Part IV
Experimental Activities
A significant amount of experimental researches have been focused on the characteriza- tion of the creep behavior of composite materials. Nevertheless, most of the creep models found in the building codes are based on the extrapolation of data from short-term exper- iments. There are still many uncertainties on how to model the phenomenon as the time scale is of the order of some decades. The prediction of the long-term behavior of viscous materials can be based on short-term experiments by using the Time-Temperature-Stress Su- perposition Principle (TTSSP). This is an extension of the Time-Temperature Superposition Principle (TTSP). The TTSSP is based on the hypothesis that every creep curve, adapted to Findley’s law, has the same shape at different temperature and stress states. However, its validity for Glass Fiber Reinforced Polymer (GFRP) pultruded beams has not been proven yet. Many experimental tests have been carried out to validate the TTSSP technique on the characterization of the creep behavior on long term of viscous materials and composite materials. Conversely, limited experiments have been addressed to determine whether such technique is also valid for GFRP pultruded beams in bending and shear-bending states.
Effectively, the characterization in the long term of the mechanical properties of GFRP pultruded beams, such as the viscoelastic modulus E(t), is really required in order to verify the Serviceability Limit State (SLS) included in every structural normative. Moreover, the long-term viscoelastic parameters, e.g. the Findley’s parameters m and n, must not be characterized by extrapolation from short-term data. In reason of that, the validation of the TTSSP principle assumes a big relevance for the design of structures made of pultruded composite elements because this technique enables the designer to characterize the long-term behavior of GFRP pultruded beams through several short-term tests.
The Chapter 8 addresses a literature review of the Time-Temperature-Stress Superpo- sition Principle (TTSSP) whereas the Chapter 9 focuses the experimental activities on the applicability of the TTSSP for GFRP pultruded beams.
The purpose of this experimental campaign is to provide the legislator and the producers of GFRP pultruded elements with a reliable method to predict the creep response in the long term, such as until 50 years. In particular, the objective is to investigate the applicability of the Time-Temperature-Stress Superposition Principle (TTSSP) for Glass Fiber Reinforced Polymer (GFRP) pultruded beams having an epoxy matrix. Actually, this technique could be used to predict the long-term behavior, such as the stiffness reduction, of GFRP pultruded elements. According to Findley’s law, the TTSSP only includes the long-term behavior in the secondary (steady-state) creep.
To this end, 4-point bending tests have been performed in the short term at different temperature and stress levels. Later, the creep data have been adapted with the Findley’s law and finally the TTSSP has been applied to predict the creep deflection until about 10 years. The applicability of the TTSSP technique has been investigated under both pure bending and shear-bending states.
The experimental tests have been carried out in the temperature controlled room of the Laboratoire Navier of Ecole des P onts − P arisT ech in Paris. The present experi- mental study is included in the research programme on composite materials developed by the Architected Materials and Structures team (AMS) of the Laboratoire Navier, Ecole des Ponts-ParisTech.
Chapter 8
Literature review
8.1 Time-Temperature-Stress Superposition Principle
Viscoelastic behavior of composite materials can be forecasted by Findley’s law, Equation (3.18), as described in Part II. In order to predict creep deflections over time, the knowledge of Findley’s parameters n and m is required. Such parameters are stress and temperature dependent and can also change as functions of the duration of the creep tests which provided their values. Thus, the mechanical behavior of composite materials in the long term should not be characterized by extrapolation of short-term test data.
Generally, the prediction of the creep response in long time, creep failure and durability of composite materials require the collecting of experimental data over a long period of time [15], which is both costly and impractical. Therefore, research and industry needed to get an
“accelerated characterization procedure” to forecast the creep response on long term through just several short-term laboratory tests. Many and different accelerated characterization procedures have been used to predict the long-term behavior of viscoelastic materials, such as polymers. Later, this approach has been also used to characterize the mechanical behavior of composite materials, in which viscoelastic behavior is due to the polymer matrix.
Herbert Leaderman in 1943 [11], writing about viscoelastic materials, stated “It appears reasonable to assume that the curves f or dif f erent temperatures are of the same shape, but that increase of temperature displaces the creep curve f or constant load to the lef t; we can then say that increase of temperature has the ef f ect of contraction the time scale”. Actually, Leaderman discovered the physics phenomenon but he did not proposed any accelerated procedure. Two years later, Tobolsky and Andrews used Learderman’s observations to form a curve which predicted the long-term behavior of viscoelastic materials.
Tobolsky introduced the horizontal and vertical shift of the short-term curves realizing the contraction of the time scale observed by Leaderman. Superposing the short time curves at long times, the “master curve”, which could predict the long-term behavior of the materials, was obtained.
This type of acceleration was called by different names including “Time Temperature Analogy”, “Method of Reduced Variables” and “Time Translation Equivalence” but only in 1980, Griffith [12] called it definitively “Time-Temperature Superposition Principle”. Since 1945 many investigations were proposed on TTSP and the method met other accelerated factors such as humidity and mostly stress levels (Stress-Time Superposition Principle) as extensions of the TTSP. Afterwards combinations between two accelerated factors were realized.
Actually, the Time-Temperature-Stress Superposition Principle (TTSSP) and the TTSP were frequently used to predict the creep and relaxation response of amorphous polymers;
subsequently, they have been also applied to predict creep failure. Since 1978, these methods have been applied to composite materials in order to characterize both their viscoelastic behavior and lifetimes through several short-term tests at different conditions [12].
According to Leaderman, the TTSSP is based on the observation that every creep curve has the same shape at different temperature and stress conditions; moreover, the increase of temperature or stress produces an acceleration of degrade, and therefore of creep response, which causes a contraction of the time. This phenomenon can be applied through a shifting of the creep curves to the right on time scale. Hence, from a data collection of short-term creep tests at elevated temperature and high stress levels, the creep response on long term at given reference temperature and reference stress can be predicted.
The TTSSP procedure requires the knowledge of the creep curves on short term; Findley’s law, Equation (3.18), is usually used to predict creep response on short time of viscoelastic materials. For a given thermomechanical condition, assuming the time exponent n to be stress and temperature independent, Findley’s equation can be rewritten as
ε(σ, T, t) = ε0(σ, T ) + m(σ, T ) · tn (8.1) Equation (8.1), following TTSSP, can be expressed in terms of the reference temperature TRand stress σR as
ε(σ, T, t) = av
ε0(σR, TR) + m(σR, TR)
t ah
n
(8.2) where the vertical and horizontal shifts are respectively
av= ε0(σ, T )
ε0(σR, TR) (8.3)
ah= ε0(σR, TR) · m(σ, T ) ε0(σ, T ) · m(σR, TR)
−n1
(8.4) It must be pointed out that the appropriate shifts are a vertical [log av] and horizontal [− log ah] shift of the log-creep strain curve plotted versus a log-time scale.
Equation (8.2) can be used to form the curve, afterwards called “master curve”, which could predict the creep response at the reference conditions (TR and σR) from the creep curves on short term, formed by Findley’s equation, at several test conditions.
Yen and Williamson have carried out several creep tests in order to utilize the TTSSP to predict the long-term creep response of unsaturated polyester reinforced with glass fibers stressed off-axis [5]; Figure 8.1 shows the relationships between the vertical and horizontal shift factors at different temperature and stress levels interpolated by Yen and Williamson from experimental data. The relationship between the vertical shift factor avand the stress at different temperature levels shows a temperature independence of the vertical shift factor.
Thanks to Equation (8.3), if the vertical shift factor is temperature-independent, the elastic response ε0 is also temperature-independent, as previously established in [5] for thermo-rheologically simple materials. Hence, if the TTSP technique is used through sev- eral short-term tests at a given stress level and different temperatures, the master curve for thermo-rheologically simple materials (TSM) can be formed by only horizontal shifting in a logarithmic time scale [5] [8] [12]; conversely, the master curve for thermo-rheologically
complex materials (TCM) also requires a vertical shifting of the data on short term because the elastic characteristics of the such materials exhibit a temperature dependence.
Figure 8.1: The relationships between the vertical log avand horizontal − log ahshift factors at different temperature and stress levels [5]
Yen and Williamson have concluded that the coupling of Findley’s equation and the TTSSP is valid for generating a master curve which predicts the long-term creep response of an off-axis composite material (polyester matrix reinforced with glass fibers) up to 3200 times of the duration of the original creep experiments [5]. They also pointed out that smooth master curves which they have fitted, represent the long-term creep response without physical aging because, as reported by Janas and McCullogh [6], the master curves in bi- logarithmic plots are upward concave as shown in Figure 8.2. In fact, in order to incorporate the physical aging effect in a master curve, creep experiments of longer duration must be taken into account.
About aging effects, Miyano et al. have studied the effect of physical aging on the time- temperature superposition principle for viscoelastic behavior of thermosetting epoxy resins [7].
Figure 8.2: Master curve obtained through TTSSP [5]
Ma et al. have planned many tests on composite laminates made of carbon-fiber- reinforced polyetheretherketone at 45° off-axis in order to evaluate their long-term behavior using the coupling between the Findley’s law and the TTSSP technique [14]. Every speci- men, at least three for each test condition, was subjected to constant load for 600 minutes at different temperature (25°C÷120°C) and stress (31.46 MPa÷188.76 MPa) levels.
Figure 8.3 shows the relationship between the two shift factors and the stress at different temperature levels; because the material was thermo-rheologically simple, there is no explicit temperature dependence of the vertical shift factor. However, as expected, the horizontal shift factor is a function of both stress and temperature. Lastly, they confirmed that the coupling of Findley’s equation and TTSSP technique is also valid for composite laminated of carbon-fiber-reinforced polyetheretherketone at 45° off-axis. Figure 8.4 shows the two master curves obtained.
Figure 8.3: The relationship between vertical and horizontal shift factors with the stress [14]
Figure 8.4: The master curves at reference conditions (85°C and 94.38 MPa) [14]
Muliana et al. [8] used Schapery’s equation and TTSP principle to predict the long-term behavior of multi-layered composite materials; since specimens were thermo-rheogically com- plex materials (TCM), master curves needed both horizontal and vertical shift factors. In particular, the vertical shifting is associated with the parameter of the nonlinear instanta- neous elastic compliance g0and its temperature dependence. The amount of vertical shifting was defined by:
av= (g0T−1)D0 (8.5)
The horizontal shift factor was equal to the log of the inverse of the time-temperature shift factor aT in Equation (3.20). In Figure 8.5, the horizontal shifting − log aT
with
the temperature is shown. Moreover, the master curve of compression creep compliance [Dc = ε(t)/σ0] of E-glass/vinylester specimens is plotted; the master curve represents creep behavior up to 500 hours and 1000 times longer than the conducted creep tests.
Figure 8.5: Master curve of uniaxial load and horizontal shift factor [8]
The TTSP principle has been widely used to predict creep and fatigue failure of com- posite materials [40] [41]. Actually, it was experimentally demonstrated that the same time-temperature shift factors used to predict the long-term viscoelastic behavior of the matrix resin can be applied in order to form the master curve for all three types of strengths (static, creep and fatigue loadings).
Christensen and Miyano have developed a procedure of accelerated testing methodology (ATM). Miyano confirmed this procedure through experimental tests on CFRP laminates composites [41] [42]. In Figure 8.6, the detailed steps are illustrated.
Figure 8.6: Procedure of accelerated testing methodology (ATM) [41]
Firstly, several creep tests at various temperatures must be carried out in order to mea- sure the viscoelastic modulus over time of the matrix. Then, time-temperature shift factors are determined experimentally by shifting the viscoelastic modulus curves at several tem- peratures into time scale to form a master curve of the modulus at a reference temperature.
Through several constant strain rate (CSR) loading tests at elevated temperatures, CSR creep strength master curve can be obtained by using the same shift factor previously cal- culated. The next step is to convert the CSR strength master curve to the creep strength master curve of the composites by the linear cumulative damage law for monotonic loading.
Fatigue tests at different temperature and stress levels with a single frequency and zero stress ratio must be conducted. Then, the fatigue strength master curve at zero stress ratio is determined by the previous tests and the same time-temperature shift factor used for the resin matrix. Finally, the fatigue strength master curve at any arbitrary frequency, stress ratio and temperature is obtained from the master curve of fatigue strength at zero stress ratio by the linear dependence of fatigue strength upon stress ratio.
The present procedure is based on three really important conditions:
• The same time-temperature shift factors are applicable for both viscoelastic behavior and strength properties of matrix resin and their composites.
• The linear cumulative damage (LCD) law is applicable to the strength by the mono- tonic loading.
• The fatigue strengths exhibit linear dependence in the stress ratio of the cyclic loadings.
Chapter 9
Applicability of the TTSSP for the characterization of the creep behavior of GFRP pultruded
beams in the long term
9.1 Introduction
The present work includes a series of nine short-term (24 hours) and one long-term (40 days) 4-point bending tests on pultruded GFRP composite beams. Because of the 4-point bending setup, the beam segment between the application points of the loads is under only bending moment (“pure bending state”), while the remain parts of the beam are subjected to combined shear force and bending moment (“shear − bending state”).
Two different applications of the TTSSP technique were applied. Firstly, the applicability of such principle was investigated in pure bending state. Deflection data were acquired at the mid-span cross section (point A) and at 80 mm from the mid-span section (point B), close to the load application point. Thus, by taking a partition of the overall beam, an equivalent beam of length 160 mm subjected to only two bending couples at both ends has been defined, as shown in Figure 9.1. Subtracting the deflections at point B from those at point A [δc(A) − δc(B)], the creep deflections at the mid-span section of the equivalent beam have been evaluated. Thanks to the equivalent beam in pure bending state, the TTSSP applicability was verified without shear effects.
Then, the deflection at the mid-span section (point A) of the overall beam subjected to 4- point bending was considered in order to investigate the applicability of the TTSSP principle on the GFRP pultruded beams including shear effects; indeed, the shear effects can alter the creep behavior in the long term. Finally, the magnitude of the shear effect was studied through a rational comparison between the results in pure bending and shear-bending states.
The characterization of creep behavior in short term is studied through Findley’s law in Equation (3.18), wherein only creep deflection is taken into account
δc(σ, T, t) = m(σ, T ) · tn(σ,T ) (9.1) where δc is the creep deflection, m is the amplitude of transient creep deflection, n is the
time exponent, σv is the stress, T is the temperature and t is the time.
According to the Time-Temperature-Stress Superposition Principle, each Findley’s curve in short-term can be shifted horizontally and vertically in order to describe the evolution creep deflections at a reference stress state σvR and temperature TR. Thus, considering the time exponent n to be stress-temperature independent, Equation (9.1) can be expressed in terms of the reference conditions (TR, σR) as follows
δc(σ, T, t) = av
"
m(σR, TR) · t ah
n(σR,TR)#
(9.2) in which the vertical shift av is equal to one because the elastic deflection is not considered and the horizontal shift ah is given by
ah=
m(σ, T ) m(σR, TR)
−n(σR,TR)1
= m(σ, T ) mR
−nR1
(9.3) where mR and nR are the parameter values at the reference conditions.
Therefore, in a log-log diagram of creep deflections versus time, the short-term curves are shifted by [− log ah]. In this way and by performing several short-term tests in various stress and temperature levels, the master curves, which could predict the creep deflections in long term, were obtained and compared with experimental creep curves in long term (40 days).
F F
a = 2'000 2'000 mm a = 2'000
F·a
F
F
M = F · a
160 mm
A B
A B
A B
A B
BENDING FORCE
SHEAR FORCE
F·a
M = F · a
80 mm
Figure 9.1: Pure bending equivalent model
9.2 Experimental programme
Several creep tests have been carried out at different environmental conditions and stress levels. Temperature levels of 26°C, 32°C and 41°C were chosen. The variation of temperature during each test was maintained within ±2% and the average temperature did not exceed
±1% of the target temperature. The temperature has been monitored through specific devices.
Three levels of loading were planned for each temperature level. Every level of loading level produced the maximum stress in the specimens that represented approximately the
26%, 35% and 45% of the ultimate strength of the tested FRP composite material. The applied loads did not exceed ±1% of the target stress level during each test.
In Table 9.1 the master plan of the experimental programme is presented. For each test condition, two specimens were tested; the duration of the short-term creep tests was 24 hours with constant applied forces. Each specimen was left at the target temperature at least 1 hour before the load application in order to assure an homogeneous temperature field.
To obtain the master curve, the previously described procedure of acceleration (TTSP) and Findley’s equation were used.
The reference conditions chosen were 26°C and 26% of ultimate stress. Indeed, in civil engineering applications, FRP composite materials are usually subjected to such stress level;
moreover, 26°C can be considered the average environmental temperature in many building sites. Finally, in order to investigate the validity of the derived master curves, two specimens were tested in the reference conditions for 40 days.
N. Test T (°C) F (N) σ(MPa)
1 26 1460 170.1 [w 26%]
2 26 1994 231.8 [w 35%]
3 26 2589 300.3 [w 45%]
4 32 1460 170.1 [w 26%]
5 32 1994 231.8 [w 35%]
6 32 2589 300.3 [w 45%]
7 41 1460 170.1 [w 26%]
8 41 1994 231.8 [w 35%]
9 41 2589 300.3 [w 45%]
Reference conditions:
Temperature 26°C
Stress 26% (170.1 MPa) Number of specimens for each test 2
Duration of creep test 24 hours Specimen length 1000 mm
Specimen width 40 mm Specimen thickness 16 mm Nominal span 600 mm Table 9.1: Master plan
9.3 Experimental setup
The creep tests were carried out on simply supported pultruded beams subjected to bend- ing. The beams were loaded in 4-point bending setup, where two vertical loads were applied symmetrically at 200 mm from the supports, thirds of the 600 mm total span. The speci- mens had rectangular cross-section 40x16 mm and length 1000 mm. Figure 9.2 shows the theoretical model of the experimental tests.
Unfortunately, there are no standards for the test methods to characterize the creep properties of real scale pultruded beams; anyway, the experimental setup was designed according to the European Standard ISO EN 14125 : 1988 on the determination of flexural properties of fiber-reinforced plastic composites [39].
100
L = 600 100
F F
a = 200 200 a = 200
b=40 h=16
800 A
Figure 9.2: Schematic representation of the experimental setup
The experimental tests were carried out on a specific custom made apparatus (Figure 9.3). The apparatus has a main framework made of standard steel U-profiles. Furthermore, 4 tubes and 4 axes improve the overall stiffness of the frame. In particular, 4 cylindrical tubes of 48 mm of diameter provide the simple support of the beams (Figure 9.4), whereas 4 axes of 30 mm of diameter support the levers, through which the loads were applied, through non-perfect hinges. The effect of friction between the holes of the levers and the axes has been taken into account.
Figure 9.3: Creep apparatus
The frame presented 13 lines in which several specimens can be tested at the same time;
6 lines of 600 mm of span and the others of 800 mm. Thus, the setup allowed to carry out many creep tests simultaneously.
The friction between the tubes, which constituted the simple supports, and the specimens was considerably reduced through a lubricant smeared in both specimens and supports. The weight of the specimens, as distributed load along the beams, was neglected because it was insignificant compared to the applied loads.
Figure 9.4: Specimens on the frame
Finally, the large deflections theory was neglected since the maximum deflection did not exceed 1/10 of the total nominal span L [39]. Moreover, Annex C demonstrates that second order effects related to large deflection and friction at the supports are negligible.
The temperature and moisture were controlled by sensors installed in the room and near the specimens. Three infra-red heaters and moisture sensor were installed in the experi- mental room (Figure 9.5); they were connected to temperature controllers in order to check the temperature during the experiments. The average temperature, in every test, did not exceed ±1% of error on the target temperature.
Figure 9.5: Devices
In each line two displacement transducers were placed at the mid-span section and at 80 mm from the mid-span section in order to record the displacements in both pure bending and shear-bending states (Figure 9.6). The accuracy of the displacement transducers was more than 0.01 mm and the errors did not exceed ±1%. The experimental data were recorded through a computer data acquisition system by the software LABV iew.
Figure 9.6: Displacement transducers
Each lever was loaded at the end by metallic plates of known weight (≈4kg), as shown in Figure 9.7. The loads were applied to the beams through hangers which connected the specimens and the load lever system. The weight of the metallic plates before the loading was led by pressure pistons. The values of the applied loads did not exceed ±1% of the target load and the time in which the loads reached their maximum value was approximately 5 seconds.
(a) The pressure pistons (b) Metallic hangers
Figure 9.7: Loading system
The loads applied to the specimens have been evaluated by considering the moment equilibrium of the lever with respect to the metallic axes; thanks to the hanger connected to the specimens and by the moment equilibrium of the lever system, the specimens were subjected to loads up to 7-8 times larger than the weight of the loading plates. Figure 9.8 shows the loading setup.
600
1060
60 60
740 370
200
Figure 9.8: Loading setup
The applied forces were calculated based on the loading lever scheme shown in Figure 9.9.
In Annex A, all the necessary mathematical manipulations are explained. From equilibrium, the final equation is given by
W L3cosα+ P L2cosα − cosγF L1cos(α + γ) −
−µ r
µ(F−W −P +µF·tgγ)
µ2+1 −F · tgγ2
+F−W −P +µF·tgγ µ2+1
2
· d= 0
(9.4)
wherein:
F = Ty is the applied load,
W is the total weight of the plates,
P is the lever weight,
d is the diameter of hinge bar,
µ= 0.024 is the average friction coefficient of the non-perfect hinge between the lever and the metal axis,
α is the angle between the lever and the horizontal axis, γ is the angle between the hanger force T and the vertical axis, L1= 50mm is the distance between the hinge and the hanger force,
L2= 203mm is the distance between the hinge and the center of mass of the lever, L3= 366mm is the distance between the hinge and the loading plates.
By using an iterative numerical solution method Equation (9.4) is solved and the effective force applied on the specimens Ty is calculated.
T W
Tx
P
Ty=F
R
γ
Rx
Ry
α Fr
L1 L2 L3
Figure 9.9: Loading lever scheme and applied forces
The horizontal force Tx applied on the specimens, which produces an axial force in the beams, was neglected because very small with respect to other applied forces. The average friction coefficient (µ = 0.024) of the non-perfect hinges between the levers and the metal axes was estimated through tests based on the dynamic theory of the simple pendulum with friction in the hinge support, in which the friction causes a certain loss of mechanical energy at every half period. In Annex B the mathematical calculations used to obtain the friction coefficient value are extensively shown. The average friction coefficient µ of all the levers was taken into account in order to calculate the real applied forces.
During every experiment, the angle of the levers α with respect to the horizontal line did not change more than 2.3◦. This variation, caused by the creep deflection over time, did not produce relevant changes in the forces F , which remained within ±1% of the target loads.
The maximum stress in the specimens was calculated by the classic theory of Navier as
σM AX = MM AX
J · h
2 (9.5)
where:
J is the inertia of the cross-section of the specimens, h is the height of the specimens and
MM AX is the maximum bending moment in the specimens.
The maximum bending moment MM AX was calculated considering the effective span after the deformation of the beam. Actually, the deflection of the specimen after the loading produces a rotation of the beam on the circular supports which decreases the effective span.
As shown in Figure 9.10, the decrease ∆ at each support is equal to
∆ = R · sin β (9.6)
therefore, the effective span considered is given by
Lef f = L − 2∆ (9.7)
where L is the initial span equal to 600 mm. The rotation β was calculated, for each load level, thanks to the classical Euler-Bernoulli’s beam theory neglecting the shear effects.
R β Δ
Figure 9.10: Decrease of span due to the rotation of the specimen
Finally, the maximum bending moment and the maximum stress have been expressed as
MM AX = F · (a − ∆) (9.8)
σM AX = 6F · (a − ∆)
b · h2 (9.9)
where:
a= 200mm is the initial distance between the supports and the applied loads, F is the single vertical applied force,
b is the width of the specimen and h is the height of the specimen.
9.4 Materials
The GFRP pultruded beams were made of glass fibers soaked in epoxy vinylester resin matrix by pultrusion process. The specimens consisted of bars of rectangular cross-section 40x16 mm and length 1000 mm, as shown in Figure 9.11. The ultimate tensile stress of the beams was evaluated to be about 663 MPa through destructive bending tests.
Figure 9.11: The specimen
The resin matrix, DERAKANE® 470 − 36S Epoxy V inylester Resin, was an epoxy novolac-based resin designed to provide exceptional thermal and chemical resistance prop- erties at higher temperatures. This resin offers a high resistance to solvents and chemicals, good retention of strength and toughness at elevated temperatures and excellent resistance to acidic oxidizing environments. It is recommended for most FRP fabrication processes such as the pultrusion. The typical properties of the resin matrix are listed in Table 9.2.
Property S.I. Test Method
Tensile Strength 90 MP a ASTM D-638/ISO 527 Tensile Modulus 3.6 GP a ASTM D-638/ISO 527 Tensile Elongation, Yield 3 ÷ 4% ASTM D-638/ISO 527 Flexural Strength 160 MP a ASTM D-790/ISO 178 Flexural Modulus 3.8 GP a ASTM D-790/ISO 178 Density 1.15 g/cm3 ASTM D-792/ISO 1183 Heat Distortion Temperature 145◦C ASTM D-648 A/ISO 75
Table 9.2: Typical properties of post-cured resin clear casting
The glass fibers, Advantex™ R25H Glass F iber, combine the excellent mechanical and electrical properties of traditional E-glass with the acid corrosion resistance of E-CR glass.
The volume fraction of fibers was 65% whereas the volume fraction of the matrix was 22%.
9.5 Results and discussion
9.5.1 Pure bending state
The creep deflection of the pultruded beams for each temperature and stress condition was recorded. Then, creep curves in pure bending state were obtained subtracting the deflection at 80 mm from the mid-span section (point B) from those at the mid-span section (point A) [δc,P B= δc(A) − δc(B)]. It must be pointed out that the creep deflections in pure bending state are related to those of the equivalent beam in Figure 9.1. The creep responses were characterized through Findley’s law as
δc(t) = m · tn (9.10)
In order to evaluate the values of the Findley’s parameters m and n, bi-logarithmic scale plot was required. In bi-logarithmic plot [log(δc) − log(t)], Findley’s equation becomes
log δc(t) = log m + n · log t (9.11) Equation (9.11) represents a straight line in the bi-logarithmic plot [log(δc) − log(t)] wherein log m is the intersection of the straight line with the ordinate axes [log(δc)] and the parameter nis the slope of the straight line.
Therefore, each experimental creep curve was plotted in bi-logarithmic scale in order to evaluate the Findley’s parameters. The values of Findley’s parameters m and n were determined by interpolation technique, neglecting the first minutes of the experimental data.
Actually, as shown in Figure 9.12, in bi-logarithmic plot the creep data were better adapted with a bilinear curve. Only the second straight line, which starts after about 30 minutes in each experiment, was considered because it predicts better the secondary creep behavior of the tested pultruded beams after 24 hours. The R-squared value of the linear regression fit of the totality of the experimental data for each short-term experiment is quite good and of the order R2= 0.924.
For each test condition, the average values of the parameters m and n between the two specimens were taken into account to obtain the average and characteristic creep curves.
Henceforward, the parameters m, n and the relative creep curves will be referred to the average values for each condition of temperature and stress level.
Log TIME (min)
Log DEFLECTION (mm)
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0
-0.5
-1
-1.5
-2
-2.5
Figure 9.12: Bilinear curve in bi-logarithmic plot [log(δc) − log(t)]
Figures 9.13-9.14 show the creep responses at different stress and temperature levels. As expected, at the same temperature, creep responses increase with increasing stress levels, as shown in Figure 9.13. Conversely, Figure 9.14 shows that the creep deflections at the same stress level, do not seem to depend strongly on the temperature at least for the lowest stress levels (26% and 35%).
Time (min)
0 200 400 600 800 1000 1200 1400
0.02 0 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
Deflection (mm)
32°C
26% (Sp.1) 26% (Sp.2) 35% (Sp.1) 35% (Sp.2) 45% (Sp.1) 45% (Sp.2)
26% (Average) 35% (Average) 45% (Average) 26% (Sp.1) 26% (Sp.2) 35% (Sp.1) 35% (Sp.2) 45% (Sp.1) 45% (Sp.2)
26% (Average) 35% (Average) 45% (Average)
26°C
Time (min)
0 200 400 600 800 1000 1200 1400
0.02
0 0.04 0.06 0.08 0.1 0.12
Deflection (mm)
Figure 9.13: Creep responses of GFRP pultruded beams at 26°C and 32°C temperature levels
A reason of this phenomenon can be the low temperature levels with respect to the Heat Distortion Temperature (HDT) of the resin matrix, which do not produce temperature dependence of the creep behavior except to the highest stress level (45%). In Figures 9.13- 9.14 the experimental data and Findley’s curves with the average values of the parameters mand n are plotted.
Time (min)
0 200 400 600 800 1000 1200 1400
0.05
0 0.1 0.15 0.2 0.25
Deflection (mm)
45%
26°C (Sp.1) 26°C (Sp.2) 32°C (Sp.1) 32°C (Sp.2) 41°C (Sp.1) 41°C (Sp.2)
26°C (Average) 32°C (Average) 41°C (Average)
Time (min)
200 400 600 800 1000 1200 1400
0.01
0
Deflection (mm)
0.02 0.03 0.04 0.05 0.06 0.07
26%
26°C (Sp.1) 26°C (Sp.2) 32°C (Sp.1) 32°C (Sp.2) 41°C (Sp.1) 41°C (Sp.2)
26°C (Average) 32°C (Average) 41°C (Average)
0
Figure 9.14: Creep responses of GFRP pultruded beams at 26% and 45% of ultimate stress
Figure 9.15 shows the variation of the parameter n with the temperature at given stress levels and its variation with the stress at given temperatures. The time exponent n remains almost constant with the stress levels, as reported by [5] [14] (Figures 3.10-3.11) for axial stress states and [16] (Figure 3.14) for bending stress state of pultruded beams.
The parameter n can be also considered temperature independent. The temperature independence of the time exponent n agrees with [14] (Figure 3.11). Conversely, in [5]
(Figure 3.10) the parameter n seems to be temperature dependent.
Stress (%)
n
15 20 25 30 35 40 45 50
0 0.1 0.2 0.3 0.4 0.5 0.6
(26°C) (32°C) (41°C) Temperature (°C)
n
20 25 30 35 40 45
0 0.1 0.2 0.3 0.4 0.5 0.6
(26%) (35%) (45%)
Figure 9.15: Variation of the time exponent n as a function of the stress and temperature levels
This aspect is reasonable because in Figure 3.10 the time exponent n exhibits a nonlinear trend starting from high temperatures, more than 70-80°C. In the present experiment, the maximum temperature reaches 41°C and in this range of temperature the time exponent exhibits a constant trend also in [5] and Figure 3.10.
As reported in Part II, in [5] and the relative plot in Figure 3.12, Findley’s parameter mis a function of both stress and temperature. Furthermore, the value of m increases with an increase of temperature and stress levels. Figure 9.16 shows the values of the parameter mas a function of the stress and temperature levels.
(26%) (35%) (45%)
(26°C) (32°C) (41°C)
Stress (%)
m
15 20 25 30 35 40 45 50
0 0.01 0.02 0.03 0.04 0.05 0.06
Temperature (°C)
20 25 30 35 40 45
m
0 0.01 0.02 0.03 0.04 0.05 0.06
Figure 9.16: Variation of the Findley’s parameter m as a function of the stress and temper- ature levels
Figure 9.17 shows a comparison between the experimental data and the theoretical Find- ley’s curves obtained through the evaluation of parameters n and m.
26% - 32°C
Time (min)
Creep deflection (mm)
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
0 0
Experimental data (Sp.1)
Experimental data (Sp.2) Findley law (Sp.1)
Findley law (Sp.2)
Findley law (Average) n = 0.214 m = 0.00805
n = 0.196 m = 0.0065
n = 0.205 m = 0.00727
(a) 26% - 32°C
35% - 32°C
Time (min)
Creep deflection (mm)
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400
0.01 0.02 0.03 0.06
0.04 0.05
0 0
Experimental data (Sp.1)
Experimental data (Sp.2) Findley law (Sp.1)
Findley law (Sp.2)
Findley law (Average) n = 0.129 m = 0.0209
n = 0.146 m = 0.0168
n = 0.138 m = 0.0188
(b) 35% - 32°C
45% - 26°C
Time (min)
Creep deflection (mm)
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400
0.02 0.06
0.04
0 0.08
0 0.1 0.12
Experimental data (Sp.1)
Experimental data (Sp.2) Findley law (Sp.1)
Findley law (Sp.2)
Findley law (Average) n = 0.198 m = 0.0267
n = 0.199 m = 0.0249
n = 0.199 m = 0.0258
(c) 45% - 26°C
Figure 9.17: Comparison between the experimental data and the theoretical Findley’s curves
The elastic deflection seems to be dependent on both stress and temperature. The stress dependence on the elastic deflection is intrinsic in the classic elastic theory and the deflection δ0 increases with an increase of the stress. Elastic deflection also increases with the temperature though the increase is less pronounced compared to the increase because of stress. Such increase is attributed to the thermo-elastic properties of the resin matrix.
Figure 9.18 shows the relationship of elastic instantaneous deflection δ0 with the tem- perature and stress levels. Anyway, the elastic deflection could be considered to be constant with the temperature without large discrepancy.
(26°C) (32°C) (41°C) (26%) (35%) (45%)
Temperature (°C)
20 25 30 35 40 45
Elastic deflection (mm)
0 1 2 3 4 5 6
Stress (%)
15 20 25 30 35 40 45 50
Elastic deflection (mm)
0 1 2 3 4 5 6
Figure 9.18: Variation of the elastic instantaneous deflection δ0in function of the stress and temperature levels
Finally, experimental data are plotted in stress percentage versus creep deflection plot for different elapsed time (12 hours and 24 hours) and temperature levels. Figure 9.19 shows a nonlinear viscoelastic behavior of the tested pultruded beams for stress levels above 35%
of ultimate stress, as described in Figure 3.5.
Creep deflection (mm)
0 0.05 0.1 0.15 0.2
Stress (%)
0 5 10 15 20 25 30 35 40 45 50
26°C (12hr) 26°C (24hr) 32°C (12hr) 32°C (24hr) 41°C (12hr) 41°C (24hr)
Figure 9.19: Stress-deflection plot at different temperature and values of elapsed time Table 9.3 lists the values of the elastic deflection δ0, the parameters m, n and the hori- zontal shift factor ahfor each temperature-stress condition.
T est conditions δ0(mm) m n ah 26% - 26°C 1.43 0.0118 0.172 1
26% - 32°C 1.97 0.00727 0.205 -
26% - 41°C 2.44 0.0118 0.213 1
35% - 26°C 2.45 0.0151 0.172 0.235
35% - 32°C 2.50 0.0188 0.138 0.0661
35% - 41°C 3.94 0.0357 0.081 0.00161
45% - 26°C 4.40 0.0258 0.199 0.0106
45% - 32°C 4.60 0.0482 0.158 0.00028 45% - 41°C 4.95 0.0491 0.192 0.000251
Table 9.3: Values of δ0, m, n and ah
The horizontal shift factors were evaluated by Equation (9.3)
ah= m(σ, T ) mR
−nR1
(9.12) where:
mR= 0.0118 the reference value of the Findley’s parameter m, nR= 0.172 the reference value of the time exponent n.
The horizontal shift factors listed in Table 9.3, show that the higher accelerations were due to the increase of the stress levels rather than to high temperatures; indeed, the accel- erated time is given by tacc= t/ah. Moreover, at the lowest stress level (26%), the shifting is not possible because there is no explicit different creep behavior at different temperature levels.
The master curve has been obtained by interpolation of the short-term curves after having being shifted with the above mentioned shift factors in bi-logarithmic scale plot, as shown in Figure 9.20.
LOG Time (min)
0 0.5 1 1.5 2 2.5 3 3.5
26% - 26°C 26% - 32°C 26% - 41°C 35% - 26°C 35% - 32°C 35% - 41°C 45% - 26°C 45% - 32°C 45% - 41°C
LOG Creep deflection (mm)
0
-0.5
-1
-1.5
-2
-2.5
(a) The short-term curves before shifting
LOG Time (min)
0 1 2 3
26% - 26°C 35% - 26°C 35% - 32°C 45% - 26°C 45% - 32°C 45% - 41°C
LOG Creep deflection (mm)
0
-0.5
-1
-1.5
-2
-2.5
4 5 6 7 8
(b) The short-term curves after shifting
Figure 9.20: Obtaining the master curve (Pure bending state)
The master curve, obtained through the horizontal shift factors shown in Table 9.3 and the present acceleration procedure, indicates a prediction of creep behavior of about 11 years based on the short-term creep tests of one day (24 hours). Figure 9.21 shows the calculated master curve.
Long-term experiment was carried out in order to validate the above obtained master curve. Two specimens were tested for 40 days at the reference test conditions (26°C and 26% of the strength).
Time (days)
500 1000 1500 2000 2500 3000
0 3500 4000
Creep deflection (mm)
0.02 0.1
0.04 0.12
0 0.14 0.16
0.06 0.08
11 years
4500
Figure 9.21: The master curve for the pure bending creep deflection of 26°C and 26% of ultimate stress
Long-term experiment
Time (days)
5 10 15 20
0 25
Creep deflection (mm)
0.03 0.06
0.04
0 0.07 0.08
0.05
0.01 0.02
35 40
30
Master curve
Figure 9.22: Comparison between theoretical master curve and long-term experimental curve (40 days)
