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Beam Elements Based on a Higher Order Theory

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Compuws & Srrucrures Vol. 58, No. 4, pp. 775-789, 1996
Copyright0 1995ElsevierScienceLtd
Pergamon
0045-7949(95)00185-9
Printed in Great Britain. All rights reserved
W45-7949/96 $9.50 + 0.00
BEAM ELEMENTS BASED ON A HIGHER ORDER
THEORY-I.
FORMULATION AND ANALYSIS OF
PERFORMANCE
R. U. Vinayak, G. Prathap,tf and B. P. Naganarayana
Structural Sciences Division, National Aerospace Laboratories, Bangalore 560 017, India
(Received 30 June 1994)
Abstract-The
flexure of deep beams, thick plates and shear flexible (e.g. laminated composite) beams and
plates is often approached through a finite element formulation, based on the Lo-Christensen-Wu (LCW)
theory. This paper is a systematic analytical evaluation of the use of the LCW higher order theory for
finite element formulation. The accuracy and other features of the computational model are evaluated by
comparing finite element method (FEM) results with available closed form classical and elasticity
solutions. Wherever possible, errors are predicted by an apriori analysis using these solutions and concepts
from an understanding of what the finite element method does.
I. INTRODUCTION
For deep beams, thick plates and for beams and
plates made of high-performance laminated composites, the classical theories based on the
Kirchhoff-Love hypothesis no longer suffice. The
Reissner and Mindlin theories [1,2] provide a firstorder improvement by accounting for the effects of
transverse shear deformation. A very large number of
papers based on these theories are available, see
Ref. [3]. Many competing first-order theories are now
available, see Refs [4,5], but most of these theories do
not account for the effects of transverse normal
strain. Also, a more refined theory is needed for
establishing more accurate stresses through the thickness, e.g. inter-laminar stresses at boundaries, discontinuities, etc. The Lo-Christensen-Wu
theory [6, 71 is
one elegant higher order theory which has found
much favor in finite element formulations [8,9]. This
can now take into account transverse normal strain
and stress effects and also allow for the computation
of inter-laminar stresses, by post-processing the FEM
results using integration of equilibrium equations.
The LCW theory has several advantages when seen
from the point of view of designing a finite element
for production run analyses, as routinely performed
in general purpose packages. It requires only a Co
formulation, unlike the C’ formulation expected for
many competing higher order theories [lo-121. The
higher order transverse shear effects and transverse
normal strain and stress effects are incorporated in
such a way that the transverse shear strains are
consistently defined in the thickness, z-direction.
Also, consistency of the discretized transverse shear
strains in the natural co-ordinate (covariant) system
can be easily implemented by assumed strain formulations for displacement type elements, so that no
locking effects appear [13,14].
Although the LCW theory has been widely used to
develop beam and plate elements, no systematic
analysis of how these elements behave has been made.
In this paper, we attempt to rationalize the performance of a higher order beam finite element based on
the LCW theory, by introducing
some recent
concepts of finite element structural analysis [ 151.
2. THE LO-CHRISTENSEN-WU (LCW)
BEAM MODEL
Figure 1 shows the general view of the beam of
rectangular cross section with length I, depth h, and
width b (not shown in the figure). External surface
load pZi(x) applied at zi and line load P:;” applied at
(xk, z,) are assumed to be acting in z direction so that
x-z plane is the plane of bending. Note that the loads
can be applied at any arbitrary surface z. The LCW
theory [6,7] expands the in-plane displacement field
U(X,z) as a cubic function in the thickness coordinate
z. The corresponding polynomial expansion for
transverse displacement w(x, z) is truncated at one
order lower than the expansion for in plane displacement. This choice leads to a consistent definition of
interpolation for the transverse shear strain with
respect to the thickness coordinate z. Defining the
displacement field in terms of the mid-surface degrees
of freedom u,, 6,, . . , w,*, we write
u(x. z) = Uo(X)+ zB,(x) + z%o*(x) + z%;(x)
tAlso, Jawaharlal Nehru Centre for Advanced Scientific
Research, Bangalore 560 012, India
fTo whom all correspondence should be addressed.
W(X, Z) =
715
wotx)+ ze,(x)+ z2wo*(x)
R. U. Vinayak et al.
176
auo/ax
4
e,+ aw,/ax
au,*jax
7.
aw,*lax +
ae, Iax
30: +
. (2f)
2w,*
ae,/ax
ae:jax
2~: +
Fig. 1. Beam under bending in x-z plane.
or
(6 > =
m%
(14
where
1 0
z
0
z*
0
0
0
z
0
22 0
WI =
{S}=(uo
1
e, e, u:
w,
(6) = (u
w$
z3]
(I,,)
e:)’
If the beam is made of layers of different isotropic
materials and/or composite materials with fiber
orientation angle of B” with respect to x axis stacked
in z direction, the stress-strain relations for a typical
layer L with reference to x-z coordinates are given by
[16-181
{s[=,if!
; i;J{$}
(lc)
W)?
(ld)
The strain field associated with eqn (la) is
au au0
=-=_+z!!%+z*?!+z3!$
‘xx ax ax
+r>”
= [8*l”[z,l{+
(3)
where 0; are condensed from the three-dimensional
orthotropic elasticity matrix (see Appendix A for
details). The total strain energy stored in the beam is
(24
(2b)
=; j- {C}T{g dx,
X
++e:+
2).
where b is the width of the beam of rectangular
and {S} is the vector of stress resultants
(2c) cross
given section
by
Again, defining the strain field in terms of the
mid-surface values Q, cm,. . . , tj:, we rewrite the
strain field as
(24
(61= L%l~~~~
(4)
(9 = (N,
N,
Q, W’
Q: Mx Mz Sx WjT.
If NL is the total number
resultants are given by
(5)
of layers, the stress
where
cxr
+> = 116, 9
Yx.2
1
1 0
0
22
0
z
0
0
23
0
0
0
1
0
0
0
z*
0
0
z
0
0
z
0
0
i
0
(W
=:j-”
L-1
= Pi
Z‘-,
I{~}
@*I”[4 1dz (4
@a)
171
Beam elements based on a higher order theory-1
The work done by the surface load pzi is
=b
I*
w,,,b) dx,
hi
where (uli w,~) is the displacement vector at z = zi,
{p} is the surface load vector. Substituting for the
displacement vector eqn (la),
W, = b
(z}T[Z];(p)
IX
dx,
(9)
where [Zli is the matrix [Z] evaluated at z = zi. Work
done by the load PC! applied at (xk, z,),
W c = bpxkwrk
*, 2,.
The total potential of the beam,
(64
= L&_lW
In eqn (6b)-(6d)
@I
re:Y=
Q:3
o
o
[
[Z,l=
Z[Z,l
LIZ: _I”= @?I
Combining
55
1
O
p
L,[z21=z2[z
I9
OF.
(12)
eqn (6a)-(6d), we write eqn (5) as
{it = [email protected]),
(7)
3. FINITE ELEMENT
I- PII
1
NNE
{z}
{i}‘[D] {E} dx.
sX
(8)
N,
0
[N] =
=
C
,=I
Ni{z)i=
IN]{&},
(13)
where NNE = number of nodes per element, Ni = ith
shape function (see Appendix B for the shape functions for linear, quadratic and cubic elements). The
element nodal displacement vector {&} and the
matrix of shape functions [N] are given by
eqn (7) into eqn (4),
U, = ;
FORMULATION
The displacement vector within an element can be
expressed in terms of the nodal degrees of freedom as
where
Substituting
(11)
Dividing the solution domain into NE elements,
L&l = Z3El,
e:,
{i?}*[Z]f{ p} dx - bP;f w:; .
-b
]
0
N,
0
0
0
0
0
0
0
0
0
.
0
.
.
.
.
0
.
0
.
.
0
. .
.
0
. .
0
0
0
0
N,
0
0
0
0
.
0
0
0
N,
0
0
0
.
0
0
0
0
N,
0
0
.
0
0
0
0
0
N,
0
.
.
.
.
0
0
0
0
0
N,
.
.
.
.
0
N NNE
R. U. Vinayak et al.
778
The strain
expressed as
vector
within
an element
{Cl = PI&
can
be
ent shape functions Ni (see Appendix B) are used for
the constrained strain components yXzO,
y h, and $,,
instead of the original shape functions N, in eqn (2~).
This allows ill-effects like locking, delayed convergence, etc. to be removed without the need to use an
artifice like reduced integration. The results reported
(14)
1
where
r
a/ax
0
0
0
0
0
0
0
0
1
0
0
0
a/ax
0
PI =
0
0
0
0
0
0
0
1
0
0
0
0
0
0
a/ax
0
0
0
0
0
a/ax
3
a/ax
0
0
0
0
0
0
0
0
0
2
0
0
0
0
alax
2
0
0
0
0
0
0
0
0
The total potential energy of the element e is
Using the principle of minimum
total potential,
[email protected]’= 0
=b
In matrix notation,
(12),
WITP%b >dx.
(15)
eqn (15) is written as
the
global
[K]‘G’{$}@) = {F}(G),
(16)
equations
are
(17)
where
e=l
WI.
a/ax
in the subsequent sections are all computed from such
consistent formulation. Two noded linear elements
(BM2), three noded quadratic elements (BM3), and
four noded cubic elements (BM4) have been developed for FEM analysis. Since the computation of
transverse
stresses from equilibrium
equations
requires at least BM3 elements, results are not presented for BM2; results are reported for BM3 in most
of the cases and for BM4 in some cases.
4. COMPUTATION OF TRANSVERSE STRESSES FROM
INTEGRATION OF EQUILIBRIUM EQUATIONS
[K](e){&} = {F,}(e).
Using eqn
obtained as
1
r=l
{F,} is the vector of line loads. Isoparametric
mapping is used to evaluate the integrals in eqn (15).
The width b of the beam is taken as unity in all the
numerical examples to follow. As observed earlier [13, 141, the elements formulated using the original Lagrangian shape functions suffer from locking,
delayed convergence and stress oscillations. Consist-
In the discussion that follows, we shall use 6,, etc.
to denote stresses computed from the FEM solutions
using the strain-displacement equations, eqn (2) and
the constitutive law, eqn (3). Terms such as 6;, fXz
denote stresses computed by integrating the equilibrium equations [see eqn (18) below]. The in plane
stress (JXcan be accurately evaluated from the computed FEM nodal displacements u0 to w$ and the
constitutive law and strain displacement relation,
eqn (2a). The FEM transverse stresses derived directly from the constitutive law and the strain displacement eqn (2b, 2c) using the same computed
displacements are not very useful. One disadvantage
is that the 4_ is accurate only to a linear order, and
fXzis accurate only to a quadratic order through the
thickness and these are therefore determined in a least
squares accurate sense of the actual strain-stress
variation through the depth. Another difficulty is that
transverse stresses have to be continuous across layer
interfaces in the case of a beam made of many layers
of different laminae, whereas transverse stresses
derived from strains using eqn (2b, c) give discontinuous stresses if the layers have different elastic moduli.
One can improve upon this by adopting a strategy
based on integration of the equations of equilibrium
for two-dimensional elasticity for each layer, and
summing up over all layers to give a more accurate
119
Beam elements based on a higher order theory-1
and realistic stress pattern. Thus, if we start with the
bending stress 6, and transverse shear stress tX,,which
are computed from the FEM displacements, using
only the constitutive laws and strain displacement
relations, we can compute improved C’, and i,, by
using the relations
a?, -- sex
-=
az
ax
(184
3 _
_--afxz
(18b)
az
ax
within each layer. Thus, after the integration and
evaluation of constants using an initial value problem
strategy, one would get a quartic variation of TXZand
a cubic variation of ~7~in the z direction. If we have
7,X= 0 and C’,= 0 at the bottom surface as the initial
values, one should be able to get the correct values of
?, and ii, at the top surface, to an order of accuracy
reflecting the accuracy inherent in the computation
of stresses such as CXand fX,,, in the FEM solution
process. We shall find that these accuracies are
maintained when we carry out the finite element
experiments later.
investigators [19-211
have
Recently,
some
suggested that (T, should be determined from the
second-order
differential equation
obtained
by
differentiating eqn (18b) further as
(18~)
This means a four-noded cubic BM4 element is
required. In our interpretation, it suffices to start with
eqn (18b), requiring only data from a BM3 model.
Also, as observed earlier, the problem is worked out
as an initial value problem, starting with one surface,
e.g. at the unloaded surface, say z = -h/2 where
ii, = 0, and at each interface, continue with the
continuity requirement on 8,. At an interface where
a load is applied, e.g. ifp is applied at z = z,, we must
=p where the superscripts (+) and (-)
have 5: -a;
indicate the values across the interface at z = z,. One
must also interpret the second “boundary” condition
at the other surface, i.e. at z = h/2, as a target to
shoot at-if this is reached, it is an indication that the
FEM solutions 5, and TX2which are used as the right
hand side of eqn (18) have been accurately obtained.
Thus in many examples, we were able to obtain this
value to accuracies up to 10-14.
5. ANALYTICAL
PREDICTIONS
FOR BENCH-MARK
The plane stress solution using the Airy stress
function approach for a laterally loaded isotropic
cantilever beam by Venkatraman and Pate1 [22] provides a useful bench-mark solution for evaluating the
accuracy and efficiency of our present finite element
models. For the configuration shown in Fig. 2, we
obtain the following expressions for the stresses (TV,
~~~and CJ;
o1 = (p/l201)[-40z3+
6{10(/ - x)~ + h2)z]
o;= -(p/l201)[5(-4z3
This conclusion was mistakenly arrived at by
accepting the notion that two boundary conditions
are available to determine the constants of the complementary solution for 8,, i.e. in the case of an
isotropic single layered beam, the boundary conditions arising from the stress components in z direction at top (z = h /2) and bottom (z = -h /2) surfaces
are available to equate to CZat these points.
The argument suffers on several counts. One, from
the strict point of view of the variational derivation
of the governing equations, it is not justified to carry
the variation (i.e. the integration by parts of the
functional) one step further, to derive an “equation
of equilibrium” such as eqn (18~). When this is done,
it implies an additional continuity of (X,/az) at
points of natural discontinuity, e.g. at lamina interfaces, which is not otherwise called for. Also, it means
that in a laminated beam with multiple layers, the
particular and complementary solutions must be set
up layer by layer and in each layer, two constants
have to be determined. At each layer therefore, two
conditions must be matched. The equating of b, at
each layer interface is called for; the second condition
is met only by equating the derivatives, and this level
of continuity is not called for, and is incorrect.
Another consideration is that with eqn (18c),
(a2*,/ax2) must be available to start the solution.
TESTS
+ 3h2z +h3)]
~~~= (p/120Z)[15(4z2 - h’)(I -x)]
(19a)
(19b)
(19c)
when the beam is of rectangular cross section of unit
width and depth h, so that I = h3/12, and is loaded
by a uniformly distributed load (u.d.1.) of intensity
p = 1.0 on the top surface, z = h/2. If we recast eqn
(19a and 19b) in terms of a dimensionless co-ordinate
q = 2z/h, we have
o, = (p/10)[(30/h2>(1 -x)~v
0: = -p[(l
+ (3~ - 5s3)]
+ rl)/2 - I(U2 - 1)/41.
(20a)
(20b)
Note that Venkatraman and Patel’s solution for ox
has a linear and a cubic variation through the
depth-here,
the terms are grouped so that they
/
/
P=I-O
llljilll,
/-.--__h______-__
/.
/
x
i
b---i
Fig. 2. Laterally loaded isotropic cantilever.
780
R. U. Vinayak et al
reveal variations in the form of the linear (q) and
cubic (3~ - 5~~) Legendre polynomials. Because of
the orthogonal nature of the Legendre polynomials,
it is easy to identify the bending moment M, as being
directly responsible for the bending stress associated
with the distribution corresponding to the linear
Legendre polynomial.
Some more interesting facts may be noted down
now for use later, when we take up some test cases
for numerical experiments. The normal stress ox due
to bending is exactly anti-symmetric even when the
applied loading is on the top surface for all h/lratios.
We shall see that in a deep beam this is not true and
this is verified in our calculations from the FEM
model. Also at the free end of a cantilever, x = I, cX
does not identically vanish to zero-the
residual
stress here corresponds to the variation according to
the cubic Legendre polynomial, and so, vanishes only
in an average integral sense. The ur shown in eqns
(19b) and (20b) is obtained for the case where the
lateral loading is on the top surface by using this as
a boundary condition. It is useful to work out what
or would have been for a case where the load is
applied at the mid-surface (z = 0); we would have for
such a case,
c:= -p[l/2+3r]/4-q3/4]
-p[-l/2+311/4-q3/4]
for -1 <q CO
forO<q
< 1. (21)
We shall need this result for our numerical examples
later.
Another set of results we shall need are least
squares fit approximations of the functions for 0, we
have in eqns (20b) and (21). We now know that finite
element displacement method solutions seek strains/
stresses in a least squares accurate sense [ 151.Thus, if
strains/stresses are computed directly from the FEM
displacement fields using the [B] and [D] matrices
[eqns (7) and (14)], these would be obtained as least
squares accurate approximations of the actual state
of stress. Thus, it will be interesting at this stage to
predict what the least-squares accurate fit, up to
linear order through the thickness are, for err from
eqns (20b) and (21) so that these can be compared
with 8:, the FEM solutions determined using eqn (2b)
and the computed displacement fields. It can be
shown that the least squares approximation a,(ls) for
the case where the load is applied on z = h/2 is
UZ(lS)= -p(1/2
+ 3r1/5)
length I = 10.0 under uniformly distributed load and
consider the bending moment M, at a station x = 5.5,
the centroid of an element whose ends lie at x = 5.0
and 6.0 in a uniform 10 element model of the beam.
We chose this point because the disturbances triggered off by discontinuity conditions at the boundaries x = 0 and x = I die out in this region. The
detailed treatment of this problem of stress oscillations is provided in Part II of this paper [23]. For
now, we shall assume that the modeling is done in
such a way that these spurious oscillations have been
filtered out within a small boundary zone. We shall
compare the bending moment M, derived from analytical theory with that obtained at the centroid of a
BM3 or BM4 element. We shall use this as the basis
for reconstituting the bending stress variation using
eqn (20a) and compare this with solutions obtained
from the FEM solution using eqn (2a), which is
also able to represent a cubic variation through the
thickness co-ordinate z.
In the problem under investigation, the variation of
bending moment IV, along the length of the beam due
to a uniformly distributed loading is quadratic. Thus,
for p = 1.0, we can compute analytically, bending
moments at x = 5.0, 5.5 and 6.0, of 12.5, 10.125 and
8.0, respectively. We also know that in a BM3
element, we can accurately represent only a linear
variation of bending moment along the length of the
element, but, in a BM4 element, we can accurately
variation
of bending
capture
a quadratic
moment [15]. Thus the BM4 element will recover the
correct bending moments for this problem everywhere along the element length, and at the element
centroid, i.e. at x = 5.5, will yield ic/ = 10.125. The
BM3 element will recover only a least squares accurate linear fit of the actual quadratic variation (see
Fig. 3)-it will show correct bending moments only
at the Gauss points corresponding to the two point
and at
Gauss integration rule, i.e. { = + l/J3;
x = 5.0, 5.5 and 6.0, it will yield computed aX of
12.4167, 10.167 and 7.9167, respectively. Thus, from
the fact that x = 5.5, a BM3 element will give an
(22)
and for the case where the load is applied on the
mid-surface is
cr,(ls) = -(3/2O)p+
(23)
Another useful result is the prediction for bending
stress and moments at any station of a beam. We
shall investigate the case of a cantilever beam of
Fig. 3. Bending moment in
. . element 6, thin beam (l/h = lo),
u.a.i. on top.
781
Beam elements based on a higher order theory-1
li;i = 10.167 and a BM4 element will give aX =
10.125, we can estimate what the variations through
the depth of Q, wiil be according to the Venkatraman
and Pate1 solution given in eqn (20a). We can show
from eqn (20a), for the case where I = 10.0, h = 1.0,
p = 1.0, at x = 5.5
o,(BM4) = 60.75q + 0.1(3~ - 5~~)
(24a)
a,(BM3) = 6l.OOq + O.l(3rj - 5r~~)
(24b)
as the bending stress variations corresponding to M,
of 10.125 and 10.167, respectively. We shall use these
results later in our section on numerical experiments.
6. NUMERICAL EXPERIMENTS
So far, we have set the stage for the evaluation of
the finite elements based on the higher order LCW
theory by deriving a priori analytical estimates for
their behavior. We shall now confirm the validity of
these predictions by performing carefully chosen
numerical experiments.
6.1. Isotropic cantilever beams
A thin beam (I = 10.0, h = 1.0) and a deep beam
(1 = h = 10.0) are considered for numerical studies.
Young’s modulus,
E = 1000.0, Poisson’s ratio,
v = 0.0 and z&Z. p = 1.0 are assumed in both the
cases. A uniform mesh of 10 elements is used unless
otherwise mentioned. All the degrees of freedom of
the cantilever beam are suppressed at x = 0, i.e.
U~=W~=8,=e,=u,*=w,*=~f=o.
In this section we shall concentrate on studying the
performance of the finite element models in predicting the stresses at points away from the ends of the
beam, in order to avoid the effects of the boundary/edge discontinuities. However, a detailed analysis
of the effects of these discontinuities and how they
can be minimized is presented in Part II of this
paper [23].
First, we shall discuss the comparative performance of BM3 and BM4 elements. Table 1 shows how
the predicted normal stresses from eqn (24a and 24b)
compare with the actual bending stresses computed
from the use of BM3 and BM4 elements based on
LCW theory for a thin beam with a uniformly
distributed load on top. The very precise agreement
between the two sets of predicted and computed
results shows that the elements based on LCW theory
perform as one can expect. Thus, the variations of
bending moment [email protected]~are predicted in the least squares
accurate sense along the length of the beam
element-the quadratic BM3 element having a linear
accuracy and the cubic BM4 element offering a
quadratic accuracy. We also see from this that the
bending stresses 5, through the depth maintain the
same cubic accuracy expected of the LCW formulation [see eqn (2a)], and this matches exactly with the
cubic variation predicted from Venkatraman and
Patel’s solution-eqn
(19a).
6.1.1. Transverse normal stress distribution. One
important improvement provided by the LCW formulation over the first-order theories is that it allows
transverse normal stresses 0, to be computed by the
post processing procedure. Here, we shall see the
accuracies involved in using LCW elements to do this.
Figure 4a depicts the distribution of transverse
normal stress ur at x = 5.5 (the centroid of the sixth
element from the root in a 10 element uniform mesh)
for a thin beam with uniformly distributed load on
top. This location is sufficiently removed from the
point x = 0 where the clamped boundary conditions
are enforced by suppressing all the seven degrees of
freedom so that the wiggles and oscillations discussed
in Part II of the paper [23] have been practically
OSI
(a)
(b)
0,s
Table 1. cx distribution at x = 5.5, thin cantilever beam
(l/h = 10) with a uniformly distributed load on top
(BM4)
tl
1.0
0.8
0.6
0.4
0.2
0
(BM3)
Predicted
eqn (24a)
Computed
b,
Predicted
eqn (24b)
Computed
b,
60.550
48.584
36.522
24.388
12.206
0
60.550
48.584
36.522
24.388
12.206
0
60.800
48.784
36.672
24.488
12.256
0
60.800
48.784
36.672
24.488
12.256
0
-I
-0’1
Fig. 4. (a) Distribution
I.,
of transverse normal stress at
x = 5.5, thin beam (I/h = lo), u.d.1. on top. (b) Distribution
of transverse normal stress at x = 5.5, thin beam (I/h = lo),
u.d.1. on top, least squares fit interpretation.
782
R. U. Vinayak et al,
eliminated. The variation of i?‘, computed by FEM
using the equilibrium eqn (18b) agrees very well with
the gz distribution obtained by the Venkatraman and
Pate1 solution, eqn (20b). Also shown in Fig. 4a are
the results obtained from a three-dimensional model
using eight-noded
brick elements from threedimensional-FEES, an inhouse package developed at
NAL [24]. An accurate representation is achieved
here-in fact 24 brick elements are used through the
depth of the beam and the symbols in Fig. 4a are
placed at element centroids where the stresses are
obtained most accurately. We see here that for a thin
beam, both the LCW and Venkatraman and Patel’s
solution are very close to each other and to the
three-dimensional FEM model results.
Figure 4b shows the least squares fit interpretation
of the present results. We see that the distribution of
5: (evaluated using the fern displacements and eqn
(2b)) is in exact agreement with the least squares fit
distribution crZ(ls) of Venkatraman and Patel’s solution. This is also seen to be the least squares fit of
both 0: and 8;. This is in line with the understanding
that finite element displacement method solutions
seek strains/stresses in a least squares accurate
sense [IS].
Next, we present the distribution of (T, for a thin
beam with uniformly distributed load at z = 0, the
mid-surface. The performance of elements based on
a higher order theory under this loading has so far
not been reported in the literature. Again a uniform
mesh with 10 elements is used. As in the case of the
thin beam loaded on the top, at x = 5.5, 6, in Fig. 5
is in good agreement with crzand so is ez with a,(ls).
oI and 6, distributions are discontinuous at z = 0, the
point where the external load p is applied. The
stresses from the present FEM and the Venkatraman
and Pate1 solution are slightly different from the
three-dimensional-FEES
predictions [24].
So far we have talked about the distribution of gz
in a thin beam. We shall now take up the case of the
deep beam. We use a uniform mesh of 10 elements.
Table 2 presents the distribution of ez at x = 5.5 for
the case when the uniformly distributed load is on the
top and the mid surfaces. The pattern of variation of
g._through the depth continues to be the same as in
Fig. 5. Distribution of transverse normal stress at x = 5.5,
thin beam (I/h = IO), u.d.1. on mid surface (z = 0).
the case of the thin beam both when the beam is
loaded on the top and on the mid surface (see
Figs 4a and 5). From Table 2, we see differences,
though small, between ci, and uz, and between CY:
and
~~(1s)unlike in the case of the thin beam (Figs 4a, 4b
and 5). The Venkatraman and Pate1 predictions for
the deep beam loaded on the top are less than the
present for -0.5 <z/h < -0.3. But in the region
-0.3 G z/h <OS the trend is the reverse, i.e. the
Venkatraman and Pate1 predictions are greater than
the present.
However,
on the top surface
ei = 17,=p = 1.0. When the uniformly distributed
load is at z = 0, the mid surface, both oz and Cz
variations are perfectly anti-symmetric in z. At z = 0
the stress distributions are discontinuous. It is also
seen from Table 2 that the absolute value of ~7;
is greater than that of 6, in the regions
-0.5 <z/h CO(-) and O(+) <z/h i 0.5. Again, at
z/h = -0.5, 0 and 0.5, the two values are the same.
Similarly, the absolute value of 6, (which is seen to
be the linear least squares fit of 8,) is greater than
a,(ls) for -0.5 < z/h CO(-) and O(+)< z/h < 0.5. At
z =o, fJ,(ls)=f?,=O.
6.1.2. In plane normal stress distribution. We now
study the variation of in plane normal stress ox.
Figure 6 shows the variation of a,h2 through the
thickness at x = 5.5 for thin (l/h = 10) and deep
(I/h = 1) beams with loads on top and mid surfaces.
As noted earlier under the section on analytical
Table 2. Variation of crzat x = 5.5 in deep beams carrying a uniformly distributed load
z/h
u.d.1. on mid surface
u.d.1. on top surface
._
0:
b:
-0.5
0.0
0.0
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.028
0.104
0.216
0.352
0.5
0.648
0.784
0.896
0.972
1.0
0.0315
0.1066
0.2147
0.3452
0.4873
0.6304
0.7639
0.8771
0.9593
1.0
u,(ls)
-0.1
6;
-0.097
0.5
0.492
1.1
1.081
01
0.0
0.028
0.104
0.216
0.352
kO.5
-0.352
-0.216
-0.104
-0.028
0.0
a;
0.0
0.040
0.120
0.230
0.360
io.5
-0.360
-0.230
-0.120
-0.040
0.0
[email protected]
5:
0.15
0.166
0.0
0.0
-0.15
-0.166
Beam elements
-0
based on a higher
-50
-30
-10
10
30
50
70
a*h’
Fig.
783
thus showing no distinction between the thin and the
deep beams. For the thin beam loaded on top, S,,
agrees very well with t,, without showing much loss
of symmetry. For the deep beam loaded on mid-surface 7, distribution shows slight deviations from 71L
but still is perfectly symmetric in z. On the other
hand, fXx2
distribution in the deep beam loaded on the
top varies significantly from t,; and there is a noticeable loss in symmetry. This again can be attributed to
the simplifying assumptions of the Venkatraman and
Pate1 solution.
5
-70
order theory-1
6. Distribution
of inplane
stress at x = 5.5, thin
(I/h = IO) and deep (I/h= 1) beams carrying u.d.1.
6.2. Simply supported composite beams
An orthotropic graphite/epoxy material with the
following material properties is considered [25]
E, = 0.25 x lo8 psi,
for bench-mark tests, o,, predicted from
the Venkatraman and Pate1 solution, is perfectly
anti-symmetric
and cannot differentiate between
loads applied at the top or mid-surface (both for thin
and deep beams). The effect of cubic variation in z
on B, and CXis seen to be negligible in the thin beam
case. Therefore a, and ~7~show a predominantly
linear variation. In the case of the deep beam
however, this effect is clearly visible. The variation
of d,h2 matches very well with a,h* when the
uniformly distributed load is on the mid surface and
is perfectly anti-symmetric; but when the load is on
top, c?,h’ is 48.6 at z = -h/2 = -5.0 and -33.8 at
z = h/2 = 5.0, thus showing a deviation from the
perfect anti-symmetry predicted by Venkatraman and
Patel. The differences in c?~and a, are due to the
simplifying assumptions made in Venkatraman and
Patel’s solution which clearly become inaccurate for
deeper beams.
E2 = E, = 0.1 x 10’ psi,
predictions
G,, = GIX= 0.5 x IO6psi
G,, = 0.2 x 106, vi2 = v,~ = v2, = 0.25,
where 1,2, 3 are the principal material directions (see
Fig. Al), E, are the Young’s moduli, G,, are the shear
moduli, and v,~are the Poisson’s ratios. The boundary
conditions imposed on the beam which carries a
sinusoidal load p = p0 sin(mnx/l) are
w = 0 at x = 0
and
x = I,
u,, = 0 at x = l/2.
Note that under such conditions, discontinuity
effects will not be seen at the supports. In the present
study p0 = 1.O,m = 1. The different cases addressed in
this section are:
(1) an orthotropic beam with fibers oriented in x
direction;
(2) a two layered laminate with directions 2 and 1
6.1.3. Transverse shear stress distribution. Finally,
aligned parallel to x in the top and the bottom layers,
we take up the study of the transverse shear stress
respectively, the layers being equally thick;
distribution. Table 3 shows the distribution of zXz
(3) a symmetric three-ply orthotropic beam with
both for thin and deep beams at x = 5.5. t,, from
direction 1 coinciding with x in the outer layers, while
eqn (1SC) is symmetric in z and parabolic in nature,
2 is parallel to x in the central layer, the layers being
Table
3. Variation
Thin beam,
of T._ at x = 5.5 for thin and deep beams
I/h= 10
Thick beam,
5,:
7,:
eqn’G9c)
u.d.1. at
z =h/2
u.d.1. at
z =o.o
0.0
0.2430
0.4320
0.5670
0.6480
0.6750
0.6480
0.5670
0.4320
0.2430
0.0
0.0
0.254692
0.450355
0.586192
0.662383
0.680083
0.641425
0.549616
0.408440
0.223256
0.0
0.0
0.243612
0.432396
0.566870
0.647407
0.674227
0.647407
0.566870
0.432396
0.243612
0.0
Tr:
zlh
eqn (19~)
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.0
2.4300
2.3200
5.6700
6.4800
6.7500
6.4800
5.6700
2.3200
2.4300
0.0
t x1
0.0
2.42994
2.31996
5.67001
6.48006
6.75008
6.48006
5.67001
2.31996
2.42994
0.0
I/h= 1
784
R. U. Vinayak er al.
equally thick. The non-dimensional
Figs 7a-9e are:
quantities used in
, [email protected], z) w’ = lOOE,h3w([/2,0)
u =hp,
Po14
.
The relationships between the maximum central
transverse deflection w’ and I/h for the three cases
considered, are shown in Figs 7a, 8a, and 9a, respectively. The present FEM slightly underestimates the
values compared to the elasticity solution [25] for
lower values of I/h in cases 2 and 3 and in case 1 these
values match the elasticity results very well for all I/h.
The Classical Plate Theory (CPT) [25] predicts very
low values of w’ which are constant for all I/h and
accurate only for the thin beam (I/h > 50). The values
predicted by Manjunatha and Kant [19,20,21] are
below the present and the elasticity results for lower
I/h, and in cases 2 and 3 these underestimates are
considerable. The solutions obtained for cases 2 and
3 from the three-dimensional FEM [26] are very close
to the elasticity values. The FEM results obtained by
Spilker [27] for case 3 agree well with the elasticity
solutions. The present and other solutions converge
to the CPT values as Z/h becomes very large
(I/h > 50).
In the discussion to follow, Figs 7b-9e show the
results from the present model, the CPT and the
elasticity theory only. The observations from other
theories are based on the quoted references.
Figures 7b, 8b, 9b show the distribution of the in
plane stress ai through the thickness when I/h = 4.
The present model predicts values very close to the
elasticity results in cases 1 and 2 (see Figs 7b and 8b).
But these results differ from the elasticity solution for
case 3 in the outer layers near the interface. Notice
that the CPT predictions are nearly linear in each
layer and are close to the least squares fit of the
elasticity solution. The predictions made in Refs
[19-211 lie between the present and the CPT results.
The three-dimensional results of Liou and Sun [26]
7
l/h
Fig. 7. (a) Variation of w’ with I/h, simply supported beam (case I), sinusoidal load on top.
(b) Distribution of inplane stress, simply supported beam (case I), sinusoidal load on top,
I/h = 4. (c) Distribution of transverse shear stress, simply supported beam (case l), sinusoidal load on top,
I/h = 4.
Beam elements based on a higher order theory--I
785
(b)
(4
45
i
Fig. 8. (a) Variation of w’ with l/h, simply supported beam (case 2), sinusoidal load on
top. (b) Distribution of inplane stress, simply supported beam (case 2), sinusoidal load on top, l/h = 4.
(c) Distribution of transverse shear stress, simply supported beam (case 2), sinusoidal load on top,
I/h = 4. (d) Distribution of transverse normal stress, simply supported beam (case 2), sinusoidal load on
top, I/h = 4.
and the results from Spilker [27] are close to the
elasticity solution. The stresses obtained for case 3 by
Engblom and Ochoa [28] are very close to the CPT
results. The differences in the stress values from the
present and the elasticity solution in case 3 decrease
as I/h increases and for I/h = 10 they are close to each
other (see Table 4 which presents the values in the top
and the bottom layers).
The distribution oft.& through the beam thickness
is shown in Figs 7c, 8c, 9c for cases l-3, respectively,
when l/h = 4. zlz in the present study is evaluated at
x = 0.0077 (where the stresses are most accurate,
since it is the centroid of the first element of a graded
mesh from the end x = 0) instead of at x = 0.0; the
error introduced by this approximation is less than
1%. In case 1, the present results follow the elasticity
solution, and show a slight loss of symmetry through
the thickness. The CPT slightly overestimates the
maximum shear stress and follows a different path. In
case 2 the present FEM results are close to the
elasticity solution in both the layers; but the models
of Manjunatha and Kant [19-211 match the elasticity
results in the top layer and are closer to the CPT
results in the bottom layer. The CPT in this case
underestimates the stresses in the top layer and
overestimates the same in the bottom layer. In case
3 the present results are close to the elasticity solution; however, the present values are slightly more
than the elasticity results in the mid layer. Also, the
maximum stress points lie in the outer layers in the
elasticity results. The present and the Lo et al. [29]
values match each other very well. Liou and Sun
predictions also show the loss of symmetry, but are
different from the elasticity solution. The values
predicted by Engblom and Ochoa are closer to the
CPT. The values from Manjunatha and Kant are in
between the CPT and elasticity solutions in the outer
layers and are close to the CPT results in the middle
layer.
The transverse normal stress CT:variations through
the thickness for cases 2 and 3 are shown in Figs 8d
and 9d. The results from the present FEM are close
786
R. U. Vinayak et al.
(a)
(b)
(4
Fig. 9. (a) Variation of w’ with I/h, simply supported beam (case 3), sinusoidal load on top.
(b) Distribution of inplane stress, simply supported beam (case 3), sinusoidal load on top, I/h = 4.
(c) Distribution of transverse shear stress, simply supported beam (case 3), sinusoidal load on top,
I/h = 4. (d) Distribution of transverse normal stress, simply supported beam (case 3), sinusoidal load on
top, I/h = 4. (e) Variation of normalized inplane displacement, simply supported beam (case 3), sinusoidal
load on top, l/h = 4.
787
Beam elements based on a higher order theory-l
Table 4. Variation of o: in a simply supported symmetric
(O/90/0) beam, sinusoidal load on top, l/h = 10
r/h
-0.5
-0.4
-0.3
-0.2
-l/6
l/6
0.2
0.3
0.4
0.5
Elasticity
(Ref. [25])
- 72.820
- 52.923
-32.830
- 16.103
- 11.282
11.590
16.410
32.830
51.282
72.820
Present
-71.958
- 50.678
-34.013
-20.806
- 16.972
16.734
20.584
33.856
50.611
72.006
CPT
(Ref. [25])
- 83.080
- 50.556
- 38.032
-25.508
-21.333
21.333
25.508
38.032
50.556
63.080
to the elasticity solution, thus confirming our argument in Section 4 in the case of the layered composite
beams, i.e. there is no need to solve the second-
order differential equation, eqn (I&). The CPT
overestimates the values slightly and follows a different path in case 2. In case 3, the CPT values are
less than the present and the elasticity results in the
region -h/2 < z < 0 and they are higher than the
present and the elasticity solutions in the upper
region.
Finally, it is essential to study the validity of the
Kirchhoff hypothesis. Figure 9e shows the variation
of u’, the normalized in plane displacement at x = 0
for case 3 where the beam is laminated. The displacement patterns from the present results are close to the
elasticity solution in the top and bottom layers; but
in the middle layer the present results predict an
almost constant displacement. The CPT predicts a
linearly varying displacement field which again is
close to the least squares fit of the elasticity solution.
The linear displacement prediction by CPT within
each layer may be reasonable, but the deformed
configuration
of the original normal cannot be
described by the Kirchhoff hypothesis, especially
when the beam is thick (I/h = 4). It is observed that
the deformed normal gradually straightens with the
increasing I/h ratio in the present and the elasticity
results which get close to the CPT values.
7. CONCLUSIONS
The effectiveness and the accuracy of the finite
element beam model based on the higher order LCW
theory is studied closely.
The results show that the integration of the first
order differential eqn (18 b) is sufficient to evaluate the
transverse normal stress accurately and the method of
solving the second-order differential eqn (1%) is
unnecessary. The variations of bending moment ii;i,
are predicted in the least squares accurate sense along
the length of the beam element-the
quadratic BM3
element having a linear accuracy and the cubic BM4
element offering a quadratic accuracy. We also see
from this that the bending stresses 6,r through the
depth maintain the same cubic accuracy expected of
the LCW formulation [see eqn (2a)], and this matches
exactly the cubic variation predicted from Venkatraman and Patel’s solution-eqn
(19a) in a thin beam.
The stress predictions in thin and deep isotropic
cantilever beams are accurate at points sufficiently
away from the boundaries. Special care must be taken
in modeling the zone near a clamped boundary-see
Part II of this paper [23] for a detailed treatment. The
loss of symmetry/anti-symmetry
in the stress predictions is clearly depicted from the present FEM results
when the loading is on the top of the cantilever beam.
The Venkatraman and Pate1 solution gives inaccurate
results in the case of the deep beams. In the case of
the simply supported composite beams the present
stress and displacement predictions match the
elasticity solution well for the cases considered.
Acknowledgement-The
authors acknowledge the encouragement and support from Dr K. N. Raju, Director and Dr
B. R. Somashekhar, Head, Structures Division, N.A.L.,
Bangalore.
REFERENCES
1 E. Reissner, The effect of transverse shear deformation
on the bending of elastic plates. J. [email protected] Mech. 12,
A69-A77 (1945).
2. R. D. Mindlin, Influence of rotatory inertia and shear
deformation on flexural motions of isotropic elastic
plates. J. appl. Mech. 18, 31-38 (1951).
3. J. N. Reddy, A review of the literature on finite element
modelling of laminated composite plates. Shock Vibr.
Digest 17, 3-8 (1985).
4. R. K. Kapania and 8. Raciti, Recent advances
in analysis of laminated beams and plates, Part I:
shear effects and buckling. AIAA J. 27, 923-934
(1989).
5. R. K. Kapania and S. Raciti, Recent advances in
analysis of laminated beams and plates, Part II: vibrations and wave propagation. AIAA J. 27, 935-946
(1989).
6. K. H. Lo, R. M. Christensen and E. M. Wu, A
higher order theory for plate deformations, Part 1:
homogeneous plates. J. appl. Mech. 44, 663-668
(1977).
I. K. H. Lo, R. M. Christensen and E. M. Wu, A higher
order theory for plate deformations, Part 2: laminated
plates. J. a$pl. iech. 44, 669-616 (1977).
8. T. Kant. D. R. J. Owen and 0. C. Zienkiewicz. A
refined higher order Co plate bending element. Comput.
Struct. 15, 177-183 (1982).
9. T. Kant and B. N. Pandya, Finite element stress analysis
of unsymmetrically laminated composite plates based
on a refined higher order theory. In: Proc. Znt. Conf. on
Composite Maierials and Struc&res (Edited by K. A. V.
Pandalai and S. K. Malhotra). Madras. vv. 373-380.
6-9, January (1988).
”
10. M. Levinson, An accurate, simple theory of the statics
and dynamics of elastic plates. Mech. Res. Commun. 7,
343-350 (1980).
11. M. V. V. Murthy, An improved transverse shear deformation theory for laminated anisotropic plates, NASA
TP 1903 (1981).
12. J. N. Reddy, A simple higher order theory for laminated
composites. J. appl. Mech. 51, 742-745 (1984).
13. P. Rama Mohan, B. P. Naganarayana and G. Prathap,
Consistent and variationally correct finite elements for
higher order laminated plate theory. TM ST 9301,
National Aerospace Laboratories, Bangalore, India
(1993).
788
R. U. Vinayak er al.
14. B. P. Naganarayana, P. Rama Mohan and G. Prathap,
Quadrilateral Co laminated plate elements based on a
higher order transverse deformation theory (in press).
15. G. Prathap, The Finite Element Method in Structural
Mechanics. Kluwer Academic Press, Dordrecht (1993).
16. R. M. Jones, Mechanics of Composite Materials.
McGraw-Hill. New York (1975).
17 B. D. Agarawal and L. J. Broutman, Analysis and
Performance of Fiber Composites, 2nd Edn. Wiley, New
York (1990).
18. S. W. Tsai and H. T. Hahn, Introduction to Composite
Materials. Technomic Publishing (1980).
19. B. S. Manjunatha and T. Kant, Different numerical
techniques for the estimation of multiaxial stresses in
symmetric/unsymmetric
composites and sandwich
beams with refined theories. J. Reinforced Plast. Com-
Fig. Al. Composite lamina with fiber orientation angle.
Assuming the beam to deform in x-z plane, cyyv,ryi, and
~~~are zero. From eqn (Al),
pos. 12, 2-37 (1993).
20 T. Kant and B. S. Manjunatha, Refined theories for
composite and sandwich beams with Co finite elements.
Comput. Struct. 33, 7555164 (1989).
21 T. Kant and B. S. Manjunatha, Higher-order theories
for symmetric and unsymmetric fiber reinforced composite beams with Co finite elements. Fin. Elem. Anal.
Des. 6, 303-320 (1990).
22. B. Venkatraman and S. A. Patel, Structural Mechanics
with Introduction to Elasticity and Plasticity. McGrawHill, New York (1970).
23 G. Prathap, R. U. Vinayak and B. P. Naganarayana,
Beam elements based on a higher order theory-II.
Boundary layer sensitivity and stress oscillations
Cornput.-St&t.
58, 791-796 (1996).
24. G. Prathao and B. P. Naaanaravana. 3D-FEESTheoretical manual, PD ST 19005, N.A.L. Bangalore,
India (1990).
25. N. J. Pagano, Exact solution for composite laminates in
cylindrical bending. J. Comput. mater. 3, 398-410
r,[email protected],
%r = 0 * (P,,
(A3)
026 036
= 0.
(A4)
Equations (A2)-(A4) together with eqn (Al) yield
(1969).
26 Wen-Jinn Liou and C. T. Sun, A three-dimensional
hybrid stress isoparametric element for the analysis of
laminated composite plates. Comput. Struct. 25,
241-249 (1987).
37
A,.
R. L. Spilker, Hybrid stress eight node elements for thin
and thick multilayered laminated plates. Int. J. numer.
Meth. Engng 18, 801-828 (1982).
28. J. J. Engblom and 0. 0. Ochoa, Through the thickness
stress predictions for laminated plates of advanced
composite materials. Znt. J. numer. Meth. Engng 21,
175991776 (1985).
Substituting eqns (AS)-(A7)
stress-strain relation reduces to
into
eqn
(Al),
the
29. K. H. Lo, R. M. Christensen
solution determination
and E. M. Wu, Stress
for higher order plate theory.
Int. J. Solids Struct. 14, 655-662
(1978).
APPENDIX A
The stress-strain relation for an orthotropic lamina with
fiber orientation angle f) as shown in eqn (Al) is [18]
(A9)
(AlO)
1
a236
Q26
- a3 Lb)
@3=Q33+Q23
(&
Q
22
_Q’
64
)
26
(All)
(Al2)
789
Beam elements based on a higher order theory--I
APPENDIX B
Table Bl gives the consistent and the inconsistent
shape functions for linear (two-noded), quadratic (three-
noded), and cubic (four-noded) beam elements. The
derived
shape
functions
are
using
consistent
variationally correct method of field redistribution
[191.
Table BI. Shape functions for linear, quadratic and cubic beam elements
Original shape
functions, N,
Element
Linear (BM2)
N,=$
Quadratic (BM3)
N,= -;S,;
Cubic (BM4)
Nz=;
N2=S,S2
Consistent shape
functions, IV,
m, = N2 = ;
N =A_!.
fl=2
’ 6 2’
2 3
Ni=;S,
m =!+i
’ 6
N, = -;S,S3Se
lV,=;-;i-;s:
N2 =;S,S,S,
i+;i
N3=;S,S2S,
N4 = -i
S,S,S,
ts, = 1 - 6; s, = 1 + i; s, = 1- 35; s, = 1 + 31; s, = 1- 3p.
[is the natural coordinate of the element varying from - 1 to + I.
2
+;s,
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