40th anniversary
SIMILARITIES AND DIFFERENCES
IN THIN-WALLED BEAMS THEORIES
by
A. PITTALUGA
Technical Bullettin N. 97 Genova, September 1986
Invited lecture at Wuhan University of Water Transportation Engineering on the calibration of their Ingekomen: I %.3.11.9.0
. INTRODUCTION
SEMILIARITIES AND DIFFERENCES IN
THIN-WALLED BEAMS ,THEORIES by
A. PITTALUGA
In the present world of widespread computational facilities
and upgraded numerical methods, such a simple -approach to structural - analysis as fa-,, beam model could seem to be dated.
-IHowever, to. understand the behaviour of a structure and to properly design it we need physical models-; not simply numerical
tools; and ,moreover., for Very complicated structures-, we still need
simplified Models to carry out the analysis- economieally.
This is probably - _the reason thin-walled bead's theories for ship design:.
Indeed, we probably Would get better results from a complete a-D finite element analysis, of the structure, but this will cost us a lot of money and iof time, Will give us: -Small- insight in the mechanics of the hull torsion, and will be quite impractical in
,
early -stages of design.
The aim of this paper .is to present\ a state-of-the-art of the theory of thin-walled beams, which is not yet completely established
since different . approaches exist, and to separate the portion of the
theory which is a direct and exact -solution of the equations
governing the elastic behaviour of bodies, from statements which
are a direct result of simplifying assumptions:
In section 3 we will present a :generalized' theory of he.O.rri
-which is exact' in the frame of the elasticity theory, but of course doesn't yield final conclusions, and in section -4 we will introduce
;
simplifying assumPtionS. and ..get practical :restiliS,
A 'cOmparison, Will be made With Vlasov.'s theory; Cfas-sica
theory 4:fid-7 "refined theory" by Kollbrunner. and Haidin.;
. STATE 'ART
,
-The. so.--called "Theory of thin-walled elastic beams" takes its -.
-name from the :fundamental work of Vlasov who, in, 1940, published Ha:. 'quite comprehensive book collecting the results, of an s.
-'entire lifetime;';; of scientific activity .devoted to, the theory r of
thin-walled structures: A second edition,- revised 'Land augmented was
-published :posthumously - in :1959: This one was translated
English and introduced in the western scientific community by the
Israel Program for Scientific .TranSlatioriS in 1961.
The
r
; fund'a:nientr. step in the theory is -'-.V1a.soxi's observation
_
`that thin walled beams " made of shells with - open ''cross
subject to beam-end loads only, .deforms without developing -Substantial .1-1.ear deformation in the shell: Middle surface and any
,distorsion of the OroSs=Sectri_O-riS in their plane ,:-when the diStorsion
of the loaded beam 'end- in sits plane is reStrained.
the -other 'hAnd these beams develop :very large deformation
of the cross-sections .outside their plane, Callee."-warping',,which is not taken into account in the Bernoulli Navier beam theOry.
In
.
-the two .fundamental hypoteSeS of no distOrsion of the. crOSS-sections in ,thir. plane and ;'-of."!:-3..n(5 shear deformation of the Middle ."Stirface. of the shell, VlasOv developed -a new beam Model
which _main 'f'etUrds. -Are:
further ', to Navier rigid; body motion cross-section 'may develop -warping out of their plane
these warpingdisplacements may be obtained by the product of a warping function, variable along the contour of the section . but
. , . .
-independent from they position along the beam a)d.S, times- a twist
r - .
fUnCtion, Which .depefidS on the position along the :7- beam only
- if this _-:warping' displacement is restrained at ne or both ''ends, the .-torsion a the 'beam is no more uniform, , ,as' assumed'', by
De Saint, Venant, And warping stresses arise, which are related to,
a new generalized longitudinal force called _"Biniomeri.e.
It must be noticed that Vlasov introduces shear stresses in
-order to fulfill the elastic equilibrium equations, in spite of having assumed no shear deformation, so what Vlasov postulates in his
theory is in fact zero shear-flexibility of the middle surface.
The second finding gives the suggestion for a generalized
method which Vlasov uses to extend his theory to closed deformable cross sections, shear deformation effects, solid sections.
This method assumes the separation of cross-sectional and along-the-axis variables, i.e. that any displacement of a point of the beam may be expressed as a sum of products of coupled functions, one independent from the position along the axis and the other independent from the position on the section.
Lack of modern computational tools prevented Vlasov from
deriving practical results from this method except in few special cases, however theories following this approach are sometimes called thin-walled beams theories, even when the walls of the beam are not at all thin.
Open cross sections are of little practical use in ship design, since ship sections are usually of a cellular type.
In closed cross-sections, the middle surface undergoes large shear strain. This shear strain has been obtained by De Saint Venant in the assumption that the section is free from longitudinal
stresses.
From the elastic equilibrium equations it follows that the sum of shear flow incoming in a single node of the section must be zero (the shear flow is the product of the shear stress times the wall thickness). By imposing the continuity condition at the nodes of the
induced warping displacements, it is possible to obtain the shear
stress associated with the pure torsion, which is sometimes called the Bredt's or the De Saint Venant's shear stress.
The so-called "classical theory of general warping torsion" based on some ideas of Benscoter (1954) and reported in Haslum and Tonnenssen (1972) extends Vlasov's theory for closed sections by assuming that the shear strain in the middle surface is non-zero, but is due to the Bredt's shear strain only.
It may be shown that this is equivalent to replace the sectional area of Vlasov's theory with the warping function of the De Saint Venant's unrestrained torsion.
An extension of this theory to thick walled and solid sections
is due. to 1. Kawai (1971) who also presented a quite attractive variational formulation of the De Saint Venant ' s problem.
The basic advantage of his approach over the classical theory is the fact that he makes use of a FEM formulation which does not require definition of circuits over the section.
The drawback of these approaches is that they disregard the warping shear stress effect over the deformation. This may lead to erroneous result especially for short beams.
Several methods have been proposed to overcome this
drawback: Westin (1980) developed and idea due to Haslum &
Stranme (1973) which consists in substituting the total shear stress to the Bredt' s shear stress in the classical theory and iterating the solution, T. Kawai (1973) included warping shear deformation in his variational equations. Both these approaches are theoretically correct, but imply an iterative solution where load effects are involved, this might be too burdensome when many loading
conditions are to be examined.
Kollbrunner and Haidin (1972) proposed to modify the classical theory by assuming that warping deformation is still proportional to the De Saint Venant ' s warping function, but the twist function is no more equal to the derivative of the torsion angle.
This additional degree of freedom allows a better adjustment
of the beam, but the hypothesis of proportionality of warping
deformation and De Saint Venant ' s warping function, which is retained, is incorrect and affects the results.
Pittaluga (1978) applied Vlasov ' s generalised method by introducing new cross-sectional functions, called "shear potentials" which allow for shear deformation and are related to the first
derivative of the bending and warping curvature. This overcomes the problem related to the warping shear strain, but the resulting differential equations are non-linear.
This requires an iterative solution, as in Westin
and in
Kawai, but, in difference of their method, cross-sectional properties are independent from the load pattern, as long as the load isapplied to the beam ends only. This might be an advantage if many loading conditions are to be considered.
Pittaluga ' s approach also provides a consistent framework for
the analysis of the distribution of the shear stresses associated with non uniform bending. The assumptions are identical to the ones of the well-known two dimensional shear-flux method, but they are embedded in an unified theory considering both bending and torsion
and associated shear stress distributions.
A similar approach has been adopted by Liebst (1984),
un-fortunately, being his report written in Danish, its diffusion has
been pretty limited.
He extends Pittaluga's approach to the case of torsional
moment variable along the span of the beam, by means of an
eigenfunction expansion of the longitudinal displacement. This
yields substantial improvements for - particular problems, such as
vibrations of the beam, and the increase in computer cost, which is proportional to the number of eigenfunctions considered, is probably
not unmanageable.
Warping restraining effects of transverse stiffeners are taken into account by almost all the theories presented here via the virtual work principle.
No further development, in the theory of prismatic beams for
linear elastic problems has been noticed up-to-now. On the contrary
considerable effort has been devoted to the extention of the theory
to other problems, such as initial stress, initial deformation, finite displacement, buckling, yielding, dynamic response.
Some of this items where dealt with by Vlasov in his book.
Recent applications are, among the others, the non-linear
theory by Moredith and Witner (1981) and the theory for pre-twisted
beams by Krenk and Gunneskov (1981).
Prismatic beams are wide spread structures in civil
engineer-ing,
but are of
little use to the naval architect, since the shipsections are varying along the hull, with frequent discontinuities. A trivial idea is to model such a variable section beam as a sequence of prismatic elements and impose continuity and
equili-brium at the junctions. This is not so easy as it could seem at a first glance, since stress and displacement distribution functions
are different for different cross sections.
The more advanced theory to cope with this problem is due to Pedersen (1982).
He proposes a model for smoothly varying hull segments which assumes that the stress and displacement distribution functions of the individual section are the same as /in the case of prismatic beams, but integrated section properties, such as areas and
moments of inertia, are allowed to vary along the beam axis. Discontinuities of the hull-beam are taken into account by imposing average equilibrium (by means of the virtual works principle) and
that the gaps due to
different warping functions are ortogonal toany of the generalized warping coordinates. However in Pedersen theory the Kollbrunner & Haidin approximation for warping shear strain is retained.
Pedersen approach is considered to be the best engineering
approach to the hull-beam analysis available at the moment.
Theoretically correct attempts to the problem of variable section beams have been proposed by Wilde (1968), who proposed a
theory for open cross-sections, and by Cazzulo (1984), who proved
the feasibility of a theoretically correct theory of variable (open and closed) cross-sections, but failed to find a practical numerical
solution.
A better understanding of the common features of the different
theories may be obtained from the synthesis in the following
sections.
3. THE GENERALIZED THEORY OF BEAMS
Let' s consider a generic elastic body and a right-hand reference system x y z. Let's define the coordinates of a point P of the body (x, y, z), the components of the point displacement along
the axis (d , d , d ), the components of strain at P (e , e
e ,T
,rx
zz xy yz, ,rY ),zthe components of stress at Pzx ( xx, yy
, z
, ) , the components of a volume load acting irtside f'geboayxy t-gz,, y gzx, g) and the components of a surface loads (px, p p ).
Let-nx, n ,z n be the direction cosines of the unit normal
the body surfXce z(positive outside), E the Young modulus of elasticity and 11 the Poisson ratio.
From the basic theory of elasticity we get:
- continuity conditions: eYY EZZ - elastic conditions: 1) i- 8 + Ezz)] ' 54)( 1+1) xx XX
rr
E z(i+ 5cY #YZ 2(i+V) XV7 i.bZX 20+0 4x xy Kyz @dx &ay a ye
&fly .@az az. @ @azsax
e x @z[EYY+ 1- 2U (exx +8,ty + 621.)1
[2zz v 1/4%-xx.+Eyy
+E2.]
on the boundary 6,c + Wx-yn + Vzx nz Px 0Y 6y 1.14 ZYZ 112 Z'xy nx PY - equilibrium conditions:inside the body
pdxx @2-xy evzx
ez
06yy
attz.
+ at-xyey OZ O X
OS=
t
a.z.): at-iy_+
_
92z
exay
6YY = +I)
- The prismatic beam assumption
Let' s now assume that the body is a prismatic beam, with the lateral surface parallel to the z axis and two end sections normal to the z axis.. Let's assume that _the beam is subject to no load parallel to the z-a.xis,. except on the end sections. Let's call "cross sections" the sections. normal to the z axis.
Let's adopt a new symbology, in which we use a vector notation for the variable laying_ in the plane of the cross-section and a scalar one for the variables along the z axis.
Let be k the unit vector normal to the cross-section; n the unit
vector normal to the section boundary (positive outside) and s the unit vector tangent to the section boundary s=n xk
Let's define the variables:
9;8
cliv
a
and the operators:
.90
.9ra.-.0.-""a
rot56 --r(X irld
pcsAy1-a0x)
; curri,
°N/Y Oy 178tch-
ir;c195 ro -gradOk x
8{VI
E. v
-- inside the section dOmain::
on the section boundary:
9753ci w
z
,The. two other 24 basic conditions are ignored,
by -assumptions on the section, deformation . The shear potentials
From equilibtiiiti conditions we have
From ,continuity and elasticity we derive:
- 2 V crw 7A1
Besides from boundary equilibrium we have:-..
and are replaced
=
These equations uniquely define the shear vector field.' as function of the scalar fields 6 and curl g
In particular by the Helrhholtz' s decomposition theorem, We can decompose r in the sumof a solenoidal vector field Vs and an irro'ta.tional vector field tg, , such that
With; these assumptions: and notations the basic coñditioñs are written as follows:
2(44-1)); az
@6
az
inside the, section.
doniain:2
tA,.
on the section boundary: U. = 0
inside the section domain: V z V =
.we introduce the shear potentials
-since = is constant along the boundary:, We choose the arbitrary .integration constant so that U.=-0 on the
boundary--U. is a vector. potential (parallel to the. z-axis). 'which , depends on the curl of the section deformation only, and is the
sblution of the Dfrichlet-like problem:
V
section boundary:.
0
n
; V,. is... a!,`,scalar potential, which ,depends on the longi.t4clindl
[
1-E121-1 9ra __Y -4- rot--..01 u.1
E hence:
where is an integration constant and cfrIci vv5 20-LPI--") rot u. -
al
az6
hence:(
A
vIv
+ 2 1-21))Once some assumption is made regarding -ti- , it is possible to solve the Dirichlet-like problem and find the vector shear potential U; then w is found by direct integration and the
scalar shear potential V is uniquely defined (a part from
integration constants).
- Generalized sectional forces
We now introduce the generalised sectional forces:
Axial force N =if6 dA
A,
Shear force -7 = rf t'ciA
1 J A
=f1A(F-IV 6d A
Kr 11) x
Our definition of the bending moment is slightly different from the standard one, which is MB = /ACT? Fs) A , hence:
MB = M x k; however, our definition will simplify the subsequent analysis.
rB and r are the distance vectors from the origin of the points B and S called respectively the section centroid and the shear
centre.
Definition of these points will be given later on.
If we recollect what we have found in the preceeding section, we note that
ws and V contain an arbitrary integration constant. If we fix such constant such that:
Bending moment Torsional moment + 2(1+v) E
[
4- (ZZZ 1-11) ZZ a VC / $ (1-I-1)) Sy ÷ cca @z az 1-20 E P:Cv 02Ws 4. V v)(4-2v) e zz oz z - clivN (i-u)1 A , hence
cl 4 1,1) - V)
Z.
(i- %.7v.
(i+v) (4-21) z
this term will be hereinafter called 611
It will also useful to fix the point B defined above in the
centroid of the section, such that ff(F-)dA rz- O.
A
This will ensure that stresses due to axial force, 6N
will cause no bending moment.
Further more
-..../f(r-"ot + 9 raciv)clA =
xi/94cl
ciA + 9-rIcdV ciAA A
by the Green's theorem:
tggrad
gradX cl&ff
I
i:cls-.11;2v Fciti A A .srad y ciA 5 A frA9radlu57::dx c:i A 9raduciA .7= CI, 9raCi U. grad j A a 9n ffAVCIA = 0 ;IL
az d A We get: or:where 115 means the line integral around the boundary of
the section. Recollecting what we have found in the
preceeding section
t4,12-1.
x1Z csi5-fey
d Aas_fivt,/
Je A 'a .4) A
Since U is constant on the contour, and cb dr = 0, the
A
vector shear potential U gives no contribution to the shear force, and
-if? 2Y -r6 clA ciA m
A A eZ
since we have assumed that no load parallel to the z axis is applied to the cross-sections, and hence:
@z
s N
laZ A
Furthermore
*)2c15
=fiA(F-r-s))476' (rot LA.+ 9-.rad v c1 A
(7- Fs)ciA
-
I/Ag---"ra 01 Lc. clAA
By the Green's theorem:
97-;c1 F-P3)6-7.$) ci A 2
dA
=2/U,
CLA Asince we have chosen the arbitrary integrator constant such that II = 0 on the boundary.
By the Stokes' theorem:
.(r;;;t vXr= (IA =
-ficuri
1-1)] ci AWe now introduce a function 633 such that:
u)5
-
-
/.6 r
on the boundary andZn '05
V cos= inside the section domain
6c1 A =
Then: ff
(1 v ) (i-t ) d A = and, by theGreen's theorem
II9-r-79 cl gi4; w5 01 A =-- -I/cos a-ml A
A
Hence:
HT 1/2 a. clAA A cid
The torsional moment is the sum of two contributions.
The first one is due to the solenoidal shear field and is independent from the longitudinal stresses and corresponds to the De Saint Venant's torsional moment: Msv lAzu.,dA The second component is the z-derivative of a new
generalised sectional force called Bimoment: 5Z flu) dd A a .1
tIT = Msv
°E;z
OzWe do not need any assumption to introduce this new generalised force, apart from the beam being prismatic and free froni any longitudinal load over its span. It has the
physical role of measuring the influence of longitudinal
stresses on torsion in a similar way as the bending
moment measures the influence of longitudinal stresses on lateral deflection.
However between bending moment and shear force exists a one to one functional relationship, whilst the torsional moment contains a constant quantity which is unrelated to longitudinal stresses and bimoment.
- The function cO5
We will now spend some more words about the function ws, which in the preceeding section we have defined on a pure geometrical base and which we have found playing a fundamental role in the definition of the bimoment.
It is easy recognised that the function Ws is the solution of the
the section domain
,
t
ecth
r
.--r3
..) B 7 0 Ct: -ro) on the boundary
=
c
Zn Ss 2
It contains an arbitrary integration constant which we fix such that ITCJs =IA = 0 , in order to avoid influence of the axial
AIA
stress 6ti on the bimoment.
Its harmonic coniugate ak*, such that 9ra44 Ws- = rot CA)5 is
therefore the solution of the Dirichlet problem: Newmann problem,
CL)
:44
(4.)S d c.r.--"rs)2+
-
5 fl the boundary .where c is an arbitrary integration constant.
We will note that if we consider a point P and we define Op such that "°°-PVn (1=e-../ we get
. es -C4..)?) 2
c
2 hence @Os, )6=(17:5)X
ds + cosi- r
01-'Os = (X5- X13) JB)-
9P) - X IS _ ("6r5-7 r )xF-- r-) PTherefore, once we have found the function C.4) referred to a given point, we can easily derive the function 0.3 referred to another
point.
The shear centre S is generally defined such that fit cos ill ciA. 0 in order to make Os ortogonal to r.
This completely define the function cOs
For a thin-walled beam we may assume that cas* takes
everywhere the value that it takes on the boundary: inside the section. domain.
It is easily recognised 'that this is the definition of Vlasov's principal sectional area for thin walled beams of open section. Hence, hereinafter we will Call sectorial coordinate the function
_ x OP.
r 'an - ; hence
For closed multiple-,-connected sections there exist many disconT
riected. boundary paths each one . with
its,
integration constant.. We -may no longer assume that cd::= FP since ' we haveto add a' -linear: variation over the thickness- to Cope with the ,different constants at the two boundaries.
s .
-Hence---"&t,s
r
r
where r, ci is the ratio. between - the difference of -boundary constants and the wall; thickness, and is unknown.
To find it, We._ remind that c45 0- for any of the . 13 D5, boundaries B, henCe -
1114 --
8 -() aiv + az G ;au. -9 16 theThis is exactly the -procedure to find the warping function in the_ classical -theory of thin-walled beams, which reinforces our
decision Of naming w after the sectorial_ coordinate. It is also possible to show that 4).5 correspond to the warping function in
De. Saint:_ s theory of torsion.
wishto stress; however, that although the derivation and the ,numerical value' of czs is identical to the warping function, we do not state up to now any relationship. with the warping- of the
or with the warping stresses. For us ca.). - is only a
geometrical funCtion without any mechanical-- meaning, which- alIow.
us to calculate the bimoment.
'- It may b0:.;found, as shown, by the Classical theory of
thin-walled beams, Were applicable, or - by the variational approach introduced by Kawai /4/, which is valid for any type Of
sections.
- Interpiediate. conclusions
Without use of any simrilifying assumption we have derived : L-a generalised theory of prismatic ' bearps. free of
-longitudinal 'loads Over their span, which- yields:
where:
gr4
=-; t1
=0
A 'az
the vector shear potential U. is found by solving U. = 0 on the boundary and V211 curl(G)
the scalar shear potential is found by solving
4. THE RIGID SECTION ASSUMPTION
;41
To proceed further, we have to make some assumption on the section displacement a
The trivial one is to assume that the section undergroes a
,rigid displacement :
-AT + (9.1 g (7" 7"-r)
where dT is the displacement of a point T of the section and ef is the rotation around it.
Hence: cur/
26 ;
9z curl .17:i
The latter is the only one assumption needed to find the vector shear potential and solve the uncostrained De Saint Venant's torsion problem.
In the rigid section assumption we get also ch./ ci= 0 .
This is not convenient for thin-walled sections, since it implies that there exist transverse stresses in the section 4xx=yy = due to Poisson's effect, which are unrealistic. We would better assume div;1= glcx+ eyy , in order to have the cross section free of Poisson's stresses.
'DV 0 on the boundary and
z V V+2 -1-evi-
S*2
= E 'a% 972 L Z P E 1-- v 1-2v ' G = 21/414-v)Hence we get: [-9-`2--tviez @z V EV + E v = -E
evis
G z2- The solenoidal shear field
s). _
where COG 'is. the -coordinate.
with
]co: 60fr
18
We see that the equations Obtained are identical to the one's obtained in the rigid section aSsuMption, with the only Change of
-* ,
E' E.
In the following analysis we will - retain assumptions making use of E instead of
It is recognised - that " section deformation, even retain-ing curl
a.
2O' .will have some effect on ws through theequation defining grad w However it is assumed that the additional displacement inguced has no effect : on the stresses':''
Having as,su.rned that curi.j= 20"'
z z
N7 = ; r;r.G&, + (a 'suitable
satisafy the boundary condition U. F
It is straightforward to find:
with . the arbitrary constant of set to zero
the
C5e1 , where
t
(r
rs
_ e
cds.]''
aA = - / (F:- rs) d A + 24
Asome mathematics this may be ,transformed
in any of the
various formulations proposed for the -torsional rigidity K .
rigId seCtiOn
- Pure bending and torsion
If we assume that longitudinal stresses are constant over the span of the beam, we get
S A 0 S
= 0
hence7- =0 ;
ti-r = Msy = GK (9.11S7. 5 @Z 3
Besides ° 6 - 0, hence V2 V = 0, which gives V = 0 since 9Z = 0 also. an. Therefore: 76. G [6^:
-
-r15)-
rv-4. cows -'Fs)X.1Z +3;c1. MsyDisregarding axial stress dN, we get
6 E SWs
where : grad w = rot
-
S31s 'az
-4. I
In the rigid section assumption °c-r
wz -
-t- erIKX (7-rr)if we choose T S. we get : grad livVscIA 0 hence :
-6
E.TT r15) E c.c) 6.1" which implies dT11 = 0' 0'111=if the torsional moment
MT is constant over the span, the twist angle e1 is constant and the warping stresses vanish. If the torsional moment MT is linearly variable, a constant bimoment arise. No other possibility exist in the assumptions of this section.
'If: we 'assuthe that -,,no load is
0 Oz
Here again hence, Ws (.&>/.
-[
.-(1) r
(r,-5z
E IL"
- the general .case
. In the gener.al case M-
/ 0 and
Bz. SV aZ
rigid section assumption, we get again: ..
V1.15. (71 Cr:7
But we can no more assume
hence; , rdia
T-
d-r hence SZv
ev+E
@2\./ . Sz 2 EI b- Pure".birnoment,_torsi6n of thin-walled beams of open section-. In this type of 'beamS"c4)-= (r7,r5). hence-a f- = 0.
5 2.
20
applied except on the end
- Which implies
V 0
q2
if we retain the rigid section assumption we get:
vti _ - - /14- =
-3ra s-ez
Ect)50 +
- r
511E,nt
If we choose T Ea:S- stresses 'due to torsion do not influence
the bending moment, since .12:(.0.57.1A= 0 the bimoment is now
52 ,i-7 where Za.)5-1;1-):d4
hencek 0 if we retain the x
potential
VB and a torsion shear potential V . In t e assumption of end-only loaded beam'ciT = 0, and again:
oVB VE3 0
E4
7rit.)
@Z '
On the contrary o-V-Vaz 0 and V VT ±.
egz
But in this case az - cO<C5'" - @z2 - 0, hence ---wiz--45KCYj;
we .find a completely consistent solution by setting anehtoin
and VT = C/Vc4) " hence V2 V + h2V=-÷-0)5 , which defines
independently from the z-coordinate, and:
6
E [w5 E Vc,, where Then: E hence: -K E laz.sThe solution is iterative since 1st S depends on V which
depends on h which in its turn depends on IA.5 .
However this iterative scheme does not depend on the load pattern,
as long as no torsional load
acts on the beam span, sothat .we may assume h2= constant.
To completely separate. torsion from bending we
need that
stresses dud to Os and Vo, do not contribute to the bending moment. To achieve this we choose the twist centre T
so. that
flA (c4)5 + \icu);c1 A = 0 with this definition the twist centre T .s
slightly shifted with respect to S, as discussed in /8/.
To find the numerical values of VB and V we ma Y directly
cen+ c Pw (9171 E. In_s 2s, sLr
r, Avui
Szt. 6K9'-solve the differential problem by a variational approach similar to Kawai's, one /4/. For thin-walled beams we may adopt an approach
similar to the classical theory:
we assume that V (V or V ) is constant along the wall thickness,
JzvB
9v Thence V21,--= osz .;
-
1/C15 con9t , integrationcons-tants are different for different branches, since we lack continuity
of al1/05 at nodes, and are obtained by requiring ..p-d5 0.
es
For Vp this approach is identical to the well-known shear
flux theory ior bending. shear.
For VT, this approach is similar to the classical theory, but
this latter aisregards the term in VZsIvT
azz"
- The classical thin-walled beam theory and the refined theory by
Kolbrunner and Haidin
The basic difference between these theories and the one shown
in the above chapter is related to the fact that the former
disregard the influence of the scalar shear potential on
logitudinal stresses.
As a consequence 6= E azWs and warping stresses are
assumed to be proportional to die principal sectional coordinate
as in Vlasov's theory. This is the consequence of the fact of not
having recognised a second component in the warping
displace-ment, which allows to relax this assumption. An interesting
consequence is that in the present theory the warping stresses
distribution does not vary along the beam, but the warping
displacement distribution does. This fact is not recognized in the
for ther theories.
In the so called "refined theory" by Kollbrunner and Haidin,
they try to overcome the lack of equilibrium consequent to the
neglected term by means of an energy balancing method, similar
to the one often adopted to find the shear area reduction
coefficients.
However they relax the proportionality of warping stresses to
the twist angle, instead of the proportionality to the sectorial
coordinate.
As a consequence they restore equilibrium in the mean, but
lack local stress equilibrium.
This drawback has been avoided in the theory presented here,
in the assumption that eYwoctit" , as in the case of end-only
loaded beams. Extention to the case of beams loaded along the
span is possible, but requires, in general, the use of a greater number of shear potentials, as in Liebst (1984).
Influence of the disregarded term on the results is not
dramatical in the case of long beams, but may increase substantially in the case of short beams, where shear deflection plays a significant role.
5. CONCLUSIONS
In the preceeding sections a generalised theory of beams has been introduced, which is exact in the frame of elasticity theory, and some assumption has been made to get practical results from it. We have seen that the assumptions are mainly related to the deformation of the cross-section, regardless of the thickness of the walls, which makes the name "thin-walled beams" not completely appropriate to this theory. However the name is retained on the basis of historical background, since some of the theories, such as Vlasov's and the classical one, requires the thickness of the wall to be small.
We have shown that this is immaterial, since this requirement is only related to the numerical method adopted to calculate
characteristic functions such as the sectorial coordinate and the shear potentials, which may be easily calculated for any type of
section, by using variational methods. We have also shown some
inconsistency in the so called "refined theory" by Kollbrunner and Haidin, which is mainly due to neglected terms in the longitudinal
displacement.
We stop here and do not proceed further to show all the implications of this theory and possibility of extention to variable section beams, stability problems etc., since our aim was to show the basics of the theory, only.
We refer the interested reader to the litterature for that.
Many other extensions of the theory are possible, e.g. how to include deformation of the section, but it is the author's opinion that the theory of beams is a useful engineering tool as long as it
-keeps, :simple, and-;-:- its complexity is substaritia.lry lower -,than,
finite element 'modelling - of the complete,-hult,
,
-- If the cOMplexity ,of our problenC---:reciiiii-e-ST-- :a. Very refined
-model,_. 'and we .ar.' "prepared to pay the cost,. 'then . we would better
change to-a FEM analysis _Which is more accurate and flexible, and
allow i4s,: to model the structure in More realistic way,
-This is the'..,:.::rea.Son why, further 'refiniment of the beam ing from a theOi.e,fiCal, .point
. revenues from the point . of view
e
-at .-reast- -in the Author' S pinion,
theory, although extremely
interest-of, view, will not -give substantial
of practical analysis pro-bre:Ms,
24
References
/1/ V .Z . Vlasov : Thin-walled elastic beams. Israel. program for
scientific translations - 1961.
/2/ S. U . Benscoter : A Theory of Torsion Bending of Multiall
Beams, Journal of Applied Mechanics - March 1954.
/3/ K. Ha slum & A. Tommensen: An Analysis of Torsion in Ship
Hulls. European Shipbuilding No. 5/6 - 1972.
/4/ T. Kawai et al. : Finite element anaslysis of thin-walled
structures based on the modern egineering theory of beams.
3rd Conf. . on Matrix Methods Struct. Mech . Wright Patterson
Air Force Base Ohio - 1971.
/5/ T. Kawai: The application of finite element methods to ship
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/6/ C.F. Kollbrunner & N. Haidin : D iinnwandige Stabe, Band 1
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/7/ A. Pittaluga : Recent development in the theory of thin-walled
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/8/ A. Pittaluga: Thin-walled beams theory and ship design
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/9/ H. Westin : Torsion of non-prismatic beam girders with special
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/10/ J. Leibst: Torsion of Container Skibe - The Technical
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JLTIMI BOLLETTNI TECNICi PUBBLICATI LAST PUBLISHED TECHNICAL BULLETINS
BT 88 FEBBRAIO 1980
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