SERIITR-253-2101UC Category: 62DE84013004

Analytical Modeling andStructural Response of aStretched- MembraneReflective Module

L. M. Murphy (SERI)D. V. Sallis (Dan-Ka Products)

June 1984 -

Prepared under Task No. 5102.31FTPNo.463

Solar Energy Research InstituteA Division of Mid west Research Institute

1617 Cole BoulevardGolden, Colorado 80401

Prepared for theU.S. Department of EnergyContract No. DE-AC02-83CH10093

TR-2101

PREFACE

The work presented in this report is part of an ongoing DOE-sponsored researcheffort to significantly improve the cost and performance of solar thermal con­centrators by the development of innovative collector concepts. Prior DOE­sponsored research has established the dramatic potential benefit"s of thestretched membrane concept. These benefits are due to the structurally effi­cient design, low weight, simple configuration, likely ease of fabrication,and potentially good optical quality. Our work is directed at increasing ourunderstanding of the structural and optical performance of the. coupledmembrane/frame problem so that we can ultimately develop an optimal stretched­membrane heliostat that can realize the full cost and performance benefits ofthis concept. This work was performed for the U.S. Department of Energy underthe general guidance of Frank. Wilkins and Martin Scheve of the Division ofSolar Thermal Technology.

Branch

Approved forSOLAR ENERGY RESEARCH INSTITUTE

o Thornton, Chieferma! Systems and Engineering Branch

annon, !1a.nagerat Research Division

iii

TR-2101S=~II_I------------------""~~

SUMltARY

Objective

To describe and develop an understanding of the optical and structural loaddeformation behavior of a uniform pressure-loaded, stretched-membrane, reflec­tive module subject to nonaxisymmetric support constraints.

Discussion

To aid in understanding the reflective module behavior, an idealized ana­lytical predictive model was developed and implemented. The stretched­membrane, reflective module is the major element of the innovative light­weight, and potentially low-cost, heliostat concept that is currently underdevelopment. The model is used to study the membrane/frame interactions anddeformation process. This report studies the responses of the membrane/framecombLnat Lon to variations in pressure and tension. It describes the varia­tions of applied lateral shear load, frame lateral deflection, and frame twistas a function of distance between the supports. The accuracy of the resultsis confirmed with the NASTRAN structural computer code. The Nastran code isalso used to provide more detailed insight on specific aspects of thedeformation process and to help establish the limits of applicability of thesimple model. A simple optical surface error model is also developed andapplied to translate frame and membrane surface deformations into quantifiableoptical accuracy measures for the assembly. This report focuses on areflec­tive module with a single structural membrane.

The pertinent structural deformation phenomena of thereflector studied and presented in this report are intendedin the optimization of the design of the stretched-membranelish appropriate design criteria for this concept, anddevelopers of this concept.

stretched-membraneultimately to helpconcept, to estab-to aid potential

R.esults and Conclusions

The analysis demonstrates the need to consider the coupled problem rather thanthe independent assessment of the frame or membrane. The findings of thisanalysis indicate that

• The membrane tension can significantly amplify out-of-plane lateral framedeformations relative to an uncompressed, laterally loaded frame. Thisamplification effect, which increases lateral frame deformation non­linearly with tension, can also magnify initial frame imperfections dueto manufacturing tolerances. However, for a given tension level thisamplification effect is constant, resulting in linear deformationincreases as the lateral load and initial imperfections are increased.

• The twist/lateral deflection coupling of the frame is significant and,depending on membrane attachment approach and design, requires a carefultrade-off between torsional stiffness and bending stiffness to minimizelateral frame deformation and hence the associated optical inaccuracies.

iv

TB-2101

• If the membrane is attached to the frame, so that its plane does not passthrough the frame shear center, an initial twist of the frame will bepresent prior to lateral pressure loading.. This initial twist canamplify out-of-plane deformations and can induce a peak bending stress inthe frame that is up to three times as great as the average compressivestress.

• For the range of tensions corresponding to single-membrane designs anti­cipated for commercial use, the axisymmetric sag of the membrane due towind-induced pressure loading leads to significantly more optical errorthan the effect due to nonsymmetric membrane deformations. These nonsym­metric surface deformations are caused by periodic support constraintsand are usually similar to a "scalloped" effect. The axisymmetriceffect, however, can be almost totally eliminated by using dual membraneswith active pressure and vacuum controls interposed between themembranes.

• Membrane attachment procedures can have a significant impact on thereflective module. Thus if the membrane is rigidly attached to the framesuch that it complements the frame-bending stiffness, lateral framedeformation can be reduced by 30% in some cases and possibly more incases of large membrane stiffnesses.

v

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TABLE OF COtrrEllTS

Nomenclature. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ix

1.0 Int roduc t ion•••••••••.•••••.•••••••.••••••..•••••••••••••••.•••.•••• 1

2.0 Model Assumptions................................................... 43.0 Frame Equilibrium••••••••• ~.................................• • • • • • • • 7

References••••.•••••••••••••••••••••••••••••••••.•••••••••••••••••••

Conclusions ••••••••••••••••••••••••••••••.••••••••••••••••••••••••••

Results ••••••••••••••••••••••••.••.•••••••••••••••••••••••••••••••••

Surface Error Model ••• ~ •••••••••••••••••••••••••••••••••••••••••••••

9

12

13

15

28

30

32

36

39

Numerical Solution Approach ••••••••••••••••••••••••••••••••••

Initial State of a Prestressed Perfect Ring ••••••••••••••••••

Frame Equilibrium Equations•••••••••••••••• ·••••••••••••••••••

Membrane Equilibrium and Corresponding Solution•••••••••••••••••••••

Solution of the Coupled Frame and Membrane Equations ••••••••••••••••

Appendix A

Appendix B

Appendix C

4.0

5.0

6.0

7.0

8.0

9.0

vi

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LIST OF FIGURES

2-1 Idealized Stretched-Membrane Reflective Module..................... 4

2-2 Frame and Membrane Cross-Section Details........................... 5

4-1 Axisymmetric and Nonsymmetric Deformation Patterns Caused byLateral Loading and Support Constraints............................ 9

7-1 Maximum Vertical Displacement of the Ring Frame Midway betweenthe Supports as a Function of Applied Lateral Pressure forSeveral Different Membrane Tensions................................ 16

7-2 Maximum Vertical Displacement of the Ring Frame Midway betweenthe Supports as a Function of Membrane Tension for SeveralApplied Lateral Pressures.......................................... 16

7-3 The Vertical Load Applied to the Frame by the Membrane at theAttachment Point as a Function of Circumferential Angle e.......... 17

7-4 Vertical Displacement of the Frame t Frame Rotation t and MembraneSlope at the Attachment Point t All as a Function of theCircumferential Coordinate S....................................... 17

7-5 The Bending and Twisting Moment on the Frame Resulting from theOut-of-Pressure Loading on the Membrane as a Function ofCircumferential Coordinate S....................................... 18

7-6 Vertical Frame Shear Resultant as a Function of CircumferentialCoordinate S....................................................... 18

7-7 Normal Membrane Deformation w as a Function of the CircumferentialCoordinate e for Several Values of the Radius •••••••••••••••••••••• 21

7-8 Normal Membrane Deformation w as a Function of Radial Dist~nce

from the Membrane Center for Two Values of S....................... 21

7-9 The Ratio of RMS Surface Errors (~2/~1) as a Function ofTension in the Membrane•••••••••••••••••••••••••••••••••••••••••••• 23

7-10 ~T and ~1 as a Function of Membrane Tension........................ 23

7-11 Effect of Torsional Rigidity on Vertical Displacement Frame........ 26

A-I Equilibrium Forces on an Incremental Element of the Frame.......... 32

A-2 A Circumferential View of the Frame Cross Section and theRotation Angle of the Frame as Well as the Relative Rotationof the Membrane to the Frame and the Horizontal.................... 34

vii

TABLE

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C-l Initial Stresses and Displacements in an InitiallyPerfect Frame•••••••••.••••••••••••••••••••••••••..•••••••••••••••••

viii

39

a

A

G

h

K

Q

R

r

u

vv

ww

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R(IOOfCLA11JllE

radius of membrane (m)

area of frame eross seetion (m2)

nth Fourier eoeffieient fo~ membrane defleetion solution

numerieal eorreetion matrix defined in Appendix B

displaeement matrix defined in Appendix B

eomponents of displaeement matrix D

Young's modulus for frame material (Pa)

total vertieal displaeement of membrane attaehment interfaee on thering/frame = vee} + p.(e) (m)

shear modulus for frame material (Pa)

vertieal distanee above the plane of the frame centroid, from whiehthe membrane is mounted (m)

half height of frame eross seetion (m)

a leal moment of inertia about the y axis of the frame eross seetion(m )

torsional eonstant of the frame eross seetion (m4)

loeal bending moment about y axis of frame (Nem)

loeal twist moment about z axis of frame (Nem)

ring eompression load (N)

uniform pressure on membrane (Pa)

vertieal load per unit length applied to the frame (N!m)

(a + p) radius to eenterline of frame (m)

radial eoordinate (m)

displaeement veetor defined in Appendix B

initial tension in membrane (N!m)

radial frame displaeement (m)

frame shear resultant (N)

out-of-plane vertieal ring deformation (m)

half-width of frame seetion (m)

membrane deformation, normal to the surfaee, in the vertieal diree­tion (m)

axisymmetrie eomponent of w (m)

nonaxisymmetrie eomponent of w (m)

eenter deformation of edge-fixed, uniformly loaded membrane (m)

ix

TR-2101S=~I Ilfl --------------------'"'''''''--''''-''........

angle with which the membrane meets the frame measuredfrom the horizontal (rad)

RMS surface slope error (rad)

circumferential coordinate (rad)

radial membrane attachment offset, measured from frame centroid (m)

angular integration variable (rad)

local surface rotation vector for the membrane (rad)

frame twist angle (rad)

total stress on frame cross section face in z direction (Pa)

stress on frame cross section face in z direction corresponding tonormal compressive load resultant (Pa)

stress on frame cross section face in z direction corresponding tobending (Pa)

a prime superscript denotes differentiation with respect to 6

and is normal to theand is parallel todirected (normal todirection

= - ~;I =r=a

displacement vector defined in Appendix B

correction vector defined in Appendix B

local coordinates measured from the centroid of an arbitrary circum­ferential (6) frame cross section; x is directed vertically upward

membrane plane; y is directed radially outwardthe membrane plane; z is circumferentiallythe cross-section plane) in the positive 6

a

~

6

p

<\J

Q

~

a

°ZN

°ZB

( ) ,

x,Y,z

+Z

+y

x

TR-2101

SECtION 1.0

INTRODUCTlmi

Tracking collectors, used to concentrate solar radiation for solar thermalapplications, currently represent the most costly part of solar thermal sys­tems, and there continues to be considerable interest in improving the costand performance of these components. The stretched-membrane heliostat concepthas the potential to significantly improve the cost and performance of helio­stat collectors, as has been established by previous DOE-sponsoredresearch [1]. Further, this research has led to the proposal of numerousinnovations. * In the stretched-membrane concept, a reflector film--which canbe metal, polymeric, or a composite--is stretched on a hollow toroidal framethat offers a structurally efficient design, low weight, a simple configura­tion, likely ease of fabrication, and a surface of potentially good opticalquality. Although stretched-membrane research was intended to improve helio­stat concentrator cost and performance for solar thermal applications, thestretched-membrane collector design approach may offer effective cost and per­formance opportunities for improving photovoltaic and solar daylighting appli­cations as well as by providing a very cost-effective tracking platform.

To realize the poten~ial of this concept, an understanding of the structuralresponse of the membrane-frame combination is needed to help in the optimiza­tion of the concep t, to establish appropriate design criteria, and to aidpotential developers of this concept. Issues such as the deformation of the'frame and the membrane, and more importantly the interaction of the frame andmembrane, under various pressure (due to wind) and tension loads need to bestudied. Idealized or simplified models of conceptual stretched-membranereflective modules are useful in developing an understanding and a definitionof how a stretched-membrane reflector should be formed.

The idealized model considered in this report, a stretched membrane mounted ona stiff support frame that in turn is supported by periodic attachments atequidistant circumferentially spaced points, is of interest relative to thedesign, evaluation, and optimization of stretched-membrane heliostats, whichhave been under development for some time [1]. To be of value this idealizedproblem should model reasonably well the real static deformation and opticalaccuracy of stretched-membrane reflectors so that the dominant physical phe­nomena are accurately considered. In the present context, optical accuracymeans the degree to which the desired membrane shape (in a macro sense) can bemaintained under specified loading conditions. If the deformed surface shapecan be predicted then the optical accuracy can be easily quantified by deter­mining the surface slope errors caused by the deformations relative to thedesired shape. Once the surface slope errors are determined, any of a numberof surface error measures can be used to quantify an average surface quality.

*At least four patent applications relating to this concept have been filed andone has been granted (i. e., U. S. Patent No.4, 425,904 issued 17 January 1984to B. Butler).

1

TR-2101S5~lr_,----------------~~

An accurate simple model also is useful to size structural members so that thedeformations of the support frame/ring and membrane are limited to acceptablelevels, especially in the preliminary design effort. Further, such a modelcan aid in the optimization of the reflector module by permitting the rapidscreening of various design approaches. Moreover, the analytical approachenhances the physical understanding of the problem by more readily pinpointingthe interaction of various phenomena such as the amplification of framelateral distortion by the membrane tension. Finally, parametric trade-offsare more easily and cost-effectively performed with the simple model than withlarge and cumbersome finite element codes such as NASTRAN [2].

Either the membrane or the support frame, when taken individually, representsan easily analyzed structure (from a deformation perspective). For membranes,solutions are readily available for both infinitesimal and large finite defor­mations [3-6], while for circular structural frames many standard proceduresare available. Numerous articles describe the response of these frames tovarious loading conditions [7-14]. The more general coupled problem, however,has not been studied extensively in the literature, though some relatedstudies have been performed [18].

The nonaxisymmetric deformation problem resulting from periodic support of theframe while under uniform lateral load is significantly more complex than theuniformly supported frame case. The problem is primarily of practicalinterest because of the structural deformation and optical accuracy issuesassociated with the stretched-membrane reflector and because the frame­membrane assembly represents initially stressed structures subjected to addi­tional incremental deformations.

An understanding of the resulting coupled frame-membrane interaction problemis of interest for the membrane parabolic dish as well as heliostat applica­tions. The frame-membrane combination is also interesting since the membraneand the frame respond in very different ways. For instance, the prestressed(tension) membrane supports the applied load by changing shape (i.e., thedeformation is independent of material properties and the thickness of themembrane), and no strains in addition to those induced by the initial tensionare induced by lateral loading. Of course with a higher order analysis themembrane experiences additional strains, and the solution depends upon thethickness of the raembrane and the e Las t Lc pr ope r t Lee of the welUl>rauematerial. The frame, which is also prestressed (in compression due to themembrane tension), deforms so that additional bending- and tWisting-inducedstrains result as the frame-membrane assembly is loaded normal to the plane ofthe membrane by wind and weight loads. * Further, the frame bending andtwisting deformation interact in a complex manner with the membrane and thenearly constant frame compressive load resultants. Small deformation theorycan be used here since deformations that are large from an optical accuracyperspective are still relatively small from a strain compatibility and struc­tural viewpoint.

2

*The plane of the membrane is represented as if it were perfectly flat in theunloaded condition. In actuality, there will always be a slight curvature ofthe membrane whether caused by a combination of focusing procedures, lateralloading due to wind and weight, or initial imperfections in the membrane-frameassembly.

TB-2101S=~II.l-------------------~

The following sections describe the model assumptions in detail and thenpresent the idealized analytical model that approximately describes the loaddeformation behavior of the stretched membrane. A.surface error model is thendeveloped to allow optical characterization of the deformed surface. Fol­lowing the development, the model is exercised on a typical stretched-membranereflective module, and the results are confirmed with the ~AST~ [2] struc­tural computer code to verify the assumptions and the applicability of themodel. The NASTRAN results are also used to help estab1ish more detailedinsight into specific aspects of the deformation process. It is noted thatultimately the verification of the model as well as the NASTRAN predictionsmust be done experimentally; such experiments are being planned by DOE.

3

TR-2 10 1S=~I.--------------------

SECTIOB 2.0

MODEL ASSUMPTIONS

Consider a circular stretched-membrane reflector support frame assembly, asshown in Figures 2-1 and 2-2, and let the following assumptions hold:

• The toroidal membrane support frame of mean radius R is supported ver­tically at three equidistant points around the circumference. These con­straints approximate the reactions of a tripod support stru~ arrangementsimilar to that found in some heliostat designs [1].

'"~Mg

(a) Perspective view - stretched-membrane reflective module;pin supported at three equidistant circumferential points

~

N

'"Ms

W

2W

(b) Top view - stretched-membrane reflective module

Figure 2-1. Idealized Stretched-Meabrane Reflective Module

4

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I

II

-~l I

w(r.S)

-'PiT 1~ -T--t--r~~1... I

I II I

TI

WW

(a) Frame and cross-secllon detail showing dIsplace­ments and the corresponding directions and appliedloading

/z

-: MdS)N(S)

(b) Perspective of frame and membrane cross sectionshowing internal load resultants and local coordinates

Figure 2-2. Fra.e and Membrane Cross-SeetioD Details

• The supports offer only a vertieal (i.e., perpendieular to the plane ofthe membrane) eonstraint; i.e., the frame is free to rotate at the sup­ports but not to translate vertieally or laterally in the radialdireetion.

• Small deformation theory is assumed for both the frame and membrane.

The membrane earries loads only in tension (00 bending eapabi lity)everywhere ineluding the region near the attaehmeot. The tensions areassumed to be eonstaot and large enough that the deformations do notlead to signifieant variations in the tension loads over the sur­faee. Furthermore, only membrane deformations normal to the surface

5

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are considered, and the membrane is assumed to be loaded normally anduniformly.

Only out-of-plane deformation and twist of the ring are considered(radial shear and radial ring deformations are ignored). Circum­ferentially compressive loads in the support frame are important asare -tihe normally considered twist, vertical shear, and moment resul­tants. The coupling of the out-of-plane deformation with the ring­compressive force must be considered, but the ring compression load isassumed to remain constant around the circumference in all cases.

• The principle of linear superposition is assumed to be valid for both thedeformation and the stress state in the frame and membrane. Thus defor­mations and stresses caused by the lateral pressure load on the membraneare superimposed on the initial prestressed and prestrained state impliedby the initial membrane tension state.

• The frame cross section is rectangular, with external width 2W and height2H, and is uniform around the circumference; wall thickness isarbitrary. *

• Displacement compatibility of the membrane at the support frame interfaceattachment is required.

• The modeL presented in the main body of this report is derived for asingle membrane that forms a plane (of a height h vertically above theplane) that in turn passes through the centroid of the toroidal frame.This model simplifies the presentation, but generalizations to allow mul­tiple membranes, each with different loadings, are easily incorporatedand are presented in Appendix A. The model also assumes a horizontaloffset, from the frame centroid, of magnitude p (in Figure 2-2 p is shownequal to W but it need not be).

*The model we developed is valid for any symmetric cross section; care must betaken to properly define the attachment dimensions. Nonsymmetric cross sec­tions with products of inertia other than zero lead to more coupling termsthan appear in the equations below.

6

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SEctION 3.0

FRAME EQUILIBR.IUM

Let the membrane of radius a have a tension To and let the deformation of theframe be described by the lateral displacement vee) and twist 4>(ar with thedirections shown in Figure 2-2. Further, let the slope of the membrane, rela­tive to the horizontal, at the frame attachment be given by aCe). The localmoment equilibrium equations in the y and z directions are then described,from the presentation in Appendix A, by:

and

o2My _ 3~lz _

ae2 esRaT , _ .!. a2v\ =0\ R aei} o , (3-1)

aHzaa- + My - aTo( 4) - alp = 0 , (3-2)

respectively, where My and Mz are the local bending and twist moments, respec­tively, about the frame centroid at the section defined by the angle e (seeFigure 2-2). The horizontal membrane attachment offset p is also measuredfrom the centroid (p = W in Figure 2-2).

It is important to note that the displacements v and ~ described here, as wellas M and Mz' are the increments caused by the lateral load on the membrane.We alsume that these are added to the values induced by the tensioned membraneduring assembly. Appendix C describes initial displacements and loads in thering for an initially perfect ring after application of the membrane.

Here, as noted earlier, h is the vertical distance above the plane formed bythe frame centroid from which the membrane is mounted. Thus, a nonzero h willresult in a constant initial twist (due to the initial membrane tension T

Q) in

the frame even when no lateral loading of the frame is considered. Appendix Apresents the derivation of these coupled moment equilibrium equations. How­ever, it is instructive to note the physical significance of the last-termgrouping in Eqs. 2-1 and 2-2. The last grouping to the right in Eq. 2-2 rep­resents the applied local twist about the z axis due to the membrane. Thelast grouping to the right in Eq, 2-1 represents two effects. The term ais aresult of the vertical loading caused by the deformed membrane, which in turnresults in shear induced out-of-pla~ be~ing (as in a beam). The second ele­ment in this grouping (involving a viae results from the moment about thelocal y axis caused by the ring compression N(e) (see Figure 2-2), which inturn is induced by the membrane tension. This last effect is analogous to theeffect of an axial compressive force on a laterally loaded straight beam.

~ can be described in terms of the displacements and the frame flexuralr1gidity Ely by the bending moment curvature change relationships [15,16]:

(3-3)

7

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Likewise, the corresponding expression for the frame twist Mz in terms of theframe torsional rigidity KG and the displacements is [15)

Mz

= KGrO~ + .!. ov1R loa R oaJ • (3-4)

The moments My and Mz can be eliminatedresulting in coupled equations for thetwist and bending of the support frame.rium equations are

by combining Eqs. 2-1 through 2-4,displacements corresponding to theThe resulting displacement equilib-

for the bending equation and

o , (3-5)

GK[A." 1 ,,]-", +-vR R

EI [ " ]+ r ; - ~ - a Top (~ - a) o , (3-6)

corresponding to the twist equation. Note that the prime superscript denotessimple total differentiation of the quantity with respect to a.

The corresponding boundary conditions for Eqs. 2-5 and 2-6 are then easilydetermined from symmetry conditions at the three supports (i.e., a = 0, 2~/3,

4~/3). Hence, from symmetry arguments only one segment of the ring DUst beconsidered (i.e., from a = 0 to a = 2~/3). Then the appropriate boundary con­ditions are

v(O) = v(;~~ = v'(O) = v'(;1t) = ~'(O) = ~,(;~) = 0.

8

(3-7)

TR-210155'1 1• 1------------------~---=.;;;~

SEaIOR 4.0

IlEHBRANE EQUILIBRIUIl AND CORRESPONDIRG SOLUTION

The linear membrane deformation solution for lateral displacement is easilydetermined from classical analysis. The solution is just the stim of twolinear solutions. The first solution corresponds to an axisymmetric problemand is obtained from Poisson's equation with homogeneous boundary condi­tions. The second linear solution corresponds to the nonaxisymmetric problemwhere the solution is obtained from Laplace's equation with nonhomogeneousboundary conditions. Physically, the second solution corresponds to a uni­formly tensioned membrane, which is unloaded laterally and has a nonzero edgedisplacement equal to the displacement of the frame (at the attachment point)(see Figure 4-1).

(a) W1 axisymmetric portion of membrane deformation

"Scalloped" ordeformed shape

(b) W2 nonsymmetric, "scalloped," membrane shape caused by support constraints

Fi.gure 4-1 • .A1d.sYJlletric and NODSy.lletric Defon.ati.on Patterns Caused byLateral Loading and Support Constraints

9

TR-21015='11111 -.Io.~.lLL.

Let the two membrane displacement solutions described above be denoted by wIand wz' respectively; then the total displacement w is

w = wI + w2 '

and the symmetric membrane deformation wI is determined from

VZw1(r , S) = - ~ 'o

(4-1)

(4-Z)

with the boundary condition

w1

( a , S) = 0 , (4-3)

where P is the uniform pressure applied to the membrane and T is the initialmembrane tension. Furthermore, the nonsymmetric membrane deformation causedby the displacement of the support frame is determined from

(4-4)

with the boundary condition

wz(a,S) = f(8) , (4-5)

where f(S) is the vertical displacement of the ring interface attachment. Thesolution to Eqs. 4-2 and 4-3 is

w = PaZ r1 _ (r)Z] (4-6)1 4To~ a '

and the solution for Wz is determined from a simple Fourier analysis given by

wz(r,S) =l ! (~)n cos nS f2n[v(~) + p~(~)]cos n~ d~ ,n n=1 a 0

(4-7)

where it. is noted that the compatibility relation for the membrane displace­ment at the frame attachment is used in the Fourier coefficient integral andis given by

wZ(a,8) = w(a,S) = v(S) + p~(S) = f(S) • (4-8)

Thus, the total solution for w is determined as in Eq. 4-1 and is given by

PaZ [ (r)Z] 1 (I)w(r,S) = - 1 - - + - I4To a n n=1 (

r ) n f2na cos nS 0 [v(~) + p~(~)] cos n~ d~ •(4-9)

Thus, the explicit dependence of the membrane solution on the frame displace­ment is shown. Further the frame Eqs. 3-5 and 3-6 depend on a(S), which isjust the slope of the membrane at the attachment point. That is, a(S) isdefined by

10

a (e) = _ ow( r , e)Ior r=a

TR-2101

Pa 1 ~=--- L n cos ne2To 1ta n=l f

21t[v(~) + p~(~}] cos n$ d~ ,

o (4-10)

which is found by differentiating Eq. 4-9.

11

TR-21015a'II��I---------------------

SECTION 5.0

SOLUTION OF THE OOOPLED FRAME ARD MEHBRABE EQUATIONS

The coupled membrane frame solution can now be obtained by the simultaneoussolution of the coupled frame displacement (Eqs. 3-5 and 3-6) and the membraneslope (Eq. 4-10).

We arrive at the simultaneous solution of Eqs. 3-5, 3-6, and 4-10 in the fol­lowing manner. First, a constant value for a(e) is chosen. A good startingvalue for a(e) is the slope corresponding to the axisymmetric solution, whichis Pa/(2To) as obtained from Eq. 4-10. Once a(e) is known the bending andtwist equations (Bqs , 3-5 and 3-6) are solved for He) and v(e). The newvalues for He) and v(e) are then substituted into Eq, 4-10, and an updatedsolution for a(e) is determined. The solution for a(e) is now substitutedback into Eqs , 3-5 and 3-6, and the process is repeated as before. Thisiterative process is .carried out until the new value and the previous valuefor He) and v(e), as well as a(e), do not differ beyond a prescribed inac­curacy limit. In practice, no more than 10 to 12 iterations are required toprovide very satisfactory convergence.

12

(6-1)

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SECTIOR 6.0

SUU'ACE ERR.OR Moon

The surfaee aeeuraey of the membrane ean be determined from the surfaee defor­mations by defining surfaee slope ehanges due to the loading. .Theae surfaeeslope changes are determined most easily from the surfaee rotation veetor asdefined in elassieal shell theory [17]. For shallow shells, using eylindriealeoordinates, and negleeting rotations about the normal surface direetions(sinee this rotation component will not lead to optical errors), the surfacerotation veetor C ean be defined to first order by

* 3w + 1 (¢w) +~ =- 3r ee + r:"\aeJ er ,

+ +where ee and e r are the unit base veetors in the a and r direetions, respee-tively. Having determined the surfaee rotation vector, the slope error andsurface aeeuraey are easily found. The most eommon surfaee error measure [19]used to define optieal aeeuraey for refleetors is the root-mean-square (RMS)average of the surface rotation C. Using the RMS average definition, ~ can bedeseribed by

~ =

1/2

(6-2)

where the surface area element dAs is given by

dAs = r dr de •

Substituting Eq. 6-1 into Eq. 6-2 results in

and

~ = l~~r + ~(~t]r dr de jl/2fsfdA

2~ a 1/2f f (VweVw)r dr deo 0 (6-3)

where Vw represents the veetor gradient of the membrane surfaee deformation w.

It is important to note that Eq, 6-3 is valid to first order for shallowshells and henee is applieable for slightly eurved or foeused membranes aswell. The only point to remember is that in Eq, 6-3 w is the inerement indisplaeement from the equilibrium position of the unloaded membrane. Suchfoeused membranes may be developed by methods as deseribed in Ref. 1.

If we rewrite Eq. 4-9 in the form

Pa2 r,w(r,S) = 4T

oL1 - lr\ 2] + r An lr)n eos ne ,\s.j n=l \:a

13

(6-~)

TR-2101

where

(6-5)

then ~ can be rewritten using Eqs , 6-5 and 6-3 and the orthogonality con­ditions in the e integration. ~ can be described by:

I(~ 2 (An~2Jl/2= 2 Wo +! n _ ,

a n=l a

where Wo is the center deflection of a uniform y loaded membrane fixed at theedge (r = a), defined by

Pa2Wo = 4T • (6-7)

o

Thus, as seen in Eq, 6-6, the surface error has two contributions--one fromthe axisymmetric uniformly loaded membrane and one from the out-of-plane sup­port frame displacements. For passive surface control (i.e., no active con­trol of the net pressure on the reflective surface) and for a reasonably stiffframe or from a low-membrane pretension (To)' the error will be dominated bythe first term, and therefore ~ can be described by

~ =: Ii (:0) . (6-8)

An even better approximation for ~ can be obtained by also considering themaximum vertical deflection between the supports Vmax' corresponding to thenonaxisymmetric deformation as well as the axisymmetric deformation, whichresults in

(6-9)

where j is the number of evenly spaced vertical supports. When the netreflective surface is actively controlled, the first term in Eq, 6-9 can beessentially eliminated. Active control might be implemented in a double mem­brane design by controlling the vacuum between the two membranes so that theaverage pressure on the reflective membrane is reduced to zero.

14

TR-ZIGI

SEctION 7.0

RESULTS

As previously noted, the primary purpose of this report is to describe thesimple analytical model and demonstrate the major structural response issuescorresponding to the stretched membrane-frame combination. An additional aimis to describe the model's characteristics and its potential capability forthe design and the prediction of the optical performance of stretched-membraneheliostat reflective modules under pressure loading. Although many. insightscan be gained into the design of such reflective modules and many meaningfultrade-offs can be presented, only a few· will be discussed here. Stabi1ityinvestigations, trade-off s£udies, and parametric variations that show designtrends and preferred technical approaches will be presented in later reports.

Figures 7-1 through 7-6 illustrate some of the model characteristics and capa­bilities. For each of these figures a single-ring configuration is used,which is characterized by the fol~owi~ geometric and sttfUctu2"al properties:R = 5.0 m, p = 0.0, EI = 1.77 x 10 N.~, GK = 0.380 x 10 Nem, h = 0, and aframe cross-section area of 1.66 x 10- m2• These properties correspond to arectangular cross section that is roughly 200 mm (2H) by 75 m (2W) J with awall thickness of 3.2 mm. Hence, for this particular frame, the p = a condi­tion is an artiface used in the analysis to study the offset sensitivities.*

The membrane used in the NASTRAN compa~isons was assumed to be 0.254-mm thickwi th a Young' s modulus of 200 x 10 Pa and a Poisson t s ratio of 0.3.Figure 7-1 shows the maximum vertical displacement of the frame between thesupports as a function of pressure. Over the range of pressures that are con­sidered, the displacement is nearly linear with pressure and is dependent onthe tension. The dependence on the tension results from the interaction ofthe membrane with the frame in two ways. The first response results as themembrane transmits the lateral pressure load to the frame thus satisfying ver­tical equilibrium.

The second interaction effect is that the membrane also induces a compressivestress in the frame, which enhances the vertical displacement of the frame.This is analogous to the effect experienced in a beam that is laterally loadedand compressed along its axis by opposing forces at both ends. It is seenthat at higher tensions, the enhancement becomes greater, as would beexpected. In fact at very high tensions, the enhancements due to the tens ionin the membrane grow quite dramatically in a nonlinear fashion as lateralpressure on the membrane is increased. Figure 7-1 shows that at low tensions,the lateral displacements approach the values predicted for rings loadedlaterally only, with no membrane as predicted by McGuiness [9]. The cor­responding frame deformations, as predicted by NASTRAN [2], are also showri inFigure 7-1. It is seen that the model and NASTRAN predictions agree quiteclosely, with the disagreement increasing with pressure. The simple modelpredictions are slightly less than those predicted by NASTRAN, and the maximumdisagreement occurs at 90 Pa and is less than 5%. For reference, 90 Pa cor­responds to a normal dynamic wind pressure (normal to the reflective surface)that would be experienced in a 1Z-m/s (27-mph) wind.

*Frames with P = a can, however, be designed.

15

TR-2101S='II.,---------------------:!:..!.!:-~u..

Figure 7-2 shows the maximum vertical displacement of the frame member as afunction of tension for several pressure levels applied to the membrane. Evenat fairly low constant pressures, there is a definite nonlinear dependence onthe membrane tension. The nonlinearity is more pronounced at the higher ten­sions and is due primarily to the load-displacement interaction in the com­pressed frame as previously discussed. The increasing nonlinearity with ten­sion is also consistent with the trends anticipated from stability considera­tions. Also, Figure 7-2 shows the NASTRAN predictions, which show quite closeagreement (as in Figure 7-1) over the range of parameters considered.Finally, the large displacements that can occur with very high tensions arenot of major concern for practical and well-designed stable systems since~ inthese cases, the frame would be designed so that the nonlinear displacementregimes and stability limits would be well beyond the normal operatingtensions required for optically accurate frame-membrane combinations.

Figure 7-3 illustrates the variation of the vertically applied load Q(9)caused by the membrane around the frame. As the membrane deforms, a verticaland a horizontal load resultant occur on the frame. As expected, the averageof the vertical load on the frame should equilibrate the lateral applied pres­sure on the membrane, as seen in Figure 7-3 where the horizontal line cor­responds to the average vertical load. It is interesting to note that thevertical load is highest at the supports and lowest midway between the sup­ports because the membrane-frame combination tends to deform between the sup­ports. The normal load is relieved in that region, resulting in a minimumenergy configuration for the system. At the supports, since there is no free­dom of vertical displacement, the structure cannot adjust to help reduce theload.

Figure 7-4 illustrates several displacement and rotational effects, all as afunction of distance between the two supports along the frame for a tension of26,250 N and a pressure of 90 Pa, The top curve marked a shows the slope ofthe membrane relative to the horizontal, at the attachment as a function of ebetween the supports. The slope of the membrane with respect to the hori­zontal is greatest at the supports and smallest at the center between the sup­ports. The second curve illustrates the displacement distribution v(9) of thebeam vertically between the two supports. The variation indicates a maximumdisplacement at the center and zero vertical displacement and slope at thesupports, which is consistent with the boundary conditions. The final curveshows the rotation of the frame ~(e) as a function of distance between the twosupports. The frame actually rotates in a positive direction at both supports(i.e., rolls outward), being unconstrained in that coordinate direction (thisis caused by the twist-bending interaction), and rotates to a minimum negativevalue in the center between the supports with zero rotation points midwaybetween. Figure 7-4 also shows the corresponding NASTRAN predictions for thevarious quantities; again quite close agreement is apparent.

The shape of the vertical displacement curve can be inferred from the boundaryconditions, along with applied loading as shown in Figure 7-3. The shape ofthe membrane attachment angle can be inferred by considering the symmetry con­ditions and the vertical displacement. The symmetry conditions imply zerovariation of a with e at and midway between the supports. The maximum ver­tical displacement of the frame in the center between the supports will resultin a minimum a at that point. The shape of the cI> curve can be inferred from

19

IR-21OJ

the boundary conditions (zero slope at the supports) and the displacement com­patibility conditions. As the frame displaces upward, it will tend to rotatein a negative direction, with a minimmn (i.e., maximum negative) rotationoccurring at the midpoint between the supports.

Figure 7-5 illustrates the predicted bending moments in the frame as a func­tion of the e position along the frame. The bending moment about the y axis,which is due primarily to the applied vertical load and the induced verticalshear, is analogous to the bending moment induced in a straight beam com­pressed at both ends and loaded laterally with a variable vertical load, suchas presented in Figure 7-3. The twisting moment, which is quite moderate com­pared to the bending moment, is induced by the interaction of bending androtation of the frame, which in turn is caused by the membrane. This effectoccurs primarily as a result of the moment curvature relation, as described byEq, 3-6. Considerations of the displacement variations in v and 41, as pre­sented in Eq. 3-7, can predict the illustrated effects. Also the shape of thetwist moment curve can be inferred from symmetry arguments (i.e., twistmoments must be zero at and midway between the ends). In general, there willbe an additional small effect caused by the membrane attachment horizontaloffset (p) from the center of rotation and the deformation of the membrane.More about this will be mentioned later. As before, the NASTRAN predictionscorresponding to the simple-solution predictions show quite good agreement.

Figure 7-6 illustrates the effect of vertical internal frame shear as a func­tion of distance between two supports. The. shear at the ends satisfies theoverall equilibrium conditions; also the shear curve is nonlinear, which iscaused by the variation of the applied vertical shear load described in Fig­ure ?-3 (see also Eq. A-6 in Appendix A).

Figures 7-7 and 7-8 demonstrate the deformation pattern predicted for the mem­brane and also demonstrate the agreement of the simplified solution with theNASTRAN predictions. Figure 7-7 is a plot of the membrane deflection w, as afunction of the circumferential coordinate. The figure presents predictionsfor several values of the radius r. Near the rim (i.e., at r = R = 5.0 m) thedeformation process follows that of the frame very closely, which is requiredby compatibility, and as the radius becomes smaller, the variation becomesattenuated. As the radius approaches zero, there is little a dependence. Theradial decay of the nonsymmetric deformation contribution, as shown in thefigure, can be inferred from Eq , 4-7 (i.e., for w2). It is noteworthy thatalthough the solution for the simple model is linear, there is extremely goodagreement with the NASTRAN solution, and the agreement becomes better as theradius becomes smaller. Even at large radii, only a 2%-3% difference isapparent.

Figure 7-8 shows the dependence of the membrane deformation on the radialcoordinate directly for two values of a.The NASTRAN predictions for thetotal membrane deformation are quite close to the simple solution results thatwould be expected from th'e previous good agreement. Additional curves shownin Figure 7-8 also illustrate how the two components corresponding to theaxisymmetric sag (i.e., wI in Eq. 4-6) of the membrane and to the nonsymmetricportion are induced by the frame deformation (i.e., w2 in Eq. 4-7).

20

TR-ZI0lS=~II_I----------------'---~-==

Physically, the symmetric portion wI corresponds to the deflection of the mem­brane due to the wind and weight load, rot with the edges fixed at zerodeflection. Conversely, the nonsymmetric portion of the deflection Wz cor­responds to zero wind and weight load rot with the membrane edge deflected toconform to the deflected frame. The breakout for these two components ofdeformation is shown for e = 0 deg and for e = 60 deg , Therefore, the non­symmetric deformation contribution results in a net average vertical displace­ment of the membrane caused by the frame deformation. This effect can bevisualized by considering the deformation of the frame between the two sup­ports in a harmonic pattern and then locating the centroid of the harmonicpattern. Since the pattern is nearly symmetric between the minimum and maxi­mum displacements, the centroid will lie approximately halfway between thedisplacement extremes. In Figure 7-8 the nonsymmetric deformation curves fore = 0 deg (minimum frame deformation) and e = 60 deg (maximum frame deforma­tions) are nearly mirror images about a plane, which is about 5.18 mm abovethe zero displacement plane. By taking the midpoint between the deformationextremes of the frame for comparison, a 5.08-mm average displacement ispredicted.

Figures 7-9 and 7-10 illustrate the measures of predicted surface deformationerror as described in Eq. 6-6. Here again, we depict the RMS error contribu­tions from the two displacement contributions along with the total error,which is the root mean square sum of the two RMS error contributions. Theerror contributions ~1 and ~Z correspond to the axisymmetric and nonsymmetricdeformations WI and wz' respectively.

Figures 7-9 and 7-10 both assume that no attempt has been made to activelycontrol the axisymmetric portion of the membrane deformation. Thus thesecurves correspond to a passive design where a prescribed initial curvature isassumed, and the incremental displacement field corresponding to the appliedpressure increment results in ~1 and ~2. Figure 7-9 illustrates the ratio of~ 2 over ~ 1 as a function of the tension in the membrane for the same framesystem considered in the previous figures. This curve is also valid for allpressure levels (or very nearly so) since both displacement contributions arefound to be nearly linear with pressure (see Figure 7-1). Figure 7-9 alsoshows that the ratio of ~2 over ~1 is very small at low tensions and increasesto significant values only as the tension becomes considerably higher. Thusthe axisymmetric contribution ~1 dominates the error process at low-to­moderate tensions for the frame and loading condition being considered here.Further, for stiffer frames ~1 will be dominant over even a larger tensionrange.

In Figure 7-10, the total RMS surface error ~T (solid curves) and ~1 (dashedcurves) is plotted as a function of tension for the same frame as mentionedabove and for two different pressure levels. The pressure levels of 30 Pa and60 Pa would correspond [1] to a wind velocity that impinges the surfacenormally at about 7 mls and 10 mis, respectively. Here again, the dominanceof ~1 as a contribution to ~T is illustrated. The total error measure, whichcorresponds to axisymmetric deformation ~ l' can be approximated quite accu­rately until the tensions exceed about 35,000 N/m (for the frame and loadingunder consideration).

22

TR-210J

It should be noted that changing the frame stiffness will not mitigate ~1 toany appreciable extent. However, ~2 can be easily decreased by increasing thestiffness of the frame to bending and twisting (EI and GK) stiffnesses. Onthe other hand, the only way the designer can decrease ~ 1 for the singlestretched-membrane concept analyzed here is to increase the tension or tooperate the stretched membrane at a lower operating pressure. If necessary,however, active control of the reflector membrane can be used to eliminate, orat least greatly reduce, the effect of ~1 at some increase in cost and com­plexity. It also should be noted that the pressures indicated on Figure 7-10are probably higher than one would design for. Hence, the error associatedwith the axisYmmetric portion could be considerably lower than this figureimplies. However, since the analysis has been performed to illustrate struc­tural response phenomena and the predictions of the model, this issue will notbe discussed further at this point blt will be relegated to a future reportthat will delve into these issues in more detail.

We will now discuss several other important effects that have been studied.First, the effect of the horizontal offset p (see Figure 2-2) is quite moder­ate for the case studied in Figures 7-3 through 7-7. Little change in framevertical displacement occurs when the value of p changes if the pressure isalso adjusted to keep the total load on the membrane constant. For example,if p is changed from 0 to 0.0381 m (which is consistent with the frame crosssection selected), the maximum vertical frame displacement for the two casesis identical to four significant figures. Increasing p to 0.1, 0.2, and 0.3 m(while holding GK and EI constant), the maximum vertical displacementdecreases from 10.15 DUn (for p = 0) to 10.14, 10.12, and 10.09 DUn, respec­tively. Vertical displacement decreases slightly because, due to the offset,the moment arm twists the frame in the opposite direction from which it wouldwant to rotate for the p = a case in the region where the vertical displace­ment is a maximum. Thus, comparing the peak rotations for the p = a case tothe p = 0.3 case, it was found that the maximum positive rotation at the sup­ports (minimum vertical displacement region) increases from 3.9 mrad to4.8 mrad, blt in the region of maximum vertical displacement the negativerotation is retarded from -3.6 mrad to -2.7 mrad.

Though we have not studied the vertical offset problem in detail, the analysisalready done indicates a number of trends. The vertical offset effect h, asshown in Figure 2-2 and as predicted with the simple analysis, is minimal (seeAppendix A, Bqs , A-10 and A-H) and has only a small nonlinear dependence.The simple model results agree with the NASTRAN results as long as the othermodel assumptions hold (primarily the assumption of constant membranetension).

To illustrate this consider a vertical offset case studied with both thesimple model and NASTRAN, where the frame is defined as corresponding to thecases studied in Figures 7-3 to 7-7. The offset h equals 100 rom; the initialuniform ring twist is 37.63 mrad for the initial tension, which we assume is26,250 N/m. For a pressure loading of P = 90 Pa, the simple linear model pre­dicts a maximum vertical displacement between the supports of 10.15 rom and amaximum rotation of 3.99 mrad, for both the offset and zero vertical offsetcases. For the zero offset case, the NASTRAN method predicts a maximum ver­tical frame displacement of 10.45 DUn and a frame rotation of 4.1 mrad. If themembrane is attached to the frame so that only radial loading is experienced

24

TR-2101S=~II.J----------------~"""""'--"'-~

at the attachment, the NASTRAN model predicts a maximum vertical frame dis­placement of 10.44 tnm and a maximum rotation of 4.27 mrad due to lateralloading. If on the other hand the membrane is assumed to be rigidly attachedto the frame circumferentially as well as radially (L, e., as in welding orbonding), then the NASTRAN model predicts a maximum vertical frame displace­ment of 8.89 mm and a maximum rotation of 3.97 mrad , Thus, the difference inthe boundary condition (i.e., attachment procedure) causes a noticeabledecrease in the maximum vertical frame displacement.

For a radial constraint only, the simple model results agree quite closelywith the NASTRAN predictions t where the membrane tensions were found to beconstant over the membrane surface to within 1.0%. For the rigid attachmentcase, the effect of the offset becomes quite noticeable because the bending ofthe frame causes additional compressive or tensile circumferential strainincrements in the membrane near the attachment. Due to bending in the fra1Jle,the circumferential strain increment in the membrane Dl1st be equal to theframe strain increment at the attachment point. In essence, the membraneincreases the effective bending resistance of the frame by the offset. Forthis case, tension nonuniformities of ±16% were observed near the frame.

It is interesting to note that, for the case studied, the rotation (in the zdirection) of the frame did not seem to affect the membrane tension very ~chas we had anticipated. It might also be noted that for very thin membranes orfor low-modulus membranes, the effect of the vertical offset should bemitigated even if the membrane were rigidly attached to the frame. Such con­siderations will be important if high-strength polymer structural membranesare used.

Similar results were obtained when the vertical offset problem was inves ti­gated with NASTRAN for the same frame as discussed above but for differentloading and support conditions. The above effects are accentuated in thiscase, where a tension of 22,750 N/m and zero pressure were used along withfour vertical concentrated loads of alternating sign placed at 90 deg inter­vals around the frame. [The primary purpose of this configuration was tostudy potential stability problems corresponding to the lowest fundamental(n = 2) mode shape.] For the zero offset and the 10o-mm offset (with radial­only constraint) cases, the maximum vertical frame displacement was predictedto be 44.8 mm and 44.7 mm, respectively. The corresponding frame rotationswere 11.56 mrad and 11.43 mrad, respectively. When the membrane was rigidlyattached, for the 100-mm offset case, the maximum vertical frame displacenwentwas predicted to be only 30.0 mm and the peak rotation was 8.28 mrad.

One response of interest that has emerged from evaluating and comparing theNASTRAN results with the results of the simple oodel is that in all caseswhere agreement is good, the membrane tension remains fairly constant thoughthe radial displacement of the frame is not predicted to necessarily remainuniform around the circumference in the NASTRAN results. Thus, the frametends to deform in such a way that the constant tension in the membrane isaccommodated, as long as additional local strains in the membrane are notinduced due to frame bending (L, e., as with the offset effect discus sedabove). It should be noted that the particular frame investigated for most ofthe comparisons in this report is fairly compliant in the radial direction, ascompared with the membrane itself. It is anticipated that if the frame ,is

25

TR-2101

stiffened significantly in the radial direction, more noticeable tension vari­ations might be experienced; these would probably lead to some higher discrep­ancies in the relative predicted deformations by the simple method and themore detailed NASTRAN approach. Hence, since radial stability does not appear(from preliminary considerations) to be an issue, low radial stiffness pre­sents no particular problems and is actually desirable from a design perspec­tive since this should help to smooth out tension variations in the membrane.

The impact that torsional rigidity has on vertical displacement was alsostudied. For the single-membrane case the impact was significant, which canbe seen intuitively from the strong twist/displacement coupling" in Eqs , 3-3through 3-6 and is demonstrated in Figure 7-11. The baseline case noted inFigure 7-11 corresponds to the case that was selected in Figures 7-3 through7-7. As is shown in Figure 7-11, for the baseline case a 50% decrease in tor­sional rigidity will increase the vertical frame deflection by about 50%; buta 100% increase in torsional rigidity will decrease the frame displacement byabout 20%, and an infinite frame torsional rigidity will decrease the verticaldisplacement by about 40% relative to the baseline case. The trend is iden­tical in form to that which would be anticipated [9] for laterally loaded butuntensioned rings.

Though stability will not be addressed in detail here since the simple modelcannot adequately elucidate many of the important stability issues, several

EE"..E

>-c:Q)

EQ)

~ 10.0Q.U)

"'0Q)

Eca...- Infinite GK

4.02.0 3.0

GK (106 N. rn-)

1.0

0.0~- ....L.. --L. L.... ....L..~\o --J

0.0

Cii~ 5.0...Q)>E::::lExca~

Figure 7-11. Effect of Torsional Rigidity on Vertical Displaee.ent Fra.e

26

TR-2101

design features and response mechanisms identified above can affect the sta­bility of the frame-membrane assembly. The most obvious response related to apotential stability problem is the amplification of out-of-plane frame defor­mations by increased membrane tension. Although catastrophic collapse of themembrane-frame (based on analysis to be presented in a later report) does notappear to be a major problem, the tendency to amplify either initial imperfec­tions (due to manufacturing tolerances) or other laterally load-induced, out­of-plane deformations at least to the point of unacceptable optical surfacequality is present at some tension level. Further, the tendency exists forthe membrane to mitigate or amplify lateral deformations in some situations.For instance, if no lateral loading of the membrane is present and the frameis initially warped out of plane, the membrane will introduce a shear load inthe frame that will tend to decrease the vertical frame deformation, but acounteracting twist effect that will tend to increase out-of-plane deforma­tions will also be present.

Other issues that have an impact on stability include dual membranes and off­set effect, which becomes more important as deformations grow. The verticaloffset effect not only becomes roore pronounced at large deformations but theinitial frame twist, induced by applying the membrane to t;he frame, alsoinduces a bending stress in the frame. The bending stress can potentia11yaffect gross buckling distortion and particularly local wall stability ofthin-walled frames as well. To get a feel for the magnitude of the prestresscaused by a vertical offset, consider the same case as presented inFigures 7-3 through 7-7. A membrane attached to the center of the frame underinvestigation would induce a nearly uniform compressive stress of about 78 MPa(10.7 ksi); however, if a single membrane were attached at a height of 100 mm(h in Figure 2-2) above the frame centerline, then an additional bendingstress of 143 MPa (20.7 ksi) would be present. Thus, for this case the peaklocal stress could be magnified three times (over the average compress ivestress) by the offset. Finally, it might also be noted that the vertical off­set effect, unless counteracted by a second rear-surface membrane, can causefurther amplification of initial vertical frame distortions by virtue of theinitial twist.

27

TB-2JOJ5='I,_,-----------------&..~loL.I.

SECTION 8.0

COIfCLUSIORS

This report has identified a number of important structural response anddesign considerations that have an impact on the deformation performance andoptical accuracy of stretched-membrane reflectors. The design considerationsthat were identified have been based on the analyses of sing1e-membranemodules. However, based on comparative analyses of the single-membrane con­cept with a vertical offset, we feel that many of the design considerationswill also be directly related to double-membrane concepts. The model alsoprovides a useful method for assessing the coupled membrane/frame problem andfor evaluating structural design approaches and system trade-off studiesrequiring optical accuracy measures.

The model and analyses presented here demonstrate the need to consider thecoupled problem rather than the independent assessment of the ring and/orframe, which becomes especially important as the membrane tension increases.From the results presented above, we see that increased tensions accentuatethe out-of-p1ane frame deformations due to the induced compressive load on theframe. This effect is important not only when considering the lateral pres­sure load and a frame with periodic supports 1:ut also when initial frameimperfections are considered because, as can be seen by analogy, theincreasing tension in the membrane will tend to amplify initial imperfectionsin the frame and thus adversely impact the optical accuracy of the assembly.

Ignoring for the moment the direct coupling between the frame and the membraneand considering the membrane to be shaped to the contour of the frame, thetwist/lateral deformation coupling of the frame is an important design con­sideration because nonuniform frame twist can, without the presence of lateralloading, induce out-of-plane frame and surface deformations. Further, a care­ful trade-off between torsional stiffness and bending stiffness of the frameis needed to minimize lateral deformations and the associated optical errors.

Another signficant consideration is the position and method of attaching themembrane to the frame. If the membrane is attached to the frame so that itsplane does not pass through the frame shear center, an initial twist of theframe will be present prior to lateral pressure loading. This initial twistcan amplify initial out-of-plane deformations and can induce a peak bendingstress, which is up to three times as great as the average compressive stress,in the frame. Further, if the membrane is rigidly attached to the frame itcan increase the effective bending stiffness of the frame and reduce the out­of-plane frame and optical distortions.

The model developed in this report provides an accurate and simple method forpredicting the structural response of single-membrane reflective modules inwhich the membrane passes through the frame shear center or in which the mem­brane is offset but attached so that only radial constraint is experienced.The model should also apply to cases in which the membrane is both offset ver­tically and rigidly attached if the membrane material is considerably morecompliant than the frame material (i.e., as with polymer membranes and steelframes). Moreover, the model provides a convenient tool for evaluating and

28

TR-21QlS=~II.'-------------------~~~

screening various designs and for system trade-off studies. Further, themodel provides a conservative estimate for deflections (i.e., it overpredicts)when there is an offset membrane that is rigidly attached to the frame.Finally, the model can be modified, as outlined in Appendix A, to considercertain dual-membrane designs.

Some cautions with respect to the simple model are noteworthy. The- mode1 isnot intended to be used for large deformation situations, stability analyses,"'(;rfor any situation where the assumptions are violated to any significantextent (though it may be worthwhile if used wisely in conjunction with othermore sophisticated stability assessments). For example, the model will giveno insight into the effect of varying tension or membrane shear, which occurswhen large frame distortions dominate. In a sense, the above cautions do notlimit the use of the model since, as noted above, a well-designed reflectivemodule will be both stable and sufficiently stiff for optical performance rea­sons and the resulting deformations will be small from a structuralperspective.

Structural response and performance issues requiring the most immediate fur­ther investigation include: a detailed assessment of the stability of theframe-membrane combination under various loading conditions and constraintconfigurations, the assessment of structural and performance benefits ofsingle- or double-membrane systems, allowable frame imperfections and theiranticipated amplifications under different loading environments, the opt imumnumber and type of frame supports, optical advantages of vacuum focusingversus laminate focusing, and nonuniform membrane surface loading effects.Further, experiments are needed to verify existing analyses and to identifywhere current procedures may be lacking. Finally, dynamic issues should beinvestigated to identify potential problems (such as low-frequency wind­induced vibrations).

29

TR-2101

SEcrIOR 9.0

REFERENCES

1. Murphy, L. M., Technical and Cost Benefits of Lightweight, Stretched­Membrane Heliostats, SERI!TR-253-1818, Golden, co: Solar Energy ResearchInstitute, May 1983.

2. Schaeffer, H. G., MSC!NASTRAN PRlMER--Static and Normal Mode Analysis: AStudy of Computerized Technology, Mount Vernon, NH: Schaeffer Analysis,Lnc , , 1979.

3. Wu, Chien H., "Large Finite Strained Membrane Problems," Quarterly ofApplied Mathematics, Vol. 36, Jan. 1979, pp. 347-359.

4. Wong, F. S., and R. T. Shield, "Large Plane Deformations of Thin ElasticSheets of Neo Hookean Material," ZAMP, Vol. 20, 1969, pp. 176-199.

5. Yang, W. H., and W. W. Feng, "Axisymmetrical Deformations of Non-LinearMembranes," J. Appl. Mech., Vol. 37, 1970, pp , 1002-1011.

6. Fenner, W. J., and C. H. Wu, "Large Plane to Surface Deformations of Mem­branes with Inclusion," J. Appl. Mech., Vol. 87, June 1981, pp. 357-360.

7. Hogan, M. B., "Circular Beams Loaded Normal to the Plane of Curvature,"J. Appl. Mech., Vol. 5, 1983.

8. Tabakman, H. D., and H. P. Valentijn, "Distortion of Circular Rings,"Mach. Des., 1964.

9. McGuiness, H., Solution of a Circular Ring Structural Problem, Tech.Rep. 32-178, Pasadena, CA: California Institute of Technology, 1961.

10. Meek, H. R., "Three-Dimensional Deformation and Buckling of a CircularRing of Arbitrary Section," J. Eng. Lnd , , 1969.

11. Blake, A., "Stress in Curved Beams," Mach. Des , , 1974.

12. Levy, Roy, "Displacements of Circular Rings with Normal Loads," Proc. Am.Soc. Civil Eng., J. Struct. Div., Vol. 88, No. I, Feb. 1962.

13. Blake, Alexander, "Deflection of a Thick Ring in Diametral Compression,"J. Appl. Mech., Vol. 26, No.2, June 1959.

14. Meek, H. R., "Three-Dimensional Deformation and Buckling of a CircularRing of Arbitrary Section," J. Eng. Ind., Vol. 91, No.1, Feb. 1969.

15. Timoshenko, S. P., Theory of Elastic Stability, Second Edition,New York, NY: McGraw Hill Book Company, 1961, pp. 313-318.

30

!B-2] 01

16. Oden, J. F., and E. A. Ripperger, Mechanics of Elastic Structures, 2ndBd , , New York, NY: McGraw Hill Book Company, 1981.

17. Mollmann, H., Introduction to the Theory of Thin Shells, New York, NY:John Wiley and Sons, 1981, pp. 160-161.

18. Mitchell, G. C., "Analysis and Stability of Floating Roofs, II Froc. Am.Soc. Civil Eng. J. Eng. Mech. Div., Vol. 99, No. EM5, Oct. 1973,pp. 1037-1052.

19. Biggs, F., and C. N. Vittatoe, The Helios Model for the Op"tical" Behaviorof Reflecting Solar Concentrators, SAND 76-0347, Albuquerque, NM: SandiaNational Laboratories, Mar. 1979.

31

TR-2IOIS=~II_)--------------------=-==-===-

APPERDfi A

FRAME EQUILIBR.IUK EQUATIOBS

Consider an incremental frame element of length Me as shown in Figure A-Iaand b where the coordinate directions are as described in Figure 2-2 in themain body of the text. First consider the moment balance for twist about thepoint 0 in the figure; then in the z direction

(Mz + ~Mz) cos ~e - Mz + (My + ~My) sin ~e + MoMe = 0 t (A-I)

ITIIIIIIIIJQ

V + I1V

..ong

V

M---NoN-,--..,

~de------=t'"de

a) Radially outward view of frame element

b) Top view of frame element

Figure A-I. Equilibrium Forces on an Incremental Element of the Frame

32

TR-210I

where Mo is the applied torque per unit length caused by the membranemeasured along the circumference of the frame centroid (see FigureThen let ~e approach zero and Eq. A-I now becomes

3MzM + My + MoR = 0 •

Consider now moments in the y direction (about point 0), then

and asA-lb).

(A-2)

(My + 6My) cos 69 - My - (Mz + 6Mz) sin 69 + (v + ~V)M9 + NAv o , (A-3)

where V is the shear resultant as defined in Figure A-I. N is the hoop com­pressive force on the fraDE caused by the radially inward membrane inducedload.

Again, letting AS approach zero results in the following equation:

oMy ovas - Mz + VR + ~ = 0 • (A-4)

It is interesting to note here that if Eq. A-4 is solved for V, contributionsin addition to those from ~ result. That is, coupling effects with the twistmoment Hz, and with the ring compressive force N, appear.

The shear resultant can be determined from the vertical equilibrium, or

v + 6V - V + QM9 = 0 , (A-S)

where Q is the vertical load per unit length caused by the membrane, whichresults· in

as 69 approaches zero.

av + QR09 o , (A-6)

By using Eq. A-6, Eq. A-4 may now be written as

o • (A-7)

It now only remains to determine Q and Mo as a function of the membrane ten­sion T , the geometry, and the deformations. First consider radial equilib­rium of the frame, which requires that

N = Toa • (A-B)

Then from Figures A-lb and A-2 it is seen that for an element of length R69,

QM9 = To~ea ,

33

TR-2101S=~II_I-------------------

_To

Circumferential view of frame-cross section

Figure A-2. A Circumferential View of the Frame Cross Section and theIlotation Angle of the Fr3lle as Well as the Relative Ilotationof the Meabrane to the Frame and the Horizontal

or

(A-9)

In determining the applied torque about the frame centroid M , the deforma­tions of the membrane relative to the frame I1IUst be considergd. Further byassuming linear superposition, only the moment increment contribution corre­sponding to the lateral loading is of interest, though there in general may besome initial applied moment about the frame due to a nonzero value of h*.This moment increment can be determined by taking the difference in themoments, induced by the membrane, before and after the lateral load is appliedto the membrane. Taking such a difference results in the applied torquemoment increment, due to the lateral load and deformation, about the framecentroid, which is given by

(A-lO)

*The initial deformation and stress state of an initially perfect ring is givenin Appendix c.

34

TR-2101

Then by taking only the first order terms. in ex and 4>, the appropriate valuefor MoR is

(A-II)

It is seen from Eq, A-lO that even if an initial applied moment about theframe is present, due to a nonzero h (of amount - hToa/R), it will iinpact theapplied moment increment caused by lateral pressure loading of the membraneonly in second order. Thus while the nonzero h and the corresponding initialstress state will impact stability considerations, it will have a negligibleeffect on the small deformation problem associated with a stable configura­tion. It should be noted, however, that the discussion above is valid only ifthe assumption of constant tension remains valid. It is anticipated that sig~

nificant inaccuracies may result as this assumption is violated.

The desired equilibrium equations can now be written by combining Eqs. A-9 andA-II with Eqs. A-7 and A-2, respectively. The resulting equations are givenby

oro , (A-I2)

a2My _ aMz _ T Ra ra _ (1 a2v) 1= 0 •oe2 oe 0 l R 08 2

(A-I3)

The last issue to be raised is related to the effect of multiple membranes.Two membranes have been suggested for focusing purposes [1] where a vacuumwould be introduced between the surfaces to induce curvature and hencefocusing. By considering the load introduced by each membrane separately theappropriate equations analogous to Eqs. A-9 and A-II are given by

2QR = a L Toiai , (A-I4)

i=1

and

where the subscriptrespectively.

MaR = ap(itlToitti - 2') . (A-IS)

i (= 1,2) corresponds to the first and second membrane,

Then the governing Eqs. A-12 and A-13 become

~:z + My + ap(itlToitti - 2') = 0 .and

02M oM 2 ( 2 ) 02:....:::I.. - _z - aR L To1ocxi + a L T i ---!. = 0 •08 2 09 i=1 i=1 0 ae 2

35

(A-I6)

(A-I7)

TR-21015=~11_'----------------~~

APPERDIX B

NUMERICAL SOLUTIOR APPROACH

There are numerous suitable approaehes that ean be fo~lowed to attain thesolution of Eqs , 3-5 and 3-6. Sinee a numerieal iterative approach wasseleeted to resolve the a(e) eoupling with Eq. 4-10, a straightforward andeffieient numerieal approaeh for the eoupled Eqs. 3-5 and 3-6 was also deemedsatisfaetory. The approaeh taken for the model presented here was to redueethe order of the two eoupled Eqs. 3-5 and" 3-6 to six first order linear dif­ferential equations. Then, the resulting set of equations ean be written inthe form

(B-1)

where

y = [v,v',v",v"', ~, ~'lT , (B-2)

(B-3)

1

and D is a square matrix of order six. The elements Di j of D are defined by

[~ ~a~ ( ~)( EI)]D43 = EI - EI - 1 + EI 1 + GK

D45 s R +~ )(Zi + pa~t)

D = - .!.. (1 + EI)63 R GK

D65 s (Zi + P~oR).D1Z = DZ3 = D34 = D56

All other terms are zero. The veetor S is defined by

S = [O,0,0,54,0'S61T , (B-4)

where

and (B-5)

RpaToa56 = - GK

The required boundary conditions at e = ° and Z~/3 for Eq. B-1 ean bedeseribed by

* * [ "'" ]Yo = Y(O) = O,O,v (O),v (O),~(O),O , (B-6)

36

TR-21Q1

and

t = t(21t) = rO0 "(21t\ '" (2'Jt\ .f.(21t\ 0]e 3 . l ' ,v 3 7' v 3 "),... 3 "), (B-])

Eq. B-1 is now easily solved by any number of proeedures. A fourth orderRunge Kutta procedure was used for the analysis presented in this report. Theonly remaining difficulty relates to the initial eonditions that' must bespeeified correctly to arrive at the solution. Thus, the initial startingveetor to must be known.

The veetor Yo cannot be eompletely specified initially since v"(O), .v"I(D),and ,(0) are unknown; nor are they arbitrary sinee they must be seleeted tosatisfy Eq. B-7 as well. It is noted that if the solution for f is unique,only one ehodee for v' '(0), v" '(0), and ,(0) will result in the first,seeond, and sixth elements in Eq. B-7 equal to zero. The seleetion of v"(O),v' "(0), and ,(0) ean be made and improved iteratively by using NewtonRaphson's approach in the following manner. Let v"(O), v"'(O), and .(0)form the eomponents of an initializing veetor zo' such that

(8-8)

then define the "end error veetor" by

Ze = [v(;,,).-: (i1t) ... tilt)Y. (B-9)

Thus, if a solution to Eg. B-1 is found, then Ze (the error vector) will beidentieally zero. When Ze La not zero, Zo must be adjusted to reduee thaterror to zero. By using Taylor's theorem we ean express the error vectorZe as a power series expanded about Zoe In terms of the eomponents ofZe (i.e., Zei' i=I,2,3), the following relation is seen to hold

= 0 , (B-10)

where ~Zok is the kth component of ~Zo. In veetor notation (B-10) becomes

(B-11)

where terms of higher order than one have been dropped and where B is a 3x3matrix defined by

B

37

(B-12)

Then solving Eq. B-11 for ~io results in

-+- -+- ( ) -+-~Zo = Zo new - Zo

or

-1 -+- (-+- )-B Ze Zo ,

TR-2101

(B-13)

-+-Thus ~. B-14 provides a simple and convenient way of modifying Zo given anerror Ze. From a practical perspectjve B is easily formulated, one column ata time by varying one element of Zo' while holding the other two elementsconstant. In practice, we found that a(9) is so well behaved that most of thetime only one (never more than two) iteration of Eq, B-14 is required toarrive at a satisfactory solution.

38

TR-2101

APPERDU C

INITIAL STATE OF A PRESTRESSED PERFECT JUNG

Elementary deformation and loads in an initially perfect ring subsequent tothe application of a membrane under uniform tension To are given in Table C-lbelow. Two cases are given. The first corresponds to the membrane beingmounted such that the plane of the membrane passes through the frame crosssection centroid (or center of twist) of the ring frame (i.e., h = 0) suchthat no twist will result. The second case corresponds to the situationwhere the plane of the membrane does not pass through the centroid (i. e. ,h * 0). The variable 0z corresponds to the stress normal to the cross sectionface as a function of x, the distance from the frame neutral axis. Thesubscripts Nand B correspond to average net compressive and bendingcomponents, respectively. Other variables are as previously described.

Table C-l. Initial Stresses and Displacementsin an Initially Perfect Fra.e

Perfect Ring Perfect RingNo Twis t Case No Twist Case

h = 0 h * 0

Applied Moment Mo 0 -hTo(a!R)

Displacementsv 0 0

4J o 0hToRa

EIRaTo RaTo

liO

--- - ""AEAE

Stresses Toa Toa°zN A A

°zB 0 - Toa (x~)Toa -(1 Xh)o"z A Toa A + I

Load ResultantsMyo 0 hToa

Mzo 0 0

Vo 0 0

N Toa Toa

39

TR-2JOlS=~II_I------------------'-~~

The initial conditions in a prestressed initially imperfect ring are more com­plex than as described above, and the final equilibrium configuration of theframe/membrane assembly (not considering lateral loading) will depend on theattachment and alignment procedure. Though the model presented in the mainbody of the report can be easily modified to study these effects, these issueswill be addressed in a later report.

40

SELECTED DISTR.IBUTIOlf LIST

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CompanyMail Stop 9A-42P.O. Box 3707Seattle, WA 98125

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41

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CompanyP.O. Box 3707Seattle, WA 98125

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Aharon RoyHead, Solar Power LaboratoryDepartment of Chemieal EngineeringBen-Gurion University of NegevP.O. Box 653Beer-Sheva 84105, Israel

TR-21015=~1'_' -----------------...,;:;,;::.:;...,;;;;.;..;;.::.

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SOLAR ENERGY RESEARCH INSTITUTE

Dave BensonMike ConnollyAl CzandernaGordon GrossBim GuptaDave JohnsonGary JorgensenKen OlsenLarry ShannonDave SimmsJohn ThorntonRick Wood

Tex WilkinsDepartment of EnergyCE-314, Room 5H-041Forrestal Building1000 Independence Avenue, S.W.Washington, DC 20585

42

Document Control 11. SERI Report No. 12. NTIS Accession No.

Page SERI/TR-253-2l0l4. Title and Subtitle

Analytical Modeling and Structural Response of aStretched-Membrane Reflective Module

7. Author(s)

L. M. Murohy9. Performing Organization Name and Address

Solar Energy Research Institute1617 Cole BoulevardGolden, Colorado 80401

12. Sponsoring Organization Name and Address

15. Supplementary Notes

3. Recipient's Accession No.

5. Publication Date

June 19846.

8. Performing Organization Rept. No.

10. Project/Task/Work Unit No.

5102-.3111, Contract (C) or Grant (0) No.

(C)

(G)

13. Type of Report & Period Covered

Technical Report14

16. Abstract (Limit: 200 words)

The optical and structural load deformation response behavior of a uniform pres­sure-loaded stretched-membrane reflective module subject to nonaxisymmetric sup­port constraints is studied in this report. To aid in the understanding of thisbehavior, an idealized analytical model is developed and implemented and pre­dictions are compared with predictions based on the detailed structural analysiscode NASTRAN. Single structural membrane reflector modules are studied in thisanalysis. In particular, the interaction of the frame-Membrane combination andvariations in membrane pressure loading and tension are studied in detail. Var­iations in the resulting lateral shear load on the frame, frame lateral support,and frame twist as a function of distance between the supports are described asare the resulting optical effects. Results indicate the need to consider thecoupled deformation problem as the lateral frame deformations are amplified byincreasing the membrane tension. The importance of accurately considering theeffects of different membrane attachment'approaches is also demonstrated.

17. Document AnalysisLDe~ri~o~ Deformation; Heliostats; Mechanical Structures; Membranes

b. Identifiers/Open-Ended Terms

C. UC Categories

62e

18. Availability StatementNational Technical Information Serviceu.S. Department of Commerce5285 Port Royal RoadSpringfield, Virginia 22161

Form No. 0069 (3-25-82)

19. No. of Pages

20. Price

52

A04