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A space-truss model for plain-weave coated fabrics
A space-truss model for plain-weave coated fabrics Norris Stubbs and Harold Fluss Department of Civil Engineering, (Received May I9 79)
Columbia University, New York, NY 10027,
USA
A coated fabric element is modelled by a stable geometrically nonlinear space truss and the problem formulated using matrix structural analysis. The solution is determined by solving the system’s equations using the secant method. Results for a range of fabric properties and boundary loadings are presented. Experimental results are presented to demonstrate the predictive capability of the model.
Introduction To cope with increasingly elaborate applications of coated fabrics in tensile and pneumatic structures, several physical models of woven fabric have been proposed to explain the biaxial mechanical response of the material.lm3 Although the objectives and theoretical approaches of these models are similar, they still differ in their geometric complexity, element constitutive properties and method of solution. Freeston er al1 modelled plain-weave fabrics with linear elastic and linear elastic-perfectly plastic, curved yarns, then solved a system of transcendental equations to determine the material biaxial response. Kawabata et ~1,~ assuming an approximate geometry and nonlinear elastic yarns, determined the biaxial response by solving a system of nonlinear algebraic equations graphically. More recently, Testa et ~1,~ using essentially the same geometry as Kawabata but including a linearly elastic isotropic coating membrane, determined the biaxial response by numerically solving a system of nonlinear algebraic equations. Mainly because of the way the problems were formulated and the methods of solution chosen, their present usefulness appears limited to explaining the biaxial response of fabrics subjected to a specified set of boundary loads and displacements. In actuality, however, fabrics sustain a variety of more complex loading conditions. First, all such structures are subjected to dynamic loadings such as wind. Also, they experience substantial temperature variations, depending on their location and the details of internal heating. Furthermore, when fabric panels are prepared for certain tensile structures, specific loading paths are necessary to produce the desired deformation. As a final example, preloaded fabric structures are often subjected to load increments whenever the internal pressure is increased to maintain the shape or to expel snow. Since it is not apparent how any of the above models can be directly used to predict the response of the material subjected to such loading conditions, there is a need to 0307-903X/80/010051-08/$02.00 0 1980 IPC Business Press Ltd
reformulate a model for fabrics which not only reproduces the biaxial behaviour of the material, but one which also provides a convenient and systematic determination of the material response to arbitrary loading conditions, all as a function of arbitrary material element properties. The geometries presented in references 2 and 3 suggest a space-truss model for coated fabrics. By using the truss formulation, effects on the response due to geometric and elastic parameters such as yarn geometry, rotations and crushing, as well as coating properties, can be routinely investigated. Furthermore, by extracting from the extensive reserve of analytic methods developed for trusses, a range of phenomena, which may include large deformations, elastic instability, behaviour under prestress, dynamic response and thermo-elastic response, may be systematically and routinely investigated. Thus the purpose of this paper is to demonstrate the feasibility of modelling coated fabrics by a space truss. If the formulation is shown to reproduce the biaxial static behaviour, then more sophisticated analytic methods may be invoked to study more complex phenomena of interest to fabric applications. To demonstrate the model, a coated fabric element is first represented by a space truss. Next, the governing equations for the three-dimensional structure are developed and solved for a variety of loading cases and a range of elastic properties. Finally, to demonstrate the practicability of the model, geometric and elastic parameters are determined and theoretical predictions compared with experimental results. In the present formulation, results will be restricted to the case of orthogonal loading. How shear effects can be included in the general response will be treated in a subsequent publication. Also, since the objective is solely to demonstrate the feasibility of modelling coated fabrics with a space truss, only results for static loadings will be given.
Appl.
Math.
Modelling,
1980,
Vol.
4, February
51
Space-truss
model
for fabrics:
N. Stubbs and
H. Fluss
The model The smallest unit of an actual fabric consisting of generally curved yarns with arbitrary cross-sections embedded in a coating material, is shown in Figure 1. The membrane stresses ul, u2 (force per unit length) are applied in the one and two directions. For future computations it is convenient to consider the load ratio CY = uz/ur of the membrane stresses. For purposes of analysis, the fabric in Figure 1 is replaced by the physical model in Figure 2. The model is obtained from Figure 1 by approximating the curved yarns with straight rods 1 to 4 and the coating membrane by the uncoupled bars 6 and 7. The rod 5, which is perpendicular to the plane of the fabric, models the flattening or crushing effect. Bars 1 and 2 have length Zr/2 while 3 and 4 have length [a/2. Bar 5 has length (tr + t2)/2 the combined radii of the yarns at the crossover point, where t is the diameter of the yarn. Bars 6 and 7 have length equal to dr and da respectively, where d is the yarn spacing. Similarly, hr and ha, the heights of the bars in the two directions, represent the crimp. Generally, the nonlinear response of the material results from a combination of material and geometric nonlinearities. To contain the complexity of the formulation, all elements will be assumed to obey Hooke’s law and any nonlinearities introduced in the overall response will be considered a consequence of geometric nonlinearities. Furthermore, the bending resistance of the members is assumed to be infinite; therefore, hinges are introduced at all the joints (l-6) to allow for rotations of the yarns. Finally, all loads will be applied at the joints. So, if u is the applied membrane
stress on one side of the fabric, then the load applied at the joint is ad. Note also that for the present formulation, loads are only applied in the plane of the fabric; no loads are applied at joints 5 and 6. Formulated in this manner, the model becomes a stable, geometrically nonlinear space truss. If rods 6 and 7 are removed, the system reduces to a model for the uncoated fabric, and the extent of initial crimp interchange is determined from the resulting stable configuration. The solution to this general structural system subjected to a variety of loads will be presented in the following pages.
Formulation Solutions for geometrically nonlinear space trusses have been developed in matrix structural analysis.4,5 Using the notation in Spillers et aZ.4the following properties are assigned to each bar i: bar deformation unit vector along bar 2. bar force A; bar area bar length Li bar axial stiffness Ki =Aici/Li Ai
(1)
The joint displacement and loads are given by 6i and Pi respectively. We define a bar-node incidence matrixN, the components of which are given by: Eii, if j is positive end of bar i
f
-Eii, if i is negative end of bar i
Nij =
(2)
0, otherwise From the joint and bar properties we may introduce the node-force, node-displacement, bar-force, and bar-deformation matrices, which are given by:
P=
ii]
6= [:-I
F=
[:I
andA=i1]
(3)
where J is the number of movable nodes, and B the number of bars. The elastic stiffness matrix for a space-truss system is given by: Figure
7
Typical
fabric element
KE = EKN = iEIKri
(4)
and if A and C represent the positive and negative nodes of a typical bar i, then: I i’ tIikini_
__ KEi = \
I -nikit?i I
7 -nikiEi -
A
I nikini-
C
I
(5)
I
represents the contribution to the stiffness matrix from that bar. Note that each KEi is a 3J x 3Jmatrix. Similarly, a geometric stiffness matrix is defined as: KG= Figure
52
2
Appl.
Math.
Modelling,
1980,
f i=l
Truss model of fabric element
Vol. 4, February
KGi
(6)
Space-truss
where the contribution
for fabrics:
N. Stubbs and H. Fluss
from each bar is:
I
,
C
A -(I
Fi
model
-!i;;,)
-(I -bigi)-
A
(,l”ii)-
C
(7)
KGi=L I
--(Ilniii,)
The matrix KGi is also a 3J x 3J and I is the identity matrix. Based on these geometric and elastic properties, equations for a discrete system whose members undergo finite rotations and axial deformations when subjected to a load change dP is given by the system’s equation: (KE + KG(F)) 6 = dP
Figure 3
(8)
P 1X
Once the bar deformation determined from: Fi = Ki Ai
Equations (8) (9) and (10) must be satisfied simultaneously to yield the required solution. Returning to the truss in Figure 2, loads in general are applied in the x-z plane at nodes 1 to 4 and the system has 10 degrees of freedom. If only the case of orthogonal loading is considered, the number of degrees of freedom reduces to 6, since there is only one degree of freedom for each joint. Under such conditions, the elastic stiffness matrix becomes:
--K6
0
Method of solution Several methods are available for the solution of nonlinear systems, the more popular being the tangent and secant
0
0
0
-K,GzxGay
-K7
0
0
-K1
0
-K,G,G,z
KE =
-KzG2&2y
K, + K4G:z -K,G,zG,
0
in which Gi, and Giy are the unit vector components
0
F2 (1 -G’,x) 0 ~z
(l-G;,);0
0
K7 + K3G2,,
0
I
‘6y_
-K,Gr,G,,
+ K2G2x2
0
0
0
0
-K,G,,GIY
P5Y
0
-K6 K6
(13) p2
To determine the model response, these values are substituted into equation (8) and the resulting system solved along with the compatability relation, equation (9) and the stress-strain relation, equation (10).
(10)
K6 + KlGlx 2
-p2
=
p4z
(9)
is known, the bar force can be
-p1
p3z
P=
the load matrix
Pl
p2x
- 1) + (2~ -_6,)*~i
finite rotations
Also, since the loads are orthogonal, becomes:
Since the rotations may be large, exact compatability equations must be used. Thus, referring to Figure 3, the equation of geometric constraint to be satisfied by each bar is given by: Ai =Li(&e~i
Bar undergoing
K5
-KxG~yGsz
0
-K,G,zG,y
0
+ K3G3, 2
0
(11)
-K5
+K4G!y
-K5
KlGly+KzGzy 2
2
+K5
of bar i. Similarly, the geometric matrix becomes:
0
0
GlxGly
F,
Ll G2xG2y
0
0
(l-o:,):
0
G3zG3y
F3 -
F2 L,
0
L3
Kg =
(12) 0
0
0
0
0
0
GlxGly
-Fl Ll
G2xG2y
-F2
(1 -GTy);
L2
+(l -G:,,% 1
Appl.
Math.
Modelling,
1980,
Vol. 4, February
53
Space-truss model for fabrics: N. Stubbs and H. Fluss
methods.5y6 Here the secant method was used to iteratively determine the solution of the following system of equations: (KGcF)
+&)(,--I)$)
= dP
n(n) =.%(,--1)~
A,,-,,>
A(n)= Wq,-,,,
n(n), &,>
04)
h(n))
(15) (16)
F (n) = K&j
(17)
where n is the iteration number. Note that equations (15) and (16) are obtained directly from equation (9). The total load P = C dP, where dP is a sufficiently small load increment. For each dP, the process was repeated until the system converged. Convergence was defined by the criterion: IF, - &_,I
001
(18)
After convergence, the system was updated by computing a new elastic stiffness matrix, KE, based on the most recent set of unit vectors. At the same time, the strains developed for that load increment are computed for each bar from the equation: Ai
(19)
Ei =-
010
b Cl08
006-
004-
002 -
Li
The total strain is determined from each increment. Results
by summing the contributions
Slrain,
P/d
Figure 5 Effect of relative coating stiffness on biaxial response. (a), OL= 0; H, = 0.5; (b), (Y = 0.5; H, = 0.5; (--_), el; t--j, 6X
and discussion
Membrane biaxial stress-strain curves for a range of load ratios, geometries and element properties, are shown in Figures 4-8. For convenience, all variables are normalized with respect to the length of member 5, the combined radii of the yarns. Thus dimensions 1, d, and h become nondimensional L, D, and H respectively. Similarly, all membrane stresses are nondimensionalized with respect
to the larger yarn stiffness K,,. Therefore, the nondimensionalized load is given by:
&kg=
CD
Y
where C is the nondimensional membrane stress. Note that the stiffnesses of the bar and the yarn are related by: K,=K,=2Ky
(21)
and assuming that the moduli and areas of the yarns are the same: K3 = K4 = KIL,IL3
(22)
If the stiffness of the coating is K,, then the stiffnesses of members 6 and 7 are related by: K, =Kg=K,L7/L6
Stmln, (%)
Figure 4 H,
54
= 0.5;
Effect of load ratio on biaxial response. (a), N = 0; (-_), E,; (- -). ~2
Appl.
Math.
Modelling,
1980,
Vol.
4, February
(23)
To provide a basis of comparison with other results published for a coated fabric, the parameter N = K6/Ky, which defines the relative stiffness of the coating, is introduced. In summary, for each case considered, the elastic properties of the fabric are specified by KY and N. The geometric properties are defined by the member lengths and unit vectors, and the loading specified by one load and the load ratio, c~. The effect of varying load ratio on the biaxial response is shown in Figure 4. For all loading cases the initial fabric geometry was the same but only the load ratio varied. As observed in experiment and from the results of other models,‘,3 increasing the load ratio has the same effect as stiffening the fabric in the transverse direction. Note also that at certain load levels and load ratio the truss model reproduces other fabric anomalies such as the negative Poisson’s ratio. The effect of relative material properties on the biaxial response is shown in Figure 5. Note that for small values
Space-truss 010
k‘0
,
II
I
1’
I
I
a
0 I;;‘;
OOE- I;;, 14
I 006-l 004-
’ I \
I
\
\ ‘1
oaz-
\\ 0
+=.
Al. Stubbs and
H. Fluss
with
experimental
results
The ultimate test of any model is the degree to which it predicts observed behaviour; thus the purpose of this section is to demonstrate the predictive capability of the space-truss model. To accomplish this end, a series of biaxial tests were performed on fabric specimens, material constants evaluated for the model, then experimental and theoretical results compared.
\\\,
I
for fabrics:
For even more general cases, it is practically feasible to model plain weave fabrics with arbitrary geometries and elastic properties by simply introducing into the model the desired elastic and geometric properties, and the appropriate boundary loads. Comparison
’
model
I
i
Y 010
I
b
cKE4 0.06 -
OO‘-
OCQ-
~
0 --
0
Stmin,
(%I
Figure 6 Effect of yarn crushing on the biaxial response. (a),N=0.1;a!=O;H,=0.5.(b),N=O.1;or=l.O;H,=0.5; (-1,
e,; (- - 4,
Experiment The experimental objective was to determine the load strain curves of several Teflon-coated fibreglass specimens subjected to a range of biaxial load ratios. Five cruciform specimens, a typical one of which is shown in Figure 9, were cut from the same sample with the warp and fill yarns parallel to the anticipated directions of loading. Since the method of strain measurement was based on the Moire’ principle, a grid was printed on to the surface of each specimen. Strains were subsequently measured by placing an analyser grid directly above the flat specimen,
E*
of parameter N, the elastic and geometric properties of the yarn govern the stress-strain response. Also in this range of values, the coating acts mainly to inhibit initial crimp interchange. For higher values of the parameter N, coating elastic properties dominate the response and the composite behaviour approaches that of an elastic plate subjected to a biaxial load. The effect of yarn flattening is shown in Figure 6. To introduce the effect, varying values of KS/K,, were inputed. A value of KS/K,, approaching infinity represents the case of no flattening and KS/KY approaching zero represents the case of no resistance to flattening. Referring to Figure 6, note that for all load levels, fabric stiffness decreases as yarn crushing becomes more dominant. For completely crushable yarns, under uniaxial loading, the response in the loaded direction is only slightly affected whereas the strain response in the transverse direction is significantly reduced. For the same conditions, for nonzero biaxial loads, the present fabric model approaches a trellis and the biaxial response becomes uncoupled.’ As shown in Figures 7 and 8, the initial fabric geometry significantly influences subsequent deformation. This is demonstrated in Figure 7 where the uniaxial stress-strain curves for the same fabric but with differing initial geometry are shown. The strain-reduction effect of decreasing yarn spacing is shown in Figure 8. Finally, to compare the response of the truss model with the model in reference 3, the results for a square fabric with N= 0.1 are compared for a range of load ratios. As seen from Figure 4b, the difference between the two results is insignificant for the range of values shown. This range of fabric behaviour has been systematically obtained by introducing the appropriate parameters into the matrix formulation and solving the system of equations.
Figure 7 (Y = 0; (-4,
Effect of initial crimp on fabric response. N = 0.1; El; (-- - 4, E2
I
I
I F 6006-
I
1 I
I f
I I I 004-
I
I \ \
-6
-4
-2
0
2
4 Strain,
Figure 8
6
a
IO
(%I
Effect of initial yarn frequency on fabric response. N = 0.1; H, 0.5;or = 0; &--_),eI; (---I, e2
Appl.
Math.
Modelling,
1980,
Vol. 4, February
55
Space-truss model for fabrics: N. Stubbs and H. Fluss
800 lbs, approximately one-half the uniaxial strength of the material. A typical load-strain curve is shown in Figure 11. Note that the response is similar to other curves reported in the literature.9 Determination
Figure
9
Typical
of material properties
To compare the model response with the observed behaviour, geometric and elastic parameters for the model must be determined. It is a straightforward matter to directly measure the geometric quantities such as h and d then express them nondimensionally. The effective elastic properties N and K,, are determined from two uniaxial tests, one in the warp and the other in the fill direction. In this procedure an approximate value for N is obtained from a test in the fill direction by comparing the ratio of the slopes at the origin and near maximum load. Then with this value of N and the actual geometry, a set of uniaxial curves are generated using the model. Next, a value of K, is selected so that the experimental points are mapped from the previously determined uniaxial load-strain curves to the corresponding theoretical load-strain curves. If the resulting fit is acceptable, the current values of N and K are selected. If not, N is adjusted and the procedure repeated. This procedure was followed for the experimentally determined uniaxial curves. The resulting fits are shown in Figure 12 along with the effective values of the elastic
test specimen
1
Figure
10 Schematic
of biaxial
tester
and by utilizing techniques such as grid rotation and analyser mismatch, the biaxial strains were determined from photographs of the resulting interference patterns8 The loading program was provided by a biaxial testing machine developed recently at Columbia University. Details of the apparatus, shown in Figure 10, are described elsewhere.8 The same procedure was used to determine each loadstrain curve. After aligning the specimen in the testing machine, the apparent strain at zero load was taken. All other measurements coincided with predetermined values of the load in the principal direction. If the principal load was less than 100 lbs (444 Newtons) strains were taken at every 25 lbs (111 Newtons). If the principal load was greater than 100 lbs (444 Newtons), strains were taken at every 100 lbs (444 Newtons). This procedure allowed for sufficiently accurate determination of all portions of the load-strain curve. Since previous measurements on the Teflon-coated material indicated that if the load is held constant the strain stabilizes after about two minutes, all loads in the present experiment were held for at least two minutes before the strain reading was taken. For all tests, the maximum principal load ranged from 600 to
56,
Appl.
Math.
Modelling,
1980,
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4, February
strain.I%1 Figure
0.010
F
2. $
w
0008
17
Uniaxial
response for Teflon-coated
fibreglass fabric
I
Figure 72 Determination of elastic parameters. Warp: (-1, theoretical; (a), experimental; Fill: (--1, theoretical; (01, experimental. N = 0.01; H, = 0.0; K, = 10 000: K, = 12 500
Space-truss model for fabrics: N. Stubbs and H. Fluss parameters. The values shown in the figure were obtained after only two iterations; better choices could be made by continuing the process. Comparison of experimental and theoretical results The value for N given in Figure 12 and the measured geometry (Or and Da) were substituted into the truss model. Then, using the effective value of K,, the experimental load-strain curves were plotted nondimensionally. These results are shown in Figures 13-15. For all three load ratios there is satisfactory agreement between the theoretical and experimental results. Note that in all cases the agreement improves at higher stress levels. This observation is expected, since the model ignores yarn bending and other effects that may compete with crimp interchange at lower load levels. At higher loads, yarn extension dominates the response. For the biaxial cases shown in Figures 13-15, the quality of the prediction depends strongly on the value of the load ratio. The predicted response in the warp direction is generally better than the predicted fill response, except for the case of (Y= 2. Also, as Q tends to 00, the predicted fill response improves substantially. For the load ratios of l/2 and 1, the model response in the fill direction overestimates the experimental response. This discrepancy can be explained by the fact that the warp yarns were assumed initially straight, although the
0
o-6
-‘4
:
0
o_/ -2
4.
.’ 4
64
8I
lo I
12 I
14 I
stmm.WI Figure 73 Theoretical and experimental comparisons. Warp: (--_), theoretical; (a), experimental; Fill: (- --), theoretical; (0). experimental. Fill load/warp load = 2
J
Figure 14 Theoretical and experimental theoretical; (a), experimental; Fill: (---I, experimental. Fill load/warp load = 1
comparisons. Warp: (-1, theoretical; (01,
strain.,%) Figure 75 Theoretical and experimental comparisons. Warp: (-1, theoretical; (a), experimental. Fill: (---I, theoretical; (01, experimental. Fill load/warp load = 0.5
actual crimp in that direction was finite, but small. This explanation is borne out by the results obtained on changing the crimp in the model from Hr = 0 (initially straight) to Hi = 0.15 (experimental upperbound value). In a typical example, the fill strain at a warp load of 0.15 (o = 1) decreased from 10.5 to 8%, while the warp strains increased from -0.8 to -1 .l%. These results further indicate the sensitivity of the biaxial response to small changes in the initial crimp. More specifically, the results also show that for a fabric with almost initially straight warp yarns, small changes in the warp crimp affect the fill response substantially more than the warp response. Finally, it should be pointed out that the agreement obtained above was based on the adjustments in only K,,(kl) and N, while in all cases K5 was taken as infinite (rigid). As suggested in the previous discussion, an even better agreement can be obtained if other parameters such as Hi, K5 and N are more carefully selected. Additional errors are also introduced in the model as a result of assuming that the fabric yarns are rods, while in fact they are generally curved. All of these assumptions and restrictions considered, the agreement between experiment and theory is quite favourable, especially when the many complex interacting mechanisms active in the response of coated fabric are taken into account _
Conclusions The fabric element is modelled here as a stable geometrically nonlinear space truss, with the objective of defining a model which not only explains the biaxial stress-strain behaviour but also provides a convenient and systematic determination of the material response to arbitrary loadings as a function of arbitrary material properties. Solutions of the truss, for a range of geometric, elastic and loading conditions, show that the model satisfactorily reproduces the observed biaxial response. Comparison with the other models reveals that no significant difference exists between the predictions, for the range of values considered here. In addition, a method of evaluating material and geometric properties has been proposed and the parameters determined from uniaxial tests successfully predict the biaxial response at an arbitrary load ratio. These results demonstrate, in terms of the ease of problem formulation and the quality of prediction that the proposed geometrically nonlinear space truss can be used as a practical model for coated fabrics.
Appl.
Math.
Modelling,
1980,
Vol. 4, February
57
Space-truss model for fabrics: N. Stubbs and H. Fluss Having demonstrated the viability of the model, the extensive reservoir of analytical methods in structural mechanics applicable to trusses may be invoked. By directly utilizing these established methods, phenomena relating to large deformations, elastic instability, prestressing, nonlinear material properties, dynamic behaviour, inelasticity, and thermoelastic analysis, in as much as they relate to fabrics, may be routinely investigated.
Acknowledgements This work was partially supported by a grant from the National Science Foundation (NSF ENG74-18096). The authors also thank the staff of the Carleton Materials Laboratory for their support and Owens-Corning Fiberglass for providing the materials.
References Air Force Materials Laboratory, Ohio, July 1967 Kawabata, S. et al. J. Text. Inst. 1963,54,323 Testa, R. B. et al. J. Eng. Mech. Div. ASCE, 1978, 104 (EM5), Proc. Paper 14070, pp. 1027-1042,1978 Spillers, W. R. et al. J. Franklin Inst. 1978, 306,5 Baron, N. and Vanketeson, M. S. J. Struct. Div. ASCE, 1971, 97 (2), 691 Przemieniecki, J. K. ‘Theory of matrix structural analysis’, McGraw Hill Book Company, New York, 1968 Haas, R. and Dietaius, A. Nat. Advis. Comm. Aeronaut. Rep. No. 16,3rd Ann. Rep., pp. 144-271, translated by K. K. Darrow, 1972
Appl.
Math.
9
Modelling,
1980,
Vol. 4, February
Stubbs, N. and Testa, R. B. Paper presented at spring meeting, Sot. Exp. Stress Anal., 20-25 May, 19’79 Skelton, J. J. Mater., JMLSA, 1971,6 (3), 656
Nomenclature A B D E F G H
J
K k L N
F t 01
Freeston, W. D. Jr. et al. Tech. Rep. AFML-TR-61-44,
58
8
A 6 E z,o
of bar number of bars yarn spacing yarn modulus bar force unit vector component yarn crimp number of joints system and element stiffness yarn length coating to yarn stiffness unit vector applied load yarn diameter load ratio bar deformation displacement strain membrane stresses Area
Subscripts E elastic geometric G bar-number i warp 1 fill 2
Columbia University, New York, NY 10027,
USA
A coated fabric element is modelled by a stable geometrically nonlinear space truss and the problem formulated using matrix structural analysis. The solution is determined by solving the system’s equations using the secant method. Results for a range of fabric properties and boundary loadings are presented. Experimental results are presented to demonstrate the predictive capability of the model.
Introduction To cope with increasingly elaborate applications of coated fabrics in tensile and pneumatic structures, several physical models of woven fabric have been proposed to explain the biaxial mechanical response of the material.lm3 Although the objectives and theoretical approaches of these models are similar, they still differ in their geometric complexity, element constitutive properties and method of solution. Freeston er al1 modelled plain-weave fabrics with linear elastic and linear elastic-perfectly plastic, curved yarns, then solved a system of transcendental equations to determine the material biaxial response. Kawabata et ~1,~ assuming an approximate geometry and nonlinear elastic yarns, determined the biaxial response by solving a system of nonlinear algebraic equations graphically. More recently, Testa et ~1,~ using essentially the same geometry as Kawabata but including a linearly elastic isotropic coating membrane, determined the biaxial response by numerically solving a system of nonlinear algebraic equations. Mainly because of the way the problems were formulated and the methods of solution chosen, their present usefulness appears limited to explaining the biaxial response of fabrics subjected to a specified set of boundary loads and displacements. In actuality, however, fabrics sustain a variety of more complex loading conditions. First, all such structures are subjected to dynamic loadings such as wind. Also, they experience substantial temperature variations, depending on their location and the details of internal heating. Furthermore, when fabric panels are prepared for certain tensile structures, specific loading paths are necessary to produce the desired deformation. As a final example, preloaded fabric structures are often subjected to load increments whenever the internal pressure is increased to maintain the shape or to expel snow. Since it is not apparent how any of the above models can be directly used to predict the response of the material subjected to such loading conditions, there is a need to 0307-903X/80/010051-08/$02.00 0 1980 IPC Business Press Ltd
reformulate a model for fabrics which not only reproduces the biaxial behaviour of the material, but one which also provides a convenient and systematic determination of the material response to arbitrary loading conditions, all as a function of arbitrary material element properties. The geometries presented in references 2 and 3 suggest a space-truss model for coated fabrics. By using the truss formulation, effects on the response due to geometric and elastic parameters such as yarn geometry, rotations and crushing, as well as coating properties, can be routinely investigated. Furthermore, by extracting from the extensive reserve of analytic methods developed for trusses, a range of phenomena, which may include large deformations, elastic instability, behaviour under prestress, dynamic response and thermo-elastic response, may be systematically and routinely investigated. Thus the purpose of this paper is to demonstrate the feasibility of modelling coated fabrics by a space truss. If the formulation is shown to reproduce the biaxial static behaviour, then more sophisticated analytic methods may be invoked to study more complex phenomena of interest to fabric applications. To demonstrate the model, a coated fabric element is first represented by a space truss. Next, the governing equations for the three-dimensional structure are developed and solved for a variety of loading cases and a range of elastic properties. Finally, to demonstrate the practicability of the model, geometric and elastic parameters are determined and theoretical predictions compared with experimental results. In the present formulation, results will be restricted to the case of orthogonal loading. How shear effects can be included in the general response will be treated in a subsequent publication. Also, since the objective is solely to demonstrate the feasibility of modelling coated fabrics with a space truss, only results for static loadings will be given.
Appl.
Math.
Modelling,
1980,
Vol.
4, February
51
Space-truss
model
for fabrics:
N. Stubbs and
H. Fluss
The model The smallest unit of an actual fabric consisting of generally curved yarns with arbitrary cross-sections embedded in a coating material, is shown in Figure 1. The membrane stresses ul, u2 (force per unit length) are applied in the one and two directions. For future computations it is convenient to consider the load ratio CY = uz/ur of the membrane stresses. For purposes of analysis, the fabric in Figure 1 is replaced by the physical model in Figure 2. The model is obtained from Figure 1 by approximating the curved yarns with straight rods 1 to 4 and the coating membrane by the uncoupled bars 6 and 7. The rod 5, which is perpendicular to the plane of the fabric, models the flattening or crushing effect. Bars 1 and 2 have length Zr/2 while 3 and 4 have length [a/2. Bar 5 has length (tr + t2)/2 the combined radii of the yarns at the crossover point, where t is the diameter of the yarn. Bars 6 and 7 have length equal to dr and da respectively, where d is the yarn spacing. Similarly, hr and ha, the heights of the bars in the two directions, represent the crimp. Generally, the nonlinear response of the material results from a combination of material and geometric nonlinearities. To contain the complexity of the formulation, all elements will be assumed to obey Hooke’s law and any nonlinearities introduced in the overall response will be considered a consequence of geometric nonlinearities. Furthermore, the bending resistance of the members is assumed to be infinite; therefore, hinges are introduced at all the joints (l-6) to allow for rotations of the yarns. Finally, all loads will be applied at the joints. So, if u is the applied membrane
stress on one side of the fabric, then the load applied at the joint is ad. Note also that for the present formulation, loads are only applied in the plane of the fabric; no loads are applied at joints 5 and 6. Formulated in this manner, the model becomes a stable, geometrically nonlinear space truss. If rods 6 and 7 are removed, the system reduces to a model for the uncoated fabric, and the extent of initial crimp interchange is determined from the resulting stable configuration. The solution to this general structural system subjected to a variety of loads will be presented in the following pages.
Formulation Solutions for geometrically nonlinear space trusses have been developed in matrix structural analysis.4,5 Using the notation in Spillers et aZ.4the following properties are assigned to each bar i: bar deformation unit vector along bar 2. bar force A; bar area bar length Li bar axial stiffness Ki =Aici/Li Ai
(1)
The joint displacement and loads are given by 6i and Pi respectively. We define a bar-node incidence matrixN, the components of which are given by: Eii, if j is positive end of bar i
f
-Eii, if i is negative end of bar i
Nij =
(2)
0, otherwise From the joint and bar properties we may introduce the node-force, node-displacement, bar-force, and bar-deformation matrices, which are given by:
P=
ii]
6= [:-I
F=
[:I
andA=i1]
(3)
where J is the number of movable nodes, and B the number of bars. The elastic stiffness matrix for a space-truss system is given by: Figure
7
Typical
fabric element
KE = EKN = iEIKri
(4)
and if A and C represent the positive and negative nodes of a typical bar i, then: I i’ tIikini_
__ KEi = \
I -nikit?i I
7 -nikiEi -
A
I nikini-
C
I
(5)
I
represents the contribution to the stiffness matrix from that bar. Note that each KEi is a 3J x 3Jmatrix. Similarly, a geometric stiffness matrix is defined as: KG= Figure
52
2
Appl.
Math.
Modelling,
1980,
f i=l
Truss model of fabric element
Vol. 4, February
KGi
(6)
Space-truss
where the contribution
for fabrics:
N. Stubbs and H. Fluss
from each bar is:
I
,
C
A -(I
Fi
model
-!i;;,)
-(I -bigi)-
A
(,l”ii)-
C
(7)
KGi=L I
--(Ilniii,)
The matrix KGi is also a 3J x 3J and I is the identity matrix. Based on these geometric and elastic properties, equations for a discrete system whose members undergo finite rotations and axial deformations when subjected to a load change dP is given by the system’s equation: (KE + KG(F)) 6 = dP
Figure 3
(8)
P 1X
Once the bar deformation determined from: Fi = Ki Ai
Equations (8) (9) and (10) must be satisfied simultaneously to yield the required solution. Returning to the truss in Figure 2, loads in general are applied in the x-z plane at nodes 1 to 4 and the system has 10 degrees of freedom. If only the case of orthogonal loading is considered, the number of degrees of freedom reduces to 6, since there is only one degree of freedom for each joint. Under such conditions, the elastic stiffness matrix becomes:
--K6
0
Method of solution Several methods are available for the solution of nonlinear systems, the more popular being the tangent and secant
0
0
0
-K,GzxGay
-K7
0
0
-K1
0
-K,G,G,z
KE =
-KzG2&2y
K, + K4G:z -K,G,zG,
0
in which Gi, and Giy are the unit vector components
0
F2 (1 -G’,x) 0 ~z
(l-G;,);0
0
K7 + K3G2,,
0
I
‘6y_
-K,Gr,G,,
+ K2G2x2
0
0
0
0
-K,G,,GIY
P5Y
0
-K6 K6
(13) p2
To determine the model response, these values are substituted into equation (8) and the resulting system solved along with the compatability relation, equation (9) and the stress-strain relation, equation (10).
(10)
K6 + KlGlx 2
-p2
=
p4z
(9)
is known, the bar force can be
-p1
p3z
P=
the load matrix
Pl
p2x
- 1) + (2~ -_6,)*~i
finite rotations
Also, since the loads are orthogonal, becomes:
Since the rotations may be large, exact compatability equations must be used. Thus, referring to Figure 3, the equation of geometric constraint to be satisfied by each bar is given by: Ai =Li(&e~i
Bar undergoing
K5
-KxG~yGsz
0
-K,G,zG,y
0
+ K3G3, 2
0
(11)
-K5
+K4G!y
-K5
KlGly+KzGzy 2
2
+K5
of bar i. Similarly, the geometric matrix becomes:
0
0
GlxGly
F,
Ll G2xG2y
0
0
(l-o:,):
0
G3zG3y
F3 -
F2 L,
0
L3
Kg =
(12) 0
0
0
0
0
0
GlxGly
-Fl Ll
G2xG2y
-F2
(1 -GTy);
L2
+(l -G:,,% 1
Appl.
Math.
Modelling,
1980,
Vol. 4, February
53
Space-truss model for fabrics: N. Stubbs and H. Fluss
methods.5y6 Here the secant method was used to iteratively determine the solution of the following system of equations: (KGcF)
+&)(,--I)$)
= dP
n(n) =.%(,--1)~
A,,-,,>
A(n)= Wq,-,,,
n(n), &,>
04)
h(n))
(15) (16)
F (n) = K&j
(17)
where n is the iteration number. Note that equations (15) and (16) are obtained directly from equation (9). The total load P = C dP, where dP is a sufficiently small load increment. For each dP, the process was repeated until the system converged. Convergence was defined by the criterion: IF, - &_,I
001
(18)
After convergence, the system was updated by computing a new elastic stiffness matrix, KE, based on the most recent set of unit vectors. At the same time, the strains developed for that load increment are computed for each bar from the equation: Ai
(19)
Ei =-
010
b Cl08
006-
004-
002 -
Li
The total strain is determined from each increment. Results
by summing the contributions
Slrain,
P/d
Figure 5 Effect of relative coating stiffness on biaxial response. (a), OL= 0; H, = 0.5; (b), (Y = 0.5; H, = 0.5; (--_), el; t--j, 6X
and discussion
Membrane biaxial stress-strain curves for a range of load ratios, geometries and element properties, are shown in Figures 4-8. For convenience, all variables are normalized with respect to the length of member 5, the combined radii of the yarns. Thus dimensions 1, d, and h become nondimensional L, D, and H respectively. Similarly, all membrane stresses are nondimensionalized with respect
to the larger yarn stiffness K,,. Therefore, the nondimensionalized load is given by:
&kg=
CD
Y
where C is the nondimensional membrane stress. Note that the stiffnesses of the bar and the yarn are related by: K,=K,=2Ky
(21)
and assuming that the moduli and areas of the yarns are the same: K3 = K4 = KIL,IL3
(22)
If the stiffness of the coating is K,, then the stiffnesses of members 6 and 7 are related by: K, =Kg=K,L7/L6
Stmln, (%)
Figure 4 H,
54
= 0.5;
Effect of load ratio on biaxial response. (a), N = 0; (-_), E,; (- -). ~2
Appl.
Math.
Modelling,
1980,
Vol.
4, February
(23)
To provide a basis of comparison with other results published for a coated fabric, the parameter N = K6/Ky, which defines the relative stiffness of the coating, is introduced. In summary, for each case considered, the elastic properties of the fabric are specified by KY and N. The geometric properties are defined by the member lengths and unit vectors, and the loading specified by one load and the load ratio, c~. The effect of varying load ratio on the biaxial response is shown in Figure 4. For all loading cases the initial fabric geometry was the same but only the load ratio varied. As observed in experiment and from the results of other models,‘,3 increasing the load ratio has the same effect as stiffening the fabric in the transverse direction. Note also that at certain load levels and load ratio the truss model reproduces other fabric anomalies such as the negative Poisson’s ratio. The effect of relative material properties on the biaxial response is shown in Figure 5. Note that for small values
Space-truss 010
k‘0
,
II
I
1’
I
I
a
0 I;;‘;
OOE- I;;, 14
I 006-l 004-
’ I \
I
\
\ ‘1
oaz-
\\ 0
+=.
Al. Stubbs and
H. Fluss
with
experimental
results
The ultimate test of any model is the degree to which it predicts observed behaviour; thus the purpose of this section is to demonstrate the predictive capability of the space-truss model. To accomplish this end, a series of biaxial tests were performed on fabric specimens, material constants evaluated for the model, then experimental and theoretical results compared.
\\\,
I
for fabrics:
For even more general cases, it is practically feasible to model plain weave fabrics with arbitrary geometries and elastic properties by simply introducing into the model the desired elastic and geometric properties, and the appropriate boundary loads. Comparison
’
model
I
i
Y 010
I
b
cKE4 0.06 -
OO‘-
OCQ-
~
0 --
0
Stmin,
(%I
Figure 6 Effect of yarn crushing on the biaxial response. (a),N=0.1;a!=O;H,=0.5.(b),N=O.1;or=l.O;H,=0.5; (-1,
e,; (- - 4,
Experiment The experimental objective was to determine the load strain curves of several Teflon-coated fibreglass specimens subjected to a range of biaxial load ratios. Five cruciform specimens, a typical one of which is shown in Figure 9, were cut from the same sample with the warp and fill yarns parallel to the anticipated directions of loading. Since the method of strain measurement was based on the Moire’ principle, a grid was printed on to the surface of each specimen. Strains were subsequently measured by placing an analyser grid directly above the flat specimen,
E*
of parameter N, the elastic and geometric properties of the yarn govern the stress-strain response. Also in this range of values, the coating acts mainly to inhibit initial crimp interchange. For higher values of the parameter N, coating elastic properties dominate the response and the composite behaviour approaches that of an elastic plate subjected to a biaxial load. The effect of yarn flattening is shown in Figure 6. To introduce the effect, varying values of KS/K,, were inputed. A value of KS/K,, approaching infinity represents the case of no flattening and KS/KY approaching zero represents the case of no resistance to flattening. Referring to Figure 6, note that for all load levels, fabric stiffness decreases as yarn crushing becomes more dominant. For completely crushable yarns, under uniaxial loading, the response in the loaded direction is only slightly affected whereas the strain response in the transverse direction is significantly reduced. For the same conditions, for nonzero biaxial loads, the present fabric model approaches a trellis and the biaxial response becomes uncoupled.’ As shown in Figures 7 and 8, the initial fabric geometry significantly influences subsequent deformation. This is demonstrated in Figure 7 where the uniaxial stress-strain curves for the same fabric but with differing initial geometry are shown. The strain-reduction effect of decreasing yarn spacing is shown in Figure 8. Finally, to compare the response of the truss model with the model in reference 3, the results for a square fabric with N= 0.1 are compared for a range of load ratios. As seen from Figure 4b, the difference between the two results is insignificant for the range of values shown. This range of fabric behaviour has been systematically obtained by introducing the appropriate parameters into the matrix formulation and solving the system of equations.
Figure 7 (Y = 0; (-4,
Effect of initial crimp on fabric response. N = 0.1; El; (-- - 4, E2
I
I
I F 6006-
I
1 I
I f
I I I 004-
I
I \ \
-6
-4
-2
0
2
4 Strain,
Figure 8
6
a
IO
(%I
Effect of initial yarn frequency on fabric response. N = 0.1; H, 0.5;or = 0; &--_),eI; (---I, e2
Appl.
Math.
Modelling,
1980,
Vol. 4, February
55
Space-truss model for fabrics: N. Stubbs and H. Fluss
800 lbs, approximately one-half the uniaxial strength of the material. A typical load-strain curve is shown in Figure 11. Note that the response is similar to other curves reported in the literature.9 Determination
Figure
9
Typical
of material properties
To compare the model response with the observed behaviour, geometric and elastic parameters for the model must be determined. It is a straightforward matter to directly measure the geometric quantities such as h and d then express them nondimensionally. The effective elastic properties N and K,, are determined from two uniaxial tests, one in the warp and the other in the fill direction. In this procedure an approximate value for N is obtained from a test in the fill direction by comparing the ratio of the slopes at the origin and near maximum load. Then with this value of N and the actual geometry, a set of uniaxial curves are generated using the model. Next, a value of K, is selected so that the experimental points are mapped from the previously determined uniaxial load-strain curves to the corresponding theoretical load-strain curves. If the resulting fit is acceptable, the current values of N and K are selected. If not, N is adjusted and the procedure repeated. This procedure was followed for the experimentally determined uniaxial curves. The resulting fits are shown in Figure 12 along with the effective values of the elastic
test specimen
1
Figure
10 Schematic
of biaxial
tester
and by utilizing techniques such as grid rotation and analyser mismatch, the biaxial strains were determined from photographs of the resulting interference patterns8 The loading program was provided by a biaxial testing machine developed recently at Columbia University. Details of the apparatus, shown in Figure 10, are described elsewhere.8 The same procedure was used to determine each loadstrain curve. After aligning the specimen in the testing machine, the apparent strain at zero load was taken. All other measurements coincided with predetermined values of the load in the principal direction. If the principal load was less than 100 lbs (444 Newtons) strains were taken at every 25 lbs (111 Newtons). If the principal load was greater than 100 lbs (444 Newtons), strains were taken at every 100 lbs (444 Newtons). This procedure allowed for sufficiently accurate determination of all portions of the load-strain curve. Since previous measurements on the Teflon-coated material indicated that if the load is held constant the strain stabilizes after about two minutes, all loads in the present experiment were held for at least two minutes before the strain reading was taken. For all tests, the maximum principal load ranged from 600 to
56,
Appl.
Math.
Modelling,
1980,
Vol.
4, February
strain.I%1 Figure
0.010
F
2. $
w
0008
17
Uniaxial
response for Teflon-coated
fibreglass fabric
I
Figure 72 Determination of elastic parameters. Warp: (-1, theoretical; (a), experimental; Fill: (--1, theoretical; (01, experimental. N = 0.01; H, = 0.0; K, = 10 000: K, = 12 500
Space-truss model for fabrics: N. Stubbs and H. Fluss parameters. The values shown in the figure were obtained after only two iterations; better choices could be made by continuing the process. Comparison of experimental and theoretical results The value for N given in Figure 12 and the measured geometry (Or and Da) were substituted into the truss model. Then, using the effective value of K,, the experimental load-strain curves were plotted nondimensionally. These results are shown in Figures 13-15. For all three load ratios there is satisfactory agreement between the theoretical and experimental results. Note that in all cases the agreement improves at higher stress levels. This observation is expected, since the model ignores yarn bending and other effects that may compete with crimp interchange at lower load levels. At higher loads, yarn extension dominates the response. For the biaxial cases shown in Figures 13-15, the quality of the prediction depends strongly on the value of the load ratio. The predicted response in the warp direction is generally better than the predicted fill response, except for the case of (Y= 2. Also, as Q tends to 00, the predicted fill response improves substantially. For the load ratios of l/2 and 1, the model response in the fill direction overestimates the experimental response. This discrepancy can be explained by the fact that the warp yarns were assumed initially straight, although the
0
o-6
-‘4
:
0
o_/ -2
4.
.’ 4
64
8I
lo I
12 I
14 I
stmm.WI Figure 73 Theoretical and experimental comparisons. Warp: (--_), theoretical; (a), experimental; Fill: (- --), theoretical; (0). experimental. Fill load/warp load = 2
J
Figure 14 Theoretical and experimental theoretical; (a), experimental; Fill: (---I, experimental. Fill load/warp load = 1
comparisons. Warp: (-1, theoretical; (01,
strain.,%) Figure 75 Theoretical and experimental comparisons. Warp: (-1, theoretical; (a), experimental. Fill: (---I, theoretical; (01, experimental. Fill load/warp load = 0.5
actual crimp in that direction was finite, but small. This explanation is borne out by the results obtained on changing the crimp in the model from Hr = 0 (initially straight) to Hi = 0.15 (experimental upperbound value). In a typical example, the fill strain at a warp load of 0.15 (o = 1) decreased from 10.5 to 8%, while the warp strains increased from -0.8 to -1 .l%. These results further indicate the sensitivity of the biaxial response to small changes in the initial crimp. More specifically, the results also show that for a fabric with almost initially straight warp yarns, small changes in the warp crimp affect the fill response substantially more than the warp response. Finally, it should be pointed out that the agreement obtained above was based on the adjustments in only K,,(kl) and N, while in all cases K5 was taken as infinite (rigid). As suggested in the previous discussion, an even better agreement can be obtained if other parameters such as Hi, K5 and N are more carefully selected. Additional errors are also introduced in the model as a result of assuming that the fabric yarns are rods, while in fact they are generally curved. All of these assumptions and restrictions considered, the agreement between experiment and theory is quite favourable, especially when the many complex interacting mechanisms active in the response of coated fabric are taken into account _
Conclusions The fabric element is modelled here as a stable geometrically nonlinear space truss, with the objective of defining a model which not only explains the biaxial stress-strain behaviour but also provides a convenient and systematic determination of the material response to arbitrary loadings as a function of arbitrary material properties. Solutions of the truss, for a range of geometric, elastic and loading conditions, show that the model satisfactorily reproduces the observed biaxial response. Comparison with the other models reveals that no significant difference exists between the predictions, for the range of values considered here. In addition, a method of evaluating material and geometric properties has been proposed and the parameters determined from uniaxial tests successfully predict the biaxial response at an arbitrary load ratio. These results demonstrate, in terms of the ease of problem formulation and the quality of prediction that the proposed geometrically nonlinear space truss can be used as a practical model for coated fabrics.
Appl.
Math.
Modelling,
1980,
Vol. 4, February
57
Space-truss model for fabrics: N. Stubbs and H. Fluss Having demonstrated the viability of the model, the extensive reservoir of analytical methods in structural mechanics applicable to trusses may be invoked. By directly utilizing these established methods, phenomena relating to large deformations, elastic instability, prestressing, nonlinear material properties, dynamic behaviour, inelasticity, and thermoelastic analysis, in as much as they relate to fabrics, may be routinely investigated.
Acknowledgements This work was partially supported by a grant from the National Science Foundation (NSF ENG74-18096). The authors also thank the staff of the Carleton Materials Laboratory for their support and Owens-Corning Fiberglass for providing the materials.
References Air Force Materials Laboratory, Ohio, July 1967 Kawabata, S. et al. J. Text. Inst. 1963,54,323 Testa, R. B. et al. J. Eng. Mech. Div. ASCE, 1978, 104 (EM5), Proc. Paper 14070, pp. 1027-1042,1978 Spillers, W. R. et al. J. Franklin Inst. 1978, 306,5 Baron, N. and Vanketeson, M. S. J. Struct. Div. ASCE, 1971, 97 (2), 691 Przemieniecki, J. K. ‘Theory of matrix structural analysis’, McGraw Hill Book Company, New York, 1968 Haas, R. and Dietaius, A. Nat. Advis. Comm. Aeronaut. Rep. No. 16,3rd Ann. Rep., pp. 144-271, translated by K. K. Darrow, 1972
Appl.
Math.
9
Modelling,
1980,
Vol. 4, February
Stubbs, N. and Testa, R. B. Paper presented at spring meeting, Sot. Exp. Stress Anal., 20-25 May, 19’79 Skelton, J. J. Mater., JMLSA, 1971,6 (3), 656
Nomenclature A B D E F G H
J
K k L N
F t 01
Freeston, W. D. Jr. et al. Tech. Rep. AFML-TR-61-44,
58
8
A 6 E z,o
of bar number of bars yarn spacing yarn modulus bar force unit vector component yarn crimp number of joints system and element stiffness yarn length coating to yarn stiffness unit vector applied load yarn diameter load ratio bar deformation displacement strain membrane stresses Area
Subscripts E elastic geometric G bar-number i warp 1 fill 2