NASA Technical Memorandum 4729

Dynamic Analysis With Stress Mode Animation

by the Integrated Force Method

Surya N. Patnaik

Ohio Aerospace Institute

Brook Park, Ohio

Rula M. Coroneos and Dale A. Hopkins

Lewis Research Center

Cleveland, Ohio

National Aeronautics and

Space Administration

Office of Management

Scientific and Technical

Information Program

1997

DYNAMIC ANALYSIS WITH STRESS MODE ANIMATION BY THE

INTEGRATED FORCE METHOD

Surya N. Patnaik

Ohio Aerospace Institute

Brook Park, Ohio 44142

Rula M. Coroneos and Dale A. Hopkins

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135

Summary

Dynamic animation of stresses and displacements, which

complement each other, can be a useful tool in the analysis and

design of structural components. At the present time only

displacement-mode animation is available through the popular

stiffness formulation. This paper attempts to complete this

valuable visualization tool by augmenting the existing art with

stress mode animation. The reformulated method of forces,

which in the literature is known as the integrated force method

(IFM), became the analyzer of choice for the development of

stress mode animation because stresses are the primary unknowns of its dynamic analysis. Animation of stresses

and displacements, which have been developed successfully

through the IFM analyzers, is illustrated in several examples

along with a brief introduction to IFM dynamic analysis. The

usefulness of animation in design optimization is illustrated

considering the spacer structure component of the International

Space Station as an example. An overview of the integrated

force method analysis code (IFM/ANALYZERS) is provided

in the appendix.

Introduction

Animation of displacement modes, which provides an overall visualization of the deformations, is an elegant tool for the

examination of the dynamic behavior of a structure. The field

of animation, however, can be improved by an augmentation in

the existing art with stress mode animation. Stress and displacement animations, which complement each other, can together

provide a comprehensive visual behavior of a structure. Consider, for example, a cantilevered beam. Its displacement

animation depicts the motion of the beam with the maximum

amplitude occurring at its free end. Fundamental stress-mode

animation, on the other hand, shows peak stress response at its

built-in end where displacements are suppressed. Both stress

and displacement animations can be useful for design, fatigue,

and fracture studies of the beam.

Displacement animation can be obtained by coupling the

popular MSC/NASTRAN stiffness code (ref. 1) with the

"Insight" animation feature of the Patran software (ref. 2).

Several other commercial codes (ASKA, ANSYS, ABACUS,

etc.) (refs. 3 to 5) can also be used for displacement animation.

Displacement formulation, as the basis ofthese codes, can pose

difficulty for stress field animation because stresses are not the

primary unknowns of its dynamic eigenvalue analysis. Furthermore, sUess prediction by the stiffness method can be liable to

accuracy deficiency, andas it turns out, dyn_c _mation of

stresses has yet to be attempted through the displacement

formulation. At this juncture, the reformulated method of

forces, which in the literature is known as the integrated force

method (IFM) (refs. 6 to 8), becomes a valuable altemate tool

for animation of the stress fields especially because stresses are

the primary unknowns of its dynamic analysis. The IFM, which

produces accurate stresses and displacements (refs. 6 and 7),

has been extended to include the dynamic animation of both

stress and displacement fields. Additionally, animation has

been used in design optimization to generate simpler structures

which can facilitate fabrication.

This paper, which discusses stress and displacement animations and their use in design optimization, is divided into five

subsequent sections. Section 2 includes the IFM equations,

dual formulation, an illustrative example, and animation procedure. Numerical examples are presented in section 3. Section 4

provides a brief summary of the use of animation in design

optimization, followed by conclusions in section 5. An overview of the IFM/ANALYZERS is given in the appendix.

Equations of the Integrated Force Method

In the integrated force method (IFM), a discretized structure

for analysis, is designated by two attributes n and m. The

number of unknown forces or force degrees of freedom (fojO is

n. Likewise, the displacement degrees of freedom (dof) is m.

Iffof= dof, then the structure is determinate (r = n- m = 0) and

its analysis is trivial. For indeterminate structures, fof exceeds

NASA TM-4729
dof and the degree of indeterminacy of the structure is

r = fof-dof = n- m > 0. The governing IFM equations are

obtained by coupling m equilibrium equations and r compatibility conditions. The equations of the integrated force method

are summarized next (refs. 6 and 7).

Static analysis equations of the IFM are as follows:

(la)

or

[S]F - P * (lb)

where [B] is the (m x n) equilibrium matrix, [C] is the (r x n)

compatibility matrix, [G] is the concatenated symmetrical

flexibility matrix of dimension (n x n), P is the m component

load vector, _iR is the r component effective initial deformation

vector, and in the absence of initial deformations it is a null

vector {;5R } - {0}, and [S] is the (n x n) governing the IFM

matrix. The matrices [B], [C], [G], and [S] are banded and have

full row ranks of m, r, n, and n, respectively. The solution of

equation (1) yields the n forces F. Forces, in other words,

can be determined without any reference to displacements

provided the structure is kinematically stable. The m displacements X, if required, can be obtained from forces by

backsubstitution as:

X- [J][G]F (2)

Here, [J] is the m x n deformation coefficient matrix defined as:

[J]=m rows of[[s]-l ] T (3)

The frequency analysis equation of the IFM without damping

is as follows:

[[S]- c°2E[M][J][G]]]F-O0 (4)

where [M] is the (m x m) mass matrix, m is the circular

frequency, and F is the force mode shape of the eigenvalue

problem.

Forces are the unknowns of the IFM vibration analysis.

Displacementmodes in IFM, if required, can be backcalculated

from forces Fusing equation (2). In other words, the IFM

provides one set of equations to determine forces (eq. (1) for

static or eq. (2) for vibration analysis), and another set for the

calculation of displacements (eq. (2)).

The dual integrated force method (IFMD) is obtained by

mapping forces into displacements (ref. 7). The basic equations

of the dual formulation neglecting initial deformations and

damping are summarized next.

Static analysis equations of the IFMD are as follows:

[D]X- P (5)

Forces can be obtained from displacements using the following

formulae:

[z - [G]-I[B] T f( (6)

Dynamic analysis equations of the IFMD are as follows:

[t,,l =o (7)

(m x m) symmetrical matrix '([D] = [BI[GI-I[B]T), \

where the

is assembled at element level.

From the displacement modes, force mode shapes can be

backcalculated using equation (6). Like IFM, the dual IFMD,

which treats displacements as the prim_ variables, has one

equation to calculate displacements (eq. (5)for static oreq. (7)

for dynamic analysis), and one equation for the dete_nation

of forces from displacements (eq. (6)). Both IFM and IFMD

provide identical solution for stresses, displacements, and

frequencies. For design and sensitivity analysis, the primal

IFM, however, has some advantage over the dual IFMD

(refs. 9 and 10).

The IFM for dynamic analysis was introduced for simple

examples well over a decade ago (ref. 8). Since then, several

improvements have been incorporated into the reformulated

method of forces such as the generation of compatibility

conditions (ref. 11), development of the dual formulation

(ref. 7), and design sensitivity calculations (ref. 9), just to

mention a few. To familiarize the reader with the IFM/IFMD

solution process, the example of a cantilevered beam is discussed in brief. The beam is 12 in. long, 2 in. deep, and 1/4 in.

thick. It is made of steel with Young' s modulus E = 30 000 ksi,

Poisson' s ratio v = 0.3. The beam, which is considered massless

for dynamic analysis, has two lumped masses of weight 5 lb

each, as shown in figure 1. For the purpose of illustration, the

beam is discretized by two QD04_05 membrane elements of

IFM/ANALYZERS (see the appendix) as shown in figure 1.

The QD04_05 is a four-node quadrilateral element and its

dof= 8 andfof= 5. The beam discretized by the two elements

has dof = 8 and fof = 10 and it is a two-degree indeterminate

structure.

2 NASA TM-4729
n \

TI

2 in.

lilz

12in. ,-

"= _ 0.00388

u slugs (51b)

• _ 0.00388

=, slugs (5 Ib)

Figure 1.---Cantilevered beam idealized by two QD04_05 membrane elements of IFM/ANALYZERS.

The frequency equations for the beam, lexplicitly written for

IFM, IFMD, and the stiffness method, are presented next. In

the IFM/ANALYZERS code, which is introduced in the

appendix, the stiffness method has been included for the purpose of comparison of numerical results.

The IFM frequency equation

for the cantilevered beam depicted in figure 1has explicit form

of equation (8).

nonzero rows in the mass matrix represent the participation of

lumped masses which are located at nodes 3 and 4. Participation of all 10fof is essential for the determination of correct

frequency and force mode shapes (refs. 7, 8, and I 0). A correct

dynamic analysis formulation cannot be obtained through any

elaborate manipulation of the equations of the classical force

method (ref. 12), because this approach does not consider the

mass or the inertia for redundant members. The four frequen-

- 1 0 -3 -1/3 0 -1 0 -3 1/3 0

0 -3 1 0 -3 0 -3 -1 0 3

0 0 0 0 0 1 0 -3 -1/3 0

0 0 0 0 0 0 -3 1 0 -3

0 0 0 0 0 1 0 3 1/3 0

0 0 0 0 0 0 3 1 0 3

1 0 3 1/3 0 -1 0 3 -1 / 3 0

0 3 1 0 3 0 3 -1 0 -3

5* -2* 0 0 5* 0 0 0 0 0

-10 * 3 * 0 0 0 5 * -2 * 0 0 5 *

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0.3 -0.06 0 -0.3 -0.06 0.3 -0.09 0 -0.31 0.02

-E a 0.02 0.8 -2.8 0.02 0.02 -0.07 0.8 -0.93 -0.11

0.3 -0.06 0 0.3 -0.06 0.3 -0.09 0 0.31 0.02

a --0.02 0.8 -2.8 -0.02 -0.02 0.07 0.8 -0.93 0.11

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

_

F=0

(8)

The last two rows in the [S] matrix are small numbers which in

equation (8) are rounded up, for example, 5* = 4.8x10 -7 and

2* = -1.6xl 0-6, etc. Also, ea = 0.0066.

The IFM eigenvalue problem for the beam is an unsymmetrical (10 x 10) set of equations. The first eight equations

(see eq. (8)) represent the equilibrium conditions, whereas the

last two equations are the compatibility conditions. The four

cies obtained for the problem are given in table I. The fundamental force mode shape along with the displacement mode

shape calculated by backsubstitution using equation (2)are

shown in table II.

The IFMD frequency equation [[D]- Eo2[M]]X = 0for the

two-element cantilevered beam depicted in figure 1 has the

explicit form of equation (9).

NASA TM-4729 3
10 6

TABLE I.mFREQUENCIES FOR A CANTILEVERED

BEAM WITH LUMPED MASS

Frequency

numbers

Frequencies in Hz

IFM IFMD Stiffness Analytical

method solution

Frequency, Hz

170.629 I70.629 353.073 168.32

2 2031.261

3 2534.278

4 9298.475

2031.261

2534.278

9298.475

2032.188

4874.846

9599.293

1052.00

2950.38

5786.00

TABLE II.DFORCE AND DISPLACEMENT MODE SHAPES FOR

FUNDAMENTAL FREQUENCY OF CANTILEVERED BEAM

IFM IFMD Stiffness method a

Force mode

shape

0.0

0.0

-3.65 x 10 -2

1.0

0.0

0.0

0.0

_3.65×10 -2

0.343

0.0

Displacement

mode shape

9.78x 10 -2

0.303

0.131

1.0

-0.131

1.0

_9.78x 10--2

0.303

Force mode

shape

0.0

0.0

-3.65 x 10- 2

1.0

0.:0

0.0

0.0

-3.65 × 10- 2

0.343

0.0

aForce mode shape cannot be determined.

Displacement Displacement

mode shape mode shape

9.78 x 10-2

0.303

0.131

1.0

-0.131

1.0

-9.78 × 10-2

0.303

9.17 xlO -2

0.315 :

0.123

1.0

-0.123

1.0

-9.18 x 10 -2

0.315

6.1 0.0

16.6

1.3

0.1

3.0

sym

--0.1 -2.6 -1.3 -3.44 0.0

4.0 -1.3 -4.5 0.0 -15.6

-1.3 -1.7 --0.1 -2.6 1.34

8.3 0.1 -7.8 1.3 -4.5

3.0 1.3 1.3 -0.1

8.3 0.1 4.0

6.1 0.0

16.6

- m2 x 10 -3

0

4

4

4

4

0

X=O (9)

There are eight equations for the dual method. The four nonzero

entries in the diagonal mass matrix in equation (9) correspond

to the lumped masses. The structure of the IFMD eigenvalue

equation appears similar to standard stiffness equations. The

differences between IFMD and the stiffness equations is examined later in this paper. The frequencies obtained by IFMD are

,given in table I. The fundamental displacement mode shape

along with the force mode, backcalculated using equation 6,

are shown in table II. Notice that the frequencies obtained as

solutions to IFM equation (4) and IFMD equation (7) agreed

with each other. Both force and displacement mode shapes

given in table II also agreed for IFM and IFMD. Henceforth, no

distinction will be made for results generated either by IFM or

IFMD.

NASA TM-4729 4
The IFM equations are unsymmetrical while those for dual

are symmetrical. Either set of equations can be used to obtain

frequency, stress, and displacement mode shapes. The symmetrical form can use popular eigenvalue solution routines

(such as DSPGV) which is readily available in the LAPACK

public domain library (ref. 13). For unsymme_cal eigenvalue

analysis, LAPACK routine DGEGV is used (ref. 13). For

static analysis, the Harwell library routines (MA28AD,

MA28CD, MA29BD, MA29CD, MA47AD, MA47BD,

MA47CD, MA47ID) are used (ref. 14). However, it is observed

that the unsymmetrical IFM version can be more useful than

IFMD in design and sensitivity analysis (ref. 10). Furthermore,

the IFM solution which provides r = n - m number of zero

frequencies and associated eigenvectors corresponding to the

r number of compatibility conditions can be used to verify

solution accuracy.

For the purpose of comparison, the stiffness analysis is also

provided for the beam. The element used for the stiffness

analysis has four nodes and eight displacement degrees of

freedom. The stiffness element uses displacement fields identical to the IFM/IFMD element QD04_05. In IFM/ANALYZERS (see the appendix) this stiffness element is also referred to

as QD04_05. The IFM/ANALYZERS, in other words, retains

the same element name for different analysis methods such as

the IFM, IFMD, and stiffness methods. The analysis method is

specified through a set of executive control keywords, which

can be IFM!, IFMD!, or STIFF!, for primal, dual, or stiffness

analysis, respectively. Generation of all elemental matrices of

the IFM, IFMD, and stiffness methods use identical numerical

integration and are programmed in the Fortran 77 language.

For the problems, the IFM, IFMD, and stiffness elements used

are considered equivalent. All numerical results have been

obtained on the Cray-YMP 8E/8128 supercomputer at the

NASA Lewis Research Center.

The stiffness method frequency equations IlK] -032 [m]]

X - 0 for the cantilever beam depicted in figure 1 have the

explicit form of equation (10). Frequency and displacement

modes obtained are presented in tables I and II. Calculation of

stress mode shapes is not readily available in the stiffness

method.

Comparison of IFM, IFMD, and

Stiffness Results

From the solution of the cantilevered beam, the following

observations can be made for the IFM, IFMD, and stiffness

methods.

Both the primal IFM and the dual IFMD provide identical

solutions for frequencies, stress, and displacement mode shapes

(see tables I and III).

Results for the integrated force method are more accurate

than those for the stiffness method. For IFMi the two-element

model of the beam produced accurate fundamental frequency

with an error of about one percent. The error in the fundamental

frequency for the stiffness method, however, is about 110 percent. The higher modal frequencies given intable I are in error

because of the crude two-element model and are not further

discussed. Force and displacement mode shapes obtained by

IFM and IFMD agreed as expected (see table II). The stiffness

method displacement mode exhibits some error even for this

modest example (see table II).

The cause of higher fundamental frequency in the stiffness

method can be observed from a comparison of equation (9) for

IFMD and equation (10) for the stiffness method. Mass matrices for both stiffness and IFMD (eqs. (9) and (10))are identical. The coefficient of the [K] matrix in stiffness equation (10)

differs from that of the [D] matrix in IFMD equation (9). For

this problem, all eight diagonal elements in matrix [K] are

bigger than the co_esponding entries in the matrix [D]. In other

words, a higher frequency for the stiffness method was anticipated because the stiffness equation (eq. (10)) is stiffer than the

IFMD equation (eq. (9)).

106

.6 0.0

17.1

0.5 -0.1 -1.9 -1.3 -4.8 0.0

0.1 4.8 -1.4 -4.3 0.0 -16.1

3.8 -1.3 -2.4 -0.1 -1.9 1.3

8.6 0.1 -8.1 1.3 -4.3

3.8 1.3 0.5 -0.1

8.6 0.1 3.8

sym 7.6 0.0

17.1

- 032 x 10 -3

-0

4

4

0

X--0 (10)

NASA TM-4729 5
TABLE HI. wRESULTS FOR AN ISOTROPIC

Model

CB_Model_l

Six/HX 08_18

CB_Model,2

Six/HX20_90

CB_Model_3

Three/HX08 18

CB_Model_4

Three/HX20 90

CB_Model_5

Three/HX08_ 18

(irregular mesh)

CANTILEVERED BEA1V

IFM/IFMD Stiffness

method

MSC/NASTRAN a

Frequency, b Hz (1,3)

1.005

1.043

1.007

0.983

1.021

1.274

1.025

1.069

1.014

1.516

1.637

1.640

1.014

0.995

2.672

1.862

1.041

1.091

2.335

1.887

1.637

1.640

1.014

0.995

2.672

1.862

1.041

1.091

2.335

1.887

aEquivalent MSC/NASTRAN elements (CHEXA with 8 nodes or

CHEXA with 20 nodes) are used.

bFrequencies are normalized with respect to strength of material

formulae.

The determination ofvon-Mises stress for three-dimensional

solid elements (of IFM/ANALYZERS), which are represented

by complete polynomials, can be quite tedious but it is easy to

calculate in a computer. The IFM/ANALYZERS provides both

displacement and stresses (including von-Mises stresses) at the

grid points and other specified locations. For completeness,

von-Mises stress calculation is illustrated for a simple rectangular membrane element QD04_05 used in example 1. The

three stress components for this membrane element can be

obtained as:

CYx = F1 + F2 ( 2 y/a )

t t

= F3 + F4 (2x/b) (12) O'y

t t

Dynamic Animation of Stress Mode Shape

Stress is a second-order tensor with six components at every

point of a solid continuum. Thus, the solid modeler solution of

the IFM/ANALYZERS has six stress components at each

node. Animations of a second-order tensor quantity is a cumbersome proposition, and it can also be difficult to visualize.

Fortunately, however, failure in a structure can be attributed to

a single quantity such as von-Mises stress, which is a scalar

function of the six stress components and can be written as:

q;xY= t

where (cyx, (Yy, _xy)are membrane stresses, (F 1, F2...... /75) are

force mode shapes for element QD04_05, t is the element

thickness, and a and b are the dimension of the QD04_05

element along the x and y coordinates, respectively. For the

membrane element, the von-Mises stress given in equation (11 )

can be calculated by substituting (o x, Oy, and Xxy) from equation

(12) and setting (c_z = "Czx- "Czy= 0).

(Yvon-M--IlI(¢Jx - Oy) 2 2 + z-Ox)21 2 2

(11)

Stress animation, in other words, represents animations of

the von-Mises stress given by equation (11). Von-Mises stress

animation is consistent with the animation of the scalar displacement magnitude in the stiffness method that is available at

the present time. Equivalent von-Mises stress is calculated for

the force eigenvector obtained from the solution of IFM

dynamic equation (4) or equation (7) for IFMD. Dynamic

animation for stress and displacement modes is obtained by

supplementing the "Insight" tool of the P3/Patran software with

force mode solutions through Patran neutral files. For example,

the animation of stress modes requires the preparation of two

neutral files. The first file should include the finite element data,

which contains the nodal coordinates and element connectivities

in the model. The second file should contain the von-Mises

stress for each node and the frequency.

Two features of P3/Patran fringe plotting and modal animating, which control time and space, are used to display the

animation results. Animations can be viewed best by using an

audio-video VCR system. A videotape, which contains the

stress and displacement animation for several examples

(ref. 15), is available and can be sent on request. In this paper,

however, only a single-frame animation depicting maximum

response can be presented. A single animation frame is

obtained by pausing the setup to approximately correspond to

peak response regime and then taking a "snap-shot", which is

available in the Silicon Graphics Indy Unix workstation used

for the purpose. Displacement animation is obtained by replacing stress results in favor of displacement mode shapes.

6 NASA TM-4729
Numerical Examples Example 2" Cantilevered Beam

Results for three examples, a cantilevered beam (example 2),

an L_Beam (example 3), and a turboprop blade (example 4)

obtained with IFM/ANALYZERS, are summarized next.

The material used for all the examples is steel with properties identical to that of example 1 with mass density

P = 0.289 lb/cu in. The frequency results were obtained for

distributed self-mass of the structure using a consistent mass

matrix formulation. The structures were discretized using an

eight-node solid element HX08_18 and a twenty-node solid

element HX20_90. The structures with principal dimensions

are depicted in figure 2. Again, solutions are obtained by using

IFM, IFMD, and the stiffness method. Results obtained by

the MSC/NASTRAN analyzer using comparableelements are

also given. For each example, a single frame from the dynamic

animation depicting the peak stress and displacement modes is

included.

Z I in.

a)_-_-_ 12 in. t

y _ ..._.,.1 in.

( X

1 in.

X

(b)

Results for the cantilevered beam, shown in figure 3,

obtained using IFM, IFMD, and the stiffness method, are

depicted in table III. Five finite element models are used for this

example. CB_Model_l has six, eight-node, HX08_18 solid

elements. CB_ Model_2 has six, twenty-node, HX20_90 solid

elements. CB_Model_3 is similar to CB_Model_l but only

three elements are used. Likewise, CB_Model_4 has three

elements but is otherwise similar to CB_Model_2. CB_Model_5

has a total of three elements, one small element near the support

and two other elements. For MSC/NASTRAN analysis, the

same models are used but IFM/ANALYZERS elements

HX08_I 8 and HX20_90 are replaced by equivalent elements.

That is, IFM/ANALYZERS elements HX08_I 8 and HX20_90

are replaced by equivalent MSC/NASTRAN elements (CHEXA

with eight nodes and CHEXA with twenty nodes, respectively).

First and third frequencies for the beam, obtained by different

methods, are given in table III (second frequency is identical to

the fundamental mode due to symmetry). Results are normalized with respect to the strength of material beam solutions. The

beam solutions are quite accurate; however, there can be some

difference in the results due to three-dimensional elasticity and

boundary restraint effects. For example, a beam modeluses one

material constant, which is Young' s modulus, whereas a threedimensional elasticity solution uses Poison's ratios in addition.

For this problem, IFM/IFMD results for the fundamental

frequency are found to be quite accurate. The six-element

models CB_Model_l and CB_Model_2 have a margin of error

of 0.5 percent and 0.7 percent, respectively. The crude threeelement models CB_Model_3 and CB_Model_4 exhibit

somewhat higher margins of error. More error is noticed for the

third mode, as anticipated, because the finite element

discretization represents a coarse mesh. Results obtained by

using the stiffness method and MSC/NASTRAN software

agreed since both are basedondisplacement formulation. For

the fundamental frequency, a higher order MSC/NASTRAN

, W" . Y

!. i_._.. _ _ element (CHEXA with twenty nodes) which is equivalent to

_._ IFM/ANALYZERS element HX20_90 produced acceptable " "_"- _ results (see results for CB_Model_2 and CB_Model 4 in

3. 9 .. Z X table III). The solution obtaind by the normal eight--node

element of the stiffness method andthe eight-node CHEXA

• MSC/NASTRAN element provide a pOor comparison with

the eight-node IFM/ANALYZERS HX08'I 8 element.

For this example, von'Mises stress and displacement

animation have been obtained by using IFM/ANALYZERS

(c) l _ 9.399 in. "l

Figure 2.---Overall geometrical dimensions of examples.

(a) Example 2: Cantilevered beam idealized by six solid

elements. (b) Example 3: Cantilevered L__beam idealized

by nine solid elements. (c) Example 4: Turboprop blade

idealized by 240 solid elements.

and Patran software. A single-flame animation picture for

fundamental displacement and stress modes is depicted in

figure 4 for CB_Model_2. In figure 4, the neutral axis of the

beam is clearly visible and maximum von-Mises stress occurs

near the support. Von-Mises stress is symmetrical with respect

to the neutral plane, which at times can be color-camouflaged.

NASA TM-4729 7
17 56 18

4 _

10 49 11 "-_'_----

(a)

Y

X

15 4 64 19 72

L/34331_ 541 5__-_ _'-- _'_ _- _ 20 80 2

29 7

65 13

(b) 73 14

Figure 3.---Cantilevered beam idealized by six HX20_90 solid elments. (a) Enlarged view of the twenty-node element.

(b) Full beam view.

Neutral plane

(a)

Figure 4.---First mode animation of cantilevered beam. (a) Stress mode. (b) Displacement mode.

8 NASA TM-4729
Thedisplacement animationflameshowsmaximumdisplacementat thefreeendof thecantilever.Thedisplacement

animationusingthestiffnessmethodorMSC/NASTRAN

code,arefoundtobesimilartoIFM/ANALYZERSresults. As

statedearlier,stressfieldanimationcannotbeobtainedbythe

displacement method.

Example3:L_Beam

AnalysisofanL_beam, showninfigure2(b)iscarriedoutby

fourdifferentmethods(IFM,IFMD,stiffness,andMSC/

NASTRAN)forfourfiniteelementmodels. LB_Model_l has

nine,eight-nodeHX08_18solidelements; LB_Model_2has

nine,twenty-nodeHX20_90solidelements; LB_Model_3is

similartoLB_Model_l exceptonlyfiveelementsareused,and

LB_Model_4, likewise,hasfiveelementsbutis otherwise

similartoLB_Model_2. Thefiniteelementmodel2forthe

L_beamis shownin figure5. Firstandthirdfrequencies

obtainedby IFM/IFMD,thestiffnessmethod,andMSC/

NASTRANarepresentedintableIV.Resultsarenormalized

withrespect totheIFM/IFMDsolutionforLB_Model_2. For

thisproblemit isobservedthatheIFM/IFMDresultsforthe

fundamental frequencyareaccuratewithamaximumerrorof

3.8percentforLB_Model_4(withfiveHX20_90elements).

IFM/IFMDresultsforthethirdmodeareacceptablewith

somewhat highererrormargins. Resultsforthestiffnessmethod

andMSC/NASTRANagreed.However,frequencyresults

obtainedusingthestiffnessmethodandMSC/NASTRAN

softwarearepoorerthancorresponding IFMsolutions. MSC/

NASTRANandstiffnessmethodperformanceisconsidered

acceptablewhenhigherorder,twenty-nodeCHEXAelements

areusedforfrequencyanalysis.

Stressanddisplacement animationusingIFM/IFMDare

obtainedfortheL_Beam, forLB_Model_2withnineHX20_90

elements. Asingleanimationframe,whichcorresponds topeak

responseforvon-Misesstressmodeanddisplacement mode,is

showninfigure6.Thedisplacement modeforthefirstfundamental frequencyoftheL_Beamrepresents motionintheplane

definedby its twolegs.Thismotioncannotbevisualized

throughaframetakenalongtheL_Beamplane.Themotion

can,however, beobservedwhenapictureistakenperpendicularto theL_Beamplaneasdepictedin figure6(b).The

displacement animationsforLB_Model_2generatedbyIFM

TABLEIV. RESULTS FOR AN ISOTROPIC L_BEAM

Model

LB_Model_l

Nine/HX08_l 8

LB_Model_2

Nine/HX20_90

IFM/IFMD Stiffness MSC/NASTRAN a

method

Frequency? Hz (1,3)

0.999 1.451 1.451

1.009 1.165 1.165

1.000

1.000

1.006

1.003

1.006

1.003

LB Model 3 1.024 i 1.720 , 1.720

vi,,g/HX08-,8 ,.003 i t.34, i 1.341

LB,Model_4 t,038 1,046 I I..046

Five/HX20_90 11016 . 1.020 [ 1,020

"Equivalent MSC/NASTRAN element (CHEXA with 8 nodes or

CHEXA with 20 nodes) is used.

hFrequencies are normalized with respect to IFM LB_Model_2.

element HX20_90, compared well with the stiffness method

results when higher order 20-node CHEXA elements were

used. The von-Mises stress animation obtained for the IFM/

ANALYZERS element HX20_90 is shown in figure 6(a). The

stress animation depicts a neutral plane along the width of the

beam which corresponds to its in-plane motion. Maximum

von,Mises stress occurs at the restrained boundary, which is

shown in red. The free end is stress,free and is shown in blue.

Variation of von'Mises stress along the beam, which peaks at

the restrained end, and is null atits free end, can be seen in

figure 6(a).

Example 4: Turboprop Blade

The turboprop blade is considered as the final example. The

blade, shown in figure 7, is discretized using eight-node solid

elements HX08_I 8. The blade has a total of 240 elements with

dof = 1620 and fof = 4320. The first eight frequencies of the

blade, obtained by IFM/IFMD, the stiffness method, and MSC/

NASTRAN analyzers, are given in table V. The fundamental

frequency of the blade, obtained by the IFMflFMD analyzer, is

41.532 Hz. The fundamental frequency obtained by the stiffness and MSC/NASTRAN analyzers agreed; however, both

methods produced higher frequencies at 76.731 Hz. The other

seven frequencies obtained by the stiffness and MSC/

NASTRAN analyzers are higher than the IFM/IFMD analyzers

by about 15 to 75 percent.

Z

1 2 3 4

Figure 5.--L_Beam idealized by nine solid elements.

NASA TM-4729 9
(a)

Neutral plane

(b)

10

Figure 6.--First mode animation of L_Beam. (a) Stress mode. (b) Displacement mode.

TABLE V. FREQUENCIES FOR AN ISOTROPIC

TURBOPROP BLADE IDEALIZED BY

240 (HX08_ 18) SOLID ELEMENTS

Mode IFM/IFMD Stiffness MSC/NASTRAN"

numbers method

1 41.532 76.371 76.371

(1.839) b (1.839)

2 217.396 386.223 386.223

(1.777) (1.777)

3 396.692 410.118 410.119

(1.150) (1.150)

4 541.273 785.97I 785.971

(1.452) (1.452)

5 809.480 1148.448 1148.448

(1.419) (1.419)

6 970.508 1465.579 1465.579

(1.510) (1.510)

7 1369.698 2094.299 2094.300

(1.529) (1.529)

8 1596.059 2801.503 2801.504

(1.755) (1.755)

"Equivalent MSC/NASTRAN elemen_ (CHEXA with

8 nodes) is used.

bQuantities in parenthesis are normalized with respect to

IFM solution.

Animation is carried out only for IFM/ANALYZERS solutions, since the stiffness and MSC/NASTRAN results for the

blade exhibited higher margins of error. A single-frame animation.for stress and displacement modes for the blades is shown

in figure 8. The blade response in the fundamental mode

represents flexure and torsion coupled motion, as can be seen

in figure 8(a). The flexure and torsion coupled response does

not depict a neutral zone. The maximum displacement amplitude occurs at the free end of the blade which also exhibits

considerable torsional motion (see fig. 8(b)). Von-Mises stress

amplitude reduces to zero at the free end from its peak value at

the built-in boundary. Stress peaks at the support of the turboprop blade while the free ends are stress free, as can be seen in

figure 8(a).

Determination of displacement and frequencies can be carded out by the displacement method, which can also be verified

by dynamic testing. Such procedures, however, do not provide

a stress state in the structure. Thus, the stress animation via the

method of forces becomes a viable tool to examine the dynamic

behavior of the structure, especially for stress response at stress

concentration regions. The stress animation tool complements

the existing displacement animation, thereby, improving the

existing visualization technique.

NASA TM-4729
;_.,....

t I . ._

Ii

_ . .,. _.

"- I ._._',' • .._ --,.

Figure 7.---Turboprop blade idealized by 240 HX08_18 solid elements.

(a)

(b)

Figure8.mFirstmode animationofturboprop blade.(a)Stressmode. (b)Displacement mode.

NASA TM--4729 11
Animation in Design Optimization

In the design of complex industrial structures with many

variables and constraints, the optimization process can quite

often be bogged down and confused, resulting in a design which

cannot be easily fabricated (ref. 16). Such deficiency can be

alleviated, to some extent, through cumbersome constraint

manipulation schemes, which require repetition of several

optimization cycles. Alleviation of the deficiency can be

attempted by coupling animation to the optimization processes.

Dynamic animation has been incorporated into the design tool

called CometBoards (ref. 17) (which is an acronym for

Comparative Evaluation Test Bed of Optimization and

Analysis Routines for the Design of Structures). The combined

animation and optimization tool has been used to generate

optimum designs for a number of industrial structural components. The usefulness of animation and optimization in design

can be illustrated by considering the simple example of a spacer

structure of the International Space Station in figure 9. The

initial configuration of the spacer structure, which has 14joints,

41 members, and weighs 500 lb, is shown in figure 9(a).

Design optimization was carried out using a finite-element

model with 307 nodes and 1835 dof (which corresponds to

8 beam elements per frame member and 2 beam elements for

the trunion (ref. 18)). This structure was designed for a number

of pseudo-static and emergency-landing load conditions. The

Longeron

trunnion _

I

I

I

77.9 77.9

Longeron

trunnion

135.5

Keel / 101.5

trunnion / 101.5

C) Location of lumped mass

(a) (weight in Ib) (b)

135.5

(c) (d) _ )

Figure 9.-- Optimization and animation coupled tool produced a lighter design for the spacer structure. (a) Manual design

has 14 joints, 41 members, and weight of 500 lb. (b) Optimum design has 13 joints, 32 members, and weight of 316.63 lb.

(c) Dynamic animation of the manual design. Notice excessive deformation for shaded member. (d) Dynamic animation of

modified configuration with open face. Notice excessive deformation for open face. (e) Dynamic animation with bracing.

The deformations of the open face are reduced.

12 NASA TM-4729
constraintsconsideredwere:stresses forallmembers, displacementlimitationsattheexteriornodesofthespacerstructurefor

theshuttledynamicenvelope, buckling,andfrequencylimitations.Thedesignwascastasanonlinearmathematical programmingproblemandwassolvedusingtwooptimizers(the

Sequential Unconstrained MinimizationTechniqueSUMTand

theSequential QuadraticProgramming techniqueoftheIMSL

libraryavailablein CometBoards). Observationsof the

dynamicanimationoftheoptimumdesignindicatedthatsome

membersaremuchmoreflexiblethanothers.Asingleanimationframefortheoptimumstructureisshowninfigure9(c).

Evenfromthissingleframeit canbeseenthatthemember

connectingnodes1and10sufferrelativelylargedeformations

comparedtoothermembers. A modifiedconfiguration, shown

infigure9(b),withfewermembers(41membersoftheoriginal

designwerereducedto32members)andfewerjoints(13

insteadof 14in theoriginalconfiguration), wasobtained

throughanexaminationof thedynamicanimationof the

component. Theoptimumdesignofthespacerwasobtained,

again,usingthedesigntoolCometBoards. Theanimationofthe

optimumdesignofthemodifiedconfigurationindicatedunacceptableexcessivedistortionforanopenfaceconnectingjoints

(2,4,11,I0)showninfigure9(d).Theopenfaceisafunctional

on-orbitrequirement fortheintegrationofthespacerstructure

totheothermodulesoftheInternational SpaceStation.This

face,however, canbetemporarilybracedinthespaceshuttle

duringlaunchbyapairofturn-buckles, whichweigh2.5lb.The

bracingcanberemovedonorbitoallowitsintegrationwiththe

spacestation:Themodifiedconfiguration, withturn-buckle

bracing,wasoptimizedagain,andasingle-frameanimationof

thebracedstructureisshownin figure9(e).Bycomparing

figure9(d)with9(e),itcanbeobservedthatheopenfacedoes

notsufferexcessivedeformation. Thefinaloptimumdesign

hasaweightof316.63lb,whichis36percentlighterthanthe

manualdesignof 500lb.Theoptimumstructurehasfewer

membersandnodeswhichfacilitatesitsmanufacturability. The

fundamental modeofthefinaldesignisabreathingtypeof

modeinwhichtheentirestructureparticipatesaasingleunit.

Inotherwords,theloadpathiswelldistributedamongst all

membersofthespacerstructure. Withouttheuseofanimation,

thegenerationofalightermanufacturable optimumstructure

wouldhavebeendifficultif notimpossible. Theanimationand

optimizationcombinedtoolhassuccessfullygeneratedan

optimummanufacturable designforthespacerstructure.

occurrances. Reviewofthefailuredatacanidentifythecritical

regionwhichcanthenberedesignedandretested. Identificationofafailure-proneregion,whichattimescaneludexisting

techniques, isimportantbothfromfinancialaswellasproject

scheduleaspects. Thedynamicanimationof stressmodes,

whichhasbeendiscussedin thispaper,complementsthe

existingtechniquesinidentifyingthefailure-proneregionsin

complexstructures. Dynamicanimationshouldbecarriedout

forthefundamental modeaswellasforhigherandotherlocal

modestoidentifyhigh-stressregions.Toisolatethosecritical

failure-proneregionsinastructure, onlytheintegratedforce

methodynamicanalysisresultsandacomputerworkstation

arerequired.

Indesignoptimization, themodificationoftheconfiguration

of acomplexstructurecanbeimprovedin morewaysthan

merelychangingits sizingvariables. A varietyof potential

configurationmodificationscanquicklybediscoveredthrough

anexaminationof thedynamicanimationof thestructure.

Scrutinyof suchcandidateconfigurationscanbeusefulin

designinganoptimumstructureforbothconfigurational and

sizingvariables. Inotherwords,optimizationaugmentedwith

animationcanbeaviabledesigntool.

Overall,theartof animationdiscussedinthispapercanbe

usefulbothin theanalysisandin thedesignof complex

structures.

Concluding Remarks

Both stress and displacement animation, which complement

each other, provide a comprehensive visual behavior of a

structure. Thus far, only displacement animation was available

through the stiffness method. The field of animation has now

been improved through the development of stress mode animation using the integrated force method of structural analysis.

IFM/ANALYZERS is the appropriate analysis tool for stress

animation because stress modes are the primary variables of its

eigenvalue analysis. Both stress and displacement animation,

obtained for a number of examples, clearly depict stress and

displacement responses, which peak at different locations in the

structure. Augmentation of animation into the design tool

CometBoards has successfully generated manufacturable

optimum designs for a number of industrial structural

components.

Usefulness of Animation in Analysis and

Design

During the qualification of industrial products through prototype testing, failures quite often occur even though modern

analysis and test methods have reduced the number of such

Lewis Research Center

National Aeronautics and Space Administration

Cleveland, Ohio, October 31, 1996

NASA TM--4729 13
Appendix--The Integrated Force Method Analysis Code (IFM/Analyzers)

The integrated force method of structural analysis, which

parallels the completed Beltrami-Michell formulation in elasticity, has been established for both finite-element discrete

analysis and analysis of elastic continua such as plates and

shells (refs. 19 and 20). The intemal forces (or stress parameters

for continua) are considered as the primary unknowns of this

method. The unknown forces are determined by solving a set of

simultaneous equations which are obtained by augmenting the

rectangular set of equilibrium equations with another rectangular system of equations expressed in terms of the same unknown forces. The augmenting system represents the strain

compatibility conditions. Displacements, if required, can be

obtained from forces by backcalculations. The variational

formulation for the integrated force method has been completed. A dual to the primary IFM has also been formulated by

mapping stresses into displacements. The dual integrated force

method (IFMD), at the equation solution stage, considers

displacements as its primary unknowns. Both IFM and IFMD

have two separate equation sets; one for the determination of

stress parameters and another for the calculation of displacements (see eqs. (1) to (7)). Both IFM and IFMD provide

identical solutions for all examples that have been solved thus

far.

A finite-element structural analysis code which incorporates

both the primal and the dual force methods, along with the

regular stiffness formulation, has been developed especially for

the analysis of airbreathing propulsion engine components.

This structural analysis code, IFM/ANALYZERS, bestows

simultaneous emphasis on stress equilibrium and strain compatibility conditions. The code has a modular organization, and

it is written in Fortran 77 language for both sequential as well

as parallel computational platforms, The IFM/ANALYZERS

currently performs linear elastic analysis of structures for

thermal and mechanical loads as well as free vibration analysis.

The element library contains over a dozen different types of

elements, some with mid-side nodes. Elements in this library

can model a continuous number of arbitrary shapes such as

airbreathing engine components. The numerical analysis segment of the code utilizes sparse unsymmetrical equation solvers of the Harwell subroutine library or, alternatively for

IFM/ANALYZERS

Figure 10.--Organization of IFM/ANALYZERS and its element library.

14 NASA TM-4729
modest problems, utilizes dense solvers of the public domain

LAPACKlibrary. IFM/ANALYZERS is developed as an analysis testbed to examine the full potential of the integrated force

methods since they hold promise of spawning new structural

analysis tools especially for engine structure applications that

can provide accurate solutions with fewer elements in the

model.

Equilibrium matrix, flexibility matrix, mass matrix, thermal

loads, and equivalent mechanical loads for each element are

developed by discretizing potential, complementary, kinetic

energies, thermal and mechanical work terms, respectively.

Interpolation based on the standard stiffness method is used to

TABLE VI.--ELEMENT LIBRARY FOR IFM/ANALYZERS

Element

name

HX0 8,18

:iHX08_33

i HX08 48

"_ HX20_57

HX20_60

HX20_90

TH04_06

TH04_ 18

TH04_21

THIO36

TH I 0 •_39

THI0_48

QD04_05

QD04_07

QD04 12

• --

QD08_I3

QD08_I5

QD08_18

TR03_03

TR03_05

TR03_07

TR06_09

TR06_I 1

TR06_I2

TS02_01

Description

Hexahedral solid element

with 8 nodes

Hexahedral solid element

with 20 nodes

Degrees of

freedom

(fo0

18

33

48

57

60

90

Degrees of

freedom

(dot3

24

24

24

60

60

60

Tetrahedral solid element

with 4 nodes

Tetrahedral solid element

6

18

21

36

12

12

12

30

approximate displacement fields. A general formulation for

stress tensor interpolation has been developed on the basis of

stress function approach. The stress polynomials thus generated represent a complete set that is free from spurious zero

energy modes and satisfies equilibrium in the element field.

The element matrices are not sensitive to the orientation of the

local coordinate systems. All elements of this library pass the

"patch tests". The organization of IFM/ANALYZERS and its

element library are depicted in figure 10 and table VI. An

element name (for example HX20_90) in IFM/ANALYZERS

provides the following information: The first two letters (HX)

describe the geometry (HX stands for hexahedral solid element). The second two numbers (20) represent the number of

nodes, here a twenty-node hexahedral eIement with sixty

displacement degrees of freedom. The last two numbers (90)

represent the force degrees of freedom of the element. The IFM/

ANALYZERS retains the same element name for different

analysis methods, such as IFM, IFMD or the stiffness method.

The analysis method is specified through executive control

keywords such as IFM !, IFMD !, or STIFF! for primal, dual, or

stiffness analysis, respectively.

The solution of a cantilevered beam, earlier used for animation as in example 2, using the IFM/ANALYZERS (HX20_90

element) as well as MSC/NASTRAN analyzer (twenty-node

CHEXA element), are given here for the purpose of illustration.

The beam geometry and discretizations are depicted in figure 3.

The cantilever beam is subjected to a concentrated load of•10 lb

with 10 nodes 39 30 which is distributed among the eight nodes at its free end.

48 " 30

_. _ Results obtained for two models (a three, element model and a

Quadrilateral membrane ! : 5 12 _ : !x-element model)are dep)cted m table VII, The closed form

element with 4 nodes i 7 12 i strength of material solution for the beam is as follows:

' , 1:2 ,,.:12 ' . :

Quadrilateral membrane 13 24 Maximum displacement: 8ti p = 2.304x10 -3 in.

element with 8 nodes 15 24 Maximum stress: CYsup = 720 psi

18 24 Fundamental frequency: f= 224.3 Hz

Triangular membrane 3 9

element with 3 nodes 5 9 The tip displacement and frequency given by the strength of

7 9 material formulae are quite accurate. However, stresses

Triangular membrane 9 18 obtained by the beam formula and three-dimensional elasticity

element with 6 nodes 11 18 analysis can differ for the problem because of boundary

12 18 restraints and the participation of Poisson's ratio.

Truss membrane element 1 6 For this problem, the tip displacement for three- and sixwith 2 nodes element models obtained by IFM/IFMD, the stiffness method,

Model

Maximum

von-Mises

stress,

psi

TABLE VII.--- ANALYSIS OF A CANTILEVER BEAM

IFM/IFMD

Maximum

disp!acementl

in 103in. (lst mode) in 103in. (lst mode) stress, in 103 in. (lst mode)

psi

Six/HX 20_90 629.7 2.270

Three/HX20_90 573•..553 ' 2.214

226.339 2.245 228.031 355.9 2.250 228.031

: . _

230.549 2.161 234.049 348.9 2.486 234.049

NASA TM-4729 15
and MSC/NASTRAN agreed well with an error of about 1.475,

2.56 and 2.343 percents, respectively. Fundamental frequency

obtained by the three different analyzers also agreed within

one-percent error. There is a remarkable difference in stress

field accuracy between the force and displacement method

analyzers. IFM/ANALYZERS stress converged to 629.7 psi

whereas MSC/NASTRAN provided a value of 355.9 psi for the

six HX20_90 elements model. Stresses by IFM/ANALYZERS

improved to 690 psi (corresponding to a 0.2 percent accuracy)

at a location 0.5 in. away from the restrained nodes when an

irregular model with seven elements was used: The von-Mises

stress contours obtained by IFM/ANALYZERS are depicted in

figure l l(a) and those obtained by MSC/NASTRAN are

depicted in figure 1l(b). From figure 11, we observe the

following:

(1) The neutral plane is well identified by both IFM and

MSC/NASTRAN analyzers.

(2) The stress patterns by IFM/IFMD and MSC/NASTRAN

also look similar along the entire volume of the beam.

(3) The peak stress value depicted by MSC/NASTRAN is

only 56.5 percent that of IFM/ANALYZERS.

(4) Only IFM/IFMD analyzers produce accurate stresses.

Stresses predicted by MSC/NASTRAN are too low.

All the analyzers (IFM/IFMD, stiffness, and MSC/

NASTRAN) can provide acceptable values for displacements

and frequencies. Stress accuracy is different between IFM/

ANALYZERS and MSC/NASTRAN analyzers. Only the IFM/

ANALYZERS can provide accurate stresses in addition to

correct displacements and frequencies.

(a_

Neutral plane

629.7 I

588.7

i_ .lli_s_i_

:ii:.:'Niii_:: _!_iiiiiiiI

547.7 _®_ll_

ii

:!

506.7

301.7 I

355.9

333.4 _!

o.8 iiill

288.2

Neutral plane

(b) 175.3 I

Figure 11.--Von_Mises stresses for a cantilevered beam. (a) IFM/ANALYZERS: maximum ervon_M = 629.7 psi with six

HX20_90 elements. (b) MSC/NASTRAN analyzer: maximum ¢rvon_M = 355.9 psi with six twenty-node CHEXA elements.

16 NASA TM-4729
References

1. MSC/NASTRAN Quick Reference Guide, Ver. 68, MacNeal-Schwendler

Corporation, Los Angeles, CA, 1994.

2. P3/PATRAN User Manual, Publication No. 903000, vol. 3, PDA

Engineering, Patran Division, Costa Mesa, CA, 1993.

3. Argyris, J.H.: ASKA-Automatic System for Kinematic AnalysismA

Universal System for Structural Analysis Based on the Matrix

Displacement/Finite Element Method. Nucl. Eng. Des., vol. 10,

Aug. 1969, pp. 441-455.

4. ANSYS User's Manual, Swanson Analysis Systems, Inc., Houston, PA,

1994.

5. ABAQUS Theory Manual, Hibbitt, Karlsson & Sorensen, Inc., Providence,

RI, 1993.

6. Patnaik, S.N.; L. Berke, L.; and Gallagher, R.H.: Integrated Force Method

Versus Displacement Method for Finite Element Analysis. NASA

TP-2937, 1990.

7. Patnaik, S.N.; Hopkins, D.A.; Aiello, R.A.; Berke, L.: Improved Accuracy

for Finite Element Structural Analysis Via a New Integrated Force

Method. NASA TP-3204, 1992.

8. Patnaik, S.N." Yadagiri, S." Frequency Analysis of Structures By Integrated Force Method, J. of Sound Vib., vol. 83, July 1982, pp. 93-109.

9. Patnaik, S.N.; Hopkins, D.A.; and Coroneos, R.M.: Structural Optimization with Approximate Sensitivities, Comput. Struct. vo158, no. 2, Jan,

1996, pp. 407-418 (NASA TM-4553).

10. Patnaik, S.N.; and Gallagher, R.H.: Gradients of Behavior Constraints and

Reanalysis Via the Integrated Force Method For Structural Analysis. Int.

J. Numer. Methods Eng., vol. 23, Dec, 1986, pp. 2205-2212.

11. Patn_k, S,N.; Berke, L." and Gallagher, R.H.: Compatibility Conditions

of Structural Mechanics for Finite Element Analysis, AIAA Journal,

vol.29, May, 1991, pp. 820-829.

12. Patnaik, S.N.: The Integrated Force Method Versus the Standard Force

Method. Comput. Struct., vol. 22, no. 2, 1986, pp. 151-163.

13. Anderson, E. et al." LAPACK Users' Guide. Second ed. Society for

Industrial and Applied Mathematics, Philadelphia, PA, 19915.

14. HARWELL Subroutine Library Specifications, Harwell Laboratory,

Oxfordshire, England, 1990.

I5. Third U.S. National Congress on Computational Mechanics. Abstracts.

Texas A & M University, College Station, TX, 1995.

16. Patnaik, S.N.; Guptill, J.D.; and Berke, L.: Merits and Limitations of

Optimality Criteria Method for Structural Optimization, NASA

TP-3373, 1993.

17. Guptill, J.D." Coroneos, R.M.; Patnaik, S.N.; Hopkins, D.A.; and Berke,

L.: CometBoards Users Manual, NASA TM-4537, 1996 .......

18. Gendy, A.S.; Patnaik, S.N." Hopkins, D.A.; and Berke, L.: "Optimization

of Space Station Components Using Code CometBoards',, Int. J. of

Computers Methods in Applied Mechanics and Engineering, vol. 129

(1996) pp. I33-149.

19. Patnaik, S.N.: Variational Energy Formulation for the Integrated Force

Method. AIAA Journal, voI. 24, 1986, pp. 129-137.

20. Patnaik, S.N.; Kaljevic, I.; Hopkins, D.A.; and Saigal, S.: "Completed

Beltrami-Michell Formulation for Analyzing Mixed Boundary Value

Problems in Elasticity", AIAA Journal, vol. 34, no. 1,1996, pp.143-148.

NASA TM-4729 17
Form Approved

REPORT DOCUMENTATION PAGE OMB No. 0704-0188

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 13. REPORT TYPE AND DATES COVERED

April 1997 I Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Dynamic Analysis With Stress Mode Animation by the Integrated Force Method

6. AUTHOR(S)

Surya N. Patnaik, Rula M. Coroneos, and Dale A. Hopkins

7. PERFORMING ORGANIZATION NAME(S)AND ADDRESS(ES)

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WU-505-63-5B

8. PERFORMING ORGANIZATION

REPORT NUMBER

E- 10077

10 ....SP ONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM-4729

11. SUPPLEMENTARY NOTES

Responsible person, Rula M. Coroneos, organization code 1470, (216) 433-5205.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category 39

This publication is available from the NASA Center for Aerospace Information, (301) 621-0390.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

Dynamic animation of stresses and displacements, which complement each other, can be a useful tool in the analysis

and design of structural components. At the present time only displacement-mode animation is available through the

popular stiffness formulation. This paper attempts to complete this valuable visualization tool by augmenting the

existing art with stress mode animation. The reformulated method of forces, which in the literature is known as the

integrated force method (IFM), became the analyzer of choice for the development of stress mode animation because

stresses are the primary unknowns of its dynamic analysis. Animation of stresses and displacements, which have been

developed successfully through the IFM analyzers, is illustrated in several examples along with a brief introduction to

IFM dynamic analysis. The usefulness of animation in design optimization is illustrated considering the spacer structure

component of the International Space Station as an example. An overview of the integrated force method analysis code

(IFM/ANALYZERS) is provided in the appendix.

14. SUBJECT TERMS

Integrated force method; Dynamic analysis and animation; Stress mode shapes

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OF REPORT

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A03

20. LIMITATION OF ABSTRACT

NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102