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Unified nonlinear elastic frame analysis
Compurers & Strucrures Vol. 60. No. I, pp. 21-30. 1996 Copyright 0 1996 Eisevier Science Ltd Printed in Great Britain. All rights reserved 0045-7949/96 $ IS.00+ 0.00
0045-7949(95)00378-9
UNIFIED
NONLINEAR
ELASTIC
FRAME
ANALYSIS
J. Petrolito and K. A. Legge School of Science and Engineering, La Trobe University, Bendigo, Bendigo, Victoria, 3550, Australia (Received
Abstract-This frames.
3 February
1995)
paper develops a unified nonlinear analysis technique for two-dimensional
Any level of nonlinear
effects can be rationally
treated,
and results automatically
structural obtained to any
specified accuracy. The procedure only requires minimum discretisation of the structure as a self-adaptive procedure is used to solve the member equations. The nonlinear solution techniques used enable either discrete equilibrium positions or the complete load-deflection response to be calculated. Copyright 0 1996 Elsevier
Science Ltd
1. INTRODUmION
cient accuracy in a nonlinear analysis generally requires that each member in the structure be subdivided into a number of elements. However, codes of practice formulate their rules on the basis of minimum discretization. Hence, subdivision of members is not compatible with the philosophy of current codes. With this in mind, the term “balanced formulation” was adopted in Ref. [ 161 to describe a nonlinear formulation that gives exact results consistent with the underlying theory using minimum discretization. Within this framework, it was shown that a beam-column analysis including the effects of bowing [17] is the most general balanced formulation available, provided it is consistently formulated. It was also noted that a consistent formulation has not always been adopted in various studies and commercial programs. Moreover, beam-column with bowing theory is restricted to small deflection problems, and hence it is not sufficiently general. However, if a full nonlinear theory is used, a finite element type of formulation that includes subdivision of individual members in the structure is required. Such an approach is only approximate, and the analyst needs to be confident that the resulting errors are within an acceptable tolerance. In addition, the amount of subdivision required for a given accuracy is uncertain, and a trial and error approach is required. Thus, there is a need to develop a unified, nonlinear formulation that is capable of allowing any level of nonlinear effects to be treated in a rational manner, and automatically provides results to a specified accuracy. The aim of this paper is to present such a unified, nonlinear elastic analysis technique for twodimensional structural frames. Any level of geometric nonlinearity may be easily treated. An adaptive scheme is used in the solution technique at the member level so that the analyst need only specify the minimum discretization required to determine the topology of the structure. Thus, despite the internal
Frame analysis is of major importance in structural engineering, and it is not surprising therefore that it is the subject of continuing research. The original classical methods of analysis are suitable for small structures. As structures increased in size and complexity classical methods became unwieldy, and approximate techniques were introduced. While these techniques are rapid, they are of variable accuracy and restricted to linear analysis [l]. Matrix methods of analysis [2] revolutionized the field of structural analysis, and freed the analyst from the limitations of the classical and approximate methods. The theory of matrix methods is now well-established, and numerous commercial programs are available to carry out such analyses. Until recently the emphasis in frame analysis has been on linear analysis, and the treatment of the destabilizing effects of axial compressive loads was done at the design stage of the individual members in the structure. However, recent changes in codes of practice such as [3] have necessitated a reappraisal of this approach. This has led to the introduction of nonlinear analysis options in commercial programs [4, 51, and a renewed interest in the formulation of nonlinear analysis. A number of methods have been developed for nonlinear elastic analysis of frames. These range from formulations based on beam-column analysis [&lo] to formulations based on finite element ideas [ 1l-l 51. Despite the level of interest in the problem, there appears to be no general consensus on the basis or formulation of such procedures. The authors have shown in a previous paper[l6] that the techniques used in a nonlinear analysis can have a significant effect on the results. In particular, the traditional frame analysis approach using minimum discretization, where each beam and column is treated as a single member, may not be suitable for some procedures and leads to errors. Hence, to achieve suffi21
22
J. Petrolito
and K. A. Legge
the displacements and rotations at each end relative to the local member axes (see Fig. 2b), which are denoted by
Similarly, the member are defined as
IYG --
end forces in local coordinates
Xc The detailed member behaviour under loading determines the relationship between v and P, which will be nonlinear in general. Without being explicit about this relationship, we assume at this stage that such a relationship exists so that we may formally write
Fig. 1. Typical frame for analysis.
PM =f(vM),
adaptivity, the results are presented for the overall individual members, as this is the most convenient format for design. The nonlinear solution techniques used enable either discrete equilibrium positions or the complete loaddeflection response to be calculated.
2.
GENERAL FORMULATION FOR NONLINEAR ANALYSIS
In this section, we present the general formulation used to solve the overall nonlinear problem. Only a minimum number of assumptions are introduced at this stage regarding the theory to be used to describe the member behaviour. For illustration, we consider the two-dimensional rigid frame shown in Fig. 1. The topology of the structure is descibed by a set of nodes connecting the individual members. We assign a set of node numbers, with typical nodes denoted i, j, k etc., and a fixed set of global reference axes X, and Y, The figure also shows typical members M, N and 0, with member M connected to nodes i and j. The movement of each node is uniquely described by two displacements and a rotation. Thus for a typical node k, the nodal displacements are #={r’;,r:,r’;>,
wherefis a nonlinear function. Moreover, we do not insist that an explicit form can be found forf. Rather, the only requirement is that P can be determined if the member properties, loading and end displacements are specified. The form off‘ will also vary depending on the modelling assumptions made for the member behaviour. Under certain assumptions, an explicit form forf can be found. Specific examples of this include linear theory and beam-column theory. The resulting forms off in these cases are well known and are given in Ref. [2]. However, we do not make use of this knowledge in the present formulation since the aim is to produce a unified, general procedure that is applicable for nonlinear analysis where an explicit form for f cannot be obtained.
(1)
where curly brackets are used to denote a column vector. Similarly, the concentrated forces that are applied at node k are denoted by
A typical member M is shown in Fig. 2, with its local axes X and Y such that it is oriented at an angle 0 to X,. The member is subjected to distributed loads px and pv along its length. We assume that the deformation of the member can be characterized by
(5)
PM 1' P1 ’
Cc)
Fig. 2. Typical member and sign convention.
Unified nonlinear elastic frame analysis It is convenient to transform the member end displacements and forces to global coordinates as shown in Fig. 2c, giving TM= T”vM, where the transformation
T=
PM = T”PM, matrix
(6)
T is given by
T,0 1
[ 0
T,’
and T, =
Since eqn (6) is an orthogonal inverse transformations are vM = (Tr)“SM, The compatibility 1) give
transformation,
PM = (Tr)?!
requirements
i;p=sp=r”
the
(9)
at node k (see Fig.
(IO)
where 7, = (U,, I&, uj} and c2 = {cd, r&, r&6).Similarly, for node k to be in equilibrium, we have
Combining eqns (St(ll) for all nodes and members gives the global set of governing equations for the structure, namely R -g(r)
= 0,
(12)
where r is a vector containing all the unknown nodal displacements, R is the corresponding vector of applied nodal forces and g is in general a nonlinear function. It is worth noting that the above formulation is equivalent to the stiffness method of frame analysis if an explicit form of the member forcedisplacement relationship (eqn (5)) is used. In this case, we have g(r) = Kr where K is the global stiffness matrix. In the linear elastic case K is independent of r, whereas for the beamcolumn case K is a function of r, and in both cases explicit forms for K can be found [2]. In the present case however, there are no a priori assumptions made about the nature of g, and thus the formulation includes the above cases merely as particular examples. Hence, the solution of the nonlinear analysis problem reduces to the solution of eqn (12). A number of solution schemes are available for this class of problem. The general details of the scheme adopted in this work are as follows:
23
(1) Set up the structural data, including topology, member characteristics, loads and supports. Initialize all unknown nodal displacements to zero. Specify a tolerance for the analysis. (2) For each member, solve eqn (5) and use eqn (6) to calculate the member end forces in global coordinates. (3) For each node, establish equilibrium equations associated with unknown displacements using equations of the form of eqn (I I). Hence, establish the overall out of balance nodal forces R -g(r) for the structure (see eqn (12)). (4) Use an update strategy to adjust the nodal displacements based on the amount of out of balance forces. (5) Check for convergence using a suitable criterion. If convergence has been achieved, the analysis is complete. If not, go to step 2 and continue. The key step in the global solution scheme is step 4, with the specific details of this step varying with the particular procedure adopted. A large number of techniques have been proposed, depending on the nature of the nonlinear problem and the goal of the analysis [18]. We can identify two general classes of problems, namely those near limit or bifurcation points and those away from such points. Accordingly, we have adopted a separate scheme for each class of problems as follows: (1) Problems requiring the determination of one equilibrium state of the structure, which is not near the first limit or bifurcation point-under these conditions, we have used a Newton-Raphson technique [19] to solve the nonlinear equations. Since we have stipulated that the solution is not close to points of instability, this approach can be expected to reliably converge under reasonable conditions. (2) Problems requiring the determination of the complete load-deflection curve including post-buckling response-a large number of techniques have been proposed for this class of problem. While most of the proposed techniques are reliable, it has been noted by Riks [20] that there is no generally optimal strategy. For the present work, we have used a continuation technique [2l], including an adaptive step procedure, to efficiently generate the loaddeflection curve. This technique reduces to a method suitable for the previous class of problems when a single step is used to apply the load. The technique is capable of traversing limit points for both loads and deflections automatically. However, the detection of bifurcation points requires the use of a perturbation technique [22]. The key feature at the member level is the solution of eqn (5) in step 2. This is a distinctive aspect of the present formulation, and is discussed in detail below. It is important to note that the global strategy is independent of the strategy adopted to solve the member problem. Thus, a wide variety of assumptions regarding the details of the member behaviour
J. Petrolito and K. A. Legge
24
strain, p, which are defined as [23]
Q>t
ds e==--1, 1:Y,v,.i
Combining
p==.
d@
(15)
eqns (13)<15) gives e = J(1 + us)’ + vi - 1,
P
~ = (1 - Us)%s- ususs (1 +u.#+v; ’
X,X,U,i
dS
Fig. 3. Beam element and sign convention can be easily explored and implemented in a straightforward manner. 3.
FORMULATION
AND SOLUTION EQUATIONS
OF MEMBER
3.1. Extensible elastica theory This is the most general model that we consider to describe the behaviour of individual members, and a detailed discussion of the theory is given in Ref. [23]. The theory includes the effects of large displacements and strains, but ignores the effect of shear deformation. Thus, it generalises linear beam theory based on the Euler-Bernoulli law [l]. Extensions of the theory that include shear deformations are available [24,25], but they will not be considered here as shear deformations are not usually significant for typical structural frames. We consider a beam whose axis is initially along the X axis with a point on the axis being specified by length S with 0 < S < L where L is the length of the beam (see Fig. 3). After deformation, the beam axis deforms into a smooth curve s. A point P with initial coordinates (X, 0) is mapped to a point P* with coordinates (x, y). The functions x(S) and y(S) define the geometry of the deformed axis of the beam. The angle between s and the X axis at P* is denoted by 4(S). The beam displacement functions u(S) and u(S) are given by
49 = x(S) -X(S),
u(S) = y(S).
(13)
An element dS = dX on the original axis is deformed into an element ds. From the geometry of Fig. 3, we have dx - = Cos 4, ds
dv - = sin 4. ds
where a subscript S denotes differentiation with respect to S. The force resultants on the beam are the force F and bending moment M, which is taken as positive if it acts anticlockwise. The force F can be expressed as F=Hi+Vj=Nn+Qt,
In this section, we consider the formulation of the governing equations of a frame member for a variety of modelling assumptions. Following this, we present the numerical method used at the member level to generate the forcedisplacement equations.
(14)
Natural strain measures for the deformation of the beam are the extensional strain, e, and the bending
(16)
(17)
where H and V are the horizontal and vertical components of F, N is the axial load and Q is the shear force. The unit vectors n and t are in the outward normal and tangential directions respectively. The force components are related by (see eqn (9))
Q=-Hsin~#~+Vcos$. The equilibrium
(18)
equations for the beam are [23]
H,+p,=O,
Vs+pv=O,
M, - Ho, + V(1 + us) = 0.
(19)
To complete the problem description requires the specification of stress-strain relationships. In principle, any stress-strain relationship consistent with the general requirements of constitutive equations [26] may be used. However, for the present work we will assume that the material is linear-elastic giving N = EAe,
M = EIp,
(20)
where E is Young’s modulus, A is the cross-section area and I is the moment of inertia. In general, the material properties could be functions of S. The theory results in a nonlinear system of ordinary differential equations of order six. Hence, three boundary conditions are required to be specified at each end of the beam, namely: (1) either u is specified or H is specified; (2) either u is specified or V is specified; and (3) either 4 is specified or M is specified. Exact solutions of the governing equations are only possible in simple cases [23,27], and numerical methods are usually required to solve practical prob-
Unified nonlinear elastic frame analysis
lems. Due to the difficulty of establishing closed form solutions for the general theory, simplifying assumptions have usually been made to achieve a tractable formulation. This approach has also been used in many prior studies on nonlinear frame analysis. However, it is not clear to what extent these ad-hoc simplifications influence the results. Moreover, there is no easy way of evaluating these effects in such approaches. An advantage of the present formulation is that modelling studies of simplified approaches becomes straightforward, as is shown below. An approach that has often been favoured in theoretical studies is the inextensible elastica theory, and a range of analytical [27-291 and numerical [30-321 solutions is available. The theory is obtained by assuming that the extensional strain is zero, and hence the deformation of the beam is only characterized by the bending strain p from eqn (IS). In addition, the axial stress-strain relationship from eqn (20) is ignored, and the axial load in the beam must be found from equilibrium (eqn (18)). However, the theory is not particularly suitable as a model for frame analysis based on a stiffness formulation. The need to enforce the inextension constraint results in an awkward numerical problem. The use of a large value for the area of the beam relative to its inertia is one possible way of enforcing the constraint, but it can lead to numerical instabilities [33]. For these reasons, we do not discuss this model further in this work. 3.2. Beam -column theory The most common simplified theory adopted for frame analysis is beam-column theory, and this approach is also typically used for stability analysis of frames [2]. In beam-column theory, it is assumed that displacements and strains are small, and that the axial strain is very much smaller than the bending strain. Thus, we assume that us<
us<<1,
cos 4 Z 1, sin 4 = 4.
(21)
With these assumptions, the strain-displacement equations (eqn (16)) reduce to e =uS++$,
p =vSS.
(22)
The second term in the equation for e represents the bowing effect on the axial strain due to bending [17], and is often ignored in the analysis. In this case, eqn (22) is replaced by e=u,,
P=u~~,
(23)
and these are the usual linear theory relationships. The assumptions also imply from eqn (18) that N=H+V&
Q=-H4+V.
(24)
25
However, beam-column theory replaces the equation for N with simply N = H. Finally, the equilibrium equations (eqn (19)) reduce to H,+p,=O, MS- Hv,+
V,+p,=O, V=O.
(25)
The last of these equations takes into account the moment induced by the horizontal force due to the relative end displacement of the member, but ignores the effect of axial shortening on the moment induced by the vertical force. Equations (20), (24), (25) and either eqn (22) or (23) represent the governing equations for the theory. Thus, we can identify two theories for beamcolumn analysis, depending on whether eqn (22) or eqn (23) is used. These theories will be denoted beamcolumn with bowing theory (using eqn (22)) and beam-column theory (using eqn (23)). 3.3. Solution of member problem We now consider the procedure for establishing the member force-displacement relationships. As a first step, we must adopt a particular model for the member behaviour, thereby establishing the governing differential equations. An important point to note is that in all cases we have a sixth-order theory for which identical boundary conditions can be specified as discussed previously. For the proposed solution scheme, we use the nodal displacements as the primary variables, and these are related to the end displacements of the member through the compatibility conditions (eqn (10)). Thus, at any given cycle of the global solution loop, we are required to solve the governing differential equations subject to known displacement boundary conditions, that is, we are solving a boundary value problem for a set of ordinary differential equations. This class of problems has been extensively studied in the numerical analysis literature, and a number of robust techniques are available [34]. The three most common techniques used are: (1) shooting and multiple shooting methods; (2) finite difference methods; and (3) collocation methods. Again, no clear consensus has emerged regarding the optimum technique to be used, but all the techniques are generally reliable. In this work, we have chosen to use a collocation method [35]. This particular method is well-established, and is also convenient since the governing equations need not be reduced to a first-order system, as is required by typical implementations of the other methods. This technique uses a collocation method in conjunction with a spline interpolation of the system variables. The routine is self-adaptive, with both the number and spacing of the collocation points being adjusted by the routine to achieve the desired
26
J. Petrolito and K. A. Legge P/10 a
t b’-p
‘.L
7
Fig. 4. Cantilever under end compression and shear. accuracy set by the analyst. This is of particular importance since the analyst need only specify the minimum topology for the structure, and still be confident of obtaining the required accuracy for the solution. In contrast, typical finite element schemes require the mesh subdivision to be made by the analyst, and error estimates are not usually automatically available. The spline interpolation ensures that all the primary variables, namely the displacements and forces, are smoothly interpolated along the element. This avoids the discontinuities in forces that typically occur with finite element discretisations. With the description of the formulation and solution of the member problem, the proposed method is complete. In summary, the method combines an iterative strategy at the global level to solve the nonlinear equilibrium equations, and a differential equation solver for the boundary value problem at the element level.
4. EXAMPLES
A number of examples are considered to demonstrate the nature and accuracy of the proposed method. The examples are used to highlight the following issues: (1) the differences and characteristics of the various nonlinear theories; (2) the convergence and accuracy of the method; and (3) the use of the method for generating high-accuracy benchmarks. The solutions are presented for three of the nonlinear theories described above, namely the beamcolumn, beam-column with bowing, and extensible elastica theories. Results are not presented for inextensible elastica theory, since as we discussed earlier, this model is not generally used for frame analysis. Note that further benchmark results are presented in [36]. For simplicity, all members in each problem are assumed to have the same properties. The relative influence of the axial and bending deformations is governed by the slenderness ratio i = L/r where L is a characteristic length of the structure and r is the radius of gyration of the cross-section. A constant P/10 1
a
b’ _P
b------L
Fig. 5. Cantilever under end torsion and shear.
NN-.--
bbobb _-ICI
xxxxx ~~~~~
“T-7 7; *oooo __I__
P
Unified nonlinear elastic frame analysis
27
Table 3. Convergence comparisons for cantilever under point loads P
Tolerance
N BC
I BC
ND”,
1.0
10-Z IO -’ IO-4 10-5 IO-6
12 12 12 12 12
2 2 2 2 2
22 31 31 32 32
2 2 2 4 8
22 30 30 30 33
IE 2 2 2 2 8
1.5
10-2 10-3 10-d 10-x 10-e
12 12 12 12 12
2 2 2 2 2
22 53 53 53 56
2 4 8 16 16
22 46 46 47 47
2 4 8 16 16
2.25
10-* IO-3 10m4 10-S 1O-6
14 14 14 14 17
2 2 2 2 4
35 37 90 78 78
10 16 40 64 64
41 101 102 106 106
10 16 20 60 50
value 1 = 100 is used in the problems. The various results quoted in the tables are identified by the subscripts BC for beam-column theory, Bow for beam-column with bowing theory, and E for extensible elastic theory. The results are quoted in terms of non-dimensional displacement and force quantities which are defined as ,-=u
62 L’
L’
where P is an applied concentrated load and p is an applied uniformly distributed load. 4.1. Cantilever
under point loads
As a first problem, we consider a cantilever under the combined effects of an end shear and either an end compressive load (see Fig. 4) or an end tensile load (see Fig. 5). Since the maximum compressive load considered is below the buckling load, the results were obtained using method I. Values of the horizontal and vertical displacements at point b and the bending moment at point a are given in Tables 1 and 2. The results quoted are accurate to the number of figures shown, and hence may be used as benchmarks for the various theories. In both cases, the only difference between the beamcolumn results and the beamcolumn with bowing results is in the horizontal displacement. For the beam-column results, the horizontal displacement is the linear result. The compressive load increases the transverse displacement due to the end shear, and this effect becomes more pronounced as the buckling load (P = n2/4 x 2.47) is approached. There are significant differences among the theories for the compressive case for high values of P. The tensile load stiffens the beam against bending, and reduces both the transverse displacement and the support bending moment. In this case there is closer agreement among the theories.
IB”W
NE
Table 3 shows the convergence characteristics of the method for varying tolerance requirements for the compressive load case. The table quotes both the number of function calls (N), which represents the number of times the nonlinear equilibrium equations (see eqn (12)) are set up, and the maximum number of subdivisions required at the element level (I) for varying tolerances (To/) and loads. The error tolerance is used to test for convergence at both the global and local levels of the iteration. At the element level, the requested tolerance is used to check the convergence of both the element displacements and forces, and as such is a more stringent convergence requirement than is usually applied in finite element approaches. It should be noted that the calculations have been done in distinct stages. A preliminary linear analysis is done in all cases. In addition, a coarse analysis based on beam-column theory is also performed for the beam-column with bowing and elastica solutions. While this strategy results in higher function calls in some cases, it avoids large corrections in the nonlinear iterations, and as such it is generally a more robust procedure.
pL/lO
PWO
Fig. 6. Two-storey frame under distributed and point loads.
J. Petrolito and K. A. Legge
28
Table 4. Results for two-storey frame under distributed and point loads B 0.1 0.2 0.3 0.4 0.5 0.6
& 1.11573 x 2.61130 x 4.73641 x 8.02341 x 1.38424 x 2.70475 x
10-2 lO-2 lO-2 lO-2 10-l 10-l
MB, - 2.29746 x -4.50643 x - 6.55220 x - 8.27483 x - 9.26234 x - 8.06308 x
lO-2 lO-2 10m2 10m2 10m2 lo-*
M,” - 2.29689 x -4.50405 x - 6.54698 x - 8.26724 x - 9.26112 x - 8.15247 x
UB”W 1.11912 x lO-2 2.62546 x lO-2 4.76992 x 1O-2 8.08647 x 1O-2 1.39421 x IO-’ 2.70624 x 10-l
GE lO-2 lO-2 1O-2 lO-2 10m2 lO-2
1.11900 x 2.62435 x 4.76397 x 8.05758 x 1.37807 x 2.56730 x
IO-’ 10m2 lO-2 IO-* IO-’ lo-’
-
HE 2.29680 x 1O-2 4.50340 X 10-l 6.54529 x lO-2 8.26662 x lO-2 9.28539 x lO-2 8.47673 x lO-2
Table 5. Convergence comparisons for two-storey frame under distributed and point loads @ = 0.2) Tolerance
NBc
I,,
N,,,
10-Z 10-j 10-4 10-s 10-h lo-’ IO-8 10-g
34 34 34 35 35 36 36 40
2 2 2 2 2 4 8 8
51 54 55 55 56 57 58 59
IBoW NE 4 4 4 4 4 8 8 16
51 55 55 55 56 57 58 59
IE 4 4 4 4 4 8 8 16
It can be seen that very few function calls and subdivisions are required to obtain typical engineering accuracy of say 1%. As expected, the amount of computation required increases as the requested tolerance increases, but significant increases only occur for high tolerances, which are not usually required in practice. In addition, the amount of computation also increases as the buckling load is approached. Thus for loads close to buckling or limit loads, it is preferable to use the continuation method.
-
~._
__L_---
_-_L---
_
Fig. 7. Toggle frame under point load. 4.2. Two-storey frame under uniform transverse load and point loads This example (see Fig. 6) considers a two-storey frame subjected to both uniformly distributed loads and point loads. Again, the results were obtained using method 1. The results quoted are the horizontal displacement and bending moment at point a, and are given in Table 4. Again, all three theories give different results, with the beam-column with bowing results being closer to the elastica results than the beam-column results. The convergence characteristics are shown in Table 5, and again confirm the rapid convergence of the method. 4.3. Toggle frame Figure 7 shows a shallow toggle frame subjected to a point load in the centre. The original problem was
2
1.8 1.6
0.6 Eiastica -
0.4
Beam-column + bowing Beam-column
--------
0.2
0
1
0.03
0.04
0.05
0.06
Normalised Displacement
Fig. 8. Results for toggle frame.
0.07
0.06
0.09
0.1
Unified nonlinear elastic frame analysis
‘i
^
iv
-I-0.2L L-
: L
1 Fig. 9. L frame under point load.
studied by Williams [37], and is often used as a test example. The frame exhibits snap-through behaviour under increasing load. Hence the generation of the complete load-deflection curve requires method 2. Due to symmetry, only one half of the frame needs to be analysed. The load-deflection curves for the various theories are shown in Fig. 8. It can be seen that beamcolumn theory is an inadequate model for this structure. It cannot predict the true response of the structure, and significantly overestimates the limit load. In contrast, beam-column with bowing theory is sufficient for the problem, and there is little difference between the results for this and the elastica theory. This contra-
29
diets the expectation [IO, 381 that bowing effects in structures are not important. 4.4. L Frame under point load
Figure 9 shows a two-member frame under a point load, which was originally considered in [39]. Again, the aim is produce the complete load-deflection response, giving the results shown in Fig. 10. It can be seen that the response of this structure is highly nonlinear, and that the structure exhibits a limit point instability. Neither beamcolumn theory nor beam-column with bowing theory can adequately predict the response except at low load values. In particular, beam-column theory predicts a significant postbuckling strength of the structure, and is thus grossly in error. The inclusion of the bowing effect avoids this erroneous prediction, but leads instead to an underestimate of the limit load. Thus, it can be concluded that elastica theory must be used to obtain reliable results for this example. 5. CONCLUSIONS
A unified approach to nonlinear elastic analysis of two-dimensional frames has been presented. The method uses a general formulation at the global level, without any a priori assumptions regarding the member behaviour. Hence, different models of member behaviour can be easily implemented and explored. The implications of various models were illustrated by a number of examples. The procedure is self-adaptive at the member level, and only the minimum topology of the structure is
20
15
IO
P 3
B E-
2
5 Beam-column + bowing Beam-column
----. --...
z”
0
-5
-10 0.2
0.4
0.6
Normalised Displacement
Fig. 10. Results for L frame.
0.8
1
J. Petrolito and IL A. Legge
30
required to be specified. Hence, the traditional frame analysis approach of minimum discretization is retained at the user level, while automatically ensuring that the required accuracy is achieved. At the computational level, the results show that engineering accuracy is achieved with the generation and solution of few equations at both the global and local levels of the method.
20.
2, 22.
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Press, Oxford
(1975).
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17. 18.
Manual, version 5 (1993). Integrated Technical Software Pty Ltd, Space&w Reference Manual, version 6.1 (1994). S. -A. Saafan, Nonlinear behaviour of structural plane frames. J. struct. Div. ASCE 89 (ST4), 557-579 (1963). J. J. Connor, R. D. Logcher and S. C. Chan, Nonlinear analysis of elastic framed structures. J. struct. Div. ASCE 94 (ST6), 1525-1547 (1968). C. Oran, Tangent stiffness in plane frames. J. struct. Div. ASCE 99 (ST6), 973-985 (1973). C. Oran and A. Kassimali, Large deformations of framed structures under static and dynamic loads. Comput. Struct. 6, 539-547 (1976). -Y. Goto and W. F. Chen, Second-order elastic analysis for frame design. J. struct. Engng ASCE 113 (ST7), 1501-1519 (1987). A. M. Ebner and J. J. Ucciferro, A theoretical and numerical comparison of elastic nonlinear finite element methods. Comput. Struct. 2, 1043-1061 (1972). M. Epstein and D. W. Murray, Large deformation in-plane analysis of elastic beams. Comput. Struct. 6, 1-9 (1976). M. H. El-Zanaty and D. W. Murray, Nonlinear finite element analysis of steel frames. J. struct. Engng ASCE 109 (ST2), 353-368 (1983). J. A. T. de Freitas and D. L. Smith, Finite element elastic beam
Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Hall, Englewood Cliffs, NJ (1983). E. Riks, On formulation of path-following techniques for structural stability problems. In: New Advances in Computational Structural Mechanics (Edited by P. Ladeveze and 0. C. Zienkiewicz). Elsevier, New York (1992). W. C. Reinboldt, Numerical Analysis of Parametrized Nonlinear Equations. Wiley, New York (1986). R. Seydel, Practical Bifiir&tion and Stability Analysis, 2nd edn. Springer. New York (1994). S. S. An&an-General solutions of plane extensible elasticae having nonlinear stress-strain laws. Q. appl. Math. 26, 3347 (1968). S. S. Antman, KirchhofI’s problem for nonlinear elastic rods. 0. appl. Math. 32. 221-240 (1974). E. Reissnk;, On one-dimensional finite-strain beam theory. ZAMP 23, 795-804 (1972). M. E. Gurtin, An Introduction to Continuum Mechanics. Academic Press, Boston, MA (1981). A. Libai and J. G. Simmonds, The Nonlinear Theory of Elastic Shells: One Spatial Dimension. Academic Press, Boston, MA (1988). R. Frisch-Fay, Flexible Bars. Butterworth, London (1962). W. Gorski, A review of the literature and a bibliography on finite elastic deflections of bars. Civ. Engng Trans. Inst Engrs, Aust. CE18, 74-85 (1976). I. Fried, Stability and equilibrium of the straight and curved elastica-finite element formulation. Comput. Meth. appl. Mech. Engng 28, 4961 (1981). B. W. Gollev, The finite element solution of a class of elastica problems. Comput. Meth. appl. Mech. Engng 46, 159-168 (1984). J. T. Holden, On the finite deflections of thin beams. Inr. J. Solids Struct. 8, 1051-1053 (1972). R. D. Cook. D. S. Malkus and M. E. Plesha. Concerts and Applications of Finite Element Analysis, 3rd edn. Wiley, New York (1989). J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 2nd edn. Springer, New York (1993). I_. Ascher, J. Christiansen and R. D. Russell, A collocation solver for mixed order systems of boundary value problems. Math. Compur. 33, 659679 (1979). J. Petrolito and K. A. Legge, Benchmarks for nonlinear elastic frame analysis. Steel Constr. AISC29(1), 2-12 (1995). F. W. Williams, An approach to the non-linear behaviour of the members of a rigid jointed plane framework with finite deflections. Q. J. Mech. appl. Math. 17, 451469 (1964). A. Kern, Effect of bowing on rectangular plane frames. J. struct. Div., ASCE, 107 (ST3), 569-574 (1981). S. L. Lee, F. S. Manuel and E. C. Rossow, Large deflection analysis and stability of elastic frames. J. Engng Mech. Div., ASCE 94 (EM2), 521-547 (1968).
19. J. E. Dennis and R. E. Schnabel,
24. 25 26. 27
28. 29.
30.
31.
32. 33.
34. 35.
36.
37.
38.
39.
0045-7949(95)00378-9
UNIFIED
NONLINEAR
ELASTIC
FRAME
ANALYSIS
J. Petrolito and K. A. Legge School of Science and Engineering, La Trobe University, Bendigo, Bendigo, Victoria, 3550, Australia (Received
Abstract-This frames.
3 February
1995)
paper develops a unified nonlinear analysis technique for two-dimensional
Any level of nonlinear
effects can be rationally
treated,
and results automatically
structural obtained to any
specified accuracy. The procedure only requires minimum discretisation of the structure as a self-adaptive procedure is used to solve the member equations. The nonlinear solution techniques used enable either discrete equilibrium positions or the complete load-deflection response to be calculated. Copyright 0 1996 Elsevier
Science Ltd
1. INTRODUmION
cient accuracy in a nonlinear analysis generally requires that each member in the structure be subdivided into a number of elements. However, codes of practice formulate their rules on the basis of minimum discretization. Hence, subdivision of members is not compatible with the philosophy of current codes. With this in mind, the term “balanced formulation” was adopted in Ref. [ 161 to describe a nonlinear formulation that gives exact results consistent with the underlying theory using minimum discretization. Within this framework, it was shown that a beam-column analysis including the effects of bowing [17] is the most general balanced formulation available, provided it is consistently formulated. It was also noted that a consistent formulation has not always been adopted in various studies and commercial programs. Moreover, beam-column with bowing theory is restricted to small deflection problems, and hence it is not sufficiently general. However, if a full nonlinear theory is used, a finite element type of formulation that includes subdivision of individual members in the structure is required. Such an approach is only approximate, and the analyst needs to be confident that the resulting errors are within an acceptable tolerance. In addition, the amount of subdivision required for a given accuracy is uncertain, and a trial and error approach is required. Thus, there is a need to develop a unified, nonlinear formulation that is capable of allowing any level of nonlinear effects to be treated in a rational manner, and automatically provides results to a specified accuracy. The aim of this paper is to present such a unified, nonlinear elastic analysis technique for twodimensional structural frames. Any level of geometric nonlinearity may be easily treated. An adaptive scheme is used in the solution technique at the member level so that the analyst need only specify the minimum discretization required to determine the topology of the structure. Thus, despite the internal
Frame analysis is of major importance in structural engineering, and it is not surprising therefore that it is the subject of continuing research. The original classical methods of analysis are suitable for small structures. As structures increased in size and complexity classical methods became unwieldy, and approximate techniques were introduced. While these techniques are rapid, they are of variable accuracy and restricted to linear analysis [l]. Matrix methods of analysis [2] revolutionized the field of structural analysis, and freed the analyst from the limitations of the classical and approximate methods. The theory of matrix methods is now well-established, and numerous commercial programs are available to carry out such analyses. Until recently the emphasis in frame analysis has been on linear analysis, and the treatment of the destabilizing effects of axial compressive loads was done at the design stage of the individual members in the structure. However, recent changes in codes of practice such as [3] have necessitated a reappraisal of this approach. This has led to the introduction of nonlinear analysis options in commercial programs [4, 51, and a renewed interest in the formulation of nonlinear analysis. A number of methods have been developed for nonlinear elastic analysis of frames. These range from formulations based on beam-column analysis [&lo] to formulations based on finite element ideas [ 1l-l 51. Despite the level of interest in the problem, there appears to be no general consensus on the basis or formulation of such procedures. The authors have shown in a previous paper[l6] that the techniques used in a nonlinear analysis can have a significant effect on the results. In particular, the traditional frame analysis approach using minimum discretization, where each beam and column is treated as a single member, may not be suitable for some procedures and leads to errors. Hence, to achieve suffi21
22
J. Petrolito
and K. A. Legge
the displacements and rotations at each end relative to the local member axes (see Fig. 2b), which are denoted by
Similarly, the member are defined as
IYG --
end forces in local coordinates
Xc The detailed member behaviour under loading determines the relationship between v and P, which will be nonlinear in general. Without being explicit about this relationship, we assume at this stage that such a relationship exists so that we may formally write
Fig. 1. Typical frame for analysis.
PM =f(vM),
adaptivity, the results are presented for the overall individual members, as this is the most convenient format for design. The nonlinear solution techniques used enable either discrete equilibrium positions or the complete loaddeflection response to be calculated.
2.
GENERAL FORMULATION FOR NONLINEAR ANALYSIS
In this section, we present the general formulation used to solve the overall nonlinear problem. Only a minimum number of assumptions are introduced at this stage regarding the theory to be used to describe the member behaviour. For illustration, we consider the two-dimensional rigid frame shown in Fig. 1. The topology of the structure is descibed by a set of nodes connecting the individual members. We assign a set of node numbers, with typical nodes denoted i, j, k etc., and a fixed set of global reference axes X, and Y, The figure also shows typical members M, N and 0, with member M connected to nodes i and j. The movement of each node is uniquely described by two displacements and a rotation. Thus for a typical node k, the nodal displacements are #={r’;,r:,r’;>,
wherefis a nonlinear function. Moreover, we do not insist that an explicit form can be found forf. Rather, the only requirement is that P can be determined if the member properties, loading and end displacements are specified. The form off‘ will also vary depending on the modelling assumptions made for the member behaviour. Under certain assumptions, an explicit form forf can be found. Specific examples of this include linear theory and beam-column theory. The resulting forms off in these cases are well known and are given in Ref. [2]. However, we do not make use of this knowledge in the present formulation since the aim is to produce a unified, general procedure that is applicable for nonlinear analysis where an explicit form for f cannot be obtained.
(1)
where curly brackets are used to denote a column vector. Similarly, the concentrated forces that are applied at node k are denoted by
A typical member M is shown in Fig. 2, with its local axes X and Y such that it is oriented at an angle 0 to X,. The member is subjected to distributed loads px and pv along its length. We assume that the deformation of the member can be characterized by
(5)
PM 1' P1 ’
Cc)
Fig. 2. Typical member and sign convention.
Unified nonlinear elastic frame analysis It is convenient to transform the member end displacements and forces to global coordinates as shown in Fig. 2c, giving TM= T”vM, where the transformation
T=
PM = T”PM, matrix
(6)
T is given by
T,0 1
[ 0
T,’
and T, =
Since eqn (6) is an orthogonal inverse transformations are vM = (Tr)“SM, The compatibility 1) give
transformation,
PM = (Tr)?!
requirements
i;p=sp=r”
the
(9)
at node k (see Fig.
(IO)
where 7, = (U,, I&, uj} and c2 = {cd, r&, r&6).Similarly, for node k to be in equilibrium, we have
Combining eqns (St(ll) for all nodes and members gives the global set of governing equations for the structure, namely R -g(r)
= 0,
(12)
where r is a vector containing all the unknown nodal displacements, R is the corresponding vector of applied nodal forces and g is in general a nonlinear function. It is worth noting that the above formulation is equivalent to the stiffness method of frame analysis if an explicit form of the member forcedisplacement relationship (eqn (5)) is used. In this case, we have g(r) = Kr where K is the global stiffness matrix. In the linear elastic case K is independent of r, whereas for the beamcolumn case K is a function of r, and in both cases explicit forms for K can be found [2]. In the present case however, there are no a priori assumptions made about the nature of g, and thus the formulation includes the above cases merely as particular examples. Hence, the solution of the nonlinear analysis problem reduces to the solution of eqn (12). A number of solution schemes are available for this class of problem. The general details of the scheme adopted in this work are as follows:
23
(1) Set up the structural data, including topology, member characteristics, loads and supports. Initialize all unknown nodal displacements to zero. Specify a tolerance for the analysis. (2) For each member, solve eqn (5) and use eqn (6) to calculate the member end forces in global coordinates. (3) For each node, establish equilibrium equations associated with unknown displacements using equations of the form of eqn (I I). Hence, establish the overall out of balance nodal forces R -g(r) for the structure (see eqn (12)). (4) Use an update strategy to adjust the nodal displacements based on the amount of out of balance forces. (5) Check for convergence using a suitable criterion. If convergence has been achieved, the analysis is complete. If not, go to step 2 and continue. The key step in the global solution scheme is step 4, with the specific details of this step varying with the particular procedure adopted. A large number of techniques have been proposed, depending on the nature of the nonlinear problem and the goal of the analysis [18]. We can identify two general classes of problems, namely those near limit or bifurcation points and those away from such points. Accordingly, we have adopted a separate scheme for each class of problems as follows: (1) Problems requiring the determination of one equilibrium state of the structure, which is not near the first limit or bifurcation point-under these conditions, we have used a Newton-Raphson technique [19] to solve the nonlinear equations. Since we have stipulated that the solution is not close to points of instability, this approach can be expected to reliably converge under reasonable conditions. (2) Problems requiring the determination of the complete load-deflection curve including post-buckling response-a large number of techniques have been proposed for this class of problem. While most of the proposed techniques are reliable, it has been noted by Riks [20] that there is no generally optimal strategy. For the present work, we have used a continuation technique [2l], including an adaptive step procedure, to efficiently generate the loaddeflection curve. This technique reduces to a method suitable for the previous class of problems when a single step is used to apply the load. The technique is capable of traversing limit points for both loads and deflections automatically. However, the detection of bifurcation points requires the use of a perturbation technique [22]. The key feature at the member level is the solution of eqn (5) in step 2. This is a distinctive aspect of the present formulation, and is discussed in detail below. It is important to note that the global strategy is independent of the strategy adopted to solve the member problem. Thus, a wide variety of assumptions regarding the details of the member behaviour
J. Petrolito and K. A. Legge
24
strain, p, which are defined as [23]
Q>t
ds e==--1, 1:Y,v,.i
Combining
p==.
d@
(15)
eqns (13)<15) gives e = J(1 + us)’ + vi - 1,
P
~ = (1 - Us)%s- ususs (1 +u.#+v; ’
X,X,U,i
dS
Fig. 3. Beam element and sign convention can be easily explored and implemented in a straightforward manner. 3.
FORMULATION
AND SOLUTION EQUATIONS
OF MEMBER
3.1. Extensible elastica theory This is the most general model that we consider to describe the behaviour of individual members, and a detailed discussion of the theory is given in Ref. [23]. The theory includes the effects of large displacements and strains, but ignores the effect of shear deformation. Thus, it generalises linear beam theory based on the Euler-Bernoulli law [l]. Extensions of the theory that include shear deformations are available [24,25], but they will not be considered here as shear deformations are not usually significant for typical structural frames. We consider a beam whose axis is initially along the X axis with a point on the axis being specified by length S with 0 < S < L where L is the length of the beam (see Fig. 3). After deformation, the beam axis deforms into a smooth curve s. A point P with initial coordinates (X, 0) is mapped to a point P* with coordinates (x, y). The functions x(S) and y(S) define the geometry of the deformed axis of the beam. The angle between s and the X axis at P* is denoted by 4(S). The beam displacement functions u(S) and u(S) are given by
49 = x(S) -X(S),
u(S) = y(S).
(13)
An element dS = dX on the original axis is deformed into an element ds. From the geometry of Fig. 3, we have dx - = Cos 4, ds
dv - = sin 4. ds
where a subscript S denotes differentiation with respect to S. The force resultants on the beam are the force F and bending moment M, which is taken as positive if it acts anticlockwise. The force F can be expressed as F=Hi+Vj=Nn+Qt,
In this section, we consider the formulation of the governing equations of a frame member for a variety of modelling assumptions. Following this, we present the numerical method used at the member level to generate the forcedisplacement equations.
(14)
Natural strain measures for the deformation of the beam are the extensional strain, e, and the bending
(16)
(17)
where H and V are the horizontal and vertical components of F, N is the axial load and Q is the shear force. The unit vectors n and t are in the outward normal and tangential directions respectively. The force components are related by (see eqn (9))
Q=-Hsin~#~+Vcos$. The equilibrium
(18)
equations for the beam are [23]
H,+p,=O,
Vs+pv=O,
M, - Ho, + V(1 + us) = 0.
(19)
To complete the problem description requires the specification of stress-strain relationships. In principle, any stress-strain relationship consistent with the general requirements of constitutive equations [26] may be used. However, for the present work we will assume that the material is linear-elastic giving N = EAe,
M = EIp,
(20)
where E is Young’s modulus, A is the cross-section area and I is the moment of inertia. In general, the material properties could be functions of S. The theory results in a nonlinear system of ordinary differential equations of order six. Hence, three boundary conditions are required to be specified at each end of the beam, namely: (1) either u is specified or H is specified; (2) either u is specified or V is specified; and (3) either 4 is specified or M is specified. Exact solutions of the governing equations are only possible in simple cases [23,27], and numerical methods are usually required to solve practical prob-
Unified nonlinear elastic frame analysis
lems. Due to the difficulty of establishing closed form solutions for the general theory, simplifying assumptions have usually been made to achieve a tractable formulation. This approach has also been used in many prior studies on nonlinear frame analysis. However, it is not clear to what extent these ad-hoc simplifications influence the results. Moreover, there is no easy way of evaluating these effects in such approaches. An advantage of the present formulation is that modelling studies of simplified approaches becomes straightforward, as is shown below. An approach that has often been favoured in theoretical studies is the inextensible elastica theory, and a range of analytical [27-291 and numerical [30-321 solutions is available. The theory is obtained by assuming that the extensional strain is zero, and hence the deformation of the beam is only characterized by the bending strain p from eqn (IS). In addition, the axial stress-strain relationship from eqn (20) is ignored, and the axial load in the beam must be found from equilibrium (eqn (18)). However, the theory is not particularly suitable as a model for frame analysis based on a stiffness formulation. The need to enforce the inextension constraint results in an awkward numerical problem. The use of a large value for the area of the beam relative to its inertia is one possible way of enforcing the constraint, but it can lead to numerical instabilities [33]. For these reasons, we do not discuss this model further in this work. 3.2. Beam -column theory The most common simplified theory adopted for frame analysis is beam-column theory, and this approach is also typically used for stability analysis of frames [2]. In beam-column theory, it is assumed that displacements and strains are small, and that the axial strain is very much smaller than the bending strain. Thus, we assume that us<
us<<1,
cos 4 Z 1, sin 4 = 4.
(21)
With these assumptions, the strain-displacement equations (eqn (16)) reduce to e =uS++$,
p =vSS.
(22)
The second term in the equation for e represents the bowing effect on the axial strain due to bending [17], and is often ignored in the analysis. In this case, eqn (22) is replaced by e=u,,
P=u~~,
(23)
and these are the usual linear theory relationships. The assumptions also imply from eqn (18) that N=H+V&
Q=-H4+V.
(24)
25
However, beam-column theory replaces the equation for N with simply N = H. Finally, the equilibrium equations (eqn (19)) reduce to H,+p,=O, MS- Hv,+
V,+p,=O, V=O.
(25)
The last of these equations takes into account the moment induced by the horizontal force due to the relative end displacement of the member, but ignores the effect of axial shortening on the moment induced by the vertical force. Equations (20), (24), (25) and either eqn (22) or (23) represent the governing equations for the theory. Thus, we can identify two theories for beamcolumn analysis, depending on whether eqn (22) or eqn (23) is used. These theories will be denoted beamcolumn with bowing theory (using eqn (22)) and beam-column theory (using eqn (23)). 3.3. Solution of member problem We now consider the procedure for establishing the member force-displacement relationships. As a first step, we must adopt a particular model for the member behaviour, thereby establishing the governing differential equations. An important point to note is that in all cases we have a sixth-order theory for which identical boundary conditions can be specified as discussed previously. For the proposed solution scheme, we use the nodal displacements as the primary variables, and these are related to the end displacements of the member through the compatibility conditions (eqn (10)). Thus, at any given cycle of the global solution loop, we are required to solve the governing differential equations subject to known displacement boundary conditions, that is, we are solving a boundary value problem for a set of ordinary differential equations. This class of problems has been extensively studied in the numerical analysis literature, and a number of robust techniques are available [34]. The three most common techniques used are: (1) shooting and multiple shooting methods; (2) finite difference methods; and (3) collocation methods. Again, no clear consensus has emerged regarding the optimum technique to be used, but all the techniques are generally reliable. In this work, we have chosen to use a collocation method [35]. This particular method is well-established, and is also convenient since the governing equations need not be reduced to a first-order system, as is required by typical implementations of the other methods. This technique uses a collocation method in conjunction with a spline interpolation of the system variables. The routine is self-adaptive, with both the number and spacing of the collocation points being adjusted by the routine to achieve the desired
26
J. Petrolito and K. A. Legge P/10 a
t b’-p
‘.L
7
Fig. 4. Cantilever under end compression and shear. accuracy set by the analyst. This is of particular importance since the analyst need only specify the minimum topology for the structure, and still be confident of obtaining the required accuracy for the solution. In contrast, typical finite element schemes require the mesh subdivision to be made by the analyst, and error estimates are not usually automatically available. The spline interpolation ensures that all the primary variables, namely the displacements and forces, are smoothly interpolated along the element. This avoids the discontinuities in forces that typically occur with finite element discretisations. With the description of the formulation and solution of the member problem, the proposed method is complete. In summary, the method combines an iterative strategy at the global level to solve the nonlinear equilibrium equations, and a differential equation solver for the boundary value problem at the element level.
4. EXAMPLES
A number of examples are considered to demonstrate the nature and accuracy of the proposed method. The examples are used to highlight the following issues: (1) the differences and characteristics of the various nonlinear theories; (2) the convergence and accuracy of the method; and (3) the use of the method for generating high-accuracy benchmarks. The solutions are presented for three of the nonlinear theories described above, namely the beamcolumn, beam-column with bowing, and extensible elastica theories. Results are not presented for inextensible elastica theory, since as we discussed earlier, this model is not generally used for frame analysis. Note that further benchmark results are presented in [36]. For simplicity, all members in each problem are assumed to have the same properties. The relative influence of the axial and bending deformations is governed by the slenderness ratio i = L/r where L is a characteristic length of the structure and r is the radius of gyration of the cross-section. A constant P/10 1
a
b’ _P
b------L
Fig. 5. Cantilever under end torsion and shear.
NN-.--
bbobb _-ICI
xxxxx ~~~~~
“T-7 7; *oooo __I__
P
Unified nonlinear elastic frame analysis
27
Table 3. Convergence comparisons for cantilever under point loads P
Tolerance
N BC
I BC
ND”,
1.0
10-Z IO -’ IO-4 10-5 IO-6
12 12 12 12 12
2 2 2 2 2
22 31 31 32 32
2 2 2 4 8
22 30 30 30 33
IE 2 2 2 2 8
1.5
10-2 10-3 10-d 10-x 10-e
12 12 12 12 12
2 2 2 2 2
22 53 53 53 56
2 4 8 16 16
22 46 46 47 47
2 4 8 16 16
2.25
10-* IO-3 10m4 10-S 1O-6
14 14 14 14 17
2 2 2 2 4
35 37 90 78 78
10 16 40 64 64
41 101 102 106 106
10 16 20 60 50
value 1 = 100 is used in the problems. The various results quoted in the tables are identified by the subscripts BC for beam-column theory, Bow for beam-column with bowing theory, and E for extensible elastic theory. The results are quoted in terms of non-dimensional displacement and force quantities which are defined as ,-=u
62 L’
L’
where P is an applied concentrated load and p is an applied uniformly distributed load. 4.1. Cantilever
under point loads
As a first problem, we consider a cantilever under the combined effects of an end shear and either an end compressive load (see Fig. 4) or an end tensile load (see Fig. 5). Since the maximum compressive load considered is below the buckling load, the results were obtained using method I. Values of the horizontal and vertical displacements at point b and the bending moment at point a are given in Tables 1 and 2. The results quoted are accurate to the number of figures shown, and hence may be used as benchmarks for the various theories. In both cases, the only difference between the beamcolumn results and the beamcolumn with bowing results is in the horizontal displacement. For the beam-column results, the horizontal displacement is the linear result. The compressive load increases the transverse displacement due to the end shear, and this effect becomes more pronounced as the buckling load (P = n2/4 x 2.47) is approached. There are significant differences among the theories for the compressive case for high values of P. The tensile load stiffens the beam against bending, and reduces both the transverse displacement and the support bending moment. In this case there is closer agreement among the theories.
IB”W
NE
Table 3 shows the convergence characteristics of the method for varying tolerance requirements for the compressive load case. The table quotes both the number of function calls (N), which represents the number of times the nonlinear equilibrium equations (see eqn (12)) are set up, and the maximum number of subdivisions required at the element level (I) for varying tolerances (To/) and loads. The error tolerance is used to test for convergence at both the global and local levels of the iteration. At the element level, the requested tolerance is used to check the convergence of both the element displacements and forces, and as such is a more stringent convergence requirement than is usually applied in finite element approaches. It should be noted that the calculations have been done in distinct stages. A preliminary linear analysis is done in all cases. In addition, a coarse analysis based on beam-column theory is also performed for the beam-column with bowing and elastica solutions. While this strategy results in higher function calls in some cases, it avoids large corrections in the nonlinear iterations, and as such it is generally a more robust procedure.
pL/lO
PWO
Fig. 6. Two-storey frame under distributed and point loads.
J. Petrolito and K. A. Legge
28
Table 4. Results for two-storey frame under distributed and point loads B 0.1 0.2 0.3 0.4 0.5 0.6
& 1.11573 x 2.61130 x 4.73641 x 8.02341 x 1.38424 x 2.70475 x
10-2 lO-2 lO-2 lO-2 10-l 10-l
MB, - 2.29746 x -4.50643 x - 6.55220 x - 8.27483 x - 9.26234 x - 8.06308 x
lO-2 lO-2 10m2 10m2 10m2 lo-*
M,” - 2.29689 x -4.50405 x - 6.54698 x - 8.26724 x - 9.26112 x - 8.15247 x
UB”W 1.11912 x lO-2 2.62546 x lO-2 4.76992 x 1O-2 8.08647 x 1O-2 1.39421 x IO-’ 2.70624 x 10-l
GE lO-2 lO-2 1O-2 lO-2 10m2 lO-2
1.11900 x 2.62435 x 4.76397 x 8.05758 x 1.37807 x 2.56730 x
IO-’ 10m2 lO-2 IO-* IO-’ lo-’
-
HE 2.29680 x 1O-2 4.50340 X 10-l 6.54529 x lO-2 8.26662 x lO-2 9.28539 x lO-2 8.47673 x lO-2
Table 5. Convergence comparisons for two-storey frame under distributed and point loads @ = 0.2) Tolerance
NBc
I,,
N,,,
10-Z 10-j 10-4 10-s 10-h lo-’ IO-8 10-g
34 34 34 35 35 36 36 40
2 2 2 2 2 4 8 8
51 54 55 55 56 57 58 59
IBoW NE 4 4 4 4 4 8 8 16
51 55 55 55 56 57 58 59
IE 4 4 4 4 4 8 8 16
It can be seen that very few function calls and subdivisions are required to obtain typical engineering accuracy of say 1%. As expected, the amount of computation required increases as the requested tolerance increases, but significant increases only occur for high tolerances, which are not usually required in practice. In addition, the amount of computation also increases as the buckling load is approached. Thus for loads close to buckling or limit loads, it is preferable to use the continuation method.
-
~._
__L_---
_-_L---
_
Fig. 7. Toggle frame under point load. 4.2. Two-storey frame under uniform transverse load and point loads This example (see Fig. 6) considers a two-storey frame subjected to both uniformly distributed loads and point loads. Again, the results were obtained using method 1. The results quoted are the horizontal displacement and bending moment at point a, and are given in Table 4. Again, all three theories give different results, with the beam-column with bowing results being closer to the elastica results than the beam-column results. The convergence characteristics are shown in Table 5, and again confirm the rapid convergence of the method. 4.3. Toggle frame Figure 7 shows a shallow toggle frame subjected to a point load in the centre. The original problem was
2
1.8 1.6
0.6 Eiastica -
0.4
Beam-column + bowing Beam-column
--------
0.2
0
1
0.03
0.04
0.05
0.06
Normalised Displacement
Fig. 8. Results for toggle frame.
0.07
0.06
0.09
0.1
Unified nonlinear elastic frame analysis
‘i
^
iv
-I-0.2L L-
: L
1 Fig. 9. L frame under point load.
studied by Williams [37], and is often used as a test example. The frame exhibits snap-through behaviour under increasing load. Hence the generation of the complete load-deflection curve requires method 2. Due to symmetry, only one half of the frame needs to be analysed. The load-deflection curves for the various theories are shown in Fig. 8. It can be seen that beamcolumn theory is an inadequate model for this structure. It cannot predict the true response of the structure, and significantly overestimates the limit load. In contrast, beam-column with bowing theory is sufficient for the problem, and there is little difference between the results for this and the elastica theory. This contra-
29
diets the expectation [IO, 381 that bowing effects in structures are not important. 4.4. L Frame under point load
Figure 9 shows a two-member frame under a point load, which was originally considered in [39]. Again, the aim is produce the complete load-deflection response, giving the results shown in Fig. 10. It can be seen that the response of this structure is highly nonlinear, and that the structure exhibits a limit point instability. Neither beamcolumn theory nor beam-column with bowing theory can adequately predict the response except at low load values. In particular, beam-column theory predicts a significant postbuckling strength of the structure, and is thus grossly in error. The inclusion of the bowing effect avoids this erroneous prediction, but leads instead to an underestimate of the limit load. Thus, it can be concluded that elastica theory must be used to obtain reliable results for this example. 5. CONCLUSIONS
A unified approach to nonlinear elastic analysis of two-dimensional frames has been presented. The method uses a general formulation at the global level, without any a priori assumptions regarding the member behaviour. Hence, different models of member behaviour can be easily implemented and explored. The implications of various models were illustrated by a number of examples. The procedure is self-adaptive at the member level, and only the minimum topology of the structure is
20
15
IO
P 3
B E-
2
5 Beam-column + bowing Beam-column
----. --...
z”
0
-5
-10 0.2
0.4
0.6
Normalised Displacement
Fig. 10. Results for L frame.
0.8
1
J. Petrolito and IL A. Legge
30
required to be specified. Hence, the traditional frame analysis approach of minimum discretization is retained at the user level, while automatically ensuring that the required accuracy is achieved. At the computational level, the results show that engineering accuracy is achieved with the generation and solution of few equations at both the global and local levels of the method.
20.
2, 22.
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Manual, version 5 (1993). Integrated Technical Software Pty Ltd, Space&w Reference Manual, version 6.1 (1994). S. -A. Saafan, Nonlinear behaviour of structural plane frames. J. struct. Div. ASCE 89 (ST4), 557-579 (1963). J. J. Connor, R. D. Logcher and S. C. Chan, Nonlinear analysis of elastic framed structures. J. struct. Div. ASCE 94 (ST6), 1525-1547 (1968). C. Oran, Tangent stiffness in plane frames. J. struct. Div. ASCE 99 (ST6), 973-985 (1973). C. Oran and A. Kassimali, Large deformations of framed structures under static and dynamic loads. Comput. Struct. 6, 539-547 (1976). -Y. Goto and W. F. Chen, Second-order elastic analysis for frame design. J. struct. Engng ASCE 113 (ST7), 1501-1519 (1987). A. M. Ebner and J. J. Ucciferro, A theoretical and numerical comparison of elastic nonlinear finite element methods. Comput. Struct. 2, 1043-1061 (1972). M. Epstein and D. W. Murray, Large deformation in-plane analysis of elastic beams. Comput. Struct. 6, 1-9 (1976). M. H. El-Zanaty and D. W. Murray, Nonlinear finite element analysis of steel frames. J. struct. Engng ASCE 109 (ST2), 353-368 (1983). J. A. T. de Freitas and D. L. Smith, Finite element elastic beam
Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Hall, Englewood Cliffs, NJ (1983). E. Riks, On formulation of path-following techniques for structural stability problems. In: New Advances in Computational Structural Mechanics (Edited by P. Ladeveze and 0. C. Zienkiewicz). Elsevier, New York (1992). W. C. Reinboldt, Numerical Analysis of Parametrized Nonlinear Equations. Wiley, New York (1986). R. Seydel, Practical Bifiir&tion and Stability Analysis, 2nd edn. Springer. New York (1994). S. S. An&an-General solutions of plane extensible elasticae having nonlinear stress-strain laws. Q. appl. Math. 26, 3347 (1968). S. S. Antman, KirchhofI’s problem for nonlinear elastic rods. 0. appl. Math. 32. 221-240 (1974). E. Reissnk;, On one-dimensional finite-strain beam theory. ZAMP 23, 795-804 (1972). M. E. Gurtin, An Introduction to Continuum Mechanics. Academic Press, Boston, MA (1981). A. Libai and J. G. Simmonds, The Nonlinear Theory of Elastic Shells: One Spatial Dimension. Academic Press, Boston, MA (1988). R. Frisch-Fay, Flexible Bars. Butterworth, London (1962). W. Gorski, A review of the literature and a bibliography on finite elastic deflections of bars. Civ. Engng Trans. Inst Engrs, Aust. CE18, 74-85 (1976). I. Fried, Stability and equilibrium of the straight and curved elastica-finite element formulation. Comput. Meth. appl. Mech. Engng 28, 4961 (1981). B. W. Gollev, The finite element solution of a class of elastica problems. Comput. Meth. appl. Mech. Engng 46, 159-168 (1984). J. T. Holden, On the finite deflections of thin beams. Inr. J. Solids Struct. 8, 1051-1053 (1972). R. D. Cook. D. S. Malkus and M. E. Plesha. Concerts and Applications of Finite Element Analysis, 3rd edn. Wiley, New York (1989). J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 2nd edn. Springer, New York (1993). I_. Ascher, J. Christiansen and R. D. Russell, A collocation solver for mixed order systems of boundary value problems. Math. Compur. 33, 659679 (1979). J. Petrolito and K. A. Legge, Benchmarks for nonlinear elastic frame analysis. Steel Constr. AISC29(1), 2-12 (1995). F. W. Williams, An approach to the non-linear behaviour of the members of a rigid jointed plane framework with finite deflections. Q. J. Mech. appl. Math. 17, 451469 (1964). A. Kern, Effect of bowing on rectangular plane frames. J. struct. Div., ASCE, 107 (ST3), 569-574 (1981). S. L. Lee, F. S. Manuel and E. C. Rossow, Large deflection analysis and stability of elastic frames. J. Engng Mech. Div., ASCE 94 (EM2), 521-547 (1968).
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