Wood and composites cantilever beam structure has gained attention among researchers in the current years due to its universal structural applications, such as bridges, aeroplane wings, buildings, and transmission towers. However, when the structure is exposed to a constant loading for a very long time, it induces a structural collapse due to creep deformation. Therefore, it is essential to understand and identify the initial creep that can lead to structural collapse. In this study, wood and composite materials exhibit the same structural material morphology which performs as anisotropic material as it majorly contributes to failure. In this study, a state-of-the-art review of creep analysis and engineering design is carried out, with particular emphasis on the creep methodology of a cantilever beam. The existing theories and creep design approaches are grouped into two analysis methods, namely experimental and numerical approaches. To be more specific, the experimental works involved two main methods, namely load-based (conventional) and temperature-based (accelerated) techniques. Selected creep test on cantilever beam structure and coupon scale of wood and composite were highlighted and proposed as the building blocks for a prospective structural creep methodology. These aids build confidence in the underlying methods while guiding future work and areas, especially for long-term service of full-scale structure. At the end, the challenges of creep behaviour description accuracy and improvement on the strength criteria in engineering design were presented.

Wood and fibre reinforced polymer (FRP) composites are anisotropic materials. It is important to note that the understanding of its mechanical behaviour when interacting with the external environment is vital in its utilisation and application. This can be observed in defected natural fibre and wood that were exposed to constant loading in a prolonged time. [1–10]. In current applications, there are many cantilever beam structures, such as aeroplane wings, cross arms in transmission towers, and bridges which have been continuously applied, causing constant stress during the operation in a long term. This phenomenon led to permanent deformation in a slow process under constant stress, which is called creep [11–14].

The study of creep behaviour of materials is crucial to guarantee the safe operation of in-service components [15,16]. Usually, creeps were evaluated experimentally by using standard tensile, creep, and compressive specimens which oblige a sufficient volume of bulk materials. Due to limited resources, it is hard to produce a standard specimen from the in-service components [17,18]. For instance, a composite cross arm may not be able to obtain enough materials for a standard specimen.

In creep calculation and analysis, extension or stress relaxation is required in predicting a material creep data and its behavioural pattern for a more extended period [19]. When constant loading is applied, changes of displacement (extension) lead to the formation of a total strain experienced by the material. However, the total strain remains the same even though the total stress would decrease in response to the material viscous characteristic—the curves of constant strain and decreasing stress rate [20]. Likewise, in the cantilever beam condition, it continues to deflect in a constant applied load, which relaxes the material due to visco-strain behaviour. Therefore, the study of viscoelasticity behaviour of a specific material is essential to provide a valid theoretical outline and background to generalise the long-term behaviour of the material. This goal assists in the development of a wide range of analytical models and equations in clarifying the time-dependent behaviour of the material.

This study is aimed at presenting a state-of-the-art review of creep methodologies or the criteria used to understand and analyse the behaviour of wood and composite cantilever structures under long-term loading. Since the subject is quite old and many published documents are already available, it is hard to conduct this review comprehensively. Additionally, many historical articles on coupon testing, such as tensile, flexural, and compression are available [21–23]. Therefore, this review places particular emphasis on the development of suitable creep methods and approach for the cantilever beam structure specifically for wood and composite. Eventually, potential complications can be solved more deliberately in the future.

2Creep fundamental and principlesCreep is a situation where a material experiences a permanent deformation due to mechanical stress effect in a long time duration, which subsequently leads to structural failure [24]. The creep effect does change the size and shape of material when exposed to long-term loading [25]. In this study, the analysis of creep contributes mainly in mechanical and civil engineering sectors by estimating the service life of components and products [26]. To be more specific, the creep study is based on two variables, which are creep strain and time to describe the mechanical phenomenon further. The two variables are dependent on each other as the creep strain is based on Hooke’s law theory [27]. According to the theory, viscoelastic materials such as wood and fibre-reinforced polymer (FRP) composites are exhibited in elastic properties until the yield limit. These materials store elastic energy during shape movement (deformation) and later revert to the original position and form after the stress is removed from the objects [28]. However, when the forces applied surpass the yield capacity of the material, the plastic properties start to express, which later lead to rupture. On the other hand, the properties of the materials also permit liquid-like behaviour when exposed to high-temperature condition. This behaviour is commonly used to translate the deformation along with time-dependent function, and it is called viscoelasticity [29,30].

Theoretically, a polymeric material needs thousands of years to deform. Therefore, it is not practical to study creep since it requires a very long time to obtain data. To obtain similar creep data in shorter time, the material is exposed to an elevated temperature to accelerate the experimental process [31,32]. This type of creep testing is called the accelerated method which implements heat to expedite the creep in a faster time to predict the lifespan of the material [33]. To be specific, the time-temperature superposition principle (TTSP) is a suitable example to explain the creep acceleration method in which several temperatures are plotted into a master curve. In this section, the prediction of creep behaviour can be determined using both conventional (time-dependent) [34] and accelerated (temperature-dependent) methods [35]. Fig. 1 shows the overview of creep testing methods.

This section describes the classification of experimental creep methodologies with their specifications. Table 1 summarises the experimented creep principles with the specifications involved.

Summary of creep principals [27,31,32,36,37].

Creep Principals | Mode | Instrumentation | Time Consumption | Specimen size | Principles |
---|---|---|---|---|---|

Conventional | Time-dependent | Simple and easy to manage | Long term period | Coupon and full-scale structure | Used creep strain and modulus |

Accelerated | Temperature-dependent | Complex and require DMA machine | Short term period | Coupon only | Used Time-Temperature Superposition (TTSP) |

A conventional method of a creep test usually implements the variation of loads capacities in order to understand the long term behaviour for most of cantilever beam structure [38]. Usually, the time taken to operate the testing in order to obtain the creep properties of the material in long-term loading is approximately 1000 h or more [39]. The condition usually takes place in a controlled environment in which the data sets are extrapolated linearly in one log cycle period [40]. Additionally, the uncertainty of extrapolation can be seen throughout the period [41]. This is one of the most reliable methods in terms of evaluation of the creep conditions and expectations, which can be represented throughout the material lifespan.

To summarise the creep load time-dependent approach, the mathematical equation is justified using the inverse of stiffness in a time-dependent function:

Where; S(t) is creep compliance, ε(t) is strain with time-dependent, and σ is applied stress.Many studies have been carried out by various researchers to analyse the creep pattern of material via conventional creep method. Slocumb et al. [42] and Lawrence [43] have conducted a compression creep test on a geosynthetic material, which is high-density polyethylene (HDPE) geonets. It was conducted with compressive force at a constant load along 103 hours whereby one log cycle was extrapolated to conclude the reduction of creep in 104 hours. Based on the results, it was concluded that the estimated creep value was shorter as compared to the service life of geonet to a certain extent. Therefore, creep test using conventional method was restricted in order to evaluate the creep of drainage material. Moreover, a study was carried out by Jarousseau and Gallo [44] to analyse the effect of geocomposite types under compressive creep test up to 500 h in shear and normal conditions. In this case, they found that there was higher reduction of thickness of around 15%–30% when the conditions were combined compared to normal stress condition applied on the specimen.

2.2Accelerated creep methodAs previously discussed in the creep principles, a TTSP model is generated in a short time at high temperature to plot the master curve. In the model, the creep properties at a specific temperature can be addressed by using a shifting parameter (at) [45,46]. According to TTSP model, a dynamic property in frequency function with reference temperature (Tr) should have a similar pattern with other nearby temperature. Therefore, a logarithmic response of time at a particular temperature S(t) can be shifted horizontally along the time axis, and it is completed to overlap with other neighbouring temperatures [47]. This shift distance along the reduced time axis is called the shift factor (at), and Eq. 2 is given as below;

Where; t is generated time steps for a temperature, and tr is the reduced time corresponding to the reference temperature Tref[48].A study conducted by Brinson et al. [49] have used the TTSP method on the graphite/epoxy laminate to develop the master curve of actual creep results in the range of 10,000 min. Apart from that, Morris and Yeow [50] carried out an experimental programme to determine the time-temperature response of the principal compliances of unidirectional graphite/epoxy laminates. The test used shift factors to assess the compliances. The study proved that the shift factors are independent towards composite fibre orientation concerning the load.

3Experimental investigationThe experimental investigations are summarised in two types of materials, which are fibre- reinforced polymer (FRP) composite and wooden material in this study. The experimental work in this section discusses the creep testing for a cantilever beam structure. A specific discussion is deliberated in terms of specific techniques and external conditions to provide a more extensive and perspicuity work. Therefore, these investigations which involve the effect of mechanical performance works dealing with conventional characterisation techniques and creep-fatigue interaction are discussed in the subsequent section. The review of relevant articles is summarised and classified based on the type of material used. Table 2 shows the summary of literature on various modes of creep test with its specification.

Overview of creep test on wooden and FRP materials.

Testing modes | Material | Method | Load | Time taken | Dimension | Temp. | Ref |
---|---|---|---|---|---|---|---|

Cantilever beam | Carbon FRP lamina and Glass FRP lamina | Double cantilever system (conventional)Implements LVDT as strain measurement | 0.2 kN(First phase:20 % and 10 % of CFRP and GFRP tensile strength respectively)(Second phase:70 % and 15 % of CFRP and GFRP tensile strength respectively) | 6 months | Both end length with 7300 mm | 20 °C (Room temp.) | [51] |

Flax fibre reinforced polymer composite | Uses additives; GTA, TEC, TBC, PEG, HBP and TDPConducted in DMA (accelerated) | Three stages of loads;5 MPa10 MPa15 MPa | Loading time is 10 minutesRecovery time at 20 minutes | Dimensions: 6 mm x 18 mm x 1.5 mm | DMA Chamber (30–150 °C with 10 °C increment for 20 minutes) | [52] | |

Glass reinforced thermoplastic (GRP) composite | Study the creep behaviour of pipe under unconditioned and preconditioned (in water for 50 °C)Test was done under submerged under water at room temperature. | (10−14 kN)Conducted the test under subsoil installation condition | – | Pipe dimensions: 250 mm (radius), 12 mm (thickness) and 300 mm (length) | 25 °C to 28 °C (Room temp. | [53] | |

Wooden fibre board | Evaluate creep based on different material fibreboard (HDF, MDF, PB)Conduct vertical creep cantilever beam Evaluated by LVDT | Load applied at 10%–20% of rupture modulus | 200 mins | Dimensions:50 mm x 290 mmThickness for each of specimens;HDF = 3.7 mm MDF = 4.6 mm PB = 5 mm | 20 °C (Room temp.) | [29] | |

Sugar maple | Evaluated by LVDT with data acquisition system (Stress Analysis Data System 5000)Implements inside chamber | Four loads;5 %25 %35 %45 % | Loading time is 10hrsRecovery time is 20hrs | Dimensions:110 mm x 25 mm x 7 mm | 30 ± 0.5 °C(Chamber) | [54] | |

Bamboo | Evaluate creep behaviour based on fibre content between of fibre content between DG and PA (Conventional)Evaluated using strain gauge and placed at top and bottom of specimen | Maximum until 8N | Three weeks | Dimensions: 70 mm x 35 mm | 25 °C to 28 °C (Room temp. | [55] | |

Tensile | Laminated composite (glass and carbon in epoxy) | Evaluated by strain gauge(Conventional) | 80 % of UTS of the material | 36 hours | Fabric laminates and glass/resin fabric laminates were cut with the size of 25 mm x 250 mm x 2.2mm | 25 °C to 28 °C (Room temp. | [56] |

Dry Chinese fir wood | Use DMA chamber to conduct isothermal creep(accelerated) | 1.3 MPa | 20 mins for each time step | Dimension:35 mm x 6 mm x 1.5 mm | 30 to 150 °C with 10 °C increment for 20 minutes | [57] | |

Flexural | Jute-PP composite | Use DMA chamber to conduct isothermal creep(accelerated) | 1 MPa for each temp. step | 60 min for each temp. step | Dimensions: 35 mm x 12.5 mm x 3 mm | 30−90 °C with 20 °C temperature increment | [58] |

Wood-plastic composite | Implements four points flexural test (Conventional)Distinguish by using different coupling agent (HDPE and MAPP) | Applied at three different load levels:20 %30 %40 % | 220 days | Dimensions: 406 mm x 50 mm x 22 mmUse span length of 355.6 mm | 20 °C (Room temp.) | [59] |

A collective of researches had been reported in many works of literature regarding the uniaxial cantilever beam test performed on fibre-reinforced polymer (FRP) material. The FRP composites were made from continuous unidirectional, multi-directional, or fibre systems integrated with randomly oriented mat. Most of these tests were performed under controlled temperature.

Mancusi et al. [51] experimented the creep behaviour of FRP lamina under sustained loads to study the service durability and limitation of the FRP lamina. The specimens were tested by using carbon FRP lamina and glass FRP lamina. The test apparatus consisted of titanium alloy (Ti–6Al–4V) and test specimens (glass FRP and carbon FRP lamina). These systems were applied with a stiff cantilever steel beam at both ends with a total length of 7300 mm and performed as lever arms flange of the titanium beam. The two suspended loads were applied at the ends of the arms symmetrically according to the load scheme with 0.2 kN force as shown in Fig. 2. The experimental works were divided into two distinct phases which include lower stresses in the lamina and stresses that were increased to a higher level. The experiment was carried out for six months and controlled in average room temperature around 20 °C. To be specific, the first phase corresponded to about 20 % of CFRP tensile strength in the first test, followed by 10 % of GFRP tensile strength in the second test. On the other hand, the second phase corresponded to about 70 % of CFRP tensile strength in the first test, followed by 15 % of GFRP tensile strength in the second test. The findings showed that under lower stress conditions, a very low creep strain was perceived in the tests for determining GFRP or CFRP. Moreover, fast mechanical response also occurred with increased stress due to severely concentrated creep at bonding interface.

Schematic diagram of Mancusi’s experimental setup [51].

Wong and Shanks [52] experimented using flax fibre composite systems with different fibre treatments and additives as shown in Fig. 3. The composite systems encompassed of biopolymers such as polylactic acid and polyhydroxybutyrate which were reinforced by short mat flax fibre with various additives. The additives were plasticisers, namely glyceryl triacetate (GTA), triethyl citrate (TEC), tributyl citrate (TBC), poly (ethylene glycol) (PEG), hyperbranched polyester (HBP) with 32 hydroxyl functionality, and thiodiphenol (TDP). A creep test was performed using three successive stress levels. The creep test was performed in a single cantilever beam structure since one end of the sample structure was fixed, while the other was free to bend. The size of the test specimens were constant at 6 mm x 18 mm x 1.5 mm. The specimens consisted of fibre which made up of short fibre mat with a fibre-matrix volume ratio of 1:1. In the experimental works, the specimens experienced three stages of stress, which were 5.0, 10.0, and 15.0 MPa. The stress was introduced by using a DMA in static mode at room temperature. In creep testing, the stress was applied for a total of 10 min and then released for recovery testing for a total of 20 min. For each composite system, at least five replications were tested and averaged to obtain more accurate and precise data. In the study, a significant amount of creep occurred at each stress level of a composite without additives. By adding plasticisers and TDP, the total creep can be reduced. However, by adding toughness modifier such as HBP the increment of creep value was affected. This occurred since plasticised TDP composites experienced a reduction in the viscous flow contribution and an increase in the viscoelastic deformation.

Schematic diagram of single cantilever beam in DMA [60].

A study was conducted by Guedes et al. [53] elaborate the effect of moisture absorption on long-term creep properties of glass-reinforced thermosetting plastic (GRP) composite pipes. In the study, experimental creep data obtained from standard test methods under ring deflection conditions were used. A subsoil installation condition was set as experimental loading condition in order to simulate the creep effect. The creep tests were performed on unconditioned and preconditioned specimens. The specimens were made up of glass fibre-reinforced polyester matrix mixed with sand using filament winding technique. The pipe wall was divided into three distinct layers. The specimen size was 250 mm in radius, 12 mm in thickness, and 300 mm in length as shown in Fig. 4. The experiment was done repeatedly for five times. The creep tests were carried out following the European standard EN 1227:1998 to determine the long-term ultimate relative ring deflection under wet conditions on GRP pipes. The test was executed in submerged underwater at room temperature with several loads from 10 kN to 14 kN. Creep tests were also performed on specimens with preconditioned water at 50 °C. The results showed that the initial stiffness of GRP pipes was little influenced by the water preconditioning. Moreover, the creep properties which were evaluated by using lateral deflection was alike either in preconditioned or unconditioned GRP pipe.

Specimen dimensions used to conduct creep test by Guedes et al. (2007) [53].

Referring Testing was executed by Basaid et al. [56] using tensile mode to study creep behaviour on laminated composites composed of glass and carbon fibre in an epoxy matrix in various matrix system. The specimens were made up of laminated composite of glass and carbon fibres using vacuum bagging. The specimens were set at a normal size of 25 mm x 250 mm x 2.2 mm. The test was performed using the Universal Machine of Zwick/Roell type 250 kN as depicted in Fig. 5. The machine was equipped with a sensor extender for trapping and a computer to monitor the extension. The test was set at a constant load of 80 % of ultimate tensile strength. In detail, the machine was installed with a high precision strain gauge. This method aided in avoiding the effect of clamping jaws as it distorted the deformation measurements. The creep test was executed at 30 h of operation to analyse the mechanical failure caused by the composite laminates. The study found that there was no occurrence of creep rupture within the 4 h of operation (short-term) for CFRP, while GFRP was ruptured as early as 1 h of test operation. The CFRP laminate experienced creep rupture after 30 h of operation, causing severe deformation at the direction of loading and forming hollows at the interface.

Creep tensile test setup by using Universal Machine of Zwick/Roell type 250 kN by Basaid at al. [56].

A study was carried out by Chandekar and Chaudhari [58] on flexural creep behaviour in short-term period using jute reinforced polypropylene resin composite. A treated weave gunny jute fabric was reinforced in the polypropylene matrix using a hot-pressed machine to form a composite laminate. The composite laminate was cut into a small coupon strip with a dimension of 35 mm x 12.5 mm x 3 mm. The laminate was later placed inside a DMA chamber to conduct accelerated isothermal creep. All stepped isothermal creep tests were started off at 30 °C–90 °C at 20 °C temperature increment. The soaking time was set at 5 min with the constant stress of 1 MPa for 60 min at each temperature step. The creep tests were done using TTSP method in three-point bending configuration along with Rheology Advantage software from TA instruments. The research found that increment in the instantaneous deformation as well as creep rate significantly increased as the temperature increased. This is attributed to the fact that the mobility of the molecular chains increases at higher temperatures. This creep behaviour at higher temperature is shifted horizontally in order to forecast the creep properties in a longer time period.

3.2Wooden material3.2.1Uniaxial cantilever beam creep testHunt et al. [29] had tested simple cantilever beam (SCB) structure that was vertically hung and fabricated from high-density fibreboard (HDF), medium-density fibreboard (MDF), and particle board (PB). The specimens were cut from rectangular plates of different thicknesses (HDF = 3.7 mm; MDF = 4.6 mm; PB = 5 mm) with coupons of 50 × 290 mm dimension. The testing was performed for 200 min within a normal room temperature around 20 °C. In the experiment, the specimens were clamped in a vertical cantilever beam whereby a perpendicular hook at the free end was set to implement the load. The load was applied until 20 mm of displacement was achieved. A laser displacement gauge was implemented to record the displacement changes that took place at the beam end. The load was measured along the time as the hook remained at the beam’s free end. A recorded beam bending stress was managed within 10%–20% based on the rupture modulus of the three materials. Fig. 6 depicts the setup of the cantilever beam test. The findings showed that the cantilever beam could be used to measure the relaxation rates of composite structure. Therefore, the study exhibited the potential of using a mechanistic approach to explain the behaviour of cantilever beam using constant displacement relaxation.

Cantilever beam test apparatus setup by Hunt el. Al (2015) [29].

An exploratory study was conducted by Segovia et al. [54] on the creep behaviour of sugar maple (Acer saccharum Marsh.) wood in cantilever beam mode setup. The test condition was controlled inside a chamber at a constant temperature of 30 ± 0.5 °C. The constant environment had a relative humidity at 37 %, 67 %, and 83 %. The experimental conditions for the test are summarised in Table 3. For the test specimen, it was fabricated in a radial position and cut into 110 mm x 25 mm x 7 mm. The setup specimens in the test apparatus were designed according to the schematic diagram in Fig. 7 in a cantilever beam position. A metal structure was installed at the end of the specimens. To hold the specimens, a dead load was applied on another end. A linear variable differential transformer (LVDT) was installed to measure deflection at the free end of the specimen. The LVDT, along with wet and dry thermocouples were connected to a data acquisition system, which is a Stress Analysis Data System 5000 that allowed data recording for every minute in real-time. The position of the LVDT was located at 70 mm from the metal jaw since it was a sufficient distance for data evaluation. Besides, the dead load location was set at 5 mm away from the LVDT. The implementation of loading and recovery periods was set at 10 and 20 h, respectively. The results showed the influence of different load levels and the moisture absorption capability affect total deflection of a cantilever beam. In this case, the higher the relative humidity, the higher the total deflection of the beam. Hence, the long term durability of sugar maple cantilever beam is highly dependent on the viscoelastic and mechanosorptive properties of the material.

Experimental condition conducted by Segovia et al. (2013) [54].

Equilibrium Moisture Content, (%) | Maximum load at failure, (g) | Applied load, (g) | Number of specimens | |||
---|---|---|---|---|---|---|

5 % | 25 % | 35 % | 45 % | |||

7 | 9910 | 490 | 2477 | 3468 | 4460 | 12 |

12 | 8856 | 443 | 2214 | 3100 | 3985 | 12 |

17 | 6750 | 337 | 1687 | 2362 | 3037 | 12 |

Schematic diagram of cantilever beam setup by Segovia et al. (2013) [54].

Armandei et al. [55] conducted a study on the variation of fibre content between Dendrocalamus giganteus (DG) and Phyllostachys áurea (PA) which affected the mechanical properties of bamboo in long-term loading. In the test, a simple bamboo cantilever beam was placed in a vertically-positioned covered place. The sample was dried and tested for three weeks. The experimental works were performed under natural moisture content and at room temperature which ranged from 25 °C to 28 °C and 52%–66% of humidity. The sample was fabricated based on the first 10 lower nodes bamboo since the internode distance was small. Therefore, all the specimens has two or three nodes along their lengths. The samples ranged between 2–3 years old. During the data collection stage, flexural strength was measured to study the load-deflection curve during the static flexure test. Also, four unidirectional strain gauges were placed at the top and bottom of the specimen to obtain the specimen’s Poisson constant. Specifically, two gauges were located at the top. The other two were located at the bottom. Each surface contained a longitudinal and a transverse gauge on a separate peace of bamboo. Fig. 8 displays dimensions and test setup for flexure creep test of bamboo. The results obtained showed that material damping of bamboo in long-term condition was highly affected by temperature.

Schematic test position and material dimensions of cantilever beam creep test conducted by Armandei et al. (2015) [55].

According to Peng et al. [57], a research was conducted on the application of the TTSP to orthotropic creep in dry Chinese fir (Cunninghamia lanceolata [Lamb.] Hook.). In the study, the investigation was done via a sequence of short-term tensile creep for longitudinal (L), radial (R), and tangential (T) specimens. The wooden specimens were prepared in the dimension of 35 mm x 6 mm x 1.5 mm. In order to ensure that the specimens are in homogeneity, these specimens were dried in a closed container with a chemical substance, namely anhydrous phosphorus pentoxide at room temperature. An isothermal creep was run in a DMA chamber (DMA 2980, TA Instrument) in tensile mode. The creep was conducted at 30 °C–150 °C with 10 °C increment for 20 min, and 20 min equilibration/recovery periods were inserted between creep segments. Dried specimens were heated to the maximum experimental temperature of 150 °C at 10 °C/min and held there for 10 min before the stress-strain sweep. The stress value of 1.3 MPa was selected. Based on the results discussed in the literature, they found that creep compliance was dependent on temperature and orthotropic directions. In this case, the creep compliance for longitudinal specimen is necessary to have additional vertical shift factor since the impact of temperature in longitudinal compliance.

3.2.3Flexural testIn a study held by Chang et al. [59], a long-term creep test was piloted in an unconditioned environment to validate the master curve obtained from short-term creep test. Ten specimens were fabricated from each wood-plastic composites (WPC) with an identical size of 406 mm x 50 mm x 22 mm. WPC has divided their formulation into two whereby FA is a simple composite of mountain pine wood and HDPE, whereas FB groups were composed of mountain pine wood with the addition of MAPP as a coupling agent and talc as additional filler. The experiment was set up using four points bending. Loading levels and stresses are summarised in Table 4. LVDT was integrated on an aluminium frame to measure the mid-span deflection of the tested specimens. The measurements were logged by using data acquisition. It was performed to study and analyse the deflection under a set of frequency. The specimens were conditioned for less than four weeks in a constant room temperature at 20 °C. A bending fixture was used to carry out the measurements as shown in Fig. 9. From the literature, the findings expressed that the effect of naturally elevated temperature during the summer periods led to significant increment of creep strain. Moreover, the comparison between conventional creep data and accelerated creep using master curve be likely to overestimate the actual creep strain of large-scale size specimen. Subsequently, the deviation was increased by time.

Schematic diagram of Chang’s experimental setup [59].

In the preliminary design stage of a specific product, the creep prediction model has to be implemented to forecast the creep strain as a function of time along with its specific stress and temperature. In most cases, the creep tests were conducted under constant load. The build-up of the creep strain along the dimensions of the test specimen supposedly affects by the increase of stress. [61]. The main contributor in this field is the empirical method which specifies the component’s creep performance. Eq. 3 depicts the numerical expression to represent the creep strain (ε) as a function of time (t), stress (σ), and temperature (T) [62]:

Based on the equation above, Q is indicated as activation energy, R is universal gas constant, A is constant, and n and m are exponent of stress and time, respectively.

Commonly, this model cannot be presented as creep curve which is divided into three distinct stages (primary, secondary, and tertiary stages of creep). Therefore, it would be essential to explain the creep behaviour by using a more complex model such as Theta projection, as mentioned by Evans et al. [63]. In order to satisfy the projection, about 20 material constants are needed to estimate the creep strain using Eq. (3). In general, this model equation is used to investigate the critical and intricate locations and product components which might lead to structural failure if the problem was not comprehended. However, this model has limitations in the uncertainty associated with the extrapolation of the data test beyond the experimental domain [64,65]. This equation is also stated in ASTM D2990 to elaborate the creep behaviour using creep compliance in the order of ‘Equation of State’.

4.2Numerical investigation4.2.1Cantilever beam conceptThe numerical investigation or mathematical verification in cantilever beam experiments have been traditionally studied by several researchers using several mechanics of materials theories. Generally, a concentrated load (P) is exerted at the free end of the cantilever beam with constant loading (Fig. 10).

Schematic diagram of cantilever beam [66].

According to Zhuang et al. [66], the effect of local deformation at the specimen and punch is negligible. Moreover, the effect of shear stress can be neglected due to the relatively small ratio of shear displacement to the overall deflection d. The statement above shows that the ratio of the height to the span should comply with Eq. 4, which is congruent with the study conducted by Beer (2012) [67].

Where; l is length of the cantilever beam and h is half the height of the beam.To be specific, o-xyz is described as three-dimensional coordinates which is projected into oxz and oyz planes to describe the bending moment of the cross-section of the beam structure. In the mechanics fundamental, the bending moment of the structure can be generated by using Eq. 5.

Where; P is the load applied to the system, l is the distance from the fixed-point to the load-point, and x is the distance from the fixed-point along the neutral axis of the beam. Moreover, z is defined as the distance from any point to the neutral surface along the z-axis. Apart from that, A is the cross-section area of the beam.By assuming that the overall deformation is small, the curvature radius of the specimen κ(x,t) and the relationship between bending strain ε(x,z,t) and κ(x,t) are obtained in Eq. 6.

Where; d″(x, t) is the second derivative of deflection d(x, t) with respect to x.4.2.2Creep numerical modelsAccording to Hao et al. [34], various numerical models had been proposed to estimate the creep properties of a material. These models are classified into two types, which are physical and empirical models. These models are implemented to interpret these parameters. Fig. 11 depicts the classification of various creep numerical models.

4.2.2.1Norton’s power lawNorton’s power law model describes the secondary creep curves to predict the creep life of a material. The model signifies the prediction of creep damage properties by examining the creep crack initiation and creep crack growth model [68].

The model explains creep behaviour as;

Where; ε˙ss denotes as the steady-state (secondary creep) strain rate, σ as applied stress, B as creep constant, and η as creep stress exponent.Eq. 8 can be attained by considering the time-dependent deformation.

The stress can be denoted as follows by substituting Eq. 7 to Eq. 8:Considering that the tensile and compressive stress comply with the creep law, the moment M of the beam structure is shown as follows:Therefore, Eq. 10 can be simplified as;

Considering the boundary conditions,

The steady-state creep rate along the neutral axis can be achieved by embedding the boundary conditions to Eq. 12 before implementing the secondary integral for x. In the expression, the minus sign shows the punch direction.

The expression deduces at loading point, x = 1;

Where; d˙ss represents the steady-state load-point displacement rate. Eq. 14 depicts the displacement rate of steady-state load-point:By substituting Eq. 14 to Eq. 13, the expression would be given is;

Eq. 14 presumes the rate of displacement, expressing a power law model dependence on the load P. Therefore, it shows that the power law is directly related to Norton’s model [69]. This is attributed due to the creep stress exponent n which is directly reverted from the rate of displacement. In Eq. 13, creep constant B can also be obtained through parameter N.

For the material that obeys the Norton’s law, the stress for a straight beam is [70];

Therefore, Eq. 15 can be rearranged into:

To simplify, the equivalent uniaxial stress σu,eq and equivalent steady-state strain rate ε˙ss,eq are rewritten based on the aforementioned equations:

The model explains eloquently that the relationship depends on the material. The corresponding equivalent uniaxial creep data can be obtained when the creep exponent n is identified. Therefore, the model exhibits that the theoretical analysis is fit for any material to comply with the creep behaviour of the cantilever beam structure.

4.2.2.2Findley’s power lawFindley’s power law is an empirical model that is used to study and examine the creep properties of a material. The model computes the prediction of the strain-time curves which completes the compliance curves [71]. However, the model applications are limited due to the direct and straightforward numerical calculation that is implemented universally to any system. This is attributed due to the lack of deliberation on dimensional changes of material in testing [72]. The dimensional changes (deformation) directly depends on the molecular mobility at the microstructure level. Besides, the external factors which include the moisture content, humidity, and temperature, exhibit shape-changing and deformations [73].

The empirical model can be explained as follows [74]:

Where; ε(t) is the creep strain, ε0 is the instantaneous initial strain, εc is the amplitude of transient creep strain, and n is the time exponent. The ε0 and εc are functions of environment and stress variables.The recovery process is exhibited when the constant stress is at time t0, which attributes to reverse creep. At time t0, the maximum deformation is accomplished:

During the recovery stage, the creep deformation is divided into two phases (recoverable strain - R(t) at time t; and non-recoverable strain - NR(t) at time t). Therefore, it can be expressed as below:

The relative creep strain, εr or creep upon loading is described as the creep strain value at each time ε(t) divided by the instantaneous initial strain value, ε0. Therefore;

Where; εr is the relative creep strain in percentage and A’ is the slope of the power law.A’ and n are parameters that can be obtained by plotting log εr versus log t, hence log A’ is obtained based on the intersection with log εr axe and n from the curve slope.

Eq. 22 is differentiated to obtain the power law creep rate, ε˙PL, as follows:

4.2.2.3Four-element burger’s modelFour-element Burger’s (FEB) model is one of the physical methods used to analyse the creep properties exhibited by a material. The model is embedded with a combination of one Maxwell unit and one Kelvin unit combined in a series. Generally, a creep strain consists of three components, which are instantaneous deformation (Maxwell spring); a viscoelastic deformation (Kelvin’s dashpot element) and viscous deformation (Kevin-Voight element) [75,76]. The combinations of these components form up a Kelvin-Voight model (Fig. 12), which describes the creep strain analysis of the whole system [77].

Schematic diagram of Kelvin-Voight model by Wong & Shanks (2008) [52].

The model can be expressed as follows [46]:

Where; ε(t) is the creep strain; εM is the elastic strain of spring, σ is the stress applied, EM and Ek are the elastic moduli of the springs, and ηk and ηM are the viscosities of the dashpots in this model.The stress applied σ at the beginning of the cantilever beam tip causes an initial cantilever tip beam displacement. The total strain ε(t) is equal to strain response as computing by the load decreases due to the recovery actions of the beam. Based on the initial maximum load, the elastic modulus of Maxwell spring can be computed. The stress is linearly increased from the fixed-point to the load contact area. Moreover, the strain response is also directly proportioned to the stress. Therefore, the average stress can be implemented in this condition [78,79]. Eq. 25 is used to compute the average stress experienced by the beam structure [80]:

The Kelvin-Voigt element induces a delayed viscoelastic response when stress σ is applied within time t. Since the beam structure experiences compression and tension at specific areas, the overall cantilever beam load is reduced over time. The total strains from each section is equal to the total initial strain.

The total strain, ε(t), equation is given in Eq. 24 and each of the element of the model is given in Eq. 25,

Where; EM and Ek are the elastic moduli of the springs, while Ed is the strain of dashpot. Moreover, C is the coefficient in terms of time.5Creep properties of the wood and composites cantilever beamThe creep properties of the cantilever beam depends on the material types, the arrangement of structure, the connection used in the structure, and the load applied on the beam. The changes in creep behaviours are also mainly attributed to the external environmental conditions, including temperature and moisture content, wind effect, and seismic movement, as being studied by several researchers [81]. Creep characterisation is done according to the ASTM standard which depends on the material used and the mode of setup.

This section elaborates the end product of a cantilever beam of each material in long-term loading conditions. Table 5 summarises the parametric studies that have been done by previous researchers on the behaviour of composites and wood cantilever beam.

Summary of various wooden and composite cantilever beam structure [51,54,82].

Material | Application | Load | Observation | Discussions |
---|---|---|---|---|

GFRP/CFRP | Retrofitting system | 1st phase: 20% (CFRP) and 10% (GFRP)2nd phase: 70 % (CFRP) and 15 % (GFRP) | Increasing the stress state towards higher levels | Creep localized at the bonding interfaceViscous properties of structural adhesives rather than to viscous properties of the lamina itself |

White Spruce wood | Wood drying | 20 % loading of bending strength | Primary and secondary creep happened and it is failed instantaneously at 300 minutes | Shell of the wood attempts to shrink when below fibre saturation point (FSP). |

Sugar maple wood | Varies of moisture content | 5 %, 25 % and 45 % of the maximum load at failure | Total deflection proved significantly greater at 17 % EMC than at the 12 % and 7 % EMC levels | Total deflection significantly increased after a relative humidity cycle.Due to mechanosorptive creep (interaction between moisture content variation and applied load) and viscoelastic creep. |

Mancusi et al. studied the double creep cantilever beam test which is used for retrofitting systems, either of carbon or glass fibres and subjected to different stress values in a regime of constant temperature. In this research, two different stress levels, which are 20 % and 10 % for CFRP and GFRP in the first phase, while 70 % and 15 % for CFRP and GFRP in the second phase were applied. In these findings, low-stress state of deficient creep strains have been noticed in the test concerning the GFRP lamina while they were negligible for CFRP lamina. Moreover, the fast changes in terms of mechanical system response were observed after increasing the stress state towards higher levels, both for GFRP and CFRP tests. This is due to the creep localised at the bonding interface and consequently due to the viscous properties of structural adhesives rather than the viscous properties of lamina itself [51].

5.2White spruce woodMoutee et al. [82] performed a study to develop a new approach to develop a model of creep behaviour of white spruce wood in a cantilever loaded concerning wood drying. They found that the proposed approach can simulate the experimental creep bending tests whereby the dry wood experienced primary and secondary creeps and failed instantaneously in 300 min at 20 % loading from the bending strength. It found out that the creep strain shows an exponential decreasing pattern along time taken. This can be described due to the moisture content of the surface layers of the board which drops below the fibre saturation point (FSP) whereby the shell (the outer portion of the board) attempts to shrink.

5.3Sugar maple woodSegovia et al. [54] reported that sugar maple wood was selected to study the creep behaviour in 7 %, 12 %, and 17 % of equilibrium moisture content (EMC), with the loading of 5 %, 25 %, and 45 % of the maximum load at failure. The study showed the plasticizing remain effect of moisture on the mechanical behaviour of sugar maple wood. The total deflection proved significantly higher at 17 % EMC level than at 12 % and 7 % EMC levels. The total deflection increased significantly after a relative humidity cycle. During the second stage of the cycle (at a higher relative humidity level), the total deflection increased rapidly. Subsequently, the total deflection continued to increase with each change in relative humidity, but in a smaller amount. The observation from the data findings showed that the creep behaviour of a wooden structure is dependable on its specific mechanosorptive and viscoelastic properties [83].

6Applications and importance of numerical and experimental creep studies6.1Experimental6.1.1Uniaxial cantilever beam testA creep deformation study can be conducted on the actual cantilever beam structure by using a uniaxial bending test. This testing model does allow the specimen to experience a pure bending stress state with load occupied at the end [84]. As it is known that most equipment operates at relatively low stress, it means that the equipment is located in the initial and secondary stages of their life cycle. Then creep properties, including initial and secondary stages should be seriously considered and evaluated. However, when large deformation happens in the initial testing stage, it is difficult to describe the stretch forming process that produces a bulge in the thin specimen, resulting in high complexity and nonlinearity [85,86]. A research conducted by Zhuang et al. stated that the cantilever beam test results of chromium-molybdenum steel corresponded well with those from the uniaxial test [87]. Generally, the uniaxial cantilever beam creep test aids the contributed in order to assess the primary and secondary creep behaviour of the material.

6.1.2Tensile creep testCreep tensile test or uniaxial tension creep test allows a single mode of applied stress in long-term deformation test. The test is implemented using dead load principle, where the stress can be adjusted. It is usually more preferred than flexural mode, since flexural test produces maximum stress localised at small regions of the specimen surface. This tends to develop a localised effect, which cannot represent the whole material’s mechanical property. On the other hand, creep tensile test can accurately measure stresses throughout the test specimen [88,89]. The tensile data attained is more reliable and suitable, especially in terms of design and modelling structures for a material engineer to develop a product.

6.1.3Flexural creep testFlexural test is essential to estimate the resistance and durability of a material with a combination of tensile and compressive modes. The test requires low shear action to ensure that the primary failure comes from tensile in the convex side or compression stress in the concave side [2,90]. This test method is beneficial to govern the deformation patterns of a viscoelastic material that is exposed to long-term static flexural stress [91]. In detail, the outcomes of this testing mode is beneficial for a design engineer to measure the isochronous stress-strain curve, the creep-to-rupture behaviour, and the relaxation properties [92]. Therefore, the flexural creep helps researchers and engineers to assess the durability of material in term of tensile, compression and shear effects.

6.2Numerical6.2.1Findley power lawFindley Power Law is a vital model to clarify the creep properties empirically in order to forecast the creep life of polymeric or wooden components. The model brought forward the description of the creep behaviour of various polymers with reasonable accuracy over a full-time scale [71]. By its nature, the model is explained and detailed by using constant independent stress, n which is generally less than 1 in value, time-independent strain, and compliant with the coefficient of time-dependent term. Therefore, the model is constructive in finding creep with a material which does not exhibit secondary creep stage, especially for a polymeric material which experiences low-stress level [93,94].

6.2.2Norton power lawIn most cases, the function of Norton’s Power Law has usually embedded the variable of material constant and stress exponent of a specific material to determine the creep performance [95]. Theoretically, the creep strain increases as the stress increases [23]. A creep study was conducted on a copper ally by Ahmad et al. [68] using a uniaxial tensile test to determine the creep behaviour of the material. The results displayed a similar pattern between the exponential curve and the experimental creep curve in the same graph. In this case, the difference between experimental and numerical in term of precision is insignificant which exhibit only 5.1 % error. Additionally, the Norton Power creep model is advantageous in perceiving service life prediction, especially for repair and replacement decisions of parts that experience creep. The model explained the creep behaviours using a combination of temperature, stress and time in order to analyse the exhaustion of a component after long-term loading [96].

6.2.3Four-element burger modelOne of the primary creep viscoelastic models is the Burger’s Model, which translate the creep behavioural prediction in terms of strain response under the constant stress of each coupled element in series [97,98]. The segregation of each element in Burger’s model does clarify more details of the creep behaviour experienced by each of the component in a product. For instance, each of the Maxwell spring, dashpot, and Kelvin unit are segregated into three terms using a numerical expression which describes their properties differently during the loading action [98]. Therefore, this model is essential to determine the practical behaviours of viscoelastic materials, especially for the effect of additive such as nanofillers on the FRP composite [99,100,109–111,101–108].

7ConclusionA cantilever beam structure is usually exposed to a constant loading for a very long time which might induce a phenomenon called structural collapse due to creep deformation. To resist this problem, a creep testing methodology has to be implemented to characterise the creep behaviours to estimate the operational lifespan of the structure. The creep analysis approaches are divided into two types, which are experimental and numerical analyses. To be more specific, the experimental works employ two main methods, which include load-based (conventional) and temperature-based (accelerated) techniques. Both methods allow the creep to be predicted by various researchers in order to understand the life expectancy of a material. Apart from that, the numerical approaches, including physical and empirical models are well needed to verify the experimental work. Both approaches rely on each other to make the outcome more accurate and successfully evaluate the behaviour of a structure during long-term constant stress.

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Uniaxial cantilever beam experiment is carried out on the wood and composite structures to thoroughly understand the behaviour at each point of the structural member. The critical points of the member during the creep exposure is evaluated.

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Uniaxial cantilever beam test also provides the structural analysis in-depth, especially in the variation value of loadings during the extended time process.

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Other types of testing modes (flexural and tensile) can be carried out to be compared with the uniaxial cantilever beam test which can replicate the same pattern of creep behaviour when relating the coupon strip size with the actual size of testing.

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Most of the wood and composite structures behave in the same character due to their anisotropic in morphological properties. Therefore, both materials can implement the same testing technique to have a broader view of understanding creep analysis.

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For numerical analysis, the cantilever beam structure does implement the mechanics of cantilever beam which involve the concept of moments, forces, and deflections.

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In the numerical analysis, the creep study is divided into two main approaches, which are the physical model and numerical model (power-law). For the physical model, the method used is a four-element Burger model, while the numerical model includes Findley’s and Norton’s law models.

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Norton’s Power Law model aids in predicting the creep damage properties by examining the creep crack initiations and creep crack growth model.

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Findley’s Power Law model performs the creep prediction using the strain-time curves, which completes creep compliance curves.

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Four-Element Burger’s Model is useful to clarify the creep properties using physical model (embedded a combination of one Maxwell unit and one Kelvin unit) which are combined in series.

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Most of the wood and composite materials experience a higher creep rate when the materials are subjected to higher temperature and humidity.

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Uniaxial cantilever beam creep test is instrumental in allowing the specimen to experience a pure bending stress state with load occupied at the end, while tensile creep test aids in understanding the creep behaviour of tensile effect whereby flexural is applied to estimate their resistance and durability with a combination of tensile and compressive modes.

The authors would like to thank Universiti Putra Malaysia (UPM) for the financial support provided through Geran Putra, UPM with VOT no. 9634000. The authors are very thankful to Department of Aerospace Engineering, Faculty of Engineering, UPM for providing space and facilities for the project. Moreover, all authors are very appreciate and thankful to Jabatan Perkhidmatan Awam (JPA) and Kursi Rahmah Nawawi for providing scholarship award and financial aids to the first author to carry out this research project.

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M. R. M. Asyraf is currently persuing his doctarate degree on Material Engineering at Universiti Putra Malaysia (UPM). He had graduated in Bachelor of Mechanical Engineering with first class honours at UPM. Asyraf's research interests are primirily in areas of mechanical behaviour of anisotropic material with emphasis on the long-term behaviour of wooden and composite material. His creep research has involved in development of creep test rig for full-scale cross arm as well as flexural creep test rig for coupon scale. Recently, he has been appointed as Associate Editor in Journal of Advanced Research in Fluid Mechanics and Thermal Sciences (ARFMTS), Scopus Indexed Journal. He has published several manuscripts in field including conference proceedings, review and research articles.

M. R. Ishak is an asscociate professor of Universiti Putra Malaysia. His research aims at understanding mechanical behaviour of composite material and he is interested investigating the natural fibre, biopolymer and biocomposite, plastic technology, manufacturing process and polymer composite design and testing. Dr. Ridzwan is also actively involve in community service, where he has been appointed as president of Persatuan Pembangunan Industri Enau Malaysia (PPIEM) recently. He has published widely in the field, with more than 150 publications, has edited several books on natural fibres and manufacturing process. He is one of the top researchers in UPM with H-Index of 31 as recorded in Scopus database.