Shear Stress-slip Relationships Research Articles

Double shooting method for FRCM reinforced systems in debonding problems

The debonding process in FRCM (Fiber Reinforced Cementitious Matrix) strengthened brittle substrates, such as concrete and masonry, is characterized by complex failure modes that include debonding at both the external and internal interfaces of the composite, as well as damage to the mortar matrix. This study addresses this problem classically as a mode-II fracture process by accounting for the one-dimensional non-linear interfacial shear stress-slip relationship combined with matrix axial non-linearity. The approach employed considers the longitudinal displacements of both the inner and outer layers of mortar, along with that of the fiber textile, as unknowns, and analytically solves the second-order ordinary differential equation (ODE) system that governs the debonding problem. Then, the boundary value problem (BVP) is transformed into an initial value problem (IVP) and a double shooting method is proposed to find displacements and stresses along the bonded length in the presence of non-linearity. The appropriate initial values of matrix layer displacements, satisfying the required boundary conditions, are efficiently determined through a two-dimensional bisection method working on rectangular and triangular patches. The softening behavior of both mortar layers and interfaces between mortar and fiber are taken into account using jagged constitutive relationships to enhance convergence. The numerical calculation speed, robustness, and key simulation parameters are thoroughly discussed. Moreover, the proposed numerical approach is compared with existing experimental data and models, on both concrete and masonry FRCM-reinforced specimens. The influence of mortar failure and interfacial strength is also explored. The model demonstrates its ability to effectively reproduce the global bond behavior observed in experimental studies, while capturing the local behavior and simulating various failure modes observed in the tests.

Semi-analytical approach for curved masonry pillars reinforced with FRCM

The paper presents a semi-analytical approach for the study of the debonding phenomenon of Fiber Reinforced Cementitious Matrix (FRCM) systems externally applied to curved masonry pillars. One of the main features of the approach consists of considering the strengthening system composed by three separated components: external mortar layer, central fiber net and internal mortar layer. These components, assumed subjected to a longitudinal state of stress, interact one each other through tangential stresses developing at the level of zero-thickness interfaces. Regarding the latter, a tri-linear shear stress-slip relationship is assumed to account for a first elastic phase, a second phase exhibiting linear softening and a third phase with a possible non-null residual strength. Since in case of curved substrates, by equilibrium, normal stresses at the interfaces between fiber and matrix arise, which modify the peak tangential resistance and the ductility, a classic Mohr-Coulomb criterion is introduced in the approach. Additionally, the central fiber net progressively transfers along the bond length the force applied at its loaded end by means of an elastic interface interposed between the internal mortar layer and the substrate, the latter assumed rigid and infinitely resistant. The longitudinal equilibrium equations written for the two mortar layers, suitably re-arranged considering the constitutive behavior of the layers, allow to deduce a field problem governed by six first order differential equations into six unknowns. The non-linearity is tackled by means of a recursive elastic numerical algorithm where the elastic modulus of the damaged materials is progressively dropped down, subdividing the bonded length into small portions where the material properties are assumed constant. For each element the solution of the field problem is known in closed form and the only variables to determine are the integration constants coming from the solution of the differential equation system. After a standard assemblage, all constants are derived imposing the boundary conditions at the extremes of the elements, which depend on the state of cracking of the matrix layer. The validation of the proposed approach is carried out with reference to recent experimental tests carried out by the Authors. The obtained results show the reliability of the approach to account for the influence of the curvature of the substrate on the debonding process of FRCM systems.

Distributed fiber optic sensing for micro- and macro-crack quantification: an interfacial-fracture-energy-based model

Clarifying the strain transfer between the host material with a crack opening displacement (COD) and the optical fiber after interfacial debonding remains a critical challenge for crack quantification. The interfacial debonding induced a triangular form strain profile around the crack due to the residual shear stresses at the interface. This study investigated the interfacial shear stress-slip relationship between the bare fiber and its coating layer through a plate splitting test. Based on that, we proposed an interfacial-fracture-energy-based analytical model to convert distributed fiber optic strains before and after interfacial debonding to CODs of micro- and macro-cracks. The accuracy of the model under a single crack was validated through plate splitting tests and the experimental data reported in the literature. Furthermore, experimental results from a four-point beam test were utilized to investigate the feasibility of the model under multiple cracks. The model, assisted by the strain superposition strategy, accurately quantified the CODs of multiple cracks before interfacial debonding but failed when interfacial debonding occurred. In addition, we proposed a simplified method based on the linear relationship between the debonded lengths and the corresponding CODs, which facilitates evaluation of the CODs after debonding.

A novel analytical model for fiber reinforced cementitious matrix FRCM coupons subjected to tensile tests

A much enhanced -with respect to existing literature- analytical model for FRCM coupons subjected to standard tensile tests is proposed. One fourth of the coupon is idealized considering two layers, matrix and fiber, subjected to a uniaxial stress state, mutually interacting at the interface through tangential stresses. The matrix is assumed elastic-perfectly fragile and the fiber linear elastic. The shear stress-slip relationship at the interface is tri-linear, with the first phase elastic, the second exhibiting linear softening followed by a third phase at possible non-null residual strength. The longitudinal equilibrium equations written for the two layers, suitably re-arranged, allow deducing a field problem governed in the three phases by a single second order linear differential equation where the independent variable is the interface slip, with solution retrievable analytically. Since the location of the points where the interface is in the different phases is not a-priori known, a discretization with small length elements is adopted. For each element the solution of the field problem is known in closed form and the only variables to determine are the integration constants coming from the solution of the differential equation. After a standard assemblage, all constants are derived imposing the boundary conditions at the extremes of the elements, which depend on the state of cracking of the matrix layer. The model is validated against two sets of experimental data on coupons tested in two different University laboratories in Italy. A satisfactory predictivity of both the global and local behavior is found.

Determination of the Matrix-Fiber Cohesive Material Law of FRCM-Concrete Joints

Fiber-reinforced cementitious matrix (FRCM) composites have been increasingly used to strengthen existing concrete and masonry structures in the last decade. Two guidelines are available for the design and construction of FRCM strengthened members: ACI 549.4R (2013) and CNR-DT 215 (2018). Both these guidelines employ the effective strain, i.e. the strain at which the composite action is lost, as key parameter for the evaluation of the capacity of FRCM strengthened members. The American guideline ACI 549.4R (2013) employs the results of clevis-grip tensile tests on FRCM coupons to determine the composite effective strain. Such strain is determined by the Italian guideline CNR-DT 215 (2018) combining the results of direct shear tests on FRCM-substrate joints and of tensile test of bare fiber textile. The effective strain is strictly related to the matrix-fiber bond behavior, which can be expressed by the interface shear stress-slip relationship, i.e. the cohesive material law (CML). The effective strain is not sufficient for a full understanding of the structural response of strengthened members, since the knowledge of the CML is needed to predict important parameters such as the crack pattern or the location where debonding occurs in beams strengthened in flexure. This paper provides a simple procedure to obtain the CML from the load response obtained by direct shear tests of FRCM-substrate joints. The procedure is discussed and applied to the case of poliparaphenilene benzobisoxazole (PBO) FRCM-concrete joints previously tested by the authors.

Characterization of OFDR distributed optical fiber for crack monitoring considering fiber-coating interfacial slip

An analytical crack-strain transfer model considering interfacial bond-slip between optical fiber and fiber coating is proposed and experimentally validated for crack monitoring of structures. Interfacial shear stress-slip relationship between the optical fiber and fiber coating was determined by using pull-out tests, and a simplified trilinear three-stage interfacial shear stress-slip model was proposed. The model is incorporated into the development of a crack-strain transfer model between the structural substrate and the optical fiber, and the fiber strain distribution is analytically obtained and compared with the strain distribution as perfect bond is assumed between the fiber and fiber coating. Experiments were also conducted to validate the crack-strain transfer model by continuously monitoring corrosion-induced concrete cracking with OFDR distributed optical fiber. Compared with strain transfer models proposed by other researchers in the literature and perfect bond model, the crack-strain transfer model considering interfacial shear stress-slip mechanism between fiber and fiber coating agrees well with experimental results, especially for large crack monitoring.

Experimental and analytical analyses of the bonding performance of CFRP–wood interface

The mechanical properties of the bonding interface between carbon fiber-reinforced polymers (CFRPs) and wood are key to study how the wood structure is strengthened by CFRPs. The interfacial bonding shear stress–slip relationship model and the interfacial bearing capacity model are important theoretical bases and practical references to investigate this strengthening mode. To study the influence of different factors on the mechanical properties of the bonding interface, and establish the bearing capacity model and the bond slip relationship model of the CFRP–wood interface, the mechanical properties of the interface between CFRPs and wood were evaluated by testing six groups of CFRP–wood specimens under double shear. The evolution of the CFRP strain and the interfacial bonding shear stress was analyzed to obtain the bond shear stress–slip relationships. Furthermore, the influence of the CFRP bond length and width on the interfacial bonding performance was discussed. The results show that an interfacial bond shear stress process was transferred from the loading end to the free end with an established effective bond length. The expressions for strain distribution in the CFRP sheets during the entire loading process were then calculated from the force analysis of the microsections of the bond interface and the regression analysis of the results. The data are in good agreement with the test curves. Furthermore, the CFRP–wood interface bonding bearing capacity model, effective bond length model, and interface bonding shear stress–slip relationship model were proposed based on the existing theoretical model and its comparison with the experimental results. The errors between the values predicted by the effective bond length model and the interface bond bearing capacity model and the experimental values are less than 18%. The curves predicted by the interface bond shear stress–slip model are in good agreement with the experimental curves.

Analytical model for shear strength of prestressed hollow-core slabs reinforced with core-filling concrete

The core-filling method, one of the shear strengthening methods of prestressed hollow-core slabs (PHCSs), has been widely applied on construction sites because of its easy reinforcement procedures. However, since there are no details for sufficient composite behavior between PHCS and core-filling concrete at the inner surface of the hollow core, interfacial cracking or separation may occur when a large shear force acts on the member. Such damage at the interface can negatively affect the shear strength of the member. In this regard, this study aims to propose a shear analysis model for PHCSs with core-filling concrete that considers the composite performance between PHCS and core-filling concrete based on previous shear test results reported by the authors. In the proposed model, the shear demands of PHCS and filled cores are calculated based on nonlinear flexural analysis, and the shear force at which one of the shear demands reaches the corresponding potential strength is regarded as the shear strength of the member. In addition, the change in the shear strength of the member according to the interfacial shear properties of the PHCS and filled core is analyzed in detail. The analysis results show that the shear strength of the members is reduced as the stiffness of interfacial shear stress–slip relationship between the PHCS and filled core decreases, and that the proposed model can quantitatively estimate the shear contributions of PHCS and filled cores.

A shear stress-slip relationship in steel fibre-reinforced concrete obtained from push-off testing

This paper presents the results of an experimental programme carried out to evaluate the shear stress-slip relationship in steel fibre-reinforced concrete. Two types of direct shear tests are modified and used. Firstly, a general, double-notched, direct shear test was modified to minimise the confinement stress provided by the restricted supports in the general test setup. Secondly, a traditional Z-type push-off test, with reinforcement crossing the pre-cracked shear plane, was used to determine the shear crack propagation in steel fibre-reinforced concrete. Two aggregates with different maximum particle sizes and two volume fractions of steel fibres were used. The modified, double-notched, direct shear test reduces the influence of the horizontal restraint forces on the supports and provides more accurate shear strength values for non-cracked shear planes. The results of push-out tests in specimens with pre-cracked shear planes, show that steel fibres contribute to reducing the crack width of the shear plane in the pre-peak shear stress branch. A model for predicting the shear stress-slip is proposed and is based on three components: shear strength of cracked concrete, shear-friction and contribution of steel fibres crossing the cracked plane. A good correlation was obtained for the experimental results and the values predicted by this model.

Fatigue behaviour of the bond interface between carbon fibre‐reinforced polymer sheets and concrete

AbstractExternally bonded carbon fibre‐reinforced polymers (CFRPs) have been applied to retrofit and strengthen civil structures. In this study, four‐point bending beams were manufactured and tested to examine the fatigue behaviour of the CFRP–concrete interface. The results indicated that the specimens exhibited debonding failure in the concrete beneath the adhesive layer under static loading. However, when cyclic loads were imposed on the small beams, debonding failure may occur in the adhesive layer. Moreover, fitting expressions were proposed to predict the shear stress–slip relationship between the CFRP sheets and concrete and the flexural strength of the CFRP‐strengthened beams under static loads, and good agreement with the test data was obtained. Finally, a fatigue life prediction model was also presented to capture the fatigue life of the CFRP–concrete interface under cyclic loads. The calculation results showed that the fatigue strength of the CFRP–concrete bond interface was approximately 65% of the ultimate load capacity.

Full-range behavior of fiber reinforced cementitious matrix (FRCM)-concrete joints using a trilinear bond-slip relationship

Interfacial debonding of fiber reinforced cementitious matrix (FRCM)-concrete joints can be considered as a mainly mode-II fracture process, a problem that can be solved by accounting for one-dimensional interfacial shear stress-slip relationships. This paper presents an analytical approach to predict the load response of FRCM-concrete joints by adopting a trilinear bond-slip relationship consisting of a linear-elastic branch, a softening branch, and a friction branch. The applied load-global slip response of FRCM-concrete joints with (relatively) long bonded length includes five stages: elastic, elastic-softening, elastic-softening-debonding, softening-debonding, and debonding stages. Closed-form solutions of the interfacial slip, shear stress, and axial stress (or strain) distribution along the bonded length are provided. The response of FRCM-concrete joints with (relatively) short bonded length is examined. The effective bond length and a critical length for the existence of the snap-back phenomenon are derived. Experimental results reported in the literature are used to calibrate the parameters needed for the analytical approach. The analytical results are then compared with experimental results and with numerical results determined using a finite difference method (FDM). Finally, the capability of determining the parameters in the trilinear bond-slip relationship using a neural network (NN) with the experimental load response as the input is investigated.

Numerical modelling of bond shear stress slip behavior of CFRP/steel composites cured and tested at elevated temperature

This paper presents a numerical model developed to predict the bond characteristics of CFRP/steel composites cured under different curing conditions and their behaviour at elevated temperatures. The measured material properties and their degradation with the temperature exposure were considered. The predicted bond performance was in a good agreement with the test results. The strain variation in the CFRP sheet was used to develop the bond shear stress-slip variations. Parametric studies were also conducted to evaluate the effects of bond line parameters on the bond shear stress-slip relationship at elevated temperature. The results indicate that the maximum bond shear stress of the joint lies in the range between 25 MPa and 28 MPa at ambient conditions, irrespective of the curing type. A rapid decrease in the maximum bond shear stress appears with exposure to the elevated temperature. Maximum shear stress reaches 10 MPa when the bond line temperature exceeds 90 °C. The elevated temperature curing, exposed temperature during service and the bond thickness notably affects on the bond slip behavior.

Direct Shear Behavior of Fiber Reinforced Concrete Elements

Improving the accuracy of load-deformation behavior, failure mode, and ultimate load capacity for reinforced concrete members subjected to in-plane loadings such as corbels, wall to foundation connections and panels need shear strength behavior to be included. Shear design in reinforced concrete structures depends on crack width, crack slippage and roughness of the surface of cracks.
 This paper illustrates results of an experimental investigation conducted to investigate the direct shear strength of fiber normal strength concrete (NSC) and reactive powder concrete (RPC). The tests were performed along a pre-selected shear plane in concrete members named push-off specimens. The effectiveness of concrete compressive strength, volume fraction of steel fiber, and shear reinforcement ratio on shear transfer capacity were considered in this study. Furthermore, failure modes, shear stress-slip behavior, and shear stress-crack width behavior were also presented in this study.
 Tests’ results showed that volume fraction of steel fiber and compressive strength of concrete in NSC and RPC play a major role in improving the shear strength of concrete. As expectedly, due to dowel action, the shear reinforcement is the predominant factor in resisting the shear stress. The shear failure of NSC and RPC has the sudden mode of failure (brittle failure) with the approximately linear behavior of shear stress-slip relationship till failure. Using RPC instead of NSC with the same amount of steel fibers in constructing the push-off specimen result in high shear strength. In NSC, shear strength influenced by the three major factors; crack surface friction, aggregate interlock and steel fiber content if present. Whereas, RPC has only steel fiber and cracks surface friction influencing the shear strength. Due to cementitious nature of RPC in comparisons with NSC, the RPC specimen shows greater cracks width.
 It is observed that the Mattock model gives very satisfactory predictions when applied to the present test results with a range of parametric variations; ranging from 0 % to 0.5 % in steel fibers content; from 0 % to 0.53 % in transverse reinforcement ratio; from 15 to 105 MPa in compressive strength of concrete. While it gives a poor prediction for a specimen with 1% steel fiber.

INVESTIGATING THE INFLUENCE OF STEEL SURFACE ROUGHNESS AND MULTIPLE CFRP LAYERS ON THE BONDING BEHAVIOUR OF A SINGLE-STRAP JOINT

Three series of steel-plate single-strap joints bonded by Carbon-Fibre Reinforced Polymer (CFRP) sheets, which are used with an adhesive material, are investigated in this study. Experimental and finite element (FE) analyses using ABAQUS software are employed for this purpose. The main objective of this study is to provide further understanding of the bonding mechanism of strengthened single-strap joints under a pure-tension load. The experimental tests specifically investigated the effects of using multiple CFRP layers with normal/high steel surface roughness on the bonding failure mode, bond-load capacity, and effective bond length. Furthermore, the FE analysis was used to establish the shear stress-slip relationships of the suggested specimens. The results showed that the bond capacity (ultimate bond load) of single-strap joint did not increased much when the CFRP bonding length pass the effective bond length (limit). The bonding capacity (ultimate bond load) of single-strap joint increased only of about 14% when increase d the CFRP layers from two to four. Moreover, the behaviour of the shear stress-slip relationship was not significantly affected by increasing the number of CFRP layers.

Effect of curing conditions on mode-II debonding between FRP and concrete: A prediction model

The rehabilitation and strengthening of concrete structures using Fiber-Reinforced Polymer (FRP) materials have been widely investigated. As a priority issue, however, the effect of curing conditions on the bonding behavior between FRP and concrete structures is still elusive. This study aims at developing a prediction model to accurately capture the mode-II interfacial debonding between FRP strips and concrete under different curing conditions. Single shear debonding experiments were conducted on FRP-concrete samples with respect to different curing time t and temperatures T. The J-integral formulation and constrained least square minimization are carried out to calibrate the parameters, i.e., the maximum slip s- and stretch factor n. The prediction model is developed based on the cohesive model and Arrhenius relationship. The experimental data are then analyzed using the proposed model to predict the debonding between FRP and concrete, i.e., the interfacial shear stress-slip relationship. A Finite Element (FE) model is developed to validate the theoretical predictions. Satisfactory agreements are obtained. The prediction model can be used to accurately capture the bonding performance of FRP-concrete structures.

Fracture Properties of CFRP–Concrete Bond Subjected to Three Environmental Conditions

AbstractThis paper presents results of a series of single shear tests of carbon fiber–reinforced polymer (CFRP) bonded concrete prisms exposed to three environmental conditions, namely, temperature cycles, wet-dry cycles, and outdoor environment, in terms of shear stress-slip relationships and fracture energies. Temperature cycles and wet-dry cycles were chosen in a manner to study the sole effects of temperature cycles and wet-dry cycles. The maximum temperature of the temperature cycles was intentionally kept below the glass transition temperature of epoxy resin. In the wet-dry cycles, specimens were exposed to varying humidity while temperatures close to ambient were maintained. Also, outdoor environmental exposure was applied to address the lack of test data on natural aging of FRP-concrete bond system. All the environmental conditions were applied for extended durations (the maximum duration of 18 months). Single shear tests (pullout test) were conducted to investigate maximum normal stress developed...

Tension-stiffening model for FRC reinforced by hybrid FRP and steel bars

This paper presents a tension stiffening model for Fiber Reinforced Concrete (FRC) tensile member reinforced by hybrid glass fiber reinforced polymer (GFRP) and steel bars. The model is developed through an explicit analytical bond formulation by considering a four-linear bond shear stress-slip relationship to simulate the bond behavior between reinforcing bars and surrounding FRC. The model is also capable of simulating both the fiber reinforcement contribution and the yielding stage of steel bar at cracked section. Additionally, a FE Model is carried out using a multi-directional smeared crack approach for modeling cracking process in FRC, and adopting interface finite elements to simulate the bond behavior between reinforcements and FRC, whose constitutive model was defined from the aforementioned bond law. Both the analytical and numerical approaches showed a good agreement with some recent experimental results on tension-stiffening in the literature. Finally, an extensive parametric study is performed by using the analytical model, and the influence of the involved parameters on the tension-stiffening and cracking behavior of hybrid GFRP/steel FRC tensile member is investigated.

Bonding Behavior of Wet-Bonded GFRP-Concrete Interface

AbstractAn experimental study was conducted to investigate the interfacial behavior between glass-fiber-reinforced polymer (GFRP) plate and cast-in-place (CIP) concrete placed against this plate—a method known as wet-bonding. The experimental program consisted of 33 single-shear specimens with various initial placement times of concrete, durations of concrete curing, and interface preparations. Control specimens using a conventional dry-bonded technique were also tested. A model was proposed that was able to predict the interfacial ultimate pullout load and local shear stress-slip relationship of a wet-bonded interface with reasonable accuracy. It is shown that (1) wet-bonded interfaces exhibit lower capacity and interfacial fracture energy than dry-bonded interfaces although the load to cause initiation of debonding is not significantly different; (2) the time of initial placement of the CIP concrete onto the wet, set, or cured resin significantly influences the ultimate pullout load and interfacial frac...

Experimental investigation and modelling of the structural behaviour of confined grouted interfaces for a new steel–concrete connection

Confined interfaces between cement grout and other materials such as embossed steel or rough concrete are formed in an innovative new steel-concrete connection for composite bridges. This article presents the experimental study of such interfaces and investigates the laws obtained from an analytical study, which describe the behaviour of those interfaces subjected to shear loading, static and cyclic, under different levels of constant normal stress. The laws describing the interface behaviour in monotonic shear loading are: (a) the relationship between the ultimate shear and the normal stress acting on the interface, i.e. the failure criterion, (b) the shear stress–slip relationship of the interface, called hereafter the constitutive law and (c) the relationship between the transverse separation (uplift) of the interface and the slip in the interface, called hereafter kinematic law. Concerning cycle shear loading, tests on interfaces enabled damage determination due to repeated loading. This damage is expressed by the development of an accumulated residual slip in the interface. Results indicate that as long as the applied shear stress in the interface, during cyclic loading, remains inferior to the elastic shear limit for static loading, the residual slip stabilizes with the number of cycles. A safe fatigue failure criterion is proposed relating the accumulated slip with the value of the slip at shear failure of the interface for static loading. The proposed laws together with the law describing the development of the confinement constitute the necessary input for a model to predict the behaviour of an innovative connection consisting of such confined grouted interfaces.

Theoretical analysis on pullout of anchor from anchor–mortar–concrete anchorage system

Theoretical studies on pullout of an anchor from an anchor–adhesive/mortar–concrete anchorage system have been carried out with the wide application of this anchorage system in civil engineering. Most of them focused on one interfacial debonding crack propagation from the loading end similar to the theoretical method on fiber pullout from matrix, or adopting a uniform interfacial shear stress model to calculate the tensile capacity of the anchor but with a limitary embedment length. Moreover, shear stresses were generally assumed uniformly along the thickness of the adhesive/mortar layer. Actually, the distributions of the shear stresses would be varied as both the thickness of the adhesive/mortar layer and the embedment length increase. Taking into account of this variation, the present study addressed the pullout of an anchor from an anchor–mortar–concrete anchorage system with two different boundary conditions for various embedment lengths analytically combining with the compatibility conditions and a simplified shear stress–slip relationship at the anchor–mortar interface. In the proposed analytical model, the variation of the shear stresses along the thickness of the mortar was gained. Additionally, the distributions of the tensile stresses in the anchor and the interfacial shear stresses along the embedment length were obtained under different loading conditions. After interfacial debonding, the sequences and probabilities of two interfacial debonding cracks developing from both ends of the system were analyzed according to the boundary conditions and the axial rigidities of both the anchor and concrete. Besides, the pullout load was expressed as a function of one or two debonding crack lengths. Then the maximum load P max was gained correspondingly, as well as the critical crack lengths, by using the theories of extremum. Results show that the obtained solutions consist with the previous work from other literature if the elastic modulus of concrete is assumed to be infinite and the anchor–mortar interface is thought as rigid before debonding. Subsequently, several fundamental structural and interfacial parameters were introduced to study their influences on the calculated results using the proposed model, such as the critical crack lengths, the initial cracking load P ini and the maximum load P max. It was found that P max increases linearly with the embedment length L, however, P ini is irrelevant to L if L is very long. P max increases monotonously with the thickness t of the mortar layer if the concrete layer keeps a constant thickness b, however, there appears a peak value in the curve of P max varying with t if the sum of b and t is invariable. Moreover, the difference between the values of P max under the two boundary conditions is marginal.