deflections of a ring due to normal loads using energy
TRANSCRIPT
Rochester Institute of TechnologyRIT Scholar Works
Theses Thesis/Dissertation Collections
1970
Deflections of a Ring Due to Normal Loads UsingEnergy Method and Stiffness Matrix MethodA.J. Rabinowitz
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Recommended CitationRabinowitz, A.J., "Deflections of a Ring Due to Normal Loads Using Energy Method and Stiffness Matrix Method" (1970). Thesis.Rochester Institute of Technology. Accessed from
DEFLECTIONS OF A RING DUE TO NORMAL LOADS USING ENERGY METHOD AND STIFFNESS MATRIX METHOD
Approved by:
by
A. J. Rabinowitz
A Thesis Submitted
in
Partial Fulfillment
of the
Requirements for the Degree of
MASTER OF SCIENCE
in
Mechanical Engineering
Prof. Name Illegible (Thesis Advisor)
Prof. William Halbleib Prof. Name Illegible Prof. Name Illegible
(Department Head)
DEPAR~ENT OF MECHANICAL ENGINEERING
COLLEGE OF APPLIED SCIENCE
ROCHESTER INSTITUTE OF TECHNOLOGY
ROCHESTER, NEW YORK
November , 1970
ABSTRACT
Deflections and reactions of a ring of1.855"
mean dia
meter, fixed at one point, simply supported at another point
and loaded normally at an arbitrary point were determined
using the energy method and the stiffness matrix method.
These results were verified experimentally and it was found
that the values of displacements and reactions obtained by
the two theoretical methods were within *4.5% and 0.57o res
pectively of experimental results. When effect of transverse
shear was included in the computations done using the energy
method it increased the values of deflections by less than
1.5% thus confirming that the effect of transverse shear is
negligible on deflections of a normally loaded ring. Finally,
the investigation established that either the energy method
or the stiffness matrix method, can be successfully used to
predict deflections and reactions of an arbitrarily normally
loaded ring.
TABLE OF CONTENTS
Page
List of Tables iv
Nomenclature v
I . Introduction 1
II. Literature Survey **
III . Statement of the Problem 5
IV. Theory 6
V. Energy Method 12
VI . Stiffness Matrix Method 23
VII. Application of the Two Methods to a Specific Ring.. 36
VIII. Experimental Verification *>U
IX. Discussion <. - **6
X. Conclusion 50
XI . Bibliography 51
XII . Appendix A 53
XIII .Appendix B 59
-in-
Table I
Table II
Table III
Table IV
Table V
Table VI
Table VII
LIST OF TABLES
Page
Deflection and Forces by Energy Method -
Transverse Shear Included 39
Deflection and Forces by Energy Method -
Transverse Shear Neglected 39
Deflection at Nodes - Stiffness Matrix
Method 41
Deflections - Experimental Method 45
Comparison of Deflections - Matrix Method
vs Experimental Method 46
Comparison of Results Obtained by Matrix,
Energy and Experimental Methods 47
Effects of Transverse Shear on Deflections. . 48
-iv-
J = polar moment of inertia
kj = stiffness coefficient, force in the i direction
due to a unit displacement in the j direction
k-f a = stiffness coefficient associated with constraintiJ
energy
s
k- = stiffness coefficient associated with strainiJ
energy
ficl = stiffness matrix, n x n matrix of stiffness
coefficients
1 = length of beam
Ml^2^3= moments
N,S,X,Y,Z, = locatiors on ringC,L
P,F^,F2,QS= normal loacb on ring
qi= generalized coordinates, displacements,i = 1, 2, . .n
fq\ - displacement vector, n xl column matrix of
Jgeneralized displacements
q^= displacement in the i direction of node a of
an element
q^= displacement in the i direction of node b of
an element
n x 1 column matrix of structural displacements,displacements at one end of an element with
respect to structural orthogonal axes
n x 1 column matrix of elemental displacements,displacements at one end of an element with
respect to an orthogonal set of axes applicable
to the particular element
Qi = generalized loads, loads in the i direction
-vi-
tt force vector, n x 1 column matrix of generalized
forces
a
Qj= force in the i direction of node a of an element
Q^= force in the i direction of node b of an element
to
n x 1 column matrix of forces and moments
applied at one end of an element with respect
to the coordinate system for the structure
n x 1 column matrix of forces and moments
applied at one end of an element with respect
to the coordinate system of the element
R = mean radius of curvature of circular beam or
ring
[Rs-e] rotation matrix which transforms a structural
system into the elemental coordinate system
s = length along curved beam
T1'T2T3= Torques
U =energy
Us= strain energy
Uc= constraint energy
UB= strain energy due to bending
Urp= strain energy due to torsion
Ups= strain energy due to transverse shear
u^= displacement in the i direction
Jut= displacement vector, n x 1 column matrix of
displacements
V;l,V2,V3=
vertical shearing forces
W = work
wl"w7=
simplifiying coefficients
-vii-
x,y,z = position variables in a three dimensional
rectangular coordinate reference system
*l~xll=
simplifying coefficients
GLj&j lft UJ - angular locations on ring
^iq = variable angular location on ring
Os , os= deflections at location S on ring
J - deflection
A. i = Lagrangian multiplier, reaction force in the
i direction
Q$ Qs = angular displacements, slope at location S on'
ring
-&$.= angular rotation: of beam due to bending moment
&& = angular rotation of beam due to twisting torque
T =shearing stress
IS = Poisson's ratio
^ =shearing strain energy per unit volume
-vm-
INTRODUCTION
A ring is universally defined as a curved bar or beam
whose cross-sectional area dimensions are small in comparison
with its radius of curvature.
Virtually every industry involved with the design of
mechanical components of structures is faced with the need
to analyze the stresses and strains in rings subjected to
various combinations of loads. In submarines, for example,
the hull is fabricated from a number of rings. These rings
must -be capable of withstanding radial forces due to shock
and water pressure. The rings also have to withstand normal
forces due to thermal expansions. Other examples of ring
applications are to be found in frameworks of aircraft and
missiles, bearings, radar, etc.
Further applications are found in the construction
industry where rings are used as supporting foundations for
reinforced concrete water tanks. Rings are also useful in
the design of high temperature piping such as used in the
engine room and reactor room of atomic submarines.
With such wide applications it becomes imperative to
know, in detail, the state of stress and strain in ring
structures.
Three methods can be used to determine displacements
of rings subjected to normal loads. One method employs
-1-
energy principles, starting with Castigliano's second theorem
(principle of least work) to find the external and internal
redundants, and then employs either Castigliano's first
theorem, or principle of virtual displacements to find the
displacements of the ring.
A second method employs stress, strain, and equilibrium
relationships to develop a differential equation. The
equation is then solved in conjunction with the appropriate
boundary cond it ions .
A third method, combines the principles and techniques
of both aforementioned methods into a matrix oriented approach
which is very useful in solving complex structures that are
difficult to handle-by exact analytical approaches. This
method, which is particularly adaptable to computer calcula
tions, is commonly referred to as the stiffness matrix
approach, or direct stiffness method, or finite element
method. It involves replacement of the continuous structure
by one composed of finite elements. The stiffness matrix
approach is relatively new; in fact it has only become
increasingly popular in the last 20 years during the advent
of the "computer age". Its popularity is particularly high
in the aviation and space industries where it is used to
analyze stresses and strains in numerous structures such as
wings and missile components.
-2-
It is the objective of this thesis to investigate the
deflections in a normally loaded ring by the classical energy
method and the modern stiffness matrix method and further,
to verify results obtained by actual experimentation.
-3-
LITERATURE SURVEY
Several authors have dealt with the problem of normally
loaded rings. Volterra, for example, used methods of harmonic
analysis to determine deflections of circular beams resting
on an elastic foundation and loaded normally. He obtained
his results in a closed form under the assumption that the
foundation reacts following the classical Winkler-Zimmerman
hypothesis. Rodriguez2analyzed normally loaded rings by
manipulating the classical equilibrium equations. Patelf
Kamel and Reddy, have analyzed normally loaded curved beams
but have not used their results to analyze rings specifically.
Levy used a combination of Castigliano's theorems and
classical techniques of solid body mechanics to derive dis
placement coefficients for circular rings loaded normally.
His analysis is limited to rings which are supported at
opposite ends of a diameter.
The above authors neglected deformations produced by
transverse shearing and/or limited their analysis to sym
metrical loading. The analysis presented herein, includes
effects of transverse shear, and deals with the use of two
approaches mentioned in the introduction in solvingsymmet-
rical as well as unsymmetrical loadings.
* Superscripts refer to the references listed in Bibliography.
-4-
STATEMENT OF THE PROBLEM
We seek the displacements at various points in a normally
loaded ring which is constrained at several points along its
circumference by fixed and/or simple supports. The points
of application of external forces are completely arbitrary -
We also wish to determine the effect of transverse shear on
the magnitude of displacements.
The results obtained by using an energy method will be
compared with those obtained by using a stiffness matrix
approach and with those obtained by experimental methods.
The theoretical basis for the solution of the problem
is given below.
-5-
THEORY
When a conservative structure is subjected to forces
which are applied gradually so that at any instant in the
loading history the structure is in static equilibrium, the
kinetic energy of the structure will be zero. All the work
W done by the forces acting through certain displacements
will be stored in the structure in the form of strain energy
U. This energy depends on the final configuration of the
structure and not on the load-path history, or the manner
7 8 9in which the final configuration is reached
',0'*'If the
force -displacement relationship for the structure under
consideration is linear, the work done by force Fi when it
causes a final displacement of Ui is given by
(*i (I2
W=\F dui
= V kiiUidui = h kiiui = h F^i (1)Jo Jo
where kii equals the stiffness coefficient of the structure,
or the force at location i due to a unit displacement |
at location i. Wh<=>n the struc+tire is ?<?td
on by a, number of forces Fi through a number of displacements,
the total strain energy can be represented by
U = k? Fi ut (2)i
which in matrix form is U = h VuV t< Ff, (3)
where f\x"l * is the transpose of -Tut.
-6-
Since $V] = [k]{u] =
ki;j Uj
U M
*fu]t
[KJ(U] = *i j
which can be written in the following form
kLL kL2 k13,
k u (4)
V = h J^i u2 u3..Ui..uJ
'li
k21 k22 k23
kln"|
k2i"k2n
k31 k32 k33--k3i--k3n,
kil ki2 ki3 kii kinr
knl kn2 kn3 kni kn'n
(4a)
If we take a partial derivative of the strain energy U with
respect to displacement u^, the only nonzero terms in the
multiplication of the right side of equation (4a) will be
terms containing ui. Also, ^since kij= k
au = :Qui
dv
aui
kij uj
or
= Fj
(5)
(6)
Equation (6) is a statement of Castigliano's first theorem.
It can also be expressed as follows:
Oil -
Qi (7)
3Qi
where qi are generalized coordinates and Q^ are generalized
forces. For an element which is loaded externally and which
is held in equilibrium by reaction forces, the U in the above
equation consists of strain energy plus contraint energy.
-7-
Equation (4) for strain energy can be expressed in
quadratic form as
s s s
us = % i j kij qi qj where kij= k
.^(8)
This form is positive definite since strain energy can assume
only positive values for any deflections, kij represents
the stiffness coefficient for the element, that is, the
generalized force at a point acting in the i direction due
to a unit displacement acting in the j direction, qi
represents generalized coordinates, that is, three transla-
tional and three rotational displacements from equilibrium
position for each node in the structure.
The constraint energy can be expressed through the
Lagrangian multiplier in a quadratic form as
uc= ^?fcij-a-iqj. <
The constraints satisfy the equation
j cij ^j= <9a>
which linearly involves generalized coordinates qi and re
presents a constraint condition. The Lagrangian multiplier
^i has the physical significance that it provides the
generalized forces of reaction in the i direction on a
constrained node. The coefficient Cij represents the
deflection at node i due to a deflection at node j.
-8-
The introduction of the constraint energy renders
equation (7) as an independent set. One realizes that if
the strain energy alone were used in equation (7), the par
tial derivatives would result in a dependent set of equations
because all of the qi would not be independent quantities.
A comparison of strain and constraint energy equations show
that they both have the same form if the Lagrangian multi
plier ^i is considered as additional unknown generalized
coordinate and which, therefore, can be called q^. Thus,
the total potential energy in the structure can be expressed
as
U =
Uc + Us
=
^Tj k?j *i <lj+ *
ij kij <U <*j=
%^-pkij qi qj do)
where kij now represents stiffness as well as constraint
coefficients, and the qi represent displacements as well as
reactions.
Differentiating equation (10) with respect to q^,
crtu
, the above equationReplacing kij by gr-: and kji by 4
becomes
-9-
oy,'
zi3
^^(M^K-U^
i K*<^
Or, restated,
or
(11)
(12)
;kiw-{o)In above, J^Kj is a matrix of stiffness coefficients for the
structure, j qv is a vector of generalized coordinates of any
point in the structure with respect to the structural co
ordinate system, andjQ\ is a vector of applied generalized
force components of which are three orthogonal forces and
three orthogonal moments applied to each point in the
structure or three mutually perpendicular rotations initially
applied to a constrained point with respect to the structural
coordinate system.
Equation (2) can also be written in matrix form as
U =
%{f} t\^\ (13)
and since Juj= [aJ\Fj
-10-
u = h
whereaij is the flexibility coefficient or the deflection
in the i direction due to a unit force in thejdirection.
If we take a partial derivative of the strain energy
as represented by equation (14), with respect to any force
Fi (with all Fi considered to be independent and withaij
=
aji) we obtain,
3U = J> aiiF-5 <15>
3Fi j3
ordu =
u (16)
aFi
which is an expression for Castigliano's second theorem.
We now proceed to apply the above general theory to
the case of a ring.
-11-
ENERGY METHOD
Consider a force P acting on a ring at angle ~ff and in
a direction normal to plane X-Y (see Figure 1).
i.OCI/5 Of roRCE p
Figure 1
The ring is fixed at point S and is simply supported at
point N. The angle X is a variable angle. Displacements
in the horizontal plane are considered to be small quantities
of a high order and are therefore neglected.
We wish to find the vertical displacement of the ring
at angle f when do lies between "2* and &.
Consider a free body diagram for the section to the
right side of fixed support S. The resultant beam is a
curved cantilever. We substitute a vertical force F^ for
the simple support. The force and the moment exerted by
-12-
the portion of the ring removed at S are represented by F2
and H2. The curved cantilever and applied forces are shown
in Figure 2.
VIEW A-A VIEH/ B-g Ytt* OC
Figure 2
From equation (16), the vertical deflection c*s at point S
equals
r c)us
<*s=0i7 <17)
where Fs= the vertical force acting at point S.
Us= the strain energy transferred to the beam.
= UB + Ut + UTS -(18)
Ug= strain energy due to bending (Figure 3) .
Vr= strain energy due to torsion (Figure 4).
Uts = strain energy due to transverse shear (Figure 5)
Strain energy due to bending is given by
-13-
Figure 3
U. r-
jmce 9ffc
2EX
(19)
Strain energy due to torsional shear is given by
Figure 4
Vr= **
s \wee 0^. =j^
(20)
-14-
Strain energy due to transverse shear is given by
vr4=pn
Flfel/RS C
- Tr dx cfm u/
From equation (21a) , the shearing strain energy per unit
volume can be expressed as
P =T3'
2G
Since shearing stress
7"
at any point y. when a vertical force
i3
(2/o)
V is acting is given by
r =
zzff- '") (21b)
the energy for the elemental volume b dy dx becomes
Att.-^C-f-**)^*
-sz.
^-1X1^^-*^^
'3.
-15-
(21c)
(21d)
(21e)
To obtain the deflection at point S due to force FLand
P acting alone (neglecting F2 and H2) it is first necessary
to assume an imaginary force Qs acting at S. The deflection
at S will then be equal to the partial derivative with
respect to Qs of the total energy in the beam (bending,
torsional and transverse shear) contributed by F^, P and the
imaginary force Qs. Thus, from equations (17), (18), (19),
(20) and (21e), the deflection at S due to force P and FL
. .,_ 10,11can be written
'as
J0 BX OC?s )fi EX OQ,4S
J^X DQS
Jo J& 0% Jfi T& QQs ]fZ&QQ (22)
where
M-l=
Qs R sintfo
M2=
Qg R sin 2& -
F-l R sin (3*D -B)
M3= Qs R sin 7& -
FL R sin (A-B) + PR sin (2-tf)
T, = Q R (1-cos ~#o )Is *
T2=
Qs R (1-cos TCo ) -
FL R (1-cos ( 7C0 -B) )
To = Q R (1-cos "tf ) -
F-, R (1-Cos ( < -B) ) +
3 s LPR (1-cos (A-D )
-16-
Vi = Qs
V2=
Qs-
FL
V3= Qs -
FL + P
Substituting Mj_, M2, M3, T]_, T2, T3, V]_, V2, V3 into equation
(22), replacing ds by R d"2fo and integrating, the resulting
deflection after setting Qs to zero is given by
Os
= "
Fl (Dl + D2 +D3 + D4) + p (D5 + D6 + D7> (23)
where
q. - J?L (-0CO4 0 -+^L*.0 + (Xoe0p-c**p/^cXc^CL-~J*<~p*<'ot\ C*-1*}2 EX V
D?.=-5^
(& c<x.^ cj>iz?^a~.u c*x.cx. _ <&j^ot^^? ~ifiotri? -*a^t?) ^"ZS1)
* -*%
Czi)
Pa = -^- /* -***
c*a2f*
*< of^ 2f- ^^ a. +o/codar1 ^{cm(<m/
0JT? I 2 ^
* -^1^-
C30)
Using the same procedure, it is found that the deflection
at S due to F^ and H2 acting alone is given by
Ss =
F2 (D8 + D9+ D10) "
H2 (DH +
D12? (31>
where
-17-
Dg=
-^ ( 4- *~ ol c+*.
cd\ (32)
fc-
-|L (g - Z^o~oc+ *-^*~~^ (33)
D,o = &$ to
0.2. - -(- C** *-^2<-
-*,) (36)
Furthermore, the slope at S due to F^ and P acting alone is
given by
^s =
Fl C13 +Diz ) + P (D15 +
D16 ) (37)
where
D/3- ELK / sfrv* J3 + oo*. ^jay^U. - oL m6~ & -sA6~ -]3si+ <* ooAeL
J (38:)
Alsoy the slope at S due to F2 and H2 acting alone is given
by
0S =
F2 (D17 + D18) -
H* (D19 + D20) (41)
where
-18-
0,7 in. &*)
D"* {^oL.i^fL^ &3)
P,| =iff
(* ***"*- c**-') (i'i)
Da - -Ar ^-^A^ (45)
We also know that
Substituting equations (23), (31), (37), (41) into equa
tions (46), (47), and using the equilibrium equation (48),
the constraints can be found to be
F = wl- p <w2) (W
W3 + Wi
F2= 1 -
FL (50)
Fi (W4). + P (W5) (51)
H2=
W-
where
-19-
Wi = Di + D2 + D3 + D^ (52)
W2=
D5 + D6 + D7 (53)
W3 = D8 + D9 + D10 (54)
W4 =
D13 + D14 (55)
W5 =
D15 + DL6 (56)
W6 = Di7 + DL8 (57)
W7 =
D19 + D20 (58)
Once the constraints F^, F2 and H2 are known, the deflection
at S is given by
Def = Xi + X2 + X3 -
Xz; . X5- X6 + X7 + X8 + X9 + Xiq
-
XL1
(59)
where ,
Xx= 55 / * M (***"
cx-cxo*) <**~>UJjt&J'oL (60)
X2=
o
ly" j Cv^T? C&<LUJ (oL 4A~. ot C&4.CC) _-^4-*/* TPc-tXi. uj^i^o*.
^i*iio 004. !? sia~P-oI -h s4*~. Tfs&u+uj (oL+s&OfCt CA*n.ot\ fr{\
Cos-2/*
c-cra.UJuj *4a~. uJ ca.cc/j + ^i, 3^to.<c/^*.A
UJ
-4-
^yt~* co cva. &* ^v^^u* ~-^a*A*z?sO*Aw (uj-^*44~.uj c&*uj}\
X o =-43^-*I *- *9^ ^-^c**&. * j>u* uj c&a. ol rcU~. oL
4- CAtttAj ( Gi--*- ^u3v. GC CASra^eti)-* *&<*,. UJ ^^.^oC^- uj vp2j
-y C~<rz. UJ ^&A*0. UJ ^OrA^-. UJ CaotA. UJ -t 6&^*v. UJ ^o&U/ ^t*-*^i*.u i<irO 4<^
a.
__ ^3~6~.UJ (s&au^Uj) j-20-
" L. .
,. xn (63)fe0* **? **+~> u* -+- &~J*f (UO + sU* UJ CaXX.(jS)\
X5 =
Y3&"i*'OL ~" ^ C4^6*^ -**^*<- + * c-c*lc ^v fee/ - 2. CajoJ* >OA~>eL
-* C01.J9 <^luj^ott. 4ti oC CJX.eC)u. ^i^ou c*x.-p ^OaUa^cc (j"</)
-+
-a^cvyS .4*/** UJ (OC ^i-*3- o< c-os. <X- +*&u%-]3 o**.<o *d***~o<.
C4^<4^fu(wf^v>tO G*ti.<o) r4J~ UJ Cjro.p*4*~?-UJ (i^O.)
-4*~ ,4C* UJ [OS -$>UUJ Ca^A-Uj) __ ^u. 13 Cd-_cO ^-c*,4<*-
2. ^X~.p cog. uj I
-+ Coq^c^ ou f OC f^U.OCtOtf.ot') -*.^c*, W&04 V^J*?~0<- (&$)
. -+ -***
"3*
-4^ <V (Ot .4^ C C**. oC *-^-* 7T c&UUJ^U^Oi
X8 = P^.. ,- 2.CO 2, Ctf3.60'4-C* 0 - 3L XH-~ C*> <r4X.tU - 3,U*lcJl4A*<A/
.4- t-a.
?*
C*X.UJ [lO + 4ju*C*JGA>4-Uj)4- ^aa^ UJ
C^X.^^ojL?'
UJ /)4-.^to*
*Z*
?*-**-> (
UJ*A*J*. UJ Cat*. UJ H- ^^m "3^ C_-cti UJ .?*--* *>
x =H^.K r_2Co^.OC- C->-.-w/ sOA-^OC *4aC~ uj (ct-sfruo(<uxet)., v
^Coq.*o - 00c to^a-**."3"*** Jd^*^ ** /c*>- ^<X. to c^rtxo)
\
21-
Xio - jg. (Fx~ F. H-Py*-,) (68)
^y^~ oljcoMjQ ^6^-tX. _j. ^n^^ty (<X--*&6.otcj~a.cc)
Co-dp C*raUj[ujsi*.OiJ C4XUJ) rl ^&a7*%J3C-4X*CaJ ^tAJ^UJ \4")
-*A^UJC^p^X~?*UJ-r4*6-*ja_^~sUj(oU+^6CAj<^O0Ov)
1
In the foregoing expressions UJ represents a specific value
of 4Q Next, we consider the theory for stiffness matrix
method .
-22-
STIFFNESS MATRIX METHOD
In order to solve for displacements using the general
theory stated in section IV with values of applied forces as
known quantities, it would be necessary to use a stiffness
matrix for the structure. Since it is very difficult to
obtain such a stiffness matrix, one prefers to obtain a
stiffness matrix for each element in the structure analyt
ically and to incorporate these stiffness matrices for the
elements into one overall matrix JKJ which can be used in
equation (12). Thus for each element, the following equation
can be written12,13
M {^ = (^ (70)
where jKeJ is a matrix of stiffness coefficients for an
element, -Tq^}is a vector of 6 deflections at each end of
an element with respect to an orthogonal set of axes appli
cable to the particular element, and fQ^ is a vector of 6
components consisting of three orthogonal forces and three
orthogonal moments applied at each end of an element with
respect to the coordinate system for the element.
The assumptions used in defining \KA are:
1) Deflections are small.
2) The physical properties of the elements
are uniform.
3) The elements are elastic.
-23-
Since equation (70) is in terms of the element coordi
nate system, its terms must be transformed into the struc
tural coordinate system in order to be consistent with
equation (12) . Using a rotation matrix which transforms
one coordinate system into another, the following equations
can be written:
[qe"J= [Rs-e][qs] (7i)
{Qe'i = [Rs-e][Qsl (72)
where[Rs_eJ
is a rotational matrix which transforms a
structural coordinate system into the element coordinate
system.
Substituting equations (71) and (72) in equation (70)
will give:
[Ke] [Rs-3 fcs}= [Rs-e]faJ\ (73)
Multiplying both sides of equation (73) by the inverse of
|_Rs-eJ whih also happens to be the transpose ofjRs_e\
will
give:
[Rs-e]*
[Ke][*.-elki = [*s-e]*
Mto
or W(qi ={Qi (7<o
where [Kg] = [Rg.e]t
[KJ [rs_^
-24-
[ksj is a matrix of either stiffness or constraint cooeff icients
for an element and will be referred to as an element stiff
ness matrix.
Since equation (74) now contains terms with respect to
the structural coordinate system, it can be substituted into
equation (12), resulting in a set of simultaneous linear
equations. The[K]matrix of equation (12) contains thefKs"l
matrices for all the elements and is called the structural
matrix.
Solution of equation *( 12) for the displacements fq^
can be obtained by employing any convenient method like
matrix inversion, Gauss Jordon method etc. The solution
may be formally written as
H- H
"l
M C75)
After the unknown displacements are obtained, the end
reactions with respect to the structural coordinate system
of each element can be determined by substituting the
appropriate displacements into equation (74).
The above theory is applicable to any type of structure,
i.e., it is general in nature. For a specific structure, one
first has to decide on the shape of elements into which the
structure will be broken up. In case of a ring, it can be
readily broken up into small curved beams or into small
straight beams. Curved elements would more closely idealize
-25-
the structure than the straight elements. However, the
stiffness matrix of a curved beam is considerably more com
plex than that of a rectangular beam. With sufficiently
large number of finite elements, a ring made up of straight
rectangular beams should closely approach the actual con
figuration. Hence for the specific problem discussed in this
thesis it was decided to break up the ring into straight
rectangular beams.
-26-
STIFFNESS MATRIX FOR RECTANGULAR BEAMS
Let us consider the rectangular beam shown in Figure 6,
Y4
NODfctfODE b
Figure 6
Nodes a and b are located at the cross-sectional centroids
of the left and right faces of the beam respectively. Dis
placements of node a and node b in relation to a reference
a bset of orthogonal axes are represented by q^
andq^
where i
varies from 1 to 6 .
qi qf and q| refer to the deflections of node a in
3. 3. A
the x, y, and z directions respectively, while q^, q5 and q6
refer to the slopes or angular rotations of node a about the
x, y and z axes respectively. Similarly, q^, q2, q3 represent
the deflections of node b in the x, y and z directions res-
-27-
pectively; and q^, q5 and q,- represent angular rotations of
node b about the x, y and z axes respectively-
Loads are represented by Qi where i varies from 1 to 6.
AAA
Loads Qi, Q2, Q3, for example, represent forces acting on
node a in the x, y and z directions respectively; and loads
Q Q Q
Q4 Q5 Q6 represent moments acting at node a about the x,
y and z axes, respectively.
To develop the stiffness matrix for the beam, let us
first consider axial loads acting on nodes a and b. The
force displacement relationships are:
Ql = AE (qi -
qj) (76)L
<?4=
kG"
(q-
q?) (77)
Ql =
_AE_ (qf-
q) (78)j
<?4=
-JSG_ (q?-
q?) (79)L
b a
(qi -
qi) represents the axial displacement of node b
in relation to node a, or compressive strain of the beam.
b a
(qi* -
q/+) represents the angular rotation of node b in
relation to node a, or torsional twist of the beam about its
longitudinal axis.
(qf "
qi) represents the axial displacement of node a
in relation to node b, or the tensile strain of the beam.
a b
(qz* - qzj) represents the angular rotation of node a in
relation to node b, or torsional twist of the beam about its
-28-
longitudinal axis.
Now application of loads at node b such that the beam
is displaced in the X-Y plane (Figure 7) leads to the follow
ing force-displacement relationships:
NOOE&
Figure 7
>ql3
****** L%*+
m AO
qJlx
ZZ(80)
where
b
q2 is the total vertical or Y deflection of node b in
relation to the structural reference coordinate system.
-29-
q2 is the initial vertical deflection of node b, which
would be equal to the vertical deflection of node a.
Lqg is the initial vertical deflection of node b con
tributed by the slope at node a. It is a product
of the length of the beam and the slope at node a.
Q2L /3EI is the vertical deflection of node b due to bending
of . the beam. That is, the vertical deflection at
node b due to a vertical force Q2 applied at node
b while node a is fixed.
QgL /2EI is the vertical "deflection of node b due to a moment
Qg applied at node b while node a is fixed.
Q2FL/AG is the vertical deflection of the beam due to trans
verse shear, which will be discussed later.
Furthermore
tf = + <?^ +^- (81)
where
q^ is the total slope at node b after application of
loads Q2 and Q*?.
q^ is the initial slope at node b which is equal to
the slope at node a.
b 2Q2L /2EI is the slope, at node b due to application of load
Q2 while node a is fixed.
QgL/EI is the slope at node b due to application of load
b
Q6 while node a is fixed.
-30-
Equations (80) and (81) may be written in the following matrix
forms :
4
att
OV
-6S2i
L
or
-6Sg
L
JlSxx
4
i-C-'C
?f-
(82)
J0^-!*Szi2L _ 6Szi
tf-
/aSz.fti 6Szgf
,4 *
L L
(83)
(84)
In the same manner as above, by considering the same
configuration but with loads on node a we find that:
GSziV
o
Q*
= 6S2ii%t 4. U Szxtt - i^jii.4.
2Szsf*
(85)
(86)
For analyzing deflection in the x-z plane, we can sub
stitute the following changes in the nomenclature of equations
(83), (84), (85), and (86):
-31-
# -#=* &-*? q;-c
which essentially implies replacing y by z and z by -y-
From efc. (83^
Q^l-^gtf + 6Stf'# -
. J2^2ii ^ jSaLgll, (87)
From e&. (w)
From eo, (s\
From c^(6^
Equations (76), (77), (78), (79), (83), (84), (85),
(86), (87), (88), (89), (90), have twelve unknown displace-
a a a a a a b b b b b bments: qf, q2 , q3, q*+, qs , qg> qi q2 > q3 34' 35' 36*
These equations can be conveniently expressed in the follow
ing matrix form where the coeff icent matrix becomes the
stiffness matrix that we are looking for:
-32-
St -*
itSzi
J*
6Szi -IZSzi SSzi
L
iZSiji -6S9.
L
-'2Sjj
L8-
L
Sts-its
^ 9ZL 2S^
65zi
L 4SZI-bSzi
L2SZ3
-St-
st
->2Sz)L8- L.
i>Sz*L4
-6Szj
L
65tf, 6Sa
L
-$M Sts
-6Syi
L2S33
6S31</S9jL
Szi~T"" 2SZ3
-6Sz
L**Sz3l
a
31
323
a
33
a
34
a
35
a
"16
4
b
32
5
b
35
<&
~a
Qi
'2
ta
<tf
qS
l n
Q?
q5
"S
<?
Q^
Q6b
In above,
st- - ae; s =ec
Lts
V(z)J_ EI(zz) C^; Ci = 1
L r^syCz)
c v _3EI , s fy(z)
Ssy(z)=
yy(zz)
Co =L+s
t ^ n1"2S
, n
sy(z); C3=
sy(z)
AGL 2
Figure 8
-33-
The above matrix equation is in effect equation (70)
and it pertains to one element in the structure being analyzed,
Each element in the structure will have a similar matrix
equation. All of these equations can be converted into forms
represented by equation (74) which in turn can be consolidated
into one giant matrix equation represented by equation (12).
The resultant stiffness matrix [ksj of equation (74), for
example, can be considered as one element of the general
stiffness matrix [k] in equation (12) ; likewise, the displace
ment matrix and applied force matrix of equation (74) for one
element can be considered as one element each of the general
displacement matrix and applied force matrix of equation (12)
respectively.
Thus, in a structure which is broken into n number of
beams, the stiffness matrix of the general equation (12)
expressing relationships of forces and displacement would be
an n x n matrix with each element in this matrix being a
12 x 12 matrix. The displacement matrix and force matrix
of equation (12) would be an n x 1 matrix and n x 1 matrix
respectively, with each element in these matrices being a
12 x 1 matrix.
To obtain displacements, equation (12) can be manip
ulated to provide:
ft} M-1
ftlEquation (91) can be solved using any of several methods
-34-
such as the Gauss Jordon method to give displacements at any
node in the structure.
-35-
APPLICATION OF THE TWO METHODS TO A SPECIFIC RING
We now proceed to apply the two methods of obtaining
displacements viz. the Energy Method and Stiffness Matrix
Method to a specific numerical example involving ring
structure.
Consider a ring which is fixed at point S (Figure 9) ,
simply supported at point L, normally loaded at point C.
The normal load P equals 1 lb. The physical properties and
dimensions of the ring are taken as follows:
D0 = Outer Diameter = 1.945 in.
D^= Inner Diameter = 1.765 in.
R = Mean Radius =.927 in.
6 2E = Modulus of Elasticity = 30 x 10 lbs/in
b = Thickness =.090 in.
n = No. of Nodes or Elements Into Which Structure
is Broken = 36
*xx= Moment of Inertia About the Element's x-x
-6 4.Axis
= 5.47 x 10 in.
yy= Moment of Inertia About the Element's y-y
-6 4Axis = 5.47 x 10 in.
Izz= Polar Moment of Inertia of Element =
10.94 x10'& in.4
f = Lateral Shear Shape Factor = 1.177
e = Torsional Shear Shape Factor = 22 x 10
-36-
A ring such as the one described above has been used as
a supporting structure for a sefaty and arming device in proximity
fuzes used in anti-aircraft projectiles, where deflections of
supporting rings are critical.
Figure 9
The lateral shear shape factor f is a dimensionless
quantity, dependent on the shape of the cross-section, which
is introduced to account for the fact that the strain, as
determined by dividing the average shear stress on a cross -
section by its shear modulus, is not uniformly distributed
over the cross -section,.
The lateral shear shape factor can be calculated from
the formula f =^-2 + 11V
wherel/ = the Poisson's ratio.
10 + (i+y)
This formula was derived by Cowper because a similar shape
coefficient derived by Timoshenko has led to unsatisfactory
-37-
results. Cowper's results agree fairly closely with other
authors who have also questionedTimoshenko'
s shear factor
and derived their own. Cowper derived his formula by inte
grating the equations of three dimensional elasticity theory.
The torsional shear shape factor e is a factor in the
well known equation for angle of twist [eq. (20)J.
eTl
eG
where for circular cross-section, e is the polar moment of
inertia J. For other cross-sectional areas, this factor will
change. For rectangular cross-sections, for example:
4ab3
16 - 3.36b
=.1406a4, if a =b
I2a*]Deflection at point X on the ring shown in Figure 9 was
obtained through the energy method by substituting the above
constants into equation (22) and solving for <j^.. To facil
itate calculation, the problem was programmed and run on an
1800 MPX operating system. The language used was Basic
Fortran. Running time for the program was nineteen seconds.
A copy of the program is contained in the appendix.
Deflection and forces found by this method, are pre
sented in Table I:
-38-
Table I
Deflection and Forces
(Energy Method - transverse shear included)
Def. (in.) at X Fi (lb.) F? (lb.) H2 (in. -lb.)
.0048279 .76019 .23980 .49055
The results when the terms that contribute to transverse
shear are removed from equation (29) are given in Table II.
TableII"
Deflection and Forces
(Energy Method - transverse shear neglected)
Def. (in.) at X Ft (lb.) F? (lb.) H9 (in. -lb.)
.0048202 .76008 .23991 .49047
To obtain a deflection at a similar point by means of the
stiffness matrix method, the mean circumference of the ring
is divided into 36 equal straight beams and the nodal points
are designated as shown in Figure 10.
-39-
Figure 10
The origin of the coordinate system for the ring is
established at node 36 with the x-y plane being coplaner
with the plane of the ring.
The (x,y) coordinates ofjth
node in relation to the
ring structure coordinate system may be expressed as:
with j varying from 1 to 36.
-40-
Node 36 is considered as a ground node. All displace
ments at this node are therefore zero. At node 13 which is
simply supported only^ the deflection in the Z direction is
zero. A force of one pound is applied at node 22.
To solve for deflections by the stiffness matrix ap
proach, an existing computer program called the Strap 3
program is used. This program, which was developed at
Eastman Kodak Company, is based on the mathematical form
ulation discussed earlier- The program consists of many
subroutines, or modules, linked together by a monitoring
system. These modules are written in Fortran IV for an IBM
360 computer. Deflections of the ring (Figure 4) obtained
using the above program are shown in the following table:
i
Table III
Deflections at Nodes - Stiffness Matrix Method
(Due to 1 lb. Force at Node 22)
Node Number Vertical Deflections (in.)
1 -.0000385
2 -.0001496
3 -.0003218
z+ -.0005348
5 -.0007661
6 -.0009907
7 -.0011720
41-
Node Number Vertical Deflections (in.)
8 -.0012850
9-.0012999
10-.0011942
H-.0009483
12-.0005536
13.0000000
14.0006903
15.0014904
16.0023595
17.0032430
18.0040942
19.0048545
20.0054827
21.0059401
22 .0061826
23.0061993
24.0060071
25.0056154
26.0050737
27.0044175
28.0036939
29.0036939
30 .0022336
-42
Node Number Vertical Deflections (in.)
31 .0015607
32 .0009930
33 .0005485
34 .0002342
35 .0000548
36 .0000000
The constraints were found to be:
Fi (lb.) F2(lb.)"
H2 (in. -lb.)
.760989 .239010 .492000
Running time for the above program was 11 seconds.
Excerpts of the program are contained in the appendix. The
entire program, which comprises over 150. pages of subprograms
is available at Eastman Kodak Company, Rochester, New York.
-43-
EXPERIMENTAL VERIFICATION
A test was conducted to verify deflections due to normal
loads on the ring analyzed by the two methods . Normal loads
were applied by a Chatillon push-pull meter, model DPP-30 at
3 locations. The experimental set-up was as shown in Figure
11.
HEAVY
OUTY VISE
SIMPLE- -SUPPORT
APPL\*0 FORCE
Figure 11
The fixed point of the ring was held rigidly by a heavy
duty vise. Points X, Y and Z and the simple support point
were located by a toolmaker's microscope (Gaertner) . The
simple support consisted of a diameter pin.
Deflections were read at points X, Y and Z upon appli
cation of a force at point C, using a Starrett Number 711
-44-
dial indicator.
Results of the experimental test were as follows:
Table IV
Deflections - Experimental Method
Experimented
Run Number
1
2
3
4
5
Average"
Normal Deflection/lb. (in. /lb.)Pt,
.0046
Pt. Y
.0013
Pt. Z
.0040 .0010 -.0010
.0042 .0010 -.0012
.0046 .0013 -.0011
.0050 .0015 -.0012
.0050 .0016 -.0014
-.0012
-45-
DISCUSSION
Deflections and Reactions
The average deflection at Pt. X, (Figure 11) which is
identical in location to node 27 in the finite matrix scheme,
was found experimentally to be .0046". Pt . Y in the experi
mental test was similar in location to a position between
node 31 and node 32 in the stiffness matrix analysis. And
Pt. Z in the experimental test was similar in location to
node 9 in the stiffness matrix analysis. A comparison of
the results for both approaches is shown in the following
table:
Table V
Comparison of Deflections - Matrix Method vs Experimental Method
Matrix Analysis Experimental Test*
Node 9 -.0012999 -.0012
Node 27 .0044175 .0046
Node 31^ .0012768 .0013
* Average value of 5 trials.
The deflections obtained by the stiffness matrix approach i
and the experimental method are thus seen to be within 4%.
The deflection and constraints found at location X on
a ring (Figure 11) by the three different methods discussed
in this thesis were as follows:
-46-
Table VI
Comparison of Results Obtained by
Matrix, Energy & Experimental Methods
Method Def. (in.) Fi (lb.) F2 (lb.) H2 (in-lb)
Experimental .0046
Stiffness Matrix .0044 .760989 .239010 .492000
Energy .0048 .76019 .23980 .49055
The above table indicates that the deflections obtained
by the energy method and the stiffness matrix method are within
4^% of the results obtained from the experimental method while
those obtained by the two theoretical methods are within 9%
of each other. Such close agreement of the results justifies
the idealization of the problem. It is of interest to note
that the energy method tends to give higher values of deflec
tion while the stiffness matrix method gives values lower thn
those obtained by the experimental method. Some variation
in results is to be expected due to lack of precise values of
the physical properties of the ring such as Young's modulus
of elasticity, Poisson's ratio etc. It is also expected
that stiffness matrix method should yield values lower than
those obtained by the energy method due to the fact that the
former method uses a foreshortened circumference resulting
from a ring made up of straight beams instead of curved beams.
The deviation in the values of deflection obtained by the
two theoretical methods can be reduced by either, using
-47-
curved beams in the stiffness matrix method while keeping the
number of nodes fixed or by employing greater number of nodes
say 100, while retaining the straight beam approximation.
The values of reactions and moments acting on the ring
obtained by the two analytical methods are in excellent
agreement, with deviation of less than one-half precent, whichi
can be attributed to the round-off and the truncation errors
in computation.
Effects of Transverse Shear
. The following table contains a comparison of the
deflections and constraints at point X as determined by the
energy method when transverse shear is included and when
transverse shear is neglected.
Table VII
Effects of Transverse Shear of Deflection (Energy Method)
Def. (in.) Fi (lb.) F2 (lb.) H2 (in. -lb.)
With Shear .0048279 .76019 .23980 .49055
Without .0048202 .76008 .23991 .49047
Shear
The above table clearly shows that the effects of trans
verse shear on deflection are less than one and a half per
cent and therefore indeed negligible. This agrees with the
observation by Seely and Smith that except for relatively
short, deep beams, the deflection caused by transverse
shear may be neglected without introducing serious error.
-U8t
Since the effects of transverse shear are so negligible
incorporation of terms accounting for it into the stiffness
matrix program becomes merely a matter of academic interest.
Comparison of the Energy Method and the Stiffness Matrix Method
Equations in chapter V for the energy method were
formulated for a specific loading configuration. Any change
in the loading scheme would necessitate reformulating the
governing equations. Also, the equations in the thesis were
formulated to predict deflection at a specific point on the
ring^ If one is interested in predicting deflection at any
arbitrary point on the ring a formidable amount of work
would indeed be required. On the other hand, equations for
the stiffness matrix method are fewer and less involved and
yet permit incorporation of any combination of loading scheme.
It has also the advantage of predicting deflections at all
the nodes almost simultaneously. It does have the disadvan
tage that it requires solution of 12 x n number of equations
where n is the number of nodes chosen, thus requiring signif-i
icant amount of computer time, especially if one needs high
accuracy.
-49-
CONCLUSION
The results of this investigation can be summerized as
follows :
(a) The energy method and the stiffness matrix method as
applied to a normally loaded ring give values of
deflections which are within 4%% of those obtained by
the experimental method.
(b) The values of reactions and moments obtained by the
two theoretical methods are within 0.5%.
(c) -Effects of transverse shear on deflection of a normally
loaded ring is negligible, less than 1^%.
(d) The energy method is suitable when computer time is at
premium and one is interested in a specific loading
configuration and in values of deflections and reactions
at a specific point on the ring.
(e) The stiffness matrix method is a versatile one capable
of incorporating various kinds of loading conditions.
It never requires expensive computer time.
-50-
BIBLIOGRAPHY
1. Enrico Volterra
2. D. A. Rodrigues
3. P. D. Patel
"Deflections of Circular Beams
Resting on Elastic Foundations
Obtained by Methods of Harmonic
Analysis". Journal of Applied
Mechanics . Vol. 19, March, 1962
"Three -Dimensional Bending of a
Ring on an Elastic Foundation".
Journal of Applied Mechanics.
Vol. 28, September, 1961.
"Analysis of Continuous Bent
Beams Loaded Out of Plane by the
Six Moment Equations". Thesis
(Ph.D. ) , Oklahoma State University,
1962.
4. M. E. Kamel
5. M. N. Reddy
6. R. Levy
7. C. Lanczos
"Influence Areas for Continuous
Beams in Space". Thesis (Ph.D) ,Oklahoma State 1963.
"Analysis of Laterally Loaded
Continuous Curved Beams".
Journal of the Structural Division,
Proceedings of the A.S.C.E.
February , 19S7.
"Displacements of Circular Rings
with Normal Loads". Journal of the
Structural Division, Proceedings of
the A.S.C.E. , February, 1962.
"The Variational Principles of
Mechanics". The University of
Toronto Press, Toronto, Ontario.
1964.
8. J. Gere and
W. Weaver
"Analysis of FramedStructures"
D. Van Nostrand Co. , Inc. Princeton,
N. T. T5F3
Moshe F. Rubinstein "Matrix Computer Analysis of
Structures", Prentice -Hall , Inc.
Englewood Cliffs, N. J.T966T"
-51-
10. S. Timoshenko
11. Seely and Smith
12. J. Ashman
13. Y. C. Fung
14. G. R. Cowper
15. John S. Archer
16. S. Timoshenko and
J. N. Goodier
17. Enrico Volterra
18. Francis B.
Hildebrand
19. R. J. Melosh
20. Eric Ressoner
Strength of Materials. Part I,3rd Edition. D. Van Norstrand
Company, 1955.
Advanced Mechanics of Materials.
John Wiley & Sons., Inc. N. Y.
N. Y. 1966.
Structural Analysis Users Manual
1969.
Foundations of Solid Mechanics.
Prentice-Hall, Inc. Englewood
Cliffs, N. J. 1965
"The Shear Coefficient in
Timoshenko 's Beam Theory".
Journal of Applied Mechanics,Vol. 33. 15ZG~.
"Consistent Matrix Formulations
for Structural Analysis UsingFinite-Element Techniques",
Vol. 3, October, 1965.
AIAAJ
Theory of Elasticity ,McGraw
Hill Book Co., Inc., New York,New York. 1951
"Bending of a Circular Beam
Resting of an Elastic Foundation",
Journal of Applied Mechanics,
March, 1952.
"Methods of Applied Mathematics".
Prentice-Hall, Inc. , Englewood
Cliffs, N. J. 1^5.
"Basis for Derivation of Matrices
for the Direct Stiffness Method".
AJAA Vol. 1, July, 1963.
"The Effect of Transverse Shear
Deformation on the Bending of
Elastic Plates". Journal of
Applied Mechanics . June, T545.
-52-
COMPUTER PBOGMM-ENFffftY METHOD
FOR CC673
0CS(CARD,1443 PRINTER)
1ST ALL
DEFLECTIONS OF NORMALLY LOADED RINGS
1 A=6.283
EL=3.840
B=2.268
W=4.712
P=l.
R=.927
E=30.E+06
AIX=5.47E-06
AJ=10.94E-06
G=11.5E+06
H=.090
C1=(P*R**3)/(2.*E*AIX)
Dl =Cl*(COS(EL)*COS( B)*( A-EL-( SI N ( A ) *COS ( A ) ) + ( COS ( EL ) *S I N ( EL ) ) ) )
1 + C 1* ( S I N ( E L ) *SI N ( B ) * ( A-E L ) -S I N ( B ) *S I N ( A ) * ( S I N ( A ) *COS ( EL ) -COS ( A )
2*S IN ( EL ) ) -S I N ( E L ) *COS( B ) *S I N ( A ) *S\E N ( A ) +S IN ( EL ) *COS ( B ) *S IN ( EL ) *"
3SIMIEL ) )
C2=(P*R**3) /( AJ*G)
D2=C2*( A-EL+SINI B ) *COS ( A ) -COS (B )"*S I N ( A ) + S I N ( EL ) -COS ( A ) -COS ( EL ) *
IS UM ( A ) + ( A/ 2 . ) * S I N ( B ) *S I N ( EL ) + ( A/ 2 . ) *COS ( B ) *COS ( EL ) + S I N ( A ) *S I N ( A ) / (
2 2.0 )*( SIN(B)*COS( EL)+COS( B)*SIN( EL) ) +{ SIN { A )*COS ( A ) ) /2. * ( COS { B ) *
3C0S(EL)-SIN(B)*SIN( EL) )-( EL/2.*SIN( B)*SIN(EL) ) -SI N ( B ) *COS ( EL ) +
"4C0S(B)*SIN(EL)-C0S( B ) *SI N ( EL ) /2.-EL*C0S ( B ) *COS (.EL ) /2 . )
D3 = { R**3/ ( 2 . #E *AIX ) ) * ( A+S I M ( A ) *COS ( A ) *S I N ( B ) * S I N ( B ) -S I N ( A ) *COS ( A ) *
1C0S ( B ) *C0 S ( B ) -S I N ( B ) * S I N ( B ) *S I N ( B ) *COS ( B ) - B+ S I N ( B ) *C0 S ( B ) * ( CO S ( B ) *
2C0S(B)-2.*SIN( A)*SIN(A) )+2.*(SlN( B) )**3*C0S(B) )
C6=R**3/(2.*AJ*G)
D5 =C6# ( 2 . * A+ ( CO S ( B ) ) *# 2* ( A+S I N( A ) *COS ( A ) ) + ( S I N ( B ) ) **2* ( A-S I M ( A ) *
lCOS(A) )+2.*C0S( B ) *SI N ( B) *( SI N( A ) ) **2-4.*C0S ( B )*SIN ( A ) +4.*SIN ( B ) *
2C0S( A)-2.*B-(C0S(B) )**2*( B+S I N ( B ) *COS ( B) )-( SIN(B) ) *#2* ( B-SI N ( B ) *
3C0S(B) )-2.*C0S( B)*( SIN(B) ) **3+4. *COS ( B ) *SIN ( B )-4.*S IN ( B ) *COS ( B ) )
C3=(P*R*H**2*( A-EL) ) /( 10.*G*AIX ).
C4= (r*h**2*( A-B) ) /( 10.*G*AIX)
D8 = ( R**3 ) / ( 2 . *E*A I X ) * ( S I N ( B ) -B*COS ( B ) +A*COS ( B ) -COS ( B ) *S I N ( A ) *COS ( A
1)
1-SIN(B)*SIN(A)*SIN( A) )
D9= ( R *'*3-) / ( 2 . *E* A I X ) * ( A* COS ( EL ) -COS ( E L ) *S I N ( A ) *COS ( A ) -S I N ( A ) *S I N
1(A)*
IS IN (EL )-EL*COS( EL) + SIN(ED)
D10=(Rl!*3) /( AJ*G)*( SIN( B) -B-(#COS ( B ) ) /2.-
1( SIM(B)*COS( B)*C0S(B))/2.-(SIN(B)*SIN(B)*SIN(B))/2.+SIN( B)*COS(B)
2-C0S( B)*SIN(B)+A-SIN(A)*COS( B ) +COS ( A ) *S I N ( B ) -S I N ( A ) +( A*COS ( B) )/2.
3+(SIN(A)*COS(A)*COS(B))/2. + (SIN(A)*SIIM(A)*SIN(B))/2.)
D11=(R**3) /(2.*AJ#G)*( 2.*A-2.#SIN( A)*COS(EL)+C0S( A) *SIN ( EL )*2.-2.
1* SI N ( A ) +A*COS ( EL ) +SI N ( A ) *COS ( A ) *COS ( EL ) + ( S IN ( A ) ) *#2*S I N ( EL )-2.
2*EL+2.*S IN (EL) -EL*CO S ( EL ) -SI N ( EL ) * ( COS ( EL ) ) **2- ( S IN ( EL ) )**3 )""'
D12= (H#*2*R*(EL-B) ) /( 10.*G*AIXJ
013= (H**2*R*( A-EL) ) /( 10.*G*AIX)
D14=(-R**2)/(2.*E*AIX )*{ B*S I N ( B ) +COS ( B )*S IN ( A) *S I N ( A ) -A* SI N ( B ) -
1SIM(B)*SIN( A)*COS( A) )
D 15= (R**2)/(2.*E*AIX)*( COS CEL)*SIN(A)*SIN(A)-A*SIN< ED-SIN (ED*
1SIN(A)*C0S(A)+EL*SIN(ED )
5-J-
PAGE (
D16=(R**2)/(2.*AJ*G)*(2.*COS(EL)-2.*COS(A)-COS(EL)*SIN(A)*SIN(A)+
SIN (EL )*(EL-A)+SIN( EL) *S I N ( A ) *COS ( A ) )D17 = ( -R**2 ) / ( 2 . *A J*G ) * ( 2. *COS ( B ) -2. *COS ( A ) + B*S I N ( B ) -COS ( B ) *SI N ( A )
1* SI N ( A ) -A#SI N ( B ) +SI N ( B ) *SI N ( A ) *COS ( A ) )018= ( R**3 ) / ( 2 . *E*A I X ) * ( A-S I N ( A ) *COS ( A ) )D 19= ( -R**2*S I N ( A ) *SI N ( A ) ) / ( 2. *E*AI X )D20= ( R**3 ) / ( 2 . *AJ*G ) * ( 2 . *A-4.*S I N ( A ) +A+S IN ( A ) *COS ( A ) )D 2 1 = ( -R ** 2 ) / ( 2 . * A J*G ) * ( 2 . * ( - C 0 S ( A ) ) - S I N ( A ) * S I N ( A ) + 2 . )
D2 2=(H**2*R*A) /( 10.*G*AIX)D 23= ( -R**2*SIN ( A ) *SI N ( A ) ) / ( 2. *E*A I X )
D24=(R*( A+SIN(A)*COS(A) ) )/(2.*E*AIX)D 25 = ( R ** 2 ) / ( 2 . *.A J *G ) * ( 2 . *C0 S ( A ) + S I N ( A ) * S I N ( A ) - 2 . )
D26=R/(2.*AJ*G)*( A-SIN ( A )*COS ( A ) )'
Y1=D8+D10+D12+D13
Y2=D9+D11+D13
Y3=D18+D20+D22
Y4=D14+D17
Y5 =D15 +D16""'""
Y6=D23+D25
Y7=D24+D26
Y8=08+D10""
Y9 =D9 +D11
Y10=D18+D20
F2=( (-Y1)+P*Y2) /(Y3+Y1)*.(-1. )
Fl=l.-F2 ..
H2=(F1*Y4+P*Y5) /Y7
F2R=( (-Y8)+P*Y9)/(Y10+Y8)*(-1. )
F1R=1.-F2R
H2R=(F1R*Y4+P*Y5) /Y7
X 1= ( F 2*R **3 ) / ( 2 . *E*AI X ) * ( COS ( W ) * ( A-S I N ( A ) *COS ( A ) ) -S I N ( W ) * ( S I N ( A ) ) *
1*2-C0S ( W ) * ( W-S IIM ( W ) *COS ( W ) ) - ( S I N ( W ) ) **3 )
X 1 1 = ( F 1 *R ** 3 ) / ( 2 . * E * A I X ) * ( C 0S ( B ) *C 0 S ( W ) * ( A - S I N ( A ) * C0S ( A ) ) - S I N ( B ) *
1C0S ( W ) * ( SI N ( A ) ) **2-S IN(W)*COS(B)*(SIN(A)) **2+S I N ( B ) *S I N ( W ) * ( A+
2S IN ( A ) *C0 S ( A ) ) -CO S ( B ) *COS ( W ) * ( W-S I N ( W ) *COS ( W ) ) +S I N ( B ) *COS ( W ) *
3(SIN(W) )**2+(SIN(W) )**3*C0S(B)-SIN(B)*SIN(W)*(W+SIN(W)*C0S(W) ) ) [X2= (P*R**3) /( 2.*E*AIX )*( COS ( ED*COS
6(W )*( A-SIN ( A)*COS( A) )-SIN( EL ) *COS ( W ) *S I N ( A ) *S IN ( A ) -S IN ( W ) *COS ( ED*
7S IN ( A ) *S I N ( A ) +S I N ( EL) *SI N ( W ) * ( A+S I N ( A ) *COS ( A ) ) -COS ( EL ) *COS ( W ) *
8 ( W -S I N ( W ) * CO S ( W ) ) + S I N ( E L ) * COS ( W ) *S I N ( W ) * S I N ( W ) + S I N ( W ) *COS (ED*
9S IN ( H ) *S I N ( W ) -S I N ( E L ) *SI N ( W ) * ( W + S I N ( W ) *COS ( W ) ) )
X3.= ( F 2*R**3 ) / ( 2 . *A J*G ) * ( 2 . *A-2 . *COS ( W ) *S I N ( A ) + 2 . *S I N ( W ) *COS ( A )
2-2 . * S I N ( A ) +C0 S ( W ) * ( A+ S I N (, A ) *COS ( A ) ) +S I N ( W ) *S I M ( A ) *S I N ( A ) -2 . *W +
3C0S ( W ) *S IN ( W ) * 2 . -2 .* SI N ( W ) *COS ( W ) +S I M ( W ) * 2 .-COS ( W ) * ( W+S I N ( W ) *
4C0S(W) )-SIN(W)*SIN( W)*SIM(W) )
X4 = ( H 2*R ** 2 ) / ( 2 . * E * A I X ) * ( C 0 S ( W ) * S I N ( A ) * S I N ( A ) - S I N ( W ) * ( A+ S I N ( A ) * C0 S
6 ( A ) ) -COS ( W ) *S IN ( W ) *SI N ( W ) +S I N ( W ) * ( W+S I N ( W ) *COS ( W ) ) )
X5 = ( F 1*R ** 3 ) / ( 2 . * A J*G ) * ( A* 2 . -2 . *COS ( W ) *S I N ( A ) +2 - *COS ( A ) *S I N ( W )
8-COS ( B ) *S I N ( A ) * 2 . +COS ( B ) *COS ( W ) * ( A+ S I N ( A ) *COS ( A ) ) +S I N ( W ) *
9C0S ( B ) *SIN ( A ) *SI N ( A ) +SI N ( B ) *S I N ( W ) * ( A-S I N ( A ) *COS ( A ) ) +S I N ( B ) *COS ( W )
1*SIN(A)*SIN(A)+2.*SIN(B)*C0S(A) )
X6=(Fl*R**3)/( 2.*AJ*G)*(-W*2.+2.*COS(W)*SIN(W)-2.*COS(W)*
2SIN(W)+2.*COS(B)*SIN(W)-COS(B)*COS(W)*(W+SIN(W)*COS(W))-SIN(W)*
3C0S(B)*SIN(W)*SIN(W)-SIN(B)*SIN(W)*(W-SIN(W)*C0S(W) ) -S IN( B)*COS ( W )
St
PAGE 03
4*S I N ( W ) * S I N ( W ) -2 .* S I N ( B ) *C0S ( W ) )
X 7= { P*R**3 ) / ( 2. * A J*G ) * ( 2 . *A-2. *C0S ( W ) *SI N ( A ) + 2 .*S I N ( W ) *C0S ( A ) -
62 . *C0 S ( E L ) * S I N ( A ) +C0 S ( ED *C0S ( W ) * ( A+S I N ( A ) *C0S ( A ) ) +S I N ( W ) *C0S ( E L ) *
7S IN ( A ) *S I N ( A ) +SI N ( EL ) *SI N ( W ) * ( A-S I N ( A ) *C0S ( A ) ) +S IN ( EL ) *C0S ( W ) *
8SIN(A)*SIN(A)+SIN( ED*C0S(A)*2. )X 8= ( P*R **3 ) / ( 2. * A J*G ) * ( -2 . *W+2. *C0S ( W ) *S I N ( W ) -2 . *S I N ( W )9*C0S ( W ) +2 . *C0 S ( EL ) * S I N ( W ) -COS ( EL) *C0S ( W ) * ( W+S I N ( W ) *COS ( W ) )1- SI M ( W ) *COS ( E L ) * S I N ( W ) *S I N ( W ) -S I N ( EL) *S I N ( W ) * ( W-S I N ( W ) *COS ( W ) )1-S I N ( EL) *C0 S ( W ) *SI N ( W ) *SI M ( W ) -2 . *S I N ( EL ) *COS ( W ) )
X9=(II2*R**2)/(2.*AJ*G)*(-C0S(A)*2.-C0S(W)*SIN(A)*SIM(A)-SIN(W)
2 ( A-S I N ( A ) *C0 S ( A ) ) + 2 . *COS ( W ) +COS (W)*SIN(W)*SIN(W)+SIN(W)*(W-
3SIN(W)*C0S(W) ) )
X 10= ( H**2*R ) / ( 10 . *G*AI X ) * ( F2-F 1+ P ) * ( A-W )
DEF=X1+X2+X3-X4-X5-X6+X7+X8-X9+X10-X11, FORMAT (F 12. 7)
X1R=X1/F2*F2R
X11R=X11/F1*F1R
X9R=X9/H2*H2R
X3R=X3/F2*F2R
X4R=X4/H2*H2R
X5R=X5/F1*F1R
X6R=X6/F1*F1R
DEFR=XlR+X2+X3R-X4Rj-X5R-X6R +X7+X8-X9R-Xll
WRITEOt 5 )F1"~ """" " '" ""
WRITE ( 3,5 )F2
WRITE (3, 5 )H2-
-
WRITE(3,5)DEF
WRITE (3,3 )F1R
WRITE (3,5 )F2R
WRI TE ( 3 , 5 )H2R' "'"
.
WRITE(3,5)DEFR
WRITE(3,5)C1
WRITE ( 3,5 )D1
WRITE (3,5 )C2"
WRITE (3,5 )l)2
WRITE (3,5 )D3
WRITE(3,5)C6
WRITE (3,5 )D5
WRITE(3,5)C3
WRITE ( 3,5 )C4
WRITE (3,5)08
"WRITE(3,5)D9"" ~~ ------
WRITE (3,5)0 10
WRITE{3, 5)011
WRITE(3, 5)012
WRITE (3,5 )D13 ",WRITE(3,5)D14
WRITE(3,5)D15
WRITE (3,5)016
WRITE(3,5)D17
WRITE(3, 5)018
WRITE(3,5)D19
WRITE(3,5)D20
-SS-
WRITE ( 3 5)021
WRITE(3,5 )D22
WRITE(3,5)D23
WRITE (3,5 )D24
WRITE(3,5)D25
WRITE(3,5 )D26
WRITE (3,5 )Y1
WRITE(3,5)Y2
WRITE ( 3,5 )Y3
WRITE(3,5)Y4
WRITE(3,5)Y5
WRITE (3, 5 )Y6
WRITE (3,5 )Y7
WRITE (3,5 )Y8
WRITE(3,5)Y9
WRITE(3,5 )Y10
WRITE(3,5)X1
WRITE(3,5)X11
WRITE(3,5)X2
WRITE (3,5
WRITE(3,5)X4
WRITE(3,5 )X5
WRITE(3,5)X6
WRITE(3,5 )X7
WRITE(3,5)X8
WRITE(3,5 )X9
WRITE(3,5)X10
CALL EXIT
END
TRIABLE ALLOCATIONS
A(R )=0000 EL(R )=0002 B(R ~)=0004 W(R )=0006
E(R )=000C AIX(R )=000E AJ(R )=0010 G(R )=0012
D1(R )=0018 C2(R )=001A D2(R )=001C D3(R )=001E
C3(R )=0024 C4(R )=0026 08(R )=0028 D9(R )=002A
D12(R )=0030 D13(R )=0032"
D 1 4 ( R )=0034 D15(R )=0036
D181R )=003C D19(R )=003E D20(R )=0040'
D2KR )=0042
D2MR )=0048 D25(R )=004A D26(R )=004C YKR )=004E
YMR )=0054 Y5(R )=0056 Y6(R )=0058 Y7(R )=005A
Y10(R )=0060 F2(R )=0062 FKR )=0064 H2(R )=0066
H2R(R )=006C XKR )=006E X1KR )=0070 X2(R )=0072
X5(R )=0078 X6(R )=007A"
X7.(R )=007C X8(R )=007E
DEF(R )=0084 X1R(R)=0086' X11R(R )=0088 X9R(R )=008A
X5R(R )=0090 X6R(R )=0092 DEFR(R )=0094.
REFERENCED STATEMENTS
1
TATEMENT ALLOCATIONS
5 =00FE 1 =0100
EATURES SUPPORTED
ONE WORD INTEGERS
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