The derivation is presented of the differential stiffness for triangular solid of revolution elements. The derivation takes into account the element rigid body rotation only, the rotation being about the circumferential axis. Internal pressurization of a pneumatic tire is used to illustrate the application of this feature

Chen, C. H. S.

English

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PRESSURE DEFORMATION OF TIRES USING DIFFERENTIAL STIFFNESS FOR
TRIANGULAR SOLID-OF-REVOLUTION ELEMENTS
By C.H.S. Chen
B.F. Goodrich Co.
Akron, Ohio
SUMMARY
This paper presents the derivation of the differential stiffness for
triangular solid of revolution elements. The derivation takes into account the
element rigid body rotation only, the rotation being about the circumferential
axis. Internal pressurization of a pneumatic tire is used to illustrate the
application of this feature.
INTRODUCTION
In the NASTRAN computer code, the rigid format No.4 is the so-called static
tanalysis with differential stiffness. This method of analysis is a first approx-
iimation to large deflection effects, The differential stiffness refers to the
geometry stiffness which is generated using the stresses in the element which
are computed from the initial linear stress analysis. The differential stiff-
ness is available for rods, beams, shear panels, plates, and conical shell ele-
ments. However, sometimes one desires that the differential stiffness be avail-
able to solid elements. This is particularly true when one deals with a
structure having thick wall yet flexible in nature. With this in mind, the
differential stiffness for a triangular solid-of-revolution element is developed
and incorporated into NASTRAN.
In the case of a ring element loaded axisymmetrically, there exists only
one rotation of the element which is the rotation about the circumferential axis.
We will assume that the stress and circumfeeential rotation are uniform through
Ithe element. Furthermore, in computing the work done by the forces in a_ n
!element we will neglect the work done in conjunction with the normal and shear
strains and the work done by the circumferential stress and strain. This is in
accord with the derivation of the differential stiffness for a triangular
membrane element presented in reference I.
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DERIVATIONOFDIFFERENTIALSTIFFNESSMATRIX
The triangular ring element in NASTRANassumesthe following displacement
field (Fig. I)
u = B1 + B2r + B3z
w = B4 + B5r + B6z
(l.a)
(l.b)
where u, w are the displacements in the direction of the r, z coordinates
respectively. Inverting the above yields
{B} : [r q]{q} (2)
where {q}T= mu I wI u2 w2 u3 w3]
{_}T: LB 1 _2 _3 B4 _5 _6 ]
Due to symmetry the only rotation of the element is the circumferential rotation
me which according to reference 1 is given as
Substituting B5 and B3 from equation (2) into equation (3) we obtain
{m} = [c]{q}
where {m}T= { _r mz mo }
-2-E o o o o
23 z23 r31 z31 r12 z12
and rij = r i - rj
(4)
zij = z i - zj
A = r12z13 - r13z12
Considering only the circumferential rigid body rotation and neglecting the
effects of the normal and shear strains, then the work done by forces in an
element of volume V reduces to
z72
W = - V r + ) (5)
_here _r and oz are the element normal stresses in the r and z directions,
"espectively. Hence the matrix of the differential stiffness coefficient for
:hat element is simply
FO 0 0 ][Kd] = V 0 0 0
L 0 0 _r+_z
(6)
We then have the differential stiffness matrix in terms of the generalized coor-
dinate {B}
[K_e] = [clT[K_][C] (7)
he above differential stiffness matrix is then transformed to grid point coor-inates by means of equation (2) as
[Kd] = [rBq]T[K_e][r_q] (8)
This geometric stiffness matrix is then added to the elastic stiffness matrix to
yield the total stiffness matrix of the structure.
APPLICATION TO PNEUMATIC TIRE INFLATION
We will apply the above formulation to the problem of inflating a pneumatic
tire. This is an example which can be considered as an axisjn_metric problem.
Fiq. 2 shows one-half of a typical meridian cross-section of a bias pneumatic
tire. Its structural elements consist of (1) rubber which spreads from the
tread area to shoulder area and to sidewall area and to the bead area, (2) casing
or carcass which is the main load carrying member, and (3) bead, which seats
ithe tire to the wheel rim. The carcass is a multi-layer composite which consists
!of a number of plies stacked together The ply is a unidirectional composite
made of many flexible high modulus cord of natural textile, synthetic polymer,
Jglass fiber embedded in and bonded to a matrix of low-modulus polymeric material
_such as rubber. The plies are usually stacking together in a pair of symmetric
bias cord angle with respect to the circumferential axis.
i
If we know the properties of the matrix and cord materials and their volume
ratios, then the properties of the composite ply in its principal directions,
directions l and 2 in Fig. 3, can be determined by the well-known Halpin-Tsai
equation, reference 2. The ply is assumed to be transversely isotropic, i.e.
Young's modulus E33 = E22 and Poisson's ratios Vl3=_12 , _^^=_23" The well-known
transformation matrix for elastic constants through a ro_ation is then applied
to obtain the ply stiffness matrix in the laminate coordinate system x, y. For
the present case, they are the meridional and circumferential directions. The
173
cord angle _ which is also the angle of rotation of the transformation is cal-
culated from the following equation known as "lift equation" in the tire
industry.
r__ _ ]_1___. = cos___..__B (9)
r c l + _ cos
where = cord angle in the body of tire which takes the molded shape
= cord angle in the initial cylinder before molding
r = radius coordinate of the location in the body of tire
rc= radius of the initial cylinder before molding
E : estimated cord strain
The total effective carcass stiffness matrix is then obtained by the method in I
reference 3. In general, the laminate can be satisfactorily represented by two- I
ply laminate of (+_) and (-_) cord orientations.
The material properties used in our example are
cord modulus Ec = 0.52376xi05 kg/cm 2 (7.45 x105 psi)
cord Poisson's ratio _c : 0.3
rubber modulus Er = 52.7278 kg/cm 2 (750 psi)
rubber Poisson's ratio _r = 0.48
Other data used are
8 = 57 degrees
rc= 21.4846 cm (8.4585 in.)
The cord strain c is assumed to be constant and its value at the crown is used.
The radius of the ply center line at the crown is 32.8308 cm (i2.9255 in.) and
the final cord angle at that point is assumed to be 32 degrees with reference
to circumferential circle. The tire is subjected to internal pressure of
1.6873 kg/cm 2 (24 psi). For simplicity, the tire is assumed to be fixed at a
section close to the bead. In all, 488 elements and 317 grid points are used
for one-half of the tire cross-section. Figure 4 shows the shape of a
deformed and undeformed tire section for the linear solution and for the
differential stiffness solution. The radial displacement of the outer ply line
using NASTRAN is 1.3785 cm (0.54277 in.) compared with 1.2217 cm (0.481 in.)
from a shell theory. The shell theory neglects the rubber in the tread,
shoulder, sidewall, and bead and only considers the carcass as a thin elastic
shell of orthotropic material. The increase in the tire outer radius due to
inflation is computed by NASTRAN to be 1.3419 cm (0.52829 in.) as compared with
1.0976 cm (0.448 in.) from measurement. The radial displacement of the same
point from NASTRAN linear solution is 1.7499 cm (0.68895 in.). The inflated
dimension of the outer carcass line in the z direction is 10.3045 cm (4.0569
174
.) from NASTRANand 9.9847 cm (3.931 in.) from shell theory. The section
_th of the tire(the maximumdimension of the tire outer profile in the z
rection) after inflation is 21.I16 cm (8.3134 in.) from NASTRANas compared
_h 20.7772 cm (8.18 in.) from the measurement. As can be seen, the inclusion
the differential stiffness results in a great improvementover the linear
lution. However, there are still discrepancies in the various results. The
screpancies can be attributed to factors such as material characterization,
sumptions on the boundary conditions, initial shape, cord angles, cord strain,
d the possibility of the residual stresses in the actual tire.
CONCLUDINGREMARKS
Wehave presented here the derivation of the differential stiffness matrix
triangular solid-of-revolution elements. Its application is illustrated
_ng the inflation of a bias tire as an example. The differential stiffness{a ring element is particularly useful when dealing with structure which has
Fckwall yet flexible enough to undergo large displacement and rotation.
example of the tire inflation demonstrates this point. The numerical
;ults in the present example are quite satisfactory.
As mentioned earlier, the derivatiun of the differential stiffness present-
in this paper is based on the information contained in the manual for Level
reference l, since it was available to this author at that time. This
_hor has learned since then that Level 15.5 (reference 4) has a refined
civation of the differential stiffness which includes the effects of the
_mal and shear strains in computing the energy. Hence, if a new derivation of
differential stiffness for a triangular ring element were madeto include
effects of the normal and shear strains and also to include the effect of
circumferential strain, it would probably give a better solution than the
_sent one.
REFERENCES
MacNeal, R.H.,ed.: The NASTRANTheoretical Manual (Level 12). NASASP-221,
1970.
Ashton, J.E., et al.: Primer on Composite Materials: Analysis. Technomic,
Stamford, Conn., 1969.
Chen, C.H.S., Tabaddor, F.: Effective Moduli of Layered Composites, J.
Spacecraft and Rocket, vol. II, no. 4, April 1974, pp. 262-264.
MacNeal, R.H.,ed.: The NASTRANTheoretical Manual (Level 15.5). NASASP-
221(01), 1972.
®
W2
U2
©
Figure i.- Triangular solid of revolution element.
Z
s_o_WALL--? SHOUL%_
CROW_
Figure 2.- Meridian cross section of a tire.
176
2Figure 3.- Cord-reinforced composite ply.
×
i??
UNDEFORMED
-//LINEAR SOLUTION
(a) Linear solution.
Figure 4.- Deformed and undeformed tire cross section.
178
UNDEFORMED
DIFFERENTIAL STIFFNESS
SOLUTION
(b) Differential stiffness.
Figure 4.- Concluded.
179


Pressure deformation of tires using differential stiffness for triangular solid-of-revolution elements

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