Bending of glass fibre-reinforced plastic (GFRP) plates on elastic supports Part I : Material Characteristics T. H. TOPPER (1), A. N. SHERBOURNE (2), V. SAARI (3). This study is concerned with the mechanical properties of a fibreglass reinforced polyester resin of laminar construction with consideration given to the application of the results to the analysis, in bending, of a plate of similar material. Tension and compression tests were performed to obtain the stress-strain relationship and ultimate stresses and strains. Because of the inherent scatter of results due to the nature of the material, average values, along with maximum and minimum limits, are determined. Flexural tests were conducted and moment-curvature relationships were obtained both theoretically and experimentally. Creep tests showed that,for loads well below the failure load, creep effects are almost negligible when the fibres are oriented to the longitudinal axis of the specimen. As the results in bending are not as scattered as those for tension and compression, this leads one to believe that they might be more appropriate for use in plate theory. INTRODUCTION The design of blast-resistant structures is complica- ted by a desire to obtain light, flexible assemblies in which ductility and energy absorption are at least as important as simple mechanical strength. The discontinuities or openings in these structures, such as, doors, windows, etc., pose special problems in attempting to devise light, rigid elements, capable of manipulation as desired for human occupancy, yet which continue to retain the flexibility under impulsive load of the overall system. As is also the case with seismic design, energy absorption can be obtained either by encouraging gross displacements in structures or elements composed of materials having moderate to low resistance under load, or conversely by using high or ultra high strength materials which continue to remain elastic, or nearly so, (') Dept. of Civil Engineering, University of Waterloo, Ontario, Canada. (2) EPF Lausanne, Switzerland and University of Waterloo. Ontario, Canada. (3) Formerly Dept. of Civil Engineering, University of Waterloo, Ontario, Canada. until fracture. In most current structural engineering usages the former is to be preferred as, not only is the capacity to absorb energy through plastic deformation substantially higher, but also impending warning of collapse or fracture is usually given through visible distortions of the structure. The capacity of a structure to absorb energy can also be modified by varying the elastic modulus of the material used to accommodate, for example, a more flexible material at a given stress level should one desire to augment the internal energy of resistance. Unfortunately, this method may have severe drawbacks in compromising the performance characteristics of the structure under normal service conditions. The present investigation involves the determination of the static response to uniform pressure of fibreglass reinforced polyester rectangular plates set in neoprene surrounds; these structural components are meant to simulate the doors of a lightweight, portable, blast resistant shelter. The static behaviour also provides a first approximation to the more complex impact analysis, where deflection response under static uniform load may be deemed to furnish certain clues to plate performance at large displacements should an "equivalence" to the blast loading pulse over 75
Transcript
Bending of glass f ibre-re inforced plastic (GFRP)
plates on elastic supports
Part I : Material Characteristics
T. H. TOPPER (1), A. N. SHERBOURNE (2), V. SAARI (3).
This study is concerned with the mechanical properties of a fibreglass reinforced polyester resin of laminar construction with consideration given to the application of the results to the analysis, in bending, of a plate of similar material.
Tension and compression tests were performed to obtain the stress-strain relationship and ultimate stresses and strains. Because of the inherent scatter of results due to the nature of the material, average values, along with maximum and minimum limits, are determined. Flexural tests were conducted and moment-curvature relationships were obtained both theoretically and experimentally. Creep tests showed that,for loads well below the failure load, creep effects are almost negligible when the fibres are oriented to the longitudinal axis of the specimen. As the results in bending are not as scattered as those for tension and compression, this leads one to believe that they might be more appropriate for use in plate theory.
INTRODUCTION
The design of blast-resistant structures is complica- ted by a desire to obtain light, flexible assemblies in which ductility and energy absorption are at least as important as simple mechanical strength. The discontinuities or openings in these structures, such as, doors, windows, etc., pose special problems in attempting to devise light, rigid elements, capable of manipulation as desired for human occupancy, yet which continue to retain the flexibility under impulsive load of the overall system.
As is also the case with seismic design, energy absorption can be obtained either by encouraging gross displacements in structures or elements composed of materials having moderate to low resistance under load, or conversely by using high or ultra high strength materials which continue to remain elastic, or nearly so,
(') Dept. of Civil Engineering, University of Waterloo, Ontario, Canada.
(2) EPF Lausanne, Switzerland and University of Waterloo. Ontario, Canada.
(3) Formerly Dept. of Civil Engineering, University of Waterloo, Ontario, Canada.
until fracture. In most current structural engineering usages the former is to be preferred as, not only is the capacity to absorb energy through plastic deformation substantially higher, but also impending warning of collapse or fracture is usually given through visible distortions of the structure.
The capacity of a structure to absorb energy can also be modified by varying the elastic modulus of the material used to accommodate, for example, a more flexible material at a given stress level should one desire to augment the internal energy of resistance. Unfortunately, this method may have severe drawbacks in compromising the performance characteristics of the structure under normal service conditions.
The present investigation involves the determination of the static response to uniform pressure of fibreglass reinforced polyester rectangular plates set in neoprene surrounds; these structural components are meant to simulate the doors of a lightweight, portable, blast resistant shelter. The static behaviour also provides a first approximation to the more complex impact analysis, where deflection response under static uniform load may be deemed to furnish certain clues to plate performance at large displacements should an "equivalence" to the blast loading pulse over
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V O L . 11 - N ~ 62 - M A T E R I A U X ET C O N S T R U C T I O N S
(. infinitesimal time, | P dt, be derived as part of the
d design method.
The plate consists of 22 layers of fibreglass strips, arranged parallel in each layer and embedded in a polyester resin. Consecutive layers are cross laminated and, consequently, the strips form an axis of orthotropy along which elastic constants may be considered as identical; the axis oforthotropy is at approximately 45 ~ to the edges of the plate. In view of the potential uncertainties of material behaviour, considering the built-up nature of the plate, the investigation of plate response is preceded by a fairly thorough examination of the gross mechanical properties of the laminated material. Accordingly, the paper is presented in two parts, the first part bearing on the determination of mechanical constants required in the large deflection plate bending analysis which, in turn, forms the latter part of the investigation.
FIBRE-REINFORCED PLASTICS
Fibre-reinforced materials are:a group of composite materials in which two or more constituents are bonded together so that the overall properties of the composite are superior to those of the constituent elements; the desirable properties are thus maintained or enhanced while the undesirable properties are suppressed. Structurally, fibre-reinforced materials can be divided into two basic categories: multiphase and lamina- ted [1].
The multiphase composite can be approximated by a quasihomogeneous continuum, i.e. locally heteroge- neous, but grossly homogeneous. When the fibres have a random orientation, the composite may be considered grossly isotropic and when they have a fixed distribution it becomes grossly anisotropic. The laminated composite consists of many layers of multiphase or homogeneous materials bonded toge- ther. It can be approximated by an inplane homogeneous and transversely heterogeneous conti- nuum.
Within recent years, interest in reinforced plastics has grown considerably and much work has been carried on towards a better understanding of their behaviour. This composite material has many advantages, principal among which, from a structural stand point,.is the high strength to weight ratio and the fact that its properties can be adjusted to suit any particular situation by varying the type and content of resin or glass, orientation of fibres, etc.
The stress-strain curve is one of the most important indicators of mechanical strength of a material and one approach has been to disregard the load-carrying capacity of the resin assuming it acts only as a binder for the reinforcement. This approach, referred to as "netting analysis" has been applied to the design of pressure vessels ([2]-[41). Hoffman [5] has approached the same problem taking into account the resistance of the resin.
Schaffer [6] approached the problem of unidirectio- nal composites as an exercise in solid mechanics. The
resin was assumed to behave elastoplastically and the reinforcement elastically. The resin was also assumed to yield first and failure occurred when the reinforcement reached its ultimate stress. As a result, the stress-strain curve became piecewise linear with a knee occurring at the yield point of the resin. The condition that the resin yields before the fibres leads to the following inequality:
Yy >Ec Y~ E,"
where Y is the yield stress, E is Young's modulus and the subscripts f and r refer to the fibres and resin respectively. By equating strains in the fibres and resin, the overall modulus for the composite then becomes:
E I = E r [ ~ E--LE, + ( 1 - ~--Z)I'
for the direction parallel to the fibres, and
E2= { Er 1-(1-(Er/Ef) . } x [0. 8247 (AI/A) 1/2 - (AI/A)])} ,
[1 - 0. 8247 (Ai/A) 1/2 (1 - (E~/E:))]
for the orthogonal direction, where A refers to the total cross-sectional area.
Tsai [1] considered the same problem, and also that of a laminated composite, by writing the constitutive equations and expressing the constants involved as functions of the geometric and material parameters. For the unidirectional composite the constitutive equation is the generalized Hooke's Law with elastic stiffness expressed in the following form:
q ; = q i ( E : , v:, 7:, E,,, v,,, 7m, R, C, K, 0),
where E is Young's modulus, 7 the specific gravity, v Poisson's ratio, R a matrix content by weight, C the contiguity factor, K the filament misalignment factor and 0 the angle of orientation of the fibres; the subscriptsfand m refer to fibres and matrix respectively. For laminated composites the constitutive equation is taken as:
where N is the stress resultant, M the stress couple, e ~ the in-plane strain, k the curvature in bending and the composite material matrix is divided into four sub- matrices such that:
f b/2
A, B, D=Ai j , Bij, Dij = (1, Z, g 2) Cij d Z . ,) -b/2
These submatrices are examined as functions of material and lamination parameters; the material parameters refer to the cij matrix of the unit plies and the lamination parameters refer to the thickness and orientation of each ply and the total number and stacking sequence of all plies. Again, the material is assumed to behave in a linear elastic maniiero Experimental verification of the theory was provided by Tsai and Azzi [7].
76
Much of the work reported to date has been experimental in nature. Krolikowski [8] studied the behaviour of a glass-reinforced polyester resin and obtained a piecewise linear stress-strain curve with two "knees" instead of the one predicted by theory. No explanation of this phenomenon was given beyond the non-homogeneous structure of the material. It was found that the modulus of elasticity, the proportional limit and tensile strength all increased proportionally with the percentage of glass fibres and decreased with increasing flexibility of the resin.
When reinforced plastics are compared with metals, one of their principal limitations is the low moduli of elasticity and hence the large deflections which may result in structures fabricated from them. They also differ in as much as the ultimate elongations of plastics are relatively low (1 to 2%) and their behaviour is relatively elastic over the entire range.
The effect of shrinkage of the resin has been investigated by Outwater and West [9] and it has been found that tensile stresses are set up which can be significant at room temperature. These stresses are maintained by the adhesive bond between resin and fibre and any breaking of this bond will provide cracks in the composite which could initiate failure under loading, These microcracks, if subjected to adverse atmospheric conditions, could reduce the overall strength of the composite. The effects of water absorption were studied by Krolikowski [8] and, while tests showed that material exposed to water for some time had a reduced strength, the action is reversible and is thus only a physical, rather than chemical, phenomenon.
Raich [10] and Barnet and Prosen [11] discussed the data obtained through experimentation. It was pointed out that composite materials are heterogeneous and thus their properties will depend upon such factors as manufacturing processes, imperfections and variation of material throughout the specimen. While the same can be said for any material, it should be realized that these are extremely important considerations when working with composites. The material in an actual structure can vary considerably from that of a test specimen and material properties are therefore known only in a gross manner. The application of standard test procedures, designed for metals, to composite materials is accordingly to be frowned upon because the general behaviour of the two materials differ greatly and hence the results must be interpreted with care.
Failure modes and strength
The conventional failure theories, such as yon Mises and Tresca, cannot be applied to fibre-reinforced plastics directly because the ultimate strength generally varies from tension to compression as well as being dependent upon fibre orientation. Marin [12] presented a variation of the yon Mises theory; taking into account the factors enumerated above, the general form of which was given as:
(0.1 - -a) 2 +(rr2 - b ) 2 +(0"3 - c ) 2
+q { (0" 1 - a ) (% - b) + (or a - b )
X ( 0 " 3 - - C ) + ( 0 . 3 - c ) (13" l - a ) } ~--- 0 "2,
T. H . T O P P E R - A . N . S H E R B O U R N E - V . S A A R I
where a, b and c are constants that can be determined by simple tests. Azzi and Tsai [13] considered unidirectio- nal composites at various angles to the applied stresses and developed a yield condition of the following quadratic form:
2f(~rl) = F ( % - %)2 + G ( G -7 0.x) 2 + H ( 0.x - 0.)2
2 ~---1, + 2 L z Z = + 2 M z Z ~ x + 2 N z x y
where the coefficients F, G, H, L, M and N are parameters characterizing the state of anisotropy and x, y and z are the axes of anisotropic symmetry; if these axes do not coincide with the co-ordinate axis, the stress components must then be transformed. Considering a two-dimensional problem, for reasons of simplicity, the failure condition given above becomes:
In 2 - n + ~2 + k 2 fi2] sin* 0
+ 2 k [3 n - 1 - 2 c~ 2 - (n - 1) f12] sin 3 0 cos 0
+[8 k 2 - n z + 2 n - l + ( 2 n + 4 k Z ) c d
+ ( n Z - 2 n + 1 -2 k 2 ) fl z]
x sin z 0 cos 2 0 + 2 k [3 - n - 2 n c~ z + (n - 1) f12]
x sin 0 cos 3 0 + [1 - n + n 2 o~ 2 q- k 2 ~2]
X COS 4 0 -~- , .
whereX= yield stress along x axis, Y= yield stress along y-axis, T=shear stress on x - y plane, c ~ = X / Y , B = X / T , n = a 2 / a l, k = r l z / % and 0=angle between co-ordinate axis and axis of anisotropic symmetry. The variation of strength with fibre orientation, O, is calculated from the above equation and agrees very well with experimental results.
Fried [14] considered the strength of the resin as the major influence in the overall strength of the composite. The resin was considered as applying a "hydrostatic" pressure on the fibres when loaded in compression and, when the resin yielded, the material failed by buckling. The stress at failure in the reinforcement was given, considering transformed sections, as:
Pt sz = Az + Ar (Fv/er)'
from which the stress in the resin became:
Er S r = S f ~ .
The above equations held only if the elastic moduli of both resin and reinforcement remained constant up to failure. In materials where the resin content is low the failure mechanism just described may not occur and other possible modes are failure of an individual filament by buckling or, if lateral support is sufficient, by "crushing" of the fibres. In either case the overall strength of the composite depends upon the strength of the resin. Another factor to be considered in the effectiveness of the lateral support is Poisson's ratio for the resin. For resins with a high ratio the lateral support would be reduced and hence filament failure would occur at a lower load. The experimental work of
77
V O L . 11 - N ~ 6 2 - M A T E R I A U X ET C O N S T R U C T I O N S
Broutman [15] showed that crack initiation, at the resin-glass interface, began at about 8 0 ~ of the ultimate stress.
Time and temperature dependence
The linear theories of visco-elastic behaviour, as applied to incompressible isotropic materials, are fairly well developed but, for anisotropic materials, know- ledge is quite limited. Biot [16] investigated the problem theoretically and Kaye and Saunders [17] provided some experimental evidence for a fibreglass resin laminate which reflected orthorhombic symmetry in its mechanical properties. The creep-compliance, J( t , 0), defined as strain per unit applied stress, varied systematically with direction of fibres in the specimen and this curve had the same shape at all times. Thus it was suggested that the creep behaviour could be described by two curves, a curve of reduced creep compliance defined as:
0" J(120, 45 ~ Jr = J ( t' ) - -~ , -4 ~ ) '
against orientation, and a curve Of reduction factor:
J(120, 45 ~
J(t , 45 ~ '
against time. It was shown that the variation of creep compliance with direction of fibres is similar in form to the variation of elastic compliance with direction in orthorhombic, anisotropic, elastic materials.
Considering loads of long duration, in contrast to the above, it was found [18] that the 1,000 hours breaking stress is approximately 60 ~ of the short term breaking stress and immersion in water during loading further reduces this to 40 ~o- The probable reason for this reduction in strength [19] is the breaking of the bond between resin and fibres.
The effects of temperature variation on the behaviour of reinforced plastics have been studied extensively. Brink [20] investigated the effect of low temperatures on several fibre-resin materials and concluded that strength at cryogenic temperatures (down to -425~ was higher than at room temperature, toughness was maintained, fatigue resistance was almost unaffected by temperature, if anything, it improved with decreasing temperature and the rate of cooling had little effect on strength. Howse and Pears [21] provided experimental data about the effects of temperature variation on the specific heat, thermal expansion and thermal conducti- vity of fibre-reinforced plastics. Kaye and Saunders [17] also pointed out that the effect of temperature is quite pronounced on the creep of a fibre-reinforced plastic and suggest that minor variations of + 1 ~ can have a sizeable effect on creep strains.
TEST PROCEDURES
Testing was undertaken to determine the mechanical properties of a laminated, continuous fibreglass- reinforced polyester with special consideration being given to the application of the results to a plate of
medium thickness fabricated from the same material and subjected to uniform normal pressure. Tension and compression tests were performed to obtain stress- strain relationships, ultimate strengths and strains and Poisson's ratio; flexural tests were carried out to determine the behaviour of" the material in bending which more closely approx:imates conditions in the plate tests. Creep tests were performed to determine the time-dependence of the material and specimens were loaded at various strain rates to observe the variation of mechanical behaviour with rate of loading.
The material used in this investigation was a cross laminated, continuous, glass strand reinforced poly- ester resin manufactured in the form of sheets. There were 22layers in the plate, each layer being approximately 0.025" (0.635 mm) thick. Within layers the strands were laid parallel to one another, while, in successive layers, the strands were orthogonal to that of adjacent layers. The thickness of the plate was approximately 0.55" (13.97 mm); the percentage of glass by volume was 60 ~o and resin 40 ~o. The plates- were fabricated specifically for testing purposes, the layers of glass being laid by hand to no specific manufacturing tolerances beyond "best possible workmanship".
A total of three plates were used for testing purposes and the results were considered separately to study the variation of behaviour from plate to plate. They were manufactured at different times by the same supplier but were supposedly of similar composition and geometry. The specimens were marked on each plate, in the desired position and orientation, and then cut out roughly with a band saw. After being milled to the final dimensions they were lightly sanded by hand to remove any loose edges. Specimens of each orientation were chosen from random positions on the plate to establish a measure of the variation o f the material throughout the plate.
TENSION
The tension specimens were cut as shown in figure 1. They generally conform to ASTM Standard D638-64T except for the grip length of the specimen. Most specimens had a nominal cross-sectional area of 0.25 square inches (161.29 mma), but some specimens of nominal area 0.50 (322.58) and 1.0 square inches (645.16 mm 2) were used to study the effect of width on the measured mechanical properties. The cross- sectional area was increased by increasing the width of the specimen, the thickness being held constant at the fabricated value of approximately 0.5" (12.7 mm). Care was taken in machining the specimens to achieve both a constant cross-sectional area in the necked portion and to maintain the sides parallel to the axis of the specimen.
The narrow specimens were tested in an Instron Universal Testing Machine for which the rate of crosshead motion in any given test could be set at a constant value. An extensometer of 1" (25.4 mm) gage length, clamped to the central portion of the specimen, measured the strain. Both strain and load were plotted automatically on an x-y plotter and the stress-strain curves were calculated from these graphs. The rate of
78 �84
T. H. T O P P E R - A. N. S H E R B O U R N E - V. S A A R I
f
2.2" .0.9". ( 55 .9mm) =(t22.9 mmr J
5" ( 76 .2 ram) rodius
X
l ~
2 . 6 " ( 6 6 mm) - I
8 . 8 " ( 2 2 3 . 5 m m )
~ 4- 4- •
Fig. 1. - Tension specimens.
cross head motion was varied to determine the effect of strain rate on material properties.
The 1" (25.4 mm) and 2" (50.8 mm) wide specimens were tested in a Tatnall, manually controlled, hydraulic machine. Load was applied in increments and then maintained while strains were read. Strain, in this case, was measured over a 2" (50.8 mm) gage length by means of a manual strain indicator which recorded the change in length between two metal tabs fastened to the specimen. For the 1" (25.4 mm) specimens, strains were read at four locations, two on each side. The average of the strain readings was used in plotting the stress-strain c u r v e s .
COMPRESSION
Compression specimens were cut as shown in figure 2; ASTM specifications (D695-63T) suggest a length of twice the nominal width for square specimens. In these tests a 2" (50.8 mm) long specimen was used to allow for installation of instruments to measure dilatation. Again, most specimens had a nominal cross- sectional area of 0.25 square inches (161.29 mm 2) but a few, used to determine the effect of width, had an area of about 0.5 square inches (322.58 mm2). In the preparation of these specimens particular care was taken to make the sides parallel to the axis of the specimen and to make the ends perpendicular to the axis to ensure uniform load distribution during testing. Initially, the specimens were capped with an epoxy resin to prevent failure by delamination at the ends of the specimen. However, when it was observed that the failure mode of capped specimens was similar to that of uncapped specimens, i. e. by delamination in the central portion, capping was discontinued and the load was applied directly to the specimen.
The small specimens were tested on the Instron Universal Testing Machine and strains were calculated from the recorder chart displacement geared to the cross-head motion. A correction for machine elonga- tion was applied to determine the actual strain of the specimen. The load, which was plotted directly, was converted to nominal stress. In the case of the larger specimens only, the modulus of elasticity was determined on the Instron machine and the remainder of the behaviour was determined using a Tatnall machine.
FLEXURE
The flexural specimens were 14" (355.6 mm) long and 0.5" (12.7 mm) and 1.0" (25,4 mm) wide. Again, the depth was the fabricated value of approximately 0.-5" (12.7 mm). In preparing these specimens, the width was controlled by making the sides parallel to the axis of the specimen to assure a constant moment of inertia along the specimen. A third-point loading system over a 12" (304.8 mm) span length with simple supports, was used and deflections were measured at the centre and third points by 0.001" (0.0254 mm) dial gages. The readings were taken at fixed time intervals of 15 seconds after the load was applied; this was necessary to obtain consistent results a t high load levels where" the deflections continued to increase under creep.
CREEP
The specimens used in the creep tests were similar to those already described for tension and compression. The width of the specimen was 0.5" (12.7 mm). Variations in thickness of the plate as supplied were such that it was necessary to measure all specimens before testing. The dimensions did not, however, vary greatly in any one specimen and after taking several measurements along the gage length, the smallest value was selected as the representative dimension.
I I . 2, �9 ( 5 0 . 8 m m )
2,, -I ( 50.8 mm )
Fig. 2. - Compression specimens.
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V O L . 11 - N ~ 6 2 - M A T E R I A U X E T C O N S T R U C T I O N S
The tension specimens were held at constant load in the Tatnall machine until failure occurred or the creep strains became negligible. The load, which was maintained manually, varied somewhat, but, in no case did this variation exceed 2 %. The strain was measured by a manual strain gage. The compression specimens were also held under constant load in the Instron machine until failure occurred or until the creep strains became negligible. Load was maintained by setting the cross-head at the slowest rate and continuously adjusting; variations again, did not exceed 2 % and strains were measured using dial gages mounted between cross-heads. One flexural creep test was performed on a 1" (25.4 mm) wide specimen with its axis oriented at 45 ~ to the fibre direction after creep, for this case was found to be pronounced in axial tests. It was performed in the same manner as the flexural tests except that the load was held constant and deflections were monitored.
DILATATION
The apparent Poisson's ratio was measured on tension and compression specimens with the fibres oriented along the axis of the specimen. The longitudinal strain was measured as in the tests previously described. The lateral deformation was measured using an extensometer connected to a strain gage bridge read out which had been calibrated in terms of deformation.
TEST RESULTS
Many theories have been proposed to describe brittle fracture phenomena [22]. Generally these theories assume that failure is initiated by stress concentration at Some critical flaw and propagates through one or more cracks issuing from this flaw. The strength of a brittle material thus depends not only upon atomic bonding but also upon the density and distribution of critical flaws and upon energy considerations in the formation and propagation of cracks. For this reason, one cannot expect two specimens of a brittle material to fail at exactly the same Stress level and statistical methods are generally used to predict a probability of fracture rather than using some average or weighted mean stress value. For example, the popular theory of Weibull [23] predicts that the relative tensile strengths of two specimens of "identical" material should be related thus:
X 1 ( V z ' ] ~/" =\vlj '
where X denotes strength, V the volume, and m is a "flaw density factor" which is assumed to be constant for a given material and independent of size. This theory is based on the greater probability of a critical flaw occurring in a larger volume and, hence, the nominal strength of a small specimen should be higher than that of a large specimen.
The distribution of flaws in brittle materials is such that there is a wide variation in nominal strength from specimen to specimen. In order to obtain an accurate picture of the distribution of strengths or the probability of failure at a given stress level, a large number of identical specimens must be tested. Bradstreet [24] pointed out that this situation is further complicated by internal errors in most tests. He has shown that care in reducing the amount of bending and torsion present in a tension test may increase the average nominal strength byas much as 20 % for brittle materials. Because of the scatter in test results and the sensitivity of these results to testing technique, values of "strength" must be interpreted with considerable care for brittle materials. This is markedly different from the case of ductile materials where plastic flow largely eliminates the effect of flaws and errors in testing technique.
Fibreglass, although it fractures in an apparently brittle manner, with little gross plastic flow, cannot be treated as a brittle material in the usual sense of the term. The glass fibres behave in a brittle manner, but the resin is capable of accommodating considerable plastic flow. When an individual fibre fails in a brittle manner the remaining fibres at that cross section absorb the load. The resin, which forms a matrix enforcing a uniform strain distribution in all fibres, suffers some plastic flow near the broken end over a length necessary to build up the force in the fibre. A critical flaw is formed when enough fibres are broken in the same region that the released elastic energy due to an extension of this region is sufficient to supply the plastic work done in extending it. Much of this work is done in pulling the ends of fibres from the resin. A crack may, however, be arrested in various ways. If it has propagated through a layer, gross bond failure between layers may result in delamination rather than transverse crack propaga- tion. Other causes of crack arrest include encountering a region of lower stress level or greater local resistance i.e., greater local ductility. Failure results when a sufficient number of local failures occur to raise the stress level in the remaining material to a point where propagation across the whole specimen occurs. BeCause the size of local failures may be quite large rather than microscopic, one occurring in a small specimen is likely to be more serious. For this reason large specimens tend to give more uniform results than small ones. In the extreme, the bending of a large plate, with the resulting non-uniform stress distribution should yield better reproducibility between tests than small test specimens.
Much experimental work has been done on fibre reinforced plastics in ordinary tension and compression testing but the results must be viewed with caution. It is necessary to know what the data or curves represent whether it is the result of one test or whether average values are being quoted; the type of material used must also be specified since this significantly affects the material properties. If a comparison of results is to be made with published data one must be certain the materials being tested are similar. In reinforced plastics, where so many parameters such as fibre orientation, fibre content, resin type, plus many others are involved, it becomes difficult to make meariingful comparisons. Also, the methods of fabrication for this type of material make it difficult to get uniformity throughout any one
80
member or between two supposedly identical pieces. Uniformity of manufacture is inherent in some processes more than in others and can be improved in all methods by closer quality control. Nonetheless, complete uniformity for this type of material is impossible.
For structures in which loads are to be sustained over long periods of time, the reduction of ultimate stress due to creep is a very important design consideration. Under loads of long duration, it has been shown experimentally ([25], [261) that the increase of strain with time is small, but the breaking stress is reduced considerably. For example, loads of the order of 103 hours reduce the breaking stress to approximately 50 % of the short term value. Experimental observations show that the creep strain consists of discrete jumps occurring throughout the creep period rather than a smooth curve of strain against time as is the case with ductile materials. Occasionally, these discrete move- ments are accompanied by audible clicks generally associated with the rupture of the glass fibres under stress. Although the mechanism of tensile creep has not been resolved, it is generally considered to be some form of failure at the glass-resin interface together with the progressive rupture of the glass fibres [27]. The strain at failure of laminates subjected to long term loading is found to be greater than in short term loading; as it also increases as the stress level decreases, this is apparently associated with ductile failure of the resin which has a more pronounced creep behaviour than the glass.
Tension and compression
Test curves for tension and compression are shown in figures 3 to 8, for 0 and 45 ~ orientations of the fibres. Each curve is the average of all tests performed and the range of all results obtained is shown on the same graph. These curves were drawn from the automatic
ZOO
150 E E
Z v u3 (,9 LIJ
IO0 ( /3
5 0
32
24
o_ x
k / x
x DENOTES FAILURE POINTS OF INDIVIDUAL TESTS
4 5 =
O _ 0 f I .. I ~ ... t
STRAIN
Fig. 3. - Average stress-strain curves in tension. Plate 1.
T. H. T O P P E R - A . N. S H E R B O U R N E - V . S A A R I
plots obtained using the lnstron machine for the narrow specimens and from calculated stress and strain values obtained from manual readings of load and deforma- tion for the larger specimens. Most of the automatic plots dropped suddenly at some load value and subsequent further increase in strain occurred at constant or slightly increased load. The point at which the load dropped initially was taken as the failure load for the specimen and the corresponding strain as the ultimate strain. For some specimens there was no sudden drop in the load and hence very high ultimate strains were recorded.
In plotting the average stress-strain curve, the average value of ultimate strain was calculated and then average stresses were calculated for strain values up to the average ultimate strain. For specimens with ultimate strain values below the average, the material was assumed to maintain a constant load between its ultimate strain and the average ultimate strain. This was done to obtain continuity in the averaged curves. This average stress-strain curve has, therefore, no physical meaning at high load levels because the shape is distorted by the non-existent assumed constant load of some specimens. However, it gives an indication of where the average falls within the given range. For small loads, the linear portion gives the average stress-strain behaviour.
For the large tension specimens, tested on the Tatnall machine, the strains could not be measured up to failure due to the loss of the metal gage tabs at high strains. In this case, the stress-strain curve was plotted as far as the readings taken and then behaviour was extrapolated using other curves. Behaviour was linear and no problems were encountered for the 0 ~ specimens; for the 45 ~ specimens, however, where the behaviour was non- linear, extrapolation was impossible and only the portion of the curve obtained could be used for comparison.
200
oa E E 150
u3 Ob I.M (Z: p.. m I00
50
X / ~ X
x DENOTES FAILURE POINTS OF INDIVIDUAL TESTS
STR~,IN
~.,,- O- QI ~.~ I ,I
F i 9. 4. - Average stress-strain curves in compression. Plate 1.
81
VOL. 11 - N ~ 62 - M A T E R I A U X ET C O N S T R U C T I O N S
20O
1501 N
E E
Z
o l ~oo
{1:
o3
50
32, ;
i
241
161
8t
ot
x DENOTES FAILURE POINTS OF INDIVIDUAL TESTS
~ X
,K
STRAIN
Fig. 5. - Average stress-strain curves in tension. P late 2.
20O
150 -- ~0
50 - -
0 --
52 r
.)41-
X DENOTES FAILURE POINTS OF INDIVIDUAL TESTS
161-
/ x 8 / yx x x
STRA IN
Fi 9. 6. - Average stress-strain curves in compression. P late 2.
ZOC
E E -%
Z
o l o1
p - o l
%
52r-
16
X DENOTES FAILURE POINTS OF INDIVIDUAL TESTS
45* 5(3 - 8 F ~
H-o,ot - I O _ O L
STRAIN
Fi 9. 7. - Average stress-strain curves in tension. Plate 3.
12 / ~ x DENOTES FAILURE POINTS ! OF INDIVIDUAL TESTS
2O0
- ~ x---
50 81 ~ e
I I I
STRAIN
Fig. 8. - Averaoe stress-strain curves in compression. P late 3.
150
E E
v
I.iJ
I- o1
Correlation of test results and theory
Although the theory based on elastic-plastic behaviour of the resin predicts a sharp knee in the stress-strain curve for reinforced plastic laminates, the behaviour may, for all practical purposes, be approximated as linear [28]. The main reason for this is that the resin, which gives rise to the knee, carries a very small share of the load. The deviation from linearity in the present tests was so small as to be almost unnoticeable and there was no distinct knee. The lack of a distinct knee, corresponding to the onset of non-
82
linearity, is probably due to the masking of yield by time dependent behaviour of the resin and to the scatter in the results.
Over the range of loading rates used 0.002"/min: (0.0508 ram/rain.) to 2.0"/min. (50.8 mm/min.) there was no significant rate effect for the 0 ~ specimens and all tests at the high rates fell within the scatter band for the lowest loading rate. Differences between tests can, therefore, be attributed to inherent scatter of the material. McAbee and Chmura [29], who studied the effects of rate of loading, concluded, on the other hand, that both the ultimate strengths and the shape of curves
T. H . T O P P E R - A . N . S H E R B O U R N E - V . S A A R I
obtained, varied somewhat with rate. They compared the conventional static rate, which produces failure in about 2 minutes, with a rate of loading high enough to produce failure in about 6 ms which is a much greater range than that covered in the present study. For the 45 ~ specimens, however, the higher rate of loading increased both the load at which the stress-strain curve became non-linear and the failure load. This change in behaviour may be attributed to a reduction in the creep of the resin, which is highly stressed in this orientation, at the high rate.
The effect of size of specimen on behaviour was also examined for a few specimens and no appreciable differences were observed. Although completely brittle materials show a decrease in strength with increasing size, it was pointed out that, for this material, this effect is partially offset because of the finite size of local failures. The present series of tests is, in any event, not definitive because the large scatter would require larger sample sizes and size differences to produce a statistically significant measure of the size effect than those employed.
Variations of test results
The plates from which the specimens were cut were manufactured separately although the manufacturing process and the geometric arrangement of the constituent materials were Supposedly similar. Imper- fections in the material and uncontrollable variations in manufacture obviously resulted in differences of strength and rigidity from one plate to another as well as from one section to another in any given plate. Imperfections were visible t o the naked eye in some specimens; these imperfections were present as voids or air bubbles in the resin, small cracks at the exterior of the specimen or misalignment of fibre orientation. It is quite probable that further imperfections were present which could not be observed either due to size of imperfection or due to their being in the interior of the specimen.
Although a statistical analysis would have given a measure of the variability of the material and manufacturing techniques, this would have required a large number of specimens from many plates which was outside the scope of the present investigation. To further complicate the problem, manufacturing of the plate has an effect on its physical properties. For instance, if average values are considered, plate 3 has almost double the ultimate strength and modulus of plate 2 with plate 1 falling somewhere in between.
Some specimens failed outside the gage length and hence this could affect, to some degree, the shape of the stress-strain curve. Since the material is brittle and failure occurs suddenly, this effect would be noticeable only at loads very close to the failure loads. For example, in some specimens in which an extensometer was used to measure strain, there was a noticeable decrease in strain just before failure due to the fact that the specimen was failing outside the gage length and the strain within the gage length was therefore relaxing. In these cases, an extrapolation was used to predict actual behaviour. Although the driving screw on the Instron is geared to the plotter, an additional correction had to be made to take into account the extension of the machine itself and, in some cases, this was as high as 30 % of the
total deformation. A small error is probably present in this correction, which was established by loading a heavy metal bar and noting the difference between chart displacement and actual deflection with load level.
Failure modes
The tensile specimens failed in a brittle manner which generally occurred suddenly accompanied by a loud snapping sound due to the release of energy in the glass fibres. Failure of the 0 ~ specimens included both tensile failure of the fibres as well as bond failure at the glass- resin interface indicated by the jagged edges present after fracture. Many fibres had also been pulled out after breaking, further indicating failure of the glass-resin bond. Failure of the layers orthogonal to the specimen axis occurred by tensile failure of the resin. Shear failures, which occurred along the layer boundaries, indicated a weak inter-layer bonding. The size of the specimen had no effect on the mode of failure. The 45 ~ specimens failed by shearing in a directionparallel to fibres in each layer such that alternate layers sheared at 90 ~ to each other. This was due to the relatively low strength of the resin as compared to the glass. In the direction perpendicular to the fracture, tensile stresses in each layer are carried entirely by the resin.
The compression specimens failed by buckling of individual lamina due to inadequate lateral support provided by the resin. This buckling effectively reduces the elastic modulus and failure strength of the material and hence the length of the specimen has a profound influence except when the specimen is sufficiently short that the buckling load is higher than the "squash" load. For the material under study this would be unreasonably small if the buckling of individual layers is calculated. These results, however, help to explain plate bending behaviour where delamination can also occur on the compressive face.
Poisson's ratio
The results obtained from these tests are indicated as follows:
VALUES OF POISSON'S RATIO
OBTAINED EXPERIMENTALLY
Tension Compression
- 0 . 4 2 - 0 . 1 6 - 0 . 1 7 - 0 . 1 2
0 . 0 0 .69 0 .56
The results are inconsistent due to the nature of the tests as the fibres in the lateral direction have low stresses and shortening must be accomplished largely through shear stresses transferred by the resin. Non-homogeneous behaviour, together with variability o f bond and relatively small width of specimen, give erratic results. Also, overall behaviour is dependent upon the different values of Poisson's ratio for the constituent elements. If only fibres parallel to the loading direction were present, the restraining effect of each component on the other would give a Poisson's ratio for the composite somewhere between those of the individual materials with the actual value depending upon such factors as
83
VOL . 11 - N ~ 62 - M A T I r : R I A U X ET C O N S T R U C T I O N S
types of materials and percentage of each. In reality, however, the materials may not act as a unit at all points because of the presence of lateral fibres and the fact that low strength of the resin combined with imperfections may result in slippage between layers.
Bending
Bending tests were conducted on six specimens from plate 2 and twelve specimens from plate 3. The width of the plate 3 specimens was varied, six being 1" (25.4 mm) wide and six being 1/2" (12.7 mm) wide. Two specimens in each group were oriented along the fibres, two at 45 ~ and two at 90 ~ . In reporting the results, the 0 and 90 ~ specimens were combined since they should have the same properties. Little work reported in the literature includes actual test results. Shand [28] showed that, for specimens of a laminate oriented at 0 and 90 ~ the load-deflection curve should be almost linear with a slight curve as the load approached the failure load. Hence, for small deflections, the moment-curvature relationship should also be linear.
Moment-curvature relationship
If a linear stress-strain curve is assumed for both tension and compression, the relationship between the applied moment and the resulting curvature of the specimen can be calculated. The development of this equation is given in appendix 1 as:
M E c 2 = T d y ,
where Ec is Young's Modulus in compression, d the
t ~ 0
x
E E
E E
Z v -r
I - -
I - - Z 2:1
LLI e,
I - -
,,:5, 0
14- 5200
IZ
24.00 I0 -
8
1600 -
6 -
4 - 800
0
/ /
UBLE MODULUS
/ / / / ,~ ,~x, S INGLE MODULUS
I t I t I
C0RVA'rURE (in "~) Imm -I 1
Fi 9. 9. - Averafe moment-curvature diagram. Plate 2.
depth of specimen, and y the distance from the top fibre to the neutral axis, such that:
y = d ~ E c Et - E t E c - E t ' and E~r t.
The moment-curvature relationships obtained from the bending tests are shown in figures 9 and 10. These curves were derived from experimental data as shown in appendix 2 and represent average values; maximum and minimum curves show the range of results. To compare the behaviour of t h e plates, moment was plotted against d 3 ~ to eliminate the effect of depth on the moment-curvature relationship. This curve is illustrated in figure 11. In plotting moment vs. curvature from experimental da ta it was assumed that the curvature of the central port ion was constant. The average deflection of the third points and the centre was used in calculating 6. Some error is present for those tests in which imperfections of the specimens caused some unsymmetrical deflection.
In order to obtain theoretical curves, the values of the physical constants must be specified. If the average values of the tension and compression tests are used, then the following values are derived.
Plate 3: - E~ = 1.44 x 106 psi (9928N/mmZ); - E, = 1.98 x 106 psi (13652N/mma); - o-c, =26,000 psi (179.2N/mm2); - tr,, = 36,000 psi (248.2N/mma).
For example, for plate 2:
y = 0 . 5 5 d,
and with the value of d taken as 0. 605:
M -~- =21,4001b. in./in. (2417.75 x 103 N-mm/mm).
From equation 11 (appendix 1) it can be shown that a tension failure results. The ultimate moment then becomes:
M= = 900 lb. in./in. (4003.2 N-mm/mm).
If it is assumed that the moduli of elasticity for both tension and compression are equal, then the general flexural formula can be used to arrive at a moment- curvature relationship:
M d 3
12'
and, using the M -~ = 25,000 lb. in./in.
values previously listed,
(2824.5 x 103 N-mm/mm)
where the modulus of elasticity in tension is used. The ultimate moment capacity then becomes 840 lb. in./in. (3736.3 N-mm/mm).
84
T. H . T O P P E R - A . N . S H E R B O U R N E - V . S A A R I
_o x 8
E E E E
6 z
- r
5 4 I-- z =3 o:2 ~ z I - z I.d
,J A ME sP c .E I ( 25.4 ram)
160C - I /2" ( 12.7 mm ) -
..~ SPECIMENS
1 2 0 C - " , INGLE MOOULUS
= o o o , o = - I--
1 I I B
~ " C U R V A T U R E ( i n - I )
( r a m -I )
F i g . 1 0 . - Average momenbeurvature diagram. Plate 3 .
The theoretical curves of single and double modulus are plotted on the same graph as the experimental curves of figures 9 and 10.
Discussion of results
The experimental results presented show "average value" curves as well as upper and lower bounds. The range is quite large for plate 3 as compared with plate 2. The curves of M - d 3 4L on'the other hand, show that the behaviour of the two plates is quite similar. Assum- ing an elastic homogeneous continuum, the value of E obtained from these curves becomes approximately 1.83 x 106 psi (12618N/mm2). This behaviour is strange in view of the tension and compression test results where large differences in stiffness between the plates is indicated. This would imply that, in bending, the variation of physical properties does not affect the behaviour as significantly as it does in tension or compression. The theoretical curves agree reasonably well with experiment. The double modulus approach is generally low and is not as good an approximation as the single modulus hypothesis. If the value of the modulus is taken as 1.83 x 10 6 psi (12618N/mm2)the single modulus curves fit the average experimental curves almost exactly. The ultimate moment, as predicted by the theory, is low for plate 2, while for plate 3 it gives a reasonable approximation.
In general, the flexure specimens failed by delamina- tion in the compression zone and fracture in the tension zone. Failure was usually sudden, a n d a l t h o u g h cracking sounds could be heard prior to actual collapse, no visible distortions were apparent; also due to the sudden nature of the failure it was difficult to determine which mechanism actually precipitated destruction. Some specimens failed by shear along a central layer boundary which indicates failure of the relatively weak interlaminar bonding under high shear stress.
Creep
The creep curves obtained in this study are shown in figures 12 to 15. It should be noted that all curves are for individual specimens and no attempt was made to run confirmatory tests.
In these tests, creep strains were negligible at small loads. As the stress level increased the creep strain as well as the time required for the completion of primaxy creep increased. If the strain reached an equilibrium point after a certain time period had elapsed, the test was stopped. When the stress level reached some critical value creep continued to take place and the strains increased until failure occurred. Due to the wide scatter of failure strengths for this material, it was impossible to say what the failure load for a particular specimen would be a priori and hence specimens were loaded arbitrarily at some "low, medium and high" stress levels and then creep behaviour observed.
For tensile creep it appears that the failure load is very close to the failure load under steady loading for the creep times considered in these tests. No specimens were tested for periods over 24 hours because, in all cases, the creep strains had either stabilized by this time or else failure had occurred.
141
r~ o x 12
i
E E I0
z
-t- I.- s
z
n-.
I - z LO
O
3 2 0 0
2 4 0 0
8 -
160(
8 0 (
0
I 0
F i g . 1 1 . - Average values o f M vs. d3q ~ curves for plates 2 and 3 .
PLATE 2
PLATE 3
I r i I I 2 4 6 8 I0
d 3 ~ ( i n 2 ) x 10 -3
I [ I I I r 2 4 6
d 3 ~ ( m m 2 )
85
V O L . 11 - N ~ 6 2 - M A T I ~ R I A U X ET C O N S T R U C T I O N S
For the compressive creep specimens the time dependence was much more noticeable and the specimens failed at a stress level about 50 ~ of the ultimate failure load under normal loading conditions. The time dependence of compression specimens can best be explained if one considers the mode of failure. To prevent delamination of the layers a high bonding force is required since the resin is relatively weak and highly time dependent: however, this bonding force is considerably reduced for long term tests.
The compression specimens oriented at 45 ~ to the fibre direction exhibited very high creep strain rates, due to the fact that the resin carried a large share of the load. Shand [28] points out that the static fatigue
characteristics of laminates may depend to a considerable degree upon the type of resin used. Hence, general statements about the creep behaviour of reinforced plastics should be avoided with due consideration being given to the nature of individual components.
One flexural creep test was done on a 45 ~ specimen and the creep behaviour was very noticeable even though the stress level was of the order of 5 ~ of the failure load estimated at normal loading rates. The plot of deflection vs. time for this specimen can be seen in figure 16. Since the present problem is not concerned with loads of extremely long duration, the test was terminated after 18 hours with creep deflections appearing to level off at this point.
Discussion of results
As can be seen from the results, the theory put forward by Kaye and Saunders [17] - tha t creep compliance varies systematically with fibre orientation, with the maximum strain rate occurring at 45 ~ holds
86
0.04
0-03
Z
=~ o02 1-- O3
0"01
6 0 0 0 psi (41.37Nfmm 2)
S 5 2 0 0 psi o ( 35 .85 N / m r n ~ )
4 0 0 0 psi o
;~ (27 .58 N / m m 2 )
t I 1 I 1 0 4 8 12 16 20
TIME ( hr )
F ig . 13. - Creep curve for tension specimen - - 4 5 ~
true for this material. Although not enough tests were performed to determine the relationship for all angles of orientation, the results show that, at loads somewhat below the ultimate fracture load, creep behaviour along the fibres is negligible once primary creep strains are taken into account, while for specimens oriented at 45 ~ to the fibres the creep strains are considerable at much lower load levels.
The appearance of fracture for creep specimens was the same as for normal loading.
0 . 0 3
0.02
Z <~ n." I.-
0.01
300 psi _ (57.23 N/ram/)
~--e6700 psi (46.2 N/ram 2)
5000 psi (34.47 N/ram 2 )
i i I l l 0 6 16 24 32 4 0
TIME ( hr )
Fi 9. 14. - Creep curve for compression specimen - 0 ~
The variation in creep behaviour between plates was not studied, the tension and compression test specimens being taken from plate 2, while the specimen for flexural creep testing was taken from plate 3.
The effect of creep on the stress-strain curve is postulated in figure 17. On this graph the stress-strain curve, obtained from the tension tests, at average loading rates, was plotted first. From the subsequent tension creep tests, the creep strain after 24 hours at various stress levels was added to the regular strain. The resulting curve shows the effects of creep on the stress- strain curve if the duration of testing is in the neighbourhood of 24 hours.
Application to plate bendin 9
Consider a plate with three planes of symmetry with respect to its elastic properties which coincide with the co-ordinate planes (sometimes called an orthotropic plate) and with stress components given by:
ax =E'~ ~ + E" ~y,
% =E'~,~y+E"~,:, "c=y = G yxy.
The behaviour under load can be described by the following differential equation:
a 4 w a 4 w d 4 w D. ~-~--2-- +2(D, + 2 D : , y ) ~ +D:, a- ~- =q,
where h 3 h 3 D,=Z,25; D,= Z,-fi ;
E " h 3 G h 3
D1 = 1---~ ; Dxy= 1--2-
and h is the thickness of the plate.
T. H. TOPPER - A. N. SHERBOURNE - V. SAARI
20
~ I 6 - E
Z 0
I - ( J ta 12 - i1 t/.I C3
- I
n r
z~s- W
4 -
0.8
0.6
0.4
0.2
~ %
OF ULTIMATE LOAD
I I I I I 0 - 0 4 8 12 16 20
TIME (hr)
Fig. 16. - Creep curve for flexural test. Plate 3.
This equation can be solved for some particular loading conditions and boundary values but the solution becomes very complicated if the problem is at all unusual. In general, approximate solutions have to be used.
The coefficients involved are E' x and E'y, the effective moduli in the co-ordinate directions; G, the torsional modulus; and E" which corresponds to Poisson's ratio in the case of isotropy. The above equation neglects shear deflections in the transverse plane which, in general, are negligible for plates of medium thickness.
200-
E 150- e Z
u J e r I"-
100-
5 0 - I_
0
~ t AVERAGE STRESS- STRAIN CURVE
RESS-STRAIN CURVE FOR ADING OF 24 HOUR DURATION
! t I I 0.01 0-02 0-05 0-04 0"05
STR A IN
Fi o. 17. - Stress-strain curves with normal and low Ioadino rates showing effects of creep.
87
V O L . 11 - N ~ 6 2 - M A T E R I A U X E T C O N S T R U C T I O N S
Medwadowski [30] has solved the problem considering such shear displacements. For fibreglass plates, large deflections are generally encountered due to the relatively low rigidity and hence membrane stresses must be considered.
If the co-ordinate axes of the plate do not coincide with the axes of etastic symmetry then a tranformation can be performed, as described by Hearmon [31] to express the elastic constants in the co-ordinate directions in terms of the elastic constants in the directions of natural symmetry.
Scatter of results
Although not too many bending tests were performed, the scatter of results is smaller than in either tension or compression. For a plate analysis, then, it would seem reasonable that values of moduli should be obtained from bending rather than tension or compression tests. The effects of variation due to imperfections would be still further reduced in a plate than in a flexural specimen due to the continuity of material and the reduced effect of local failure. Considering the pIot of M vs. d a ~/' it is seen that, despite the differences in modfili from tension to compression, the behaviour in bending is almost identical. The argument was subsequently reinforced by the test plates which, in bending, had almost identical load-deflection curves indicating that scatter was reduced in the gross plate well below that found in smaller test pieces. In order to compare results of loading tests, the rates should be relatively equal, since the creep phenomenon has a pronounced effect on the deflection, particularly at loads approaching the failure load.
Physical constants
The tension and compression tests have considerable scatter in the results. Obtaining a true value of the moduli is further complicated by the severity of buckling in the compression test. This raises the question as to whether results from tension and compression tests can be applied to bending problems with any degree of accuracy. Comparison of theoretical and experimental curves for the moment-curvature relationships give reasonable and safe approximations for design purposes when a single modulus is used. The modulus obtained from the bending tests, however, gives the most consistent results and since the single modulus theory gives a reasonable fit to the experimental results itl would seem reasonable to use this theory with the value of E obtained from the bending tests.
The modulus in shear, G, was not determined experimentally, but Baer and Corten [32] suggest a value of 1/4.1 to 1/4.3 of E for this type of material.
Variation of material and geometry
For general loading conditions, the isotropic plate is most suitable since it can sustain the ultimate stress in any direction. If, however, the conditions of loading are
88
known before hand to be of a certain type and magnitude, then maximum stresses will occur only at certain points and in certain directions in the plate. It becomes economical, therefore, to strengthen only these portions to the required degree, other sections being designed according to prevailing maximum stress values. Thus, composites are quite useful in this respect.
For a plate with fibres oriented at 90 ~ to each other, the properties are similar in these directions and, for uniform loading, these directions should be symmetri- cally located about the co-ordinate axes to obtain optimum results. The strength of the material itself is directly proportional to the percentage of glass but for any particular loading condition, the overall strength of the plate depends also upon the orientation of the reinforcement.
With the composition of the material under study it is probable that strength could be increased by changing the geometry of the plate. Were the fibres alternated in individual layers, rather than in layers of certain thickness, there would be a better bond which might reduce lateral buckling of the layers. The material would also be more uniform and scatter of data would be minimized.
APPENDIX 1
Development of moment-curvature relationship
Let Et., Ec=Young's modulus in tension and compression; et, ~ , at, a~=tensile and compressive strain and stress; et,,, e~.,,, %,, % , = m a x i m u m tensile and compressive strain and stress; et=, e~=, at=, cry= = ultimate tensile and compressive strain and stress; d = depth of specimen; y = distance from top surface to neutral axis
Assume that the stress-strain curves in both tension and compression are linear to failure.
The strain distribution will be linear with distance from the neutral axis and the stress distribution on either side of the axis will be linear as shown in figure 1 A.
Force equilibrium:
X F x = O or C = T,
from which:
Now:
ac, . d -- y
Crtm Y
~rcm = ~c E,
firm = g't E t . 3 But, from plane strain:
(1)
(2)
ec Y D (3) e t d - y "
From (1) and (2):
e c E c d - y
e, E t y
From (3) and (4):
y E c _ d - y
d - y E t y
o r :
y 2 = ~ (d - y ) 2 .
Expanding equation (5):
( 1 - - ~ ) Y 2 + ( 2 ~ - d 3 , - ~ / E~
o r :
(4)
e c O ' c m
oxis
~t % m
Fi 9 . 1 A . - S t r e s s and strain distribution.
(5)
(6)
It should be noted that if E t = E c then equation (6) assumes an indeterminate form but the solution in this case is trivial and can be obtained using equation (5).
Now, suppose we have a known applied moment, M e :
. M e = M i
= 2_dC 3
dy 3 cr,~
from which:
- d 3 E c ec,
3 M e (7) ec y dEc
Let us consider the curvature of the concave side of the member rather than the neutral surface although the difference between the two will be quite small for small deflections. The reason for doing this is that the results can be checked against experimental values and where readings are usually taken at exposed surfaces.
Then strain:
% (8)
T . H . T O P P E R - A. N. S H E R B O U R N E - V. SAARI
and if, only small deflections are considered, then 1 >> ~y such that:
ec - q~y. (9)
From equations (7) and (9), the moment-curvature relationship can be obtained in the form:
M Er - 3 dy2' (10)
which can be seen to be linear.
To establish the ultimate moment capacity of the section, a failure criterion must be assumed. From the experiments performed the most usual mode of failure was a sudden delamination on the compression side and a tension failure on the bot tom surface. Hence, it would seem likely that failure occurs when the stress at the outermost fibre reaches its ultimate value either in tension or compression.
From equations (1) and (6):
at,,, _ Ec - ~ Et
a . . ~ E c E~ - E t "
Hence, if:
E c - E,f~E, ~cu - - <
ff tu gN/~c E t - - E t '
a compression failure will result, and if: (11)
from ffcu
Gtm fftu
then a balanced failure will occur with both tension and compression sides yielding simultaneously. Otherwise a tension failure will result. Once the model of failure is determined the ultimate moment capacity of the section can be obtained from equation (7).
MU ~ ~ (~ctn'
where acre is the compressive stress at moment of failure and from equation (10) the curvature at this moment will be:
~r
APPENDIX 2
Derivation of moment-curvature diagrams from experi- mental results
Consider a member with constant applied moment, M o, along its length. The curvature, being directly proportional to the bending moment, will also be constant and form the arc of a circle of radius R (fig. 2A).
89
V O L . 11 - N ~ 6 2 - M A T I ~ R I A U X ET C O N S T R U C T I O N S
y
R x
Fi 9. 2 A. - Elastic curve.
Consider the deflection at the centre as 6 and the length of the m e m b e r as I. The equa t ion of the elastic curve becomes:
x2 q.-y2=R 2, (1)
when x=O, y = R , x=I/2, y = R - 6 .
Subst i tu t ion into (1) leads to :
462+12 t R= 8 ~ ;
1 86 } = R - 4 3 2 +~2 �9
(2)
F o r small deflections l 2 ~ 4 62.
So Equat ion (2) can be writ ten, as an app rox ima t ion :
86 ~b= 7 .
F o r a par t icu lar p rob lem l is known and 6 can be measured for any value of moment ; the momen t - curvature d iag ram can thus be calculated.
A C K N O W L E D G E M E N T S
This work was carr ied out in the Depa r tmen t of Civil Engineering of the Univers i ty of W a t e r l o o under cont rac t to the Emergency Measures Organ iza t ion of the Government of Canada .
R E F E R E N C E S
[1] TSAI S. W. - Structural behaviour of composite materials. NASA CR-71, Washington, D.C., July, 1964.
[2] KITZMILLER A. H., DE HAVEN C. C., YOUNG R. E. - Design Considerations for spirally 91ass-reinforced filament wound structures as rocket inert parts. ARS Preprint 983-59, 1959.
[3] ZICKEL J. - Isotensoid pressure vessels. ARS Journal; 32, 1962.
[4] READ W. S. - Equilibrium shapes for pressurized fibreglass domes. ASME Paper 62-AV-4, 1962.
[5] HOFFMAN O. -- Stresses a~d deformation in filament reinJbrced structures. IAS Paper 62-26, 1962.
[6] SHAFFER B. W. - Stress-strain relations of reinforced plastics parallel and normal to their internal filaments. AIAA Journal, Vol. 2, No. 2, February 1964.
[7] TSAI S. W., AzzI V. O. - Elastic moduli of laminated anisotropic composites. Experimental Mechanics, Vol. 5, No. 6, June 1965.
[8] KROLIKOWSKI W. - Stress-strain characteristics of glass fibre reinforced polyester. SPE Journal, Vol. 20, No. 9, September 1964.
[9] OUTWATER J. O., WEST D. C. - Stress distribution in the resin of reinforced plastics. Modern Plastics, September 1961.
[10] RATCH Jr. H. - What's wrong with data on reinforced plastics:' Materials in Design Engineering, May 1961.
[I 1] BARNET R. R., PROSEN F. P. - Validity of mechanical tests Jbr glass reinforced plastics. Materials Protection, June 1964.
[12] MARTIN J. - Theories of strength for combined stresses - and non-isotropic materials, Journal of the Aeronautical Sciences, Vol. 24, April 1957.
[13] AzzI V. D., TSAI S. W. - Anisotropic strength of composites, Experimental Mechanics, Vol. 5, No, 9, September 1965.
[14] FRIED N. - Role of resin in reinJbrced plastics. Modern Plastics, September 1963.
[ 1 5 ] BROUTMAN L. J. - Failure mechanisms Jbr filament reinJorcedplastics. Modern Plastics, Vol. 42, April 1965.
[16] BIOT M. A. - Theory of stress-strain relations in anisotropic visco-elasticity and relaxation phenomena. Journal of Applied Physics, Vol. 25, No. 11, Novem- ber 1954.
[17] KAYE A., SAUNDERS D. W. - - Creep in an anistropic medium. British Journal of Applied Physics, Vol. 15, 1964.
[18] Engineering design properties of glass reinforced plastics. British Plastics, Vol. 38, No. 2, February 1965.
[19] BERSHTEIN V. A., GLIKMAN L. A. - Mechanism of delayed failure in polyester ftbreglasses, Soviet Physics- Solid State, Vol. 5, No. 8, February 1964. Translated from Fizika Tverdogo Tela, Vol. 5, No. 8, August 1963.
[20] BRINK N. O. - Mechanical behaviour of reinforced plastics at cryogenic temperatures. SPE Journal, Vol. 20, No. 10, October 1964.
[21] HOWSE Jr. P. T., PEARS C. D. - Thermal properties of reinforced plastics, Modern Plastics, Vol. 39, Septem- ber 1961.
[22] BARNET R. L. - Review of structural design techniques for brittle components under static loads. Contract AF 33(657)-8339, ARF Report, No. 8259, May 1963.
[23] WEIBULL W. -- A statistical theory of the strength of materials. Royal Swedish Academy of Engineering Sciences, No. 151, 1939.
[24] BRADSTREET S. W. - Principles affecting high strength to density-composites with fibres or flakes. Technical Documentary Report, No. MLTDR64-85, May 1964.
[25] SONNEBORN R. H. - Fibreglass reinforced plastics. Reinhold Publishing Corp., New York, 1954.
[26] VAN ECHO J., REMELY G. R., SIMMONS W. F. - High temperature creep-rupture properties of glass fabric plastic laminates. Dept. AF, WADC 53-491, Decem- ber 1953.
STEEL D. J. - The creep and stress-rupture of reinforced plastic's. Transactions, J. Plastics Inst., October 1965.
[271
90
[28] SHAND E. B. - Glass engineering handbook, McGraw- Hill Book Co. Inc., New York, 1958.
[29] MCABEE E., CHMURA M. - Effects of high rates compared with static rates of loadin9 on the mechanical properties of 91ass reinforced plastics. 16th Annual Technical and Management Conference, Reinforced Plastics Division, The Society of the Plastics Industry Inc., Paper 13-D, 1961.
T. H . T O P P E R - A . N . S H E R B O U R N E - V . S A A R I
[30] MEDWADOWSKI S. J . . - Refined theory of elastic orthotropic plates. A.S.M.E. Paper No. 58-APM-16, September 1958.
[31] HEARMON R.F.S. - An introduction to applied anisotropic elasticity. Oxford University Press, 1961.
[32] BAER E. -- Engineering design for plastics. Reinhold Publishing Corp., Chap. 14, New York, 1964.
Rs
Flexion des plaques de mati6re plastique renforc6es de fibres pos6es sur des supports 61astiques. - On ~tudie les propri~t~s m~caniques d'une r~sine polyester renforc~e de fibres de verre et de structures laminaires, en ayant en rue l'application des r~sultats d l'~tude en flexion d'une plaque d'un matOriau similaire.
Les essais de traction et de compression ont it~ r~alis~s afin d'obtenir la relation contrainte/dOformation ainsi que les contraintes et ddformations ultimes. La nature du
matiriau dlterminant une dispersion des r~sultats, on a f ix i des valeurs moyennes comportant des limites maximales et minimales. On a r~alisi des essais deflexion rant th~oriques qu'exp~rimentaux, et obtenu des relations moment-courbure. Les essais de fluage montrent que pour des charges nettement au-dessous de la charge de rupture les effets du fluage sont presque nigligeables si les fibres sont disposies longitudinalement dans l'iprouvette. Comme les rdsultats en flexion sont moins dispersis que ceux obtenus en traction et en compression, on peut supposer qu'ils se pr~tent mieux Otre utilisis dans la th~orie des plaques.
