alternatively as

F e= ]-IuilTK e { ui} q- IuilTLeI~ e -I- ~'eTHe(~e

v~ U e -~_

2[ujJ

uj

(4)

tuff

where L e = KeN ~ and H ~ = N~rKeN~. The total strain energy F in the structure is the sum of all the element contributions and is a homogeneous quadratic function in terms of the global displacement vector u and the vector of plastic hinges dp

I v~ I

In terms of the global or nodal displacements, the element displacement vector becomes Ui

1

1

(5)

F = ~ urKu + uTL& + ~ $TH~ b

Differential changes in the kinematic variables yield

Vi

Oi + 4'e~

dF

=

OF du +

OF

(6)

d ~ = rdu - xddp

U e ~.

uj and we identify the internal nodal forces r and the internal forces or conjugate forces x acting on the hinges. We set

vj

oj + 4~ OF

It is more convenient to separate the nodal and element contributions to the element displacements as follows

Vi Oi

Ue =

in order to obtain the forces (moments) applied by the structure to the hinges rather than the forces applied by the hinges to the structure. It follows that both r and x are homogeneous linear functions of u and ~b

0 0 ]

Ui

+

uj

0

0

1

0

0

0

oo

v~

0

Or

o,1,

4,~J

1

r = Ku + L~

(7)

- x = Lru + H ~

(8)

Equilibrium requires that the internal forces at the nodes be equal to the external loads, p, applied at the nodes and therefore we can rewrite equation (7) as

which can be written as ue={Ui}+NedS

(1)

uj

The element strain energy, F e can now be computed as

(9)

Ku + L & = p

The rate of change of the plastic rotations at the hinges is governed by a rigid plastic relation as shown in Figure 4. Moment M

F'=~1

ueTKeu

e

(2) Mp-

1

Rotation ~b

°'I , "I /S I

q---I

,

, ui

Figure 3 Element after displacement

I L Uj

~

1

' -Mp

Figure 4 Assumed moment-rotation relationship

Internal variables for plastic analysis of plane frames: C. P. Roth et al.

216

We assume the existence of a dissipation function D, such that OD

x=__ o6

(10)

Ku. + L 6 . = p .

D is convex in the cases of perfect plasticity an.d hardening; D --> 0 for all cases with D = 0 if and only if 6 = 0. It follows that the derivatives of D are discontinuous at the origin, and that, since D will generally be the sum of independent dissipation functions associated with individual hinges, derivatives of D may be discontinuous along lines which are radial in the ~b space. F i g u r e 5 depicts the contribution of an individual hinge to the total dissipation. Identifying the components of x as being the moments at hinges (i.e. positions in the structure where the moments M have actually exceeded the plastic yield moment), we have for the positive components with M > 0 and th > 0 that M=Mp+

vals At which are not necessarily equal. We seek to satisfy equations (14) and (15) only at the end of these intervals. Thus at time t,, after n intervals have elapsed, these equations are written as

(11)

sO

and for the negative components with M < 0 and ~b < 0 that

(17) In order to trace the behaviour of the frame under varying load, we introduce a series of steps fi, t2, t3 Assuming that at step tn_~ we have a solution to equations (16) and (17), we now seek a solution at step t,. Using the relationship .

.

.

.

6 . = 6.-1 + ± 6 we can rewrite equation (16) as Ku. + L6.-1 + LA6

= Pn

(18)

and hence we can express the displacements as lg n

M=-Alp+

(16)

=

U E

--

K-ILA6

(12)

s+

where where s is a linear hardening (s > 0) or softening (s < 0) parameter as indicated in F i g u r e 4. We can thus write the components of x as

Xi=+_Mp+sf'oSdt

(13)

where t denotes time. If we regard the load applied to the structure as a function of time, p = p ( t ) , u = u ( t ) , 6 = 6 ( 0 , and t~ = d r / d t . Since the problems are rate independent, the parameter t measures the order of events, rather than real time. The initial condition 6 ( 0 ) will be taken to be zero indicating that there are no plastic hinge rotations at the start of an analysis. The governing equations can be written as Ku(t) + Lr(t) Lru(t)+Hr(t)

= p(t)

(14)

= _ { 0 ~ } , b (,)

(15)

u.~ = K - ' ( p .

- Lr._,)

The only unknown in equation (18) is the change in the plastic hinge rotations during the step. In order to evaluate it, we use equation (17) and substitute u, as follows LT(ue~ - K-1LA6) + H6.-~ + H A 6 = x.

This equation can be simplified to ZA6

+ I x , = - L ~u ,e - H 6 , - 1

(19)

with Z = H - L r K - ~ L and where all the unknown quantities are grouped on the left-hand side of the equation. Equations (18) and (19) form the basis of the numerical algorithm which is employed. Firstly we use equation (18) to compute u~e and obtain an estimate for the conjugate force (moments) vector x e" = - L r u ~ - H 6 . _ ,

Incremental analysis In order to set up a numerical procedure which determines the response of the structure to a given loading program p ( t ) , we need to divide the time domain into discrete interDissipation D

Rotation

Figure 5 Assumed dissipation function

We then start with an iterative process where we solve for x, and A 6 by recognizing that each row in the set of algebraic equations of equation (19) represents a possible plastic hinge in the structure. There are effectively two unknowns, namely the moment and the plastic rotation at each hinge position. By making use of the fact that when the estimated moment at a possible hinge position does not exceed the yield value the incremental hinge rotation is zero, and when the estimated moment does exceed the yield value both moments and incremental hinge rotations are unknown, but linearly related by x~ = + M~ + s6._1 + sA+ In the case where the component of A 6 is zero, the

Internal variables for plastic analysis of plane frames: C. P. Roth appropriate column of Z is replaced with the corresponding column from the identity matrix and the component of x, treated as an unknown. When the component of At~ is unknown, the appropriate entry on the right-hand side of the equation is adjusted by subtracting +_Mp+ sc~,_l, and the Z matrix is adjusted by adding s to the corresponding diagonal term. The solution to this set of equations must provide a consistent set of results. By this we mean that where the components of At~ are zero, the computed value of conjugate force components must satisfy the yield condition. If it does not, we compute a new estimated conjugate force vector

x~s,

=

LTUE -

H(+,,_I+ A~)

and iterate until we obtain consistent results. Once we have Ad#, we substitute into equation (18) to compute u,. Implicit in the formulation and algorithm described above is the possibility of plastic hinges forming in all of the elements connected to a node. This will result in a local mechanism being formed at the node and the solution algorithm will fail. To overcome this problem, it is necessary to prevent the plastic rotation of any one of the possible hinges. This means that the increment in plastic rotation must be set to zero during the load step. It is important to choose the hinge position which will provide consistent results at the end of the step. We have based our choice of this hinge location on the ratio actual moment at hinge yield moment at hinge At nodes where the estimated moments of all the possible hinge positions exceed the yield values, the hinge location for which this ratio has the lowest value is prevented from yielding by setting its plastic rotation to zero. In all examples that were run it was found that this approach gave satisfactory results.

Numerical examples The following examples have been selected to demonstrate the capabilities of the program. The first two are taken from the STRUPL Users Manual 4, the third from Heyman9, and the final one from Neal 2. The results obtained with this method are precisely the same as those obtained in the references cited. In example 1, load factor versus displacement curves were compared with STRUPL results for two cases. In the first case the material behaviour was assumed to be elastic perfectly plastic and in the second elastic plastic with linear hardening. In the second example, detailed results were available only for the perfectly plastic case and in each of the remaining examples only the collapse load factors assuming perfect plasticity were available for purposes of comparison. In these cases, the frames were also analysed assuming softening and hardening behaviour. These analyses are intended only to illustrate the qualitative behaviour of the structures and thus in the final two examples dimensionless quantities are used.

et al.

217

area I = 0.000067 m 4 and Young's modulus E = 210 GPa. The plastic yield moment for the members is Mp = 260 kNm and where hardening is assumed to take place, the gradient s = 3000 kNm/rad. In Figure 7 the load factor is plotted against the horizontal displacement of node 2 and the vertical displacement of node 3 for the perfectly plastic and the hardening case.

Two-storey rectangular frame (from STRUPL Users ManuaP) The frame analysed in this example is illustrated in Figure 8. The structure contains four different steel elements namely, the lower columns, the lower beam, the upper columns and the upper beam, all with a Young's modulus of 200 GPa. The properties for these elements are given in Table 1. The plastic hinges, numbered in their order of formation, are also shown in Figure 8. Table 2 lists the load factors at which the hinges are formed. A softening case (s negative in Table 2), a perfectly plastic case (s = 0), and a hardening case (s indicated as positive in Table 2) were investigated. It was found that the structure collapsed at the formation of the seventh hinge, and since the structure is statically indeterminate to the sixth degree, the failure mechanism is regular. A regular mechanism is defined as one that forms with n + 1 hinges when there are n degrees of indeterminacy. An example of a similar two-storey frame with an irregular failure mechanism is discussed by Heyman 9. This structure was also analysed and the collapse mode and load factor correctly predicted. The load factor versus displacement curves depicted in Figure 9 are coincident with the results obtained with STRUPL for the perfectly plastic case and consistent with what one would intuitively expect for the hardening and softening cases. The displacement under a given load factor is least for the hardening case and greatest for the case with the steepest softening gradient. The hardening case will not produce a clear collapse load in this model, but theoretically will continue to resist load indefinitely.

Two-bay rectangular frame (from Heyman 9) The structure analysed in this example is shown in Figure 10. The yield moment for the beams is 80, while that for the columns is 50. In this case a regular mechanism once again forms and the load factors coinciding with the formation of each hinge are tabulated in Table 3. The results are plotted in Figure 11 and the plastic collapse factor of 1.44 for the perfectly plastic analysis coincides with the value obtained by Heyman.

135kN

j90kN 3m

-JL

Single-storey rectangular frame (from STRUPL Users Manual 4) The frame depicted in Figure 6 consists of steel sections with cross-sectional area A = 0.02 m 2, second moment of

Figure 6

Single-storeyframe

Internal variables for plastic analysis o f plane frames: C. P. Roth

218 2.5

.... A

/,

2

)

//

1.5

i

I, 0.5

[~

:.A"

2

8

u.

.9 °

Node3V

40

i I 6o 8o Displacement (mm)

..... Node2H

100

120

STRUPLV

•

140

)?"4,"

//

20

]

STRUPLH

[~

Node3V

t,,j ~ .....

40

60 80 100 Displacement (mm)

........ N o d e 2 H

•

STRUPLV

120

•

140

STRUPLH

I

Results f o r s i n g l e - s t o r e y f r a m e

Table 1

Section p r o p e r t i e s f o r t w o - s t o r e y frame

Element

Lower Lower Upper Upper

•

1 0.5

F.=oe.i. 4 20

1.5

.....~.~k° ~

/

2.5

,,

/

0,

Figure 7

et al.

columns beam columns beam

Iz

Ax

Mp

s

m4

m2

kNm

kNm/rad

76.95 151.41 60.56 105.24

768.33 1214.75 981.33 707.43

22.2 85.1 17.2 48.9

18 kN

x x x x

10 6 10 -6 10 -6 10 6

4.73 4.94 3.79 4.17

34 kN

x x x x

Rectangular frame with distributed load (from NeaF ) The frame shown in Figure 12 is subjected to a uniformly

18 kN

9 kN

I 3.5

40 kN

68 kN

;

8

18 kN ..

40 kN

2

~7

6

Z7

3

5.5

Figure 8

10 3 10 3 10 -3 10 3

distributed vertical load of magnitude 1 load unit per unit length, and a horizontal point load of 1 unit. The yield moment is 1 throughout. In this case the position of the maximum moment in the element with the distributed load is not known at the outset and it would be necessary to increase the number of elements to locate this position with any measure of accuracy. In our case the analysis was done in two stages. First the frame was loaded until hinge 2 was about to form. The position of maximum moment (i.e. where the hinge would form) could then be calculated assuming a parabolic moment distribution along the beam. In the second stage a node was placed at this position to enable a hinge to form. The results for this example are tabulated in Table 4 and plotted in Figure 13. The collapse factor of 2.96 for the perfectly plastic case coincides with the value obtained by Neal 2.

Two-storey frame

Conclusions Table2

Results f o r t w o - s t o r e y f r a m e

Hinge,

Load factor

s

1 2 3 4 5 6 7

-ve

0

+ve

2.16 2.31 2.32 2.32 2.32 2.32 2.32

2.16 2.33 2.36 2.39 2.42 2.52 2,60

2.16 2.34 2.37 2.44 2.49 2.82 3.02

In this paper an internal variable formulation for the plastic analysis of plane frames has been presented. The results obtained using this method are in agreement with the results obtained using other more established methods. These comparisons have been somewhat restricted in that the problem of softening has received little attention in the past and examples cited in the literature deal mainly with perfectly plastic materials. The formulation provides a suitable method for analysing plane frames where the material behaviour can be elastic plastic with the postyield response ranging from softening to hardening. The ability to cope with perfectly plastic behaviour makes it a suitable method for implementation

Internal variables for plastic analysis of plane frames: C. P. Roth et al. 3.5

3.5

f !

3

3'

/ -

J

2.5

2.r

I 1.5

1.5

1

1

0.5

0..r

ot

50

219

100

150

200

0,

3oo

250

A m

50

Maximum Horizontal Displacement(mm)

I- P ~ " " - * - " m ' ° °

.... ~ " ~ ' ~

100

150

200

250

300

MaximumVertical Displacement(ram)

's=uP'~l

I-- "'~"=-*-'~°'~

~"

''~P'~'l

Figure 9 Results for two-storey frame

1.5units l

0.5units

1 unit

T

1.81units

t 0

i

i

2

7"

,

200

,,

Figure 12 Frame with distributed load

200

Figure 10 T w o - b a y frame Table3 Results for two-bay frame Hinge,

Load factor

S

-0.004

0

+0.004

1.04 1.11 1.25 1.25 1.29 1.29 1.29

1.04 1.12 1.31 1.33 1.39 1.42 1.44

1.04 1.12 1.33 1.37 1.50 1.52 2.06

2.5

2.5

2

2

,~ 1.5

1.5

1

/

1, 0.5

/

/

0.1

1 0.5

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Max Horizontal DisplacementNormalised

J--B- Perfect Plas

I

Hardening --X-- Softening

Figure 11 Results for t w o - b a y frame

0.9

f ~_.to.-.----X

/

/

/

..4.

/

C 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Max Vertical Displacement(NormaliNd)

-43- Perfect Plas t

Hardening --X.- Softening

I

Internal variables for plastic analysis of plane frames: C. P. Roth et al.

220 3.5

3.5

3

p

J

2.5

3

2/.

2.5

~,, 1.5

1.5

0.5

0.5

/ 0

~/

/

0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 Max Hodzontal Displacement Normalised

J--B- Pedeot Plas

:

Hardening

--)~-. Softening

C

0.9

0

I

0.1

l -e-

0.2 0.3 0.4 0.5 0.6 0.7 0.8 Max Vertical Displacement (Normalised)

Perfect Ptas .-I-- Hardening

--X-. Softening

I

0.9

I

Figure 13 Results for frame with distributed load

Table4

Results for frame with distributed loads

Hinge, s

1 2 3 4

Load factor -0.05

0.0

+0.05

2.21 2.24 2.86 2.92

2.21 2.25 2.89 2.96

2.21 2.25 2.92 3.00

into programs linked to steel codes where collapse load factors are generally sought. Although we have concentrated predominantly on proportional loading in the examples, the incremental nature of the method makes it applicable to the more general cases where the structural loads are applied in any sequence. An advantage of the formulation is that the nonlinear behaviour of the frame is restricted to the possible hinge positions and this makes the resulting solution procedures particularly efficient; the iterative process required for the nonlinear solution involves only the internal variables and their conjugate forces. The formulation can be modified to include axial and shear forces and it can also be extended to three dimensions. In order to do this additional internal variables are required at member ends and the relationship

between them and their conjugate forces needs to be defined. A further extension to include large displacements in the formulation is possible, but in this process the advantage of having the nonlinear behaviour restricted to the internal variables would be lost.

References 1 Baker, J. F., Home, M. R. and Heyman, J. The steel skeleton, Cambridge, University Press, 1956 2 Neal, B. G. The plastic methods of structural analysis, John Wiley, New York, 1963 3 Baggett, J., Martin, J. B. and Reddy, B. D. 'Elastic-plastic deformations in plane frames', Civil Engr in South Africa, 1977, 19, 89-93 4 Cohn, M. Z., Erbatur, F. and Franchi, A. STRUPL 1: User's Manual, University of Waterloo Press, 1982 5 Maier, G, 'On structural instability due to strain-softening', Proceedings of International Union of Theoretical and Applied Mathematics Symposium on Instability of Continous Systems. Berlin, Springer Vetlag, 1969, pp. 411-417 6 Ba2ant, Z. P. 'Instability, ductility and size effect in strain-softening concrete', J. Engng Mech. Div., ASCE 1976, 102, 331-343 7 Darvall, P. and Mendis, P. A. 'Elastic-plastic-softening analysis of plane frames', J. Struct Engng, ASCE 1985, 111, 871-887 8 Martin, J. B. 'An internal variable approach to the formulation of finite element problems in plasticity', in Physical nonlinearities in structural analysis (Ed. J. Hult and J. Lemaitre), Springer-Verlag, 1980, pp. 165-176 9 Heyman, J. Beam and framed structures, Pergamon Press, Oxford, 1974