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A nonlinear theory of general thin-walled beams
1
Nonlinear Mathematical Theories and Formulation Aspects
A NONLINEAR THEORY OF TiENERAL THIN-WALLED BEAMS
Meredith Engineering, 33170 Glen Vaiiey Drive, Farmington Hills, MI 48018. U.S.A. and E. A. WXTNBZR Department of Aeronauftcs and Astronautic& M~~~~ts Cambridge, MA 02139,U.S.A.
Institute
ofTangy,
(Received 18 April 1980) Abemet-A eompkte and consistent theory is formulated to r&c&e the large&dktion ehtttia-pltstic behaviot of thin-walk3 beams 5Farbkrary initial shape and sWbjc@d t5 arbitrary trikusientkradings.The nonlinear beam theory is derived from a wekkcumeated nonlinear shdi the5ry through an appiicrtion of Hamilton’s Principlc. By interpreting elastical beam theory displacement fidds asthe lower order terms in a more general series expansion of deformation modes which describe the bcbayi5r oftis beam CIOIIBtion, a higher arder beam theory based on such an expansion can be developed. The theory r*n be 5ufiki~~ti~ ~soast5incorpoirttcfhteffensof~o~~o~a~dierortion oadcdlapseiardditiont5thtcom~~e range of behavior ndy asso&& with beam theory ~~~~0~ extetlsi5r&bctt&g tzmtwew shtnr d&rmation, torsion, and warping). The generalized equations of motion ttss5cSabd with this s&s or modal technique am shown to be of the same form 88 tic generalizedequations of shdl theory from Which they ware derived. These nonlinear equations of motion for the beam are east into an approximate form suitable for imdementation on a dirritnleomDuter. The resulting prqram, MENTOR-3 is aP@ied to treat a nottlinear re&ttse and impact p>bitm. _
In recent years the concern over nuclear power plant safety and vehicle crashworthiness has speeded the development of sophisticate& ana&tical tools and computer codes required by the analyst to predict the noniinea~ b&avior of safety related structures during abnormal or upset conditicms. It has been demonstrated
J
that when suffiticnt care and attmtion is paid in developing the nonlinear analysis the predictians can be shown to bc rdiabk and accurate to within the order of aoqted a~~tioR. Therein, the engineer must determine the appropriate level of sophistication or precision which must be incorporated into the analysis in order for the predictions to be useful and meaningful. The three basic theories of structural mechanics
available to the engineering ana@% include bemn, shelf and general J-dimensional continuum theory. Each of these “matzo-mechanics” type theories incorporates a particular level of approximation which is clearly evident in its derivation from the fundamentaf prin&pies of physics. Traditiona continuum theory involves the Ieast amount ~fappro~tio~ and is normally considered to be a valid basis for the development of the more specialized theories of shells and beams. A measure which forms a suitable basis for disaping between beam, shell, and solid structures
lies in th&r relative phrsica dimensions fadditional criteria concerned v&h material properties, and smoothness of geometry and loading may also be addressed). For instance, shells are distinguishable from solids by their having one physical dimension, their thickntss, small in comparison with their Other dimensions. Beams on the other hand have two diiensions Oh& Cross sectiob measurements) Small in comparison with the remaining dimension (span). It is
I
t SOLXR
THICK-W&t&ED a~ r--IGcRoss-SECTION BEAM ---GaaAW Fig 1. Inter-relation between theories for solids, she& and ,7IIN-WLW
beams. evident
therefore, that one should in principle be able to connect and derive the equations for hesun% sk&
and 3-dimensional continuum (solids) by taking advantage of these basic factors which serve to distinguish each structure. Such a relationship is illustrated in Fig. 1
for smoothly contoured and loaded homogeneous stnretures, Some of the paths disoernable in Fig. 1 have already been developed. The theoretical developments ass&ated with paths A, B, C and D are available in [l-4]. For instance, in [l] advantage is taken of the fact that the thickness of a shell is thin in comparison with its other dimensions to reduce a 34irnensional mntinuum theory to one de&& on a 24mensional subspace or reference surface. This is accomplished by assuming within the cOntext of Hamilton’s Principle that the deformation through the shell thickness is of a known functional form (truncated series of ddotmation modes) whose parameters are de&d on the shell rekence surface. Unfortunately, it is often prohibitivdy expen sive to treat many problems of engineering interest by
4
D
MEREDITH
and E. A
employing a 3-dimensional continuum analysis or even the less-costly shell analysis. This is particularly true for structures in which at least one. and more likely two, of its physical dimensions is “thin” with respect to its other dimensions. Many investigators [i7] have studied the behavior of such doubly-thin structures and have contributed to the modem engineering theory of general beams. Most of these fo~ulations were developed to treat the behavior of beams whose cross-sections deformed in accordance with classical beam theory assumptions (translation, bending and torsion) or incorporated the first order effects of warping and transverse shear deformation. Several attempts to incorporate higher order effects in beams have been reported [g]: however, most are restricted to elastic beam material behavior and specific types of deformation fields. Of specific interest in the present paper is the nonlinear response of general 3-dimensional thin-walled beams. This field has also received considerable attention [S-11]; however, again the published results are generally restricted to small strain elastic behavior or incorporate a multitude of kinematic assumptions or constraints. It is nevertheless important to note that in each of the above-mentioned shell and beam forrmrlations, the displacement field of the structure along its thin dimension(s) is postulated in terms of generalized displacements deflned along the remaining dimension(s). By reducing the infinite number of degrees of freedom along the thin dimension to a suitable and finite num ber (these normally are just the ~pla~ents of the structure at its reference surface or reference axis), great economies in expenditure and effort are derived while maintaining suitable engineering accuracy. The present paper extends this basic concept to develop a thin-walls beam theory from shell theory by imposing an assumed displacement field aIong the shell reference surface coordinate (r~)which lies within the cross sectional face of the beam (Fig. 2). This reduces the shell theory to a beam theory (path E in Fig. I) wherein the parameters (displacement field coefhcients) which describe the structural defo~tion are evaluated or defined on a reference line (the beam axis). Such a formulation for thin-walled beams has a number of distinct advantages, the most significant of which is the orderly overlapping of the domains of applicability of shell and beam theory. In addition, since no restriction is applied to the type or number of independent degrees of freedom incorporated in the description of the dis-
Y’
/
5
Fig. ;. ‘Position vector to a 3-dimensional thin-walled beam. $Qandard tensor summation conventions are used throughout. Greek minus&es range over the values 1and 2 whereas Latin minuscules range over the values 1. ? and 3.
WITMER
placement field of the beam across its cross section, the analyst is no longer confined by the deformation restrictions of classical beam theory (which may be interpreted as the lower order terms m a more general series expansion). By the judicious insertion of additional mdependent degrees of freedom into the assumed displacement field of the cross section. effects such as cross-section distortion and collapse. normally associated with shell theory. may readily be incorporated mto the “beamtype” analysis. it is perhaps worthwhile to note at this stage that because each deformation mode is treated as being independent (at least in an incremental sense). the present procedure may also be Interpreted as a “generalized nonlinear mcremental modal technique” as applied to a beam-type structure. Also note that in the limit, as an infinite number of independent degrees of freedom is employed in the beam cross section displacement field, the beam solution should converge to the shelltheory solution as the truncation error approaches zero. In the following sections. the generalized equations which describe the arbitrarily large-deflection elasticplastic response of a thin-walled beam are derived from a thin-walled shell theory through an application of Hamilton’s Principle into which the above-mentioned beam displacement field is introduced. It will be shown that the generalized equations of motion suitable for describing the behavior of a beam are similar in form to the equations which describe the behavior of a shell and of a 3-dimensional continuum; a similar analogy exists in any modal or generalized Ritz of finite-element analysis. The fi~te-di~erence method is employed to approximate the spatial derivatives along the span of the beam. The resulting approximate equations of equilibrium are then integrated in time by means of a central-difference procedure to obtain the transient response of the beam. This analysis has been incorporated into the MENTOR-3 computer code which is available for treating structures such as shells, beams, or combinations thereof. This program has been employed to analyze a variety of problems wherein the effects of large~p~~ment e~ti~pI~tic behavior associated with the overall deformation of the structure are complicated by the presence oflocal cross sectional collapse.
An outline of the general development leading to the formulation of the general modal equations of motion suitable for a thin-walled beam is presented in this section. A more comprehensive and detailed develop ment may be found in El. 31. Geonrerry and position vector of the beam A general 34imensional body can be described in terms of the position vector R to any point within the body. Each such point can be uniquely defined by its i:’ intrinsic coordinates? (which, for each material point, are invariant in time). The position vector may be described in terms of its coordinates YJ in Cartesian space as: R= YJi I I yJ(<‘)i. I. In general, the positions YJ are a function of the three intrinsic coordinates <’ used to identify the material point. Consider now a beam whose cross section is comprised ofthin-walled members. It is often advantageous
A nonlinear theory of general thin-walled beams when treating thin structures to employ the concept of a reference surface which may, but need not be, the mid-surface of the structure. The reference surface of such a shell-type structure is defined by its position vector R, which can be described in terms of the two intrinsic coordinates cl, e2 associated with material points on the reference surface (Fig. 2) Rs= Yjj
(2)
Next, a third intrinsic coordinate c is introduced which is associated with the direction which is initially normal to the reference surface and is needed to describe the position of points located through the thickness of the structure. For thin structures. one employs the Kirchhoff hypothesis to describe the deformed shape through the thickness as: R=R(&=R(?,
sented by a series of generalized displacements ferred to Cartesian directions) u,= “$, +&Yr”‘%.
(re-
(9)
Consequently, one can describe the position vector to any point through the beam thickness as R=R,+IN, = rs + i
+A”~ Y(“% + CN,
II=1
=[Yt+UVfJi,.
(10)
The base vectors to the deformed beam geometry are then given by
r2, i) (3)
= r Y:+IN:(c’, r21j, = Yj({‘, 52. i)i,
When describing a thin-walled beam which is treated as a thin-shell constrained in some fashion to behave as a beam, it is advantageous to employ the coordinates s, q and 5 which are defined to exist along the beam reference axis, lateral cross section coordinate (on the reference surface), and through the shell thickness, respectively (Fig. 2). The position vector to the beam reference axis, R, can be described in terms of its axial coordinate s as: R,=Rb(s)=
5
Y{(S)ij
(4)
In the present analysis, the reference axis has no particular significance other than being a convenient means of describing the position of the beam in space. Next, the position vector R, to any point on the reference surface of the thin-walled beam may be d&bed by RS= Y$, s)ij
(Sal
and since this same reference surface may be described in terms of the shell intrinsic coordinates as R,= Yj(tl, e2)ij
(32352
E=E+i$]ik,
G,=N$,
(11)
Expressions for the changes in the base vectors and the associated incremental strain tensor are available in [3]. The equations of modal equilibrium and the associated boundary conditions The equilibrium equations and boundary conditions for a thin-walled beam can be obtained from an application of Hamilton’s Principle as applied to a shell upon whose reference surf%z a beam-type displacement field is imposed. Hamilton’s Principle for a shill reads [l-3]
s
@H,+GH,+GH,)dt=O. (12) f Neglezting transverse shear and rotary inertia terms through the shell thickness, the various terms in Hamilton’s Principle read : SHi=6T+6W+6V,
(5b)
it is convenient to set, without loss of any generality {‘=?j
<2=s.
(6)
The position vector R, to the deformed reference surface will now be related to the beam displacement field. First, the position vector to the deformed beam axis is given by R,==r,+u, = Y{(t2)i,
m,=w%-WI,,
(7)
where r, is the position vector to the undeformed beam axis, and II*is the displacement field vector of the beam reference axis. Consider now the position vector to points away from the beam axis. Since shell theory will be employed to describe the behavior of the beam through the wall thickness, it suffices to describe the position vector to the deformed reference surface in order to describe fully the geometry of the deformed beam. If rs represents the position vector to the undeformed reference surface, then
dH,=@W,-WI,,
x(-h’,,$)jd
(15)
Consider now the satisfaction of Hamilton’s Principle within the interior of the beam. Setting at each time R,=r,+u,. (81 instant The reference surface displacement vector II, is repre6H,+dH,=O (164
6 one obtains the fotlowing
D.
interiorcondition
MEREDITH
:
and E. A. WrTMEa iaterai beam bounds moments (tiz’‘1 and applied values. Under conditions associated interior of the beam pendent 6 Yc”“)
the condition that the she&type twists @I”) are equal to their either condition the equilibrium with satisfying eqn (16) in the become (for arbitrary and inde-
_%ta.nlYtm)k =:$$i
-A,
+ Cit
(21)
wh.i& have the same form (but one less dimension) as the generalized equation of sheil theory [l]. Attention will now be focused on the beam end-edge boundary conditions. Returning to Hamilton’s Principle as applied to a thin structure, the beam edge work terms along the boundary C2P constant may be written
;@I.,,=[
I, ~~~~:,-i2~~~~+(~~~~-~zz~ x(-N$$&$j)d
(22)
Employing eqn (17b) and (18b) in eqn (22) leads to
Then%if one sets the shelf-type moments (ri?‘) and twist (tii”‘f on the boundary edge to their prescribed values, the boundary edge work term becomes P &H,dt= (N;;~,,-N$c~YJ~‘~~ (24) fI , which implies. at each time ins’mnt. the following conditions
J
N=” i”)= %%$A, or
dYr’=O.
(25)
The MENTOR-3 computer code employs a Fouriertype series expansion of deformation modes to describe the behavior of the beam reference surface crowsecti~n as a function of the lateral reference surface coordinate 5 f. ‘This displacement field was chosen since it decouples the general&d mass term Mim,n)for many geometries.
(Rotanon. Distortion, ate)
where the “effective late& moment arm” L is introduced as an aid in relating the generatied beam *MC-
7
A nonlinear theory of general thin-walled beams
tions to the classical beam moment and torsion terms. The user may set L to unity if desired. 3. FINITE-DIFFERENCEFORMULATION
Approximate solutions to the variational statement of equilibrium (eqn 12) may be obtained by a variety of techniques (finite element, finite-difference, etc.). In tbe present formulatios the differential equations of equilibrium (eqn 21) are cast into finite-differenceform, resulting in a set of algebraic equations suitable for numerical ~te~tion in time. In general, the centraldifference formulae having truncation errors of the order AZ are used to represent the spatial and the temporal derivatives. A method for joining structures which are modeled as finite-di!Terencegrids has also been devised. Details of the techniques are available in [3].
The analysis which has been described in the previous sections has been incorporated into the MENTOR-3 computer program. This program has been applied to various example problems to compare predictions with those from other computer codes and/or expetimental results for largeddiection, elastic-plastic dy namic rv and to eXaminethe impact interaction process involving co&l@ structures [33. An example to demonstrate the feasibility ofemploying MENTOR3 to treat problems of generic interest is included below.
Advantage was taken of symmetry along two axes, and the quarter section ofthe piped elbow was modeled. &&h shell and beam predictions WM obtained. For the shell and initial beam analysis, a 17 x 18 mesh was employed. The beam analysis incorporated 11 deformtion modes across the pipe cross section. Due to the 6ne axial mesh size, the stable time increment for the higher order modes cf the beam theory was considerably smaller than the shell theory stable time increment (the beam stable time increment typically diminish&s faster with decreasing mesh size than the corresponding shell theory stable time in~~t). Con~~tly, the generalized masses of the big&r order beam modes were artificially increased to permit the use of the same time increment as that employed in the shell analysis : this was not expected to have a significant influence on the lower frequency response which dominates the solution. One benefit of employing a beam modaI theory analysis is the ability to employ a coaase axial mesh in conjunction with a finer ~~~~ meah and not have to be concerned with a circumferential time stability criterion. Therefore, a coarser mesh beam analysis was performed; this also permitted the use of a larger execution time increment. Typical impact deformation patterns are illustrated in Fig. 3. The predictions display the local oil-canning indentation
phenomena associated with typical elbow
crush. As can be readily observed, under these impact
Impact of an elbow with attached pipe against a rigid barrier MENTOR-3 was employed to treat the pipe elbow impact example presented’in [12]. Therein, the 3500 in&cc (8890czn/sec)impact of a whipping pipe against a stationary rigid barrier was m&led as an elbow with straight kngtb of pipe attacimd at each _-d to represent the additional eB!.ctivepiping length and mass which must be stopped via the local impact intemction. The differences in boundary conditions, fluid momentum change, and the dfect of the changing elbow geometry on the forcing function was ignored. This model was intended to demonstrate the classical unrestrained pipe whip impact problem in a conf&uration which would be easier to model ex~rn~~ly. A 1.5D 90”24 in. SCH 80 elbow was employed in this example. The following material parameters were employed: Mass density =0.738 x lo- 3 Ib-sec2/in.4 (0.8x 1O-s kg-se$/nn*) Poisson’s ratio =0.3
The stress-strain coordinates used to model the uniaxial stress-strain curve (reflecting conditions at typical operating t~~ature) were: d psi a kg/cm2 sin/in 26,000 4zMo 65,000
1828 2988 4570
0.001 0.025 0.15
1/Dsec
0.316 0.316 0.316
VP
x:; 0.3 (Cl
mars* Ewm
Pradiceion
Fig. 3. Elbow deformations foncquartcr
model).
D.
MEREDITH
and E. A. WITMER
SheI4 ( fine mesh) Beam Cflre mestl) -----
0
01
02
03
04
@earn kcarse
05
06
i 07
I 08
.O 09
Time x IO‘'set
Fig. 4. Impact force time histones.
conditions, most of the deformation occurs locally around the region of impact. The two end pipes served to stiffen the elbow to some extent in the latter stages of crush and their defo~atio~ were relatively small compared to the severity of the local deformation and indentation of the elbow. Both the fine and coarse mesh beam predictions for the present example compared favorably with the shell predictions. The 11 beam modes reasonably reproduced the severe axial and circumferential curvature changes experienced by the crushing elbow. The beam impact forces (Fig. 4) compared reasonably well with the shell theory prediction. The generai amplitude, period, and characteristics of the impact forces were similar, although the beam theory predictions demonstrated a somewhat greater fluctuation in the smaller high frequency oscillations which were superimposed on the main loading cycle. Employing the coarser beam mesh with the larger time increment did not significantly degrade the prediction.
The present beam formulation represents a general nonlinear dynamic incremental modal technique sunable for analyzing the transient response of thin-walled structures. The theoretical-experimental results showed generally good correlation for shells and beams subjected to prescribed transient forces, initial velocities and impacts [3). Predictions for impacts of flexible
structures with rigid missiles also show good correlation with experiment [3]. Predictions for impacts between flexible structures, as illustrated in Fig 5, appear plausible. The present technique has been applied successfully to predict the static and dynamic crush of pipe elbows. MENTORS also has been applied to predict the nonlinear response of piping systems incorporating elbows in which the inplanar and out-of-plane elbow rigidity dominates the structural response [3].
The present modal fo~~tion unifies the concepts of sheil and beam theories to form a hybrid theory capable of incorporating, to any desirable degree, the salient features of either theory. This general formulation provides the framework for a unified theory of structural mechanics (Fig. I 1.Such a unified approach can serve as a valuable teaching aid for students first being exposed to structural mechanics since it places each of the basic theories of apphed mechanics into clearer focus and proper perspective. The basic formulation presented herein has already been employed to develop a nonlinear theory for beams of solid cross section [4]. The concept of employing an incremental modal solution along one coordinate can be extended to employing an assumed modal solution along the remaining coordinates, resulting in a nonlinear incremental modal analysis theory for shells and solids.
; FORCE
Fig. 5. MENTOR-3 general impact capability
A nonlinear theory of general thin-walled beams
Similarly, alteruate displacement fields may be investigated for use in the current thin-walled beam formulation. The generaked beam theory incorporated herein offers several distinct computational advantages over traditional shell and beam formulations including: -Elimination of stability requirements associated with coordinate directions other than theaxial direction (an advantage somewhat offset by the need to consider the stability of the axial propagation of the higher order beam deformation modes). -Ability to tailor the accuracy of the analysis tb suit the problem by incorporating only those modes considered necessary for an adequate solution. -Ease of joining beams to shells either axially or laterally. -Ease of input, modeling, and interpretation of results. Of major interest to the nuclear power industry is the in~c~on between pipes (pressure vessels) and contained or impinging fluids. The MENTOR-3 computer code is ideally suited for incorporating or interfacing with a fluid-dynamics program which would determine the response of the fluid. Although conceived primarily to treat beams with crushable cross-sections, the technique introduced herein constitutes a new and attractive approach to nonlinear structural analysis. Acknowledgemenr-This paper is dedicated to the memory of Dr. John Pee& who provided the inaction for tbe present investigation.
1. S. D. Pirotin. L. Mormo and J. W. Leech, Finite-difference analysis for predicting iarge elastic-plastic transient deformations of vanable thickness Kirchboff, soft-
9
bonded thin and transverse shear deformable thicker shells. BRL CR 315 (MITASRLTR 152-3) (Sept. 1976). 2. MENTOR-l Computer Program Manual. Meredith Engineering, 91 Sycamore Road, Mdrose, Massachusetts. 3. MENTOR-3 Computer Program Manual. Meredith Engineering, 91 Sycamore Road, Mdrose, Massachusetts. 4. D. Meredith, Devdopment of a 3-dimmsional solid cross section beam theory. Meredith Engineering Internal Document, Meredith Engineering, Melrose, Massachusetts. 5. V. 2. Vlasov, Thin-Walled Elastic Beams, 2nd Edn. Israel Program for Scientific Translations, Jerusalem, lsrael (19&i). 6. S. Timosbenko and J. N. Goodier, Theory oj Elasticity, 2nd Edn. McGraw-Hill, New York (1951). 7. K. Wash&u, Some considerations on a naturally curved and twisted slender beam. J. Math. Phys. 43(2), 11l-l 16 119641. 8. i. T-s. Wang and J. N. Dickson, Elastic beams of various orders. AIAA, pp, 535-537 (May 1979). 9. A. Rosen and P. Friedman, The nonlinear behavior of elastic slender straight beams undergoing small strains and moderate rotations. J. Appl. Mech. 46, 161-167 11979). 10. S. D. Pirotin and G. H. East, Jr., Large-deBection, elastioplastic response of piping: experiment, analysis and apljiication. Paper F3/1, 4th SMiRT Cm& San Fran&. (Aug. 197;). 11. E. A. Witmer. F. Merlis and R. L. Suilker. Exnerimental tiansient and permanent deformaion &ii& of steelsphere-impacted ofimpulsively-loaded aluminum beams with damped ends. NASA CR 134922, MIT ASRL TR 1.54-11 (Oct. 1975) 12. D. Meredith and E. A. Witmer, Computer code for predicting the dynamic response of high energy piping, pressure vessels, and shell structures to transient loads and impacts. ASME Paper 78-PVP-33, Presented at Joint ASME/CSME Pressure Vesselsand Piping Co@, Montreal. Canada (June 1978).
Nonlinear Mathematical Theories and Formulation Aspects
A NONLINEAR THEORY OF TiENERAL THIN-WALLED BEAMS
Meredith Engineering, 33170 Glen Vaiiey Drive, Farmington Hills, MI 48018. U.S.A. and E. A. WXTNBZR Department of Aeronauftcs and Astronautic& M~~~~ts Cambridge, MA 02139,U.S.A.
Institute
ofTangy,
(Received 18 April 1980) Abemet-A eompkte and consistent theory is formulated to r&c&e the large&dktion ehtttia-pltstic behaviot of thin-walk3 beams 5Farbkrary initial shape and sWbjc@d t5 arbitrary trikusientkradings.The nonlinear beam theory is derived from a wekkcumeated nonlinear shdi the5ry through an appiicrtion of Hamilton’s Principlc. By interpreting elastical beam theory displacement fidds asthe lower order terms in a more general series expansion of deformation modes which describe the bcbayi5r oftis beam CIOIIBtion, a higher arder beam theory based on such an expansion can be developed. The theory r*n be 5ufiki~~ti~ ~soast5incorpoirttcfhteffensof~o~~o~a~dierortion oadcdlapseiardditiont5thtcom~~e range of behavior ndy asso&& with beam theory ~~~~0~ extetlsi5r&bctt&g tzmtwew shtnr d&rmation, torsion, and warping). The generalized equations of motion ttss5cSabd with this s&s or modal technique am shown to be of the same form 88 tic generalizedequations of shdl theory from Which they ware derived. These nonlinear equations of motion for the beam are east into an approximate form suitable for imdementation on a dirritnleomDuter. The resulting prqram, MENTOR-3 is aP@ied to treat a nottlinear re&ttse and impact p>bitm. _
In recent years the concern over nuclear power plant safety and vehicle crashworthiness has speeded the development of sophisticate& ana&tical tools and computer codes required by the analyst to predict the noniinea~ b&avior of safety related structures during abnormal or upset conditicms. It has been demonstrated
J
that when suffiticnt care and attmtion is paid in developing the nonlinear analysis the predictians can be shown to bc rdiabk and accurate to within the order of aoqted a~~tioR. Therein, the engineer must determine the appropriate level of sophistication or precision which must be incorporated into the analysis in order for the predictions to be useful and meaningful. The three basic theories of structural mechanics
available to the engineering ana@% include bemn, shelf and general J-dimensional continuum theory. Each of these “matzo-mechanics” type theories incorporates a particular level of approximation which is clearly evident in its derivation from the fundamentaf prin&pies of physics. Traditiona continuum theory involves the Ieast amount ~fappro~tio~ and is normally considered to be a valid basis for the development of the more specialized theories of shells and beams. A measure which forms a suitable basis for disaping between beam, shell, and solid structures
lies in th&r relative phrsica dimensions fadditional criteria concerned v&h material properties, and smoothness of geometry and loading may also be addressed). For instance, shells are distinguishable from solids by their having one physical dimension, their thickntss, small in comparison with their Other dimensions. Beams on the other hand have two diiensions Oh& Cross sectiob measurements) Small in comparison with the remaining dimension (span). It is
I
t SOLXR
THICK-W&t&ED a~ r--IGcRoss-SECTION BEAM ---GaaAW Fig 1. Inter-relation between theories for solids, she& and ,7IIN-WLW
beams. evident
therefore, that one should in principle be able to connect and derive the equations for hesun% sk&
and 3-dimensional continuum (solids) by taking advantage of these basic factors which serve to distinguish each structure. Such a relationship is illustrated in Fig. 1
for smoothly contoured and loaded homogeneous stnretures, Some of the paths disoernable in Fig. 1 have already been developed. The theoretical developments ass&ated with paths A, B, C and D are available in [l-4]. For instance, in [l] advantage is taken of the fact that the thickness of a shell is thin in comparison with its other dimensions to reduce a 34irnensional mntinuum theory to one de&& on a 24mensional subspace or reference surface. This is accomplished by assuming within the cOntext of Hamilton’s Principle that the deformation through the shell thickness is of a known functional form (truncated series of ddotmation modes) whose parameters are de&d on the shell rekence surface. Unfortunately, it is often prohibitivdy expen sive to treat many problems of engineering interest by
4
D
MEREDITH
and E. A
employing a 3-dimensional continuum analysis or even the less-costly shell analysis. This is particularly true for structures in which at least one. and more likely two, of its physical dimensions is “thin” with respect to its other dimensions. Many investigators [i7] have studied the behavior of such doubly-thin structures and have contributed to the modem engineering theory of general beams. Most of these fo~ulations were developed to treat the behavior of beams whose cross-sections deformed in accordance with classical beam theory assumptions (translation, bending and torsion) or incorporated the first order effects of warping and transverse shear deformation. Several attempts to incorporate higher order effects in beams have been reported [g]: however, most are restricted to elastic beam material behavior and specific types of deformation fields. Of specific interest in the present paper is the nonlinear response of general 3-dimensional thin-walled beams. This field has also received considerable attention [S-11]; however, again the published results are generally restricted to small strain elastic behavior or incorporate a multitude of kinematic assumptions or constraints. It is nevertheless important to note that in each of the above-mentioned shell and beam forrmrlations, the displacement field of the structure along its thin dimension(s) is postulated in terms of generalized displacements deflned along the remaining dimension(s). By reducing the infinite number of degrees of freedom along the thin dimension to a suitable and finite num ber (these normally are just the ~pla~ents of the structure at its reference surface or reference axis), great economies in expenditure and effort are derived while maintaining suitable engineering accuracy. The present paper extends this basic concept to develop a thin-walls beam theory from shell theory by imposing an assumed displacement field aIong the shell reference surface coordinate (r~)which lies within the cross sectional face of the beam (Fig. 2). This reduces the shell theory to a beam theory (path E in Fig. I) wherein the parameters (displacement field coefhcients) which describe the structural defo~tion are evaluated or defined on a reference line (the beam axis). Such a formulation for thin-walled beams has a number of distinct advantages, the most significant of which is the orderly overlapping of the domains of applicability of shell and beam theory. In addition, since no restriction is applied to the type or number of independent degrees of freedom incorporated in the description of the dis-
Y’
/
5
Fig. ;. ‘Position vector to a 3-dimensional thin-walled beam. $Qandard tensor summation conventions are used throughout. Greek minus&es range over the values 1and 2 whereas Latin minuscules range over the values 1. ? and 3.
WITMER
placement field of the beam across its cross section, the analyst is no longer confined by the deformation restrictions of classical beam theory (which may be interpreted as the lower order terms m a more general series expansion). By the judicious insertion of additional mdependent degrees of freedom into the assumed displacement field of the cross section. effects such as cross-section distortion and collapse. normally associated with shell theory. may readily be incorporated mto the “beamtype” analysis. it is perhaps worthwhile to note at this stage that because each deformation mode is treated as being independent (at least in an incremental sense). the present procedure may also be Interpreted as a “generalized nonlinear mcremental modal technique” as applied to a beam-type structure. Also note that in the limit, as an infinite number of independent degrees of freedom is employed in the beam cross section displacement field, the beam solution should converge to the shelltheory solution as the truncation error approaches zero. In the following sections. the generalized equations which describe the arbitrarily large-deflection elasticplastic response of a thin-walled beam are derived from a thin-walled shell theory through an application of Hamilton’s Principle into which the above-mentioned beam displacement field is introduced. It will be shown that the generalized equations of motion suitable for describing the behavior of a beam are similar in form to the equations which describe the behavior of a shell and of a 3-dimensional continuum; a similar analogy exists in any modal or generalized Ritz of finite-element analysis. The fi~te-di~erence method is employed to approximate the spatial derivatives along the span of the beam. The resulting approximate equations of equilibrium are then integrated in time by means of a central-difference procedure to obtain the transient response of the beam. This analysis has been incorporated into the MENTOR-3 computer code which is available for treating structures such as shells, beams, or combinations thereof. This program has been employed to analyze a variety of problems wherein the effects of large~p~~ment e~ti~pI~tic behavior associated with the overall deformation of the structure are complicated by the presence oflocal cross sectional collapse.
An outline of the general development leading to the formulation of the general modal equations of motion suitable for a thin-walled beam is presented in this section. A more comprehensive and detailed develop ment may be found in El. 31. Geonrerry and position vector of the beam A general 34imensional body can be described in terms of the position vector R to any point within the body. Each such point can be uniquely defined by its i:’ intrinsic coordinates? (which, for each material point, are invariant in time). The position vector may be described in terms of its coordinates YJ in Cartesian space as: R= YJi I I yJ(<‘)i. I. In general, the positions YJ are a function of the three intrinsic coordinates <’ used to identify the material point. Consider now a beam whose cross section is comprised ofthin-walled members. It is often advantageous
A nonlinear theory of general thin-walled beams when treating thin structures to employ the concept of a reference surface which may, but need not be, the mid-surface of the structure. The reference surface of such a shell-type structure is defined by its position vector R, which can be described in terms of the two intrinsic coordinates cl, e2 associated with material points on the reference surface (Fig. 2) Rs= Yjj
(2)
Next, a third intrinsic coordinate c is introduced which is associated with the direction which is initially normal to the reference surface and is needed to describe the position of points located through the thickness of the structure. For thin structures. one employs the Kirchhoff hypothesis to describe the deformed shape through the thickness as: R=R(&=R(?,
sented by a series of generalized displacements ferred to Cartesian directions) u,= “$, +&Yr”‘%.
(re-
(9)
Consequently, one can describe the position vector to any point through the beam thickness as R=R,+IN, = rs + i
+A”~ Y(“% + CN,
II=1
=[Yt+UVfJi,.
(10)
The base vectors to the deformed beam geometry are then given by
r2, i) (3)
= r Y:+IN:(c’, r21j, = Yj({‘, 52. i)i,
When describing a thin-walled beam which is treated as a thin-shell constrained in some fashion to behave as a beam, it is advantageous to employ the coordinates s, q and 5 which are defined to exist along the beam reference axis, lateral cross section coordinate (on the reference surface), and through the shell thickness, respectively (Fig. 2). The position vector to the beam reference axis, R, can be described in terms of its axial coordinate s as: R,=Rb(s)=
5
Y{(S)ij
(4)
In the present analysis, the reference axis has no particular significance other than being a convenient means of describing the position of the beam in space. Next, the position vector R, to any point on the reference surface of the thin-walled beam may be d&bed by RS= Y$, s)ij
(Sal
and since this same reference surface may be described in terms of the shell intrinsic coordinates as R,= Yj(tl, e2)ij
(32352
E=E+i$]ik,
G,=N$,
(11)
Expressions for the changes in the base vectors and the associated incremental strain tensor are available in [3]. The equations of modal equilibrium and the associated boundary conditions The equilibrium equations and boundary conditions for a thin-walled beam can be obtained from an application of Hamilton’s Principle as applied to a shell upon whose reference surf%z a beam-type displacement field is imposed. Hamilton’s Principle for a shill reads [l-3]
s
@H,+GH,+GH,)dt=O. (12) f Neglezting transverse shear and rotary inertia terms through the shell thickness, the various terms in Hamilton’s Principle read : SHi=6T+6W+6V,
(5b)
it is convenient to set, without loss of any generality {‘=?j
<2=s.
(6)
The position vector R, to the deformed reference surface will now be related to the beam displacement field. First, the position vector to the deformed beam axis is given by R,==r,+u, = Y{(t2)i,
m,=w%-WI,,
(7)
where r, is the position vector to the undeformed beam axis, and II*is the displacement field vector of the beam reference axis. Consider now the position vector to points away from the beam axis. Since shell theory will be employed to describe the behavior of the beam through the wall thickness, it suffices to describe the position vector to the deformed reference surface in order to describe fully the geometry of the deformed beam. If rs represents the position vector to the undeformed reference surface, then
dH,=@W,-WI,,
x(-h’,,$)jd
(15)
Consider now the satisfaction of Hamilton’s Principle within the interior of the beam. Setting at each time R,=r,+u,. (81 instant The reference surface displacement vector II, is repre6H,+dH,=O (164
6 one obtains the fotlowing
D.
interiorcondition
MEREDITH
:
and E. A. WrTMEa iaterai beam bounds moments (tiz’‘1 and applied values. Under conditions associated interior of the beam pendent 6 Yc”“)
the condition that the she&type twists @I”) are equal to their either condition the equilibrium with satisfying eqn (16) in the become (for arbitrary and inde-
_%ta.nlYtm)k =:$$i
-A,
+ Cit
(21)
wh.i& have the same form (but one less dimension) as the generalized equation of sheil theory [l]. Attention will now be focused on the beam end-edge boundary conditions. Returning to Hamilton’s Principle as applied to a thin structure, the beam edge work terms along the boundary C2P constant may be written
;@I.,,=[
I, ~~~~:,-i2~~~~+(~~~~-~zz~ x(-N$$&$j)d
(22)
Employing eqn (17b) and (18b) in eqn (22) leads to
Then%if one sets the shelf-type moments (ri?‘) and twist (tii”‘f on the boundary edge to their prescribed values, the boundary edge work term becomes P &H,dt= (N;;~,,-N$c~YJ~‘~~ (24) fI , which implies. at each time ins’mnt. the following conditions
J
N=” i”)= %%$A, or
dYr’=O.
(25)
The MENTOR-3 computer code employs a Fouriertype series expansion of deformation modes to describe the behavior of the beam reference surface crowsecti~n as a function of the lateral reference surface coordinate 5 f. ‘This displacement field was chosen since it decouples the general&d mass term Mim,n)for many geometries.
(Rotanon. Distortion, ate)
where the “effective late& moment arm” L is introduced as an aid in relating the generatied beam *MC-
7
A nonlinear theory of general thin-walled beams
tions to the classical beam moment and torsion terms. The user may set L to unity if desired. 3. FINITE-DIFFERENCEFORMULATION
Approximate solutions to the variational statement of equilibrium (eqn 12) may be obtained by a variety of techniques (finite element, finite-difference, etc.). In tbe present formulatios the differential equations of equilibrium (eqn 21) are cast into finite-differenceform, resulting in a set of algebraic equations suitable for numerical ~te~tion in time. In general, the centraldifference formulae having truncation errors of the order AZ are used to represent the spatial and the temporal derivatives. A method for joining structures which are modeled as finite-di!Terencegrids has also been devised. Details of the techniques are available in [3].
The analysis which has been described in the previous sections has been incorporated into the MENTOR-3 computer program. This program has been applied to various example problems to compare predictions with those from other computer codes and/or expetimental results for largeddiection, elastic-plastic dy namic rv and to eXaminethe impact interaction process involving co&l@ structures [33. An example to demonstrate the feasibility ofemploying MENTOR3 to treat problems of generic interest is included below.
Advantage was taken of symmetry along two axes, and the quarter section ofthe piped elbow was modeled. &&h shell and beam predictions WM obtained. For the shell and initial beam analysis, a 17 x 18 mesh was employed. The beam analysis incorporated 11 deformtion modes across the pipe cross section. Due to the 6ne axial mesh size, the stable time increment for the higher order modes cf the beam theory was considerably smaller than the shell theory stable time increment (the beam stable time increment typically diminish&s faster with decreasing mesh size than the corresponding shell theory stable time in~~t). Con~~tly, the generalized masses of the big&r order beam modes were artificially increased to permit the use of the same time increment as that employed in the shell analysis : this was not expected to have a significant influence on the lower frequency response which dominates the solution. One benefit of employing a beam modaI theory analysis is the ability to employ a coaase axial mesh in conjunction with a finer ~~~~ meah and not have to be concerned with a circumferential time stability criterion. Therefore, a coarser mesh beam analysis was performed; this also permitted the use of a larger execution time increment. Typical impact deformation patterns are illustrated in Fig. 3. The predictions display the local oil-canning indentation
phenomena associated with typical elbow
crush. As can be readily observed, under these impact
Impact of an elbow with attached pipe against a rigid barrier MENTOR-3 was employed to treat the pipe elbow impact example presented’in [12]. Therein, the 3500 in&cc (8890czn/sec)impact of a whipping pipe against a stationary rigid barrier was m&led as an elbow with straight kngtb of pipe attacimd at each _-d to represent the additional eB!.ctivepiping length and mass which must be stopped via the local impact intemction. The differences in boundary conditions, fluid momentum change, and the dfect of the changing elbow geometry on the forcing function was ignored. This model was intended to demonstrate the classical unrestrained pipe whip impact problem in a conf&uration which would be easier to model ex~rn~~ly. A 1.5D 90”24 in. SCH 80 elbow was employed in this example. The following material parameters were employed: Mass density =0.738 x lo- 3 Ib-sec2/in.4 (0.8x 1O-s kg-se$/nn*) Poisson’s ratio =0.3
The stress-strain coordinates used to model the uniaxial stress-strain curve (reflecting conditions at typical operating t~~ature) were: d psi a kg/cm2 sin/in 26,000 4zMo 65,000
1828 2988 4570
0.001 0.025 0.15
1/Dsec
0.316 0.316 0.316
VP
x:; 0.3 (Cl
mars* Ewm
Pradiceion
Fig. 3. Elbow deformations foncquartcr
model).
D.
MEREDITH
and E. A. WITMER
SheI4 ( fine mesh) Beam Cflre mestl) -----
0
01
02
03
04
@earn kcarse
05
06
i 07
I 08
.O 09
Time x IO‘'set
Fig. 4. Impact force time histones.
conditions, most of the deformation occurs locally around the region of impact. The two end pipes served to stiffen the elbow to some extent in the latter stages of crush and their defo~atio~ were relatively small compared to the severity of the local deformation and indentation of the elbow. Both the fine and coarse mesh beam predictions for the present example compared favorably with the shell predictions. The 11 beam modes reasonably reproduced the severe axial and circumferential curvature changes experienced by the crushing elbow. The beam impact forces (Fig. 4) compared reasonably well with the shell theory prediction. The generai amplitude, period, and characteristics of the impact forces were similar, although the beam theory predictions demonstrated a somewhat greater fluctuation in the smaller high frequency oscillations which were superimposed on the main loading cycle. Employing the coarser beam mesh with the larger time increment did not significantly degrade the prediction.
The present beam formulation represents a general nonlinear dynamic incremental modal technique sunable for analyzing the transient response of thin-walled structures. The theoretical-experimental results showed generally good correlation for shells and beams subjected to prescribed transient forces, initial velocities and impacts [3). Predictions for impacts of flexible
structures with rigid missiles also show good correlation with experiment [3]. Predictions for impacts between flexible structures, as illustrated in Fig 5, appear plausible. The present technique has been applied successfully to predict the static and dynamic crush of pipe elbows. MENTORS also has been applied to predict the nonlinear response of piping systems incorporating elbows in which the inplanar and out-of-plane elbow rigidity dominates the structural response [3].
The present modal fo~~tion unifies the concepts of sheil and beam theories to form a hybrid theory capable of incorporating, to any desirable degree, the salient features of either theory. This general formulation provides the framework for a unified theory of structural mechanics (Fig. I 1.Such a unified approach can serve as a valuable teaching aid for students first being exposed to structural mechanics since it places each of the basic theories of apphed mechanics into clearer focus and proper perspective. The basic formulation presented herein has already been employed to develop a nonlinear theory for beams of solid cross section [4]. The concept of employing an incremental modal solution along one coordinate can be extended to employing an assumed modal solution along the remaining coordinates, resulting in a nonlinear incremental modal analysis theory for shells and solids.
; FORCE
Fig. 5. MENTOR-3 general impact capability
A nonlinear theory of general thin-walled beams
Similarly, alteruate displacement fields may be investigated for use in the current thin-walled beam formulation. The generaked beam theory incorporated herein offers several distinct computational advantages over traditional shell and beam formulations including: -Elimination of stability requirements associated with coordinate directions other than theaxial direction (an advantage somewhat offset by the need to consider the stability of the axial propagation of the higher order beam deformation modes). -Ability to tailor the accuracy of the analysis tb suit the problem by incorporating only those modes considered necessary for an adequate solution. -Ease of joining beams to shells either axially or laterally. -Ease of input, modeling, and interpretation of results. Of major interest to the nuclear power industry is the in~c~on between pipes (pressure vessels) and contained or impinging fluids. The MENTOR-3 computer code is ideally suited for incorporating or interfacing with a fluid-dynamics program which would determine the response of the fluid. Although conceived primarily to treat beams with crushable cross-sections, the technique introduced herein constitutes a new and attractive approach to nonlinear structural analysis. Acknowledgemenr-This paper is dedicated to the memory of Dr. John Pee& who provided the inaction for tbe present investigation.
1. S. D. Pirotin. L. Mormo and J. W. Leech, Finite-difference analysis for predicting iarge elastic-plastic transient deformations of vanable thickness Kirchboff, soft-
9
bonded thin and transverse shear deformable thicker shells. BRL CR 315 (MITASRLTR 152-3) (Sept. 1976). 2. MENTOR-l Computer Program Manual. Meredith Engineering, 91 Sycamore Road, Mdrose, Massachusetts. 3. MENTOR-3 Computer Program Manual. Meredith Engineering, 91 Sycamore Road, Mdrose, Massachusetts. 4. D. Meredith, Devdopment of a 3-dimmsional solid cross section beam theory. Meredith Engineering Internal Document, Meredith Engineering, Melrose, Massachusetts. 5. V. 2. Vlasov, Thin-Walled Elastic Beams, 2nd Edn. Israel Program for Scientific Translations, Jerusalem, lsrael (19&i). 6. S. Timosbenko and J. N. Goodier, Theory oj Elasticity, 2nd Edn. McGraw-Hill, New York (1951). 7. K. Wash&u, Some considerations on a naturally curved and twisted slender beam. J. Math. Phys. 43(2), 11l-l 16 119641. 8. i. T-s. Wang and J. N. Dickson, Elastic beams of various orders. AIAA, pp, 535-537 (May 1979). 9. A. Rosen and P. Friedman, The nonlinear behavior of elastic slender straight beams undergoing small strains and moderate rotations. J. Appl. Mech. 46, 161-167 11979). 10. S. D. Pirotin and G. H. East, Jr., Large-deBection, elastioplastic response of piping: experiment, analysis and apljiication. Paper F3/1, 4th SMiRT Cm& San Fran&. (Aug. 197;). 11. E. A. Witmer. F. Merlis and R. L. Suilker. Exnerimental tiansient and permanent deformaion &ii& of steelsphere-impacted ofimpulsively-loaded aluminum beams with damped ends. NASA CR 134922, MIT ASRL TR 1.54-11 (Oct. 1975) 12. D. Meredith and E. A. Witmer, Computer code for predicting the dynamic response of high energy piping, pressure vessels, and shell structures to transient loads and impacts. ASME Paper 78-PVP-33, Presented at Joint ASME/CSME Pressure Vesselsand Piping Co@, Montreal. Canada (June 1978).