Analytical Solutions For Stresses Research Articles

Analytical solution for the interfacial stress and energy release rate at failure initiation of the three-point bending test (ISO 14679:1997)

Determining adhesive failure in adhesively bonded joints is still challenging in the field. While the three-point bending test (3PBT) (ISO 14679:1997) has been helpful in identifying critical forces and displacements related to bond strength, it only allows for a qualitative assessment of the bond line. Recently, a new quantitative methodology has been developed to determine critical stress and fracture toughness using the coupled stress–energy criterion and the 3PBT. However, these assessments require a significant effort using semi-analytical or finite element (FE) analysis. Therefore, the present study proposes a reliable set of analytical equations to determine peel and shear stress distributions using a weak interface formulation and 1D Euler–Bernoulli approach. More precisely, a particular numerical method computes the integration constants from these equations. A new method has been proposed for calculating the interfacial stiffness in peel and shear mode, based on the material and geometrical parameters of the geometry. This method differs from the previous approach, where the interfacial stiffness was calibrated from the experimental behavior of the test. The whole approach has been validated through experimental and numerical analysis, including the costly 3D FE analysis. An analytical expression for interfacial energy release is suggested, developed following the works of Fraisse and Schmit on J-integral assessment of sandwich-type overlaps and depending on the applied force and a rotation, which could be experimentally measured. Therefore, this work is significant progress in determining bond strength using a simple mechanical test and equations applicable to various industries.

Anisotropic functionally graded nano-beam models and closed-form solutions in plane gradient elasticity

This study delves into the investigation of exact analytical solutions for the plane stress and displacement fields within linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on solving the Helmholtz equation, which encompasses a second-order non-homogeneous linear partial differential equation in plane gradient elasticity theory, utilizing polynomial series-type solutions. The analysis centers on the utilization of gradient Airy stress functions to derive stress fields in the gradient theory. The research yields closed-form analytical solutions for Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five distinct types of two-dimensional functionally graded cantilever beams with various boundary conditions: a cantilever anisotropic nano-beam subjected to a concentrated force at the free end, a cantilever anisotropic nano-beam under a uniform load, a simple anisotropic nano-beam under a uniform load, a propped cantilever anisotropic nano-beam under a uniform load, and a fixed end cantilever anisotropic nano-beam under a uniform load. General analytical solutions for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions are provided. The study showcases significant strain gradient size effects at the nano-scale through the derived analytical solutions for anisotropic beams. Additionally, it demonstrates that the strain gradient theory results for the limit case of the gradient coefficient c precisely align with results for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, real-world applications are discussed, considering stress and displacement fields in real anisotropic materials such as TiSi2 single crystals and orthotropic materials, which are special cases of anisotropic materials like wood and epoxy as documented in the literature.

Optimisation of material composition in functionally graded plates for thermal stress relaxation using statistical design support system

Abstract This study addresses the optimisation of material composition in a functionally graded plate for thermal stress relaxation, subjected to through-thickness thermal gradients, with the aim of minimising a stress utilisation ratio. We simplify the problem by approximating the functionally graded plate as a multi-layered plate. Material compositions in individual layers are optimised using a statistical design support system (SDSS), incorporating design of experiments and mathematical programming techniques. The volume fractions of ceramic constituent in the respective layers are considered as design variables, and an analytical solution for transient thermal stresses is utilised to evaluate the objective function. The optimisation results obtained using the SDSS are compared with those from a genetic algorithm (GA) to validate the applicability of the proposed method. Our findings indicate that the SDSS replicates the ceramic volume fraction distribution optimised by the GA, while significantly reducing optimisation time.

Root anatomy and biomechanical properties: improving predictions through root cortical and stele properties

PurposeQuantifying the stability of individual plants or their contribution to soil reinforcement against erosion or landslides requires an understanding of the tensile properties of their roots. This work developed a new analytical model to understand the tensile stress–strain behaviour of a single root axis, which is the first to incorporating root anatomical features, in order to reduce the existing uncertainty in predictions.MethodsThe root was modelled as a linear elastic stele connected to a surrounding linear elastic cortex by means of a linear elastic stele–cortex interface. By solving for force equilibrium, an analytical solution for the full tensile stress–strain behaviour — including any intermediate brittle failures of the stele, cortex and/or interface — was obtained. This model was compared to tensile tests and laser ablation tomography for maize roots.ResultsThe new modelling approach demonstrated that the root tensile strength is fully determined by the strength of the stele alone, which was an order of magnitude larger than that of the cortex while also 3–4 times stiffer. The reduction in root stiffness beyond the yield point was linked to continuing fracturing of the cortex and debonding along the stele–cortex interface. A larger proportion of the variation in experimentally measured biomechanical characteristics could be explained compared to root diameter power-law fitting methods typically applied in the literature.ConclusionStele and cortex biomechanical properties are substantially different, affecting the tensile behaviour of plant roots. Accounting for these anatomical traits increased the accuracy root biomechanical properties from tensile tests.

Influence of principal stress orientation on stress distribution and plastic zone evolution of rock surrounding tunnels

During the excavation of tunnels, the principal stress orientation changes, with a significant impact on the stress distribution, mechanical properties, and plastic zone evolution of rocks surrounding tunnels, causing severe deformation control, and monitoring problems in the stability of tunnels. Currently, biaxial compression tests were conducted to explore the influence of principal stress orientation on mechanical properties of rocks surrounding tunnels. The analytical solution of stress and the model of plastic zone of rocks considering the principal stress orientation and the distance from the excavation boundary were established to reveal the failure mechanism of surrounding rock under different principal stress orientations. With an increase in the angle between the principal stress orientation and the long axis of tunnel, the maximum tangential stress around the tunnel gradually changed to the minimum horizontal principal stress direction, and its value gradually increased, leading to quicker failure of surrounding rocks, and reducing the strength enhancement effect of the same section size of the tunnel. However, the increase in the angle reduced the damage range and the range of the plastic zone around the tunnel and caused the plastic zone to gradually approach the bottom and roof. The maximum depth of the plastic zone remained parallel to or nearly parallel to the minimum horizontal principal stress direction. When the principal stress orientation was kept constant, the maximum depth of the plastic zone shifted to the minimum horizontal principal stress direction with an increase in the vertical principal stress.

Effect of external prestressing on the special shear lag behavior of precast box girder bridges

The shear lag behavior exhibited by precast segmental box girders (PSBGs) under concentrated external prestressing differs significantly from that observed in box girders subjected to vertical loads. To study the special shear lag behavior of PSBGs under external prestressing, a theoretical analytical method was developed based on the classical energy variational principle. In this analysis, the effect of external prestressing was decomposed into axial compression and bending effects through the equivalent load method. For each of these effects, appropriate longitudinal warping displacement modes were assumed, and the total energy functional expressions of the system under these two effects were further established. The governing differential equations for the two effects were derived using the principle of energy variation. Analytical solutions for the normal stress of the beam section under axial compression and bending effects were deduced, and corresponding boundary conditions were obtained. Subsequently, the normal stress of the section under the two effects was superimposed to obtain the comprehensive normal stress distribution curve of the beam section under external prestressing. The reliability of the theoretical analysis method was validated through a comparison of the theoretical analysis results with experimental and numerical analysis results.

Seepage Actions and Their Consequences on the Support Scheme of Deep-Buried Tunnels Constructed in Soft Rock Strata

The stability of deep soft rock tunnels under seepage conditions is of particular concern. Aiming at thoroughly discussing seepage actions and their consequences on the support schemes of such structures, the host rocks of the Weilai Tunnel situated in the Guangxi province of China are used as the research subject. Emphasis is placed on adequately examining the seepage conditions, stresses, displacements and plastic zone radii along the surrounding rocks of such tunnels, taking into consideration the Mogi–Coulomb strength criterion and the elastic-plastic theory. Explicitly, this article proposes analytical solutions for stresses, displacements and plastic radii around deep tunnels in soft rocks under seepage conditions by considering the aforesaid criterion and nonlinear elastoplastic approaches. Subsequently, based on the strain-softening model, the coupled actions of seepage and softening on the rocks surrounding the tunnel are studied. In order to investigate the effects of relevant influencing factors on tunnel stability, parametric studies are thoroughly examined. According to the results, it is revealed that the support scheme of deep soft rock tunnels must be of the highest resistance possible to better decrease the plastic zone and the tangential stress along the host rocks. Moreover, throughout the surrounding rocks, the dissemination of pore water pressure is strongly affected by the uneven permeability coefficient under anisotropic seepage states. The combined effects of softening and seepage are very dangerous for the surrounding rocks of deep-buried tunnels. It is also shown that the seepage pressure substantially affects the plastic radii and tunnel displacements. Under high seepage pressure, the surface displacements of the tunnel are excessive, easily exceeding 400 mm. To better guarantee the reasonable longevity of such tunnels, the long-term monitoring of their support structures with reliable remote sensors is strongly recommended.

Exact analytical solution for deep tunnels in viscoelastic–plastic rock considering the actual loading path

The rock rheology and time process of tunnel construction make the mechanical responses of rock a function of time and are closely related to the loading path, however, these factors are not properly or correctly considered in analytical studies. The analytical study for deep tunnels in rheological rock is performed in this study, strictly taking into account the viscoelastic‒plastic characteristic of surrounding rocks and the actual loading path during excavation and supporting stages.The plane-strain problem of a circular tunnel in infinite plane under hydraulic far-field stress is simplified, with the time-dependent supporting pressure exerted at a specific time to consider the effect of delay installation of yielding support. The solving procedure as well as the time-dependent analytical solution of displacement, stress, and plastic radius in the excavation and supporting stages are presented in detail when the rocks satisfy the Unified strength theory and perfect plasticity. The analytical solutions agree very well with the numerical results, where the time procedure of tunnel excavation and support is simulated in the numerical model. Furthermore, the proposed solution can predict well the long-term deformation of the Shangxinzhai tunnel, which validates its applicability in real engineering. It is found that the tunnel deformation is underestimated if the unloading stage is not correctly considered. Based on the analytical solution, the influence of the supporting time, start time of the yielding stage and viscosity coefficient on the time dependence of tunnel convergence and stresses is investigated.

Supporting Structure of Steel Corrugated Plate-Mold Bag Concrete and Its Application in a Circular Shaft

To cater to the problems of a poor working environment, a large amount of rebound, and the high energy consumption of sprayed concrete in tunnel engineering, this paper proposes a new support scheme with a steel corrugated plate combined with mold bag concrete. Analytical solutions of stresses for steel corrugated plate and mold bag concrete were deduced based on the thin-walled cylinder, and then their strength safety factors were presented. Subsequently, a series of numerical simulations were conducted to investigate the mechanical performance of the third ring of the main structure based on the ‘load-structure’ 3D model. The numerical results were verified using the classical theoretical analysis and the proposed model, and then parametric studies were performed through the numerical method. Finally, field tests in a circular shaft were carried out to verify the feasibility of the structure and process and the engineering effect. The results show that the combination of steel corrugated plate and mold bag concrete is feasible and can realize rapid support technology for underground engineering, which provides a new idea for the supporting technology of underground engineering such as tunnels.

Mechanism of surrounding rock failure in impact stress and in-situ stress in circular tunnel

With depth increase of many mines, the damage of surrounding rock by high ground stress cannot be ignored under the blasting excavation method. In view of the strong disturbance of surrounding rock caused by dynamic excavation and unloading under high initial rock stress conditions, based on the elastic unloading theory, the analytical solution of dynamic excavation and unloading stress is given using the residue theorem and Laplace inverse transform. The stress field distribution under the coupling effect of blasting load and initial rock stress is described. By making a rock model and implementing biaxial loading using a drop hammer to simulate the impact load, radial cracks generated by the impact stress wave and circumferential cracks generated by unloading during excavation were captured, and the stress change curve during excavation was monitored. The monitoring curve and theoretical curve have consistency in trend. Under the coupling effect of blasting load and initial stress, surrounding rock will generate tensile stress and broken, and the unloading effect of initial stress occurs after the blasting load, which causes tensile damage to the surrounding rock. The stress field curve plotted by the theory explains well the crack propagation caused by the coupling effect of blasting load and initial stress.

Theoretical analysis of anchorage-seepage coupling effect of the surrounding rock stability in deep buried abandoned chambers

How to ensure the safety of abandoned mine resources, scientifically develop and utilize abandoned mine resources, and promote the transformation of resource-exhausted mining areas have become an important issue in the field of energy and environment in the world today. Aiming at the stability of the surrounding rock in deep closed/abandoned mine chamber, the mechanical model of the surrounding rock under the coupling effect of anchorage and seepage field was proposed. Considering the elastic brittleness degradation and plastic dilatancy effect of rock mass, the analytical solutions of stress and displacement of rockbolt-seepage-surrounding rock coupling system were respectively deduced, and the accuracy of the results were verified. Based on the analytical results, the evolution law of stress and displacement of the surrounding rock under the combined action of seepage field and anchorage effect were further revealed, and a new quantitative design method of rockbolt parameters was proposed. Results show that the influence of rockbolt spacing and rod diameter on the mechanical field is obvious, while the rockbolt length and pre-tension load is small. Dense, short rockbolt with larger diameter should be used in the surrounding rock of deep chamber. The influence of seepage on the displacement of the surrounding rock is very significant. The more serious the seepage is, the more obvious the control effect of rockbolt on the displacement is. Appropriately increasing the density and diameter of rockbolt can effectively reduce the displacement of the surrounding rock.

Out-Plane Buckling of Arches with Variable Cross-Section

The variable cross-section arch is widely used in practical engineering because of its beautiful arc and excellent mechanical properties. However, there is still no systematic and comprehensive study on the out-plane buckling of variable cross-section arches. In view of this, this paper is focused on the elastic analytical research of out-plane buckling of arches with variable cross-sections under a uniformly distributed radial local load. The pre-buckling and out-plane buckling behavior of a variable cross-sectional arch under an external load is quite different from that of an arch with a uniform cross-section. Castigliano’s second theorem is used to establish pre-buckling force method equilibrium equations for variable cross-sectional arches under a uniformly distributed radial local load, and corresponding analytical solutions of normal stress, axial compression, and the bending moments are obtained. Based on the energy method and the Ritz method, analytical solutions of the critical load for the elastic out-plane buckling of arches with variable cross-sections are derived. Comparisons with ANSYS results indicated that the analytical solutions are able to accurately predict the pre-buckling internal forces and critical out-plane buckling load of variable cross-section arches subjected to a uniformly distributed radial local load. It is found that the internal forces and the out-plane buckling load of an arch are significantly affected by the variation of cross-sectional height. As the ratio of the arch’s cross-sectional height increases, the bending moment decreases, and the axial force and critical out-plane buckling load increase. Analytical solutions of pre-buckling internal force and critical out-plane buckling load problems for arches with variable cross-sections have a wider significance since they can provide an effective explicit analytic method for the optimal design of arch structures.

Damage effect and progressive failure mechanism of sandstone considering different flaw angles under dynamic loading

In many engineering applications, it is essential to have information about rocks that inherently contain pre-existing flaws under dynamic loading conditions. Dynamic impact tests are conducted on samples with varying flaw angles using the split Hopkinson pressure bar (SHPB) test system and the Digital Image Correlation system (DIC). The characteristics of the samples after dynamic loading, including dynamic strength, energy dissipation, and fractal fracture, are compared and analyzed. As the flaw angle increases, the peak stress and strain exhibit a typical V-shaped pattern, reaching the minimum value at 30°, and the initial initiation position shifts from the flaw tips to the middle of the flaw. Failure modes can be divided into three modes depending on the flaw angle. The progressive failure process, taking into account the heterogeneity of the rock, is demonstrated by developing an elastic damage constitutive model that uses dynamic compression and tensile tests to parameterize it. As the flaw angle increases, the initial damage zone also moves from the flaw tips to the middle of the flaw. Failures around the hole with redistributed stress are observed, and the failure mechanisms can be explained with the aid of strain energy density (SED). Using fracture mechanics, the analytical solution of stress around the flaw is provided, and the variation of crack initiation angle, stress distribution, and energy dissipation under different flaw angles is theoretically explained, which is in good agreement with the experimental and simulated results.

Hygrothermal analysis of a semi-infinite cylinder: an integral transform approach

PurposeThe purpose of this paper is to examine the impact of moisture and temperature changes on the behavior of a semi-infinite solid cylinder made of T300/5208 composite material. This study aims to provide analytical solutions for temperature, moisture and thermal stress through the de-coupling technique and the method of integral transforms. Both coupled and uncoupled cases are considered.Design/methodology/approachThis study investigates the hygrothermo-elastic response of a semi-infinite solid circular cylinder using an integral transform technique that includes Hankel and Fourier transforms. The cylinder is subjected to prescribed sources, and a numerical algorithm is developed for the numerical computation of the results. The goal is to understand how the cylinder responds to changes in temperature and moisture.FindingsThe paper presents an analytical solution for temperature, moisture and thermal stress in a semi-infinite solid cylinder obtained through the use of an integral transform technique. The study focuses on a graphite fiber-reinforced epoxy matrix composite material (T300/5208) and discusses the coupled and uncoupled effects of temperature, moisture and thermal stress on the material. The results of the transient response hygrothermo-elastic field are presented graphically to provide a visual representation of the findings.Research limitations/implicationsThe research presented in this article is primarily hypothetical and focused on the analysis of mathematical models.Originality/valueTo the authors' best knowledge, this study is the first to investigate the hygrothermal effect in a semi-infinite circular cylinder. Additionally, the material properties used in the analysis are both homogenous and isotropic and independent of both temperature and moisture. These unique aspects of the study make it a novel contribution to the field.

Investigation on bonding behavior of basalt fiber reinforced polymer (BFRP) sheet reinforced concrete beam

Although basalt fiber reinforced polymer (BFRP) sheet offers promising benefits including environmental friendliness and excellent mechanical performance in concrete reinforcement, its full potential is still hindered by the lack of understanding about their bonding mechanism. Therefore, the comprehensive investigation of the bonding behavior of BFRP sheet reinforced concrete is necessary. This work aims to address the research gap between the analytical design and industrial application of BFRP sheet reinforced concrete due to the complicated bonding behavior. Analytical method, experimental measurement, and numerical simulation are employed to comprehensively investigate the bonding behavior of BFRP sheet reinforced concrete. According to the mechanical model and assumptions of the reinforcement, analytical solutions for interfacial stress and strain are derived. In addition, four-point bending test are conducted on specimens, and multi-view digital image correlation (DIC) technique is used to analyze the deformation on concrete part and BFRP sheet simultaneously. Based on analytical and experimental results, finite element model (FEM) is applied to simulate mechanical behavior of BFRP sheet reinforced concrete beam, and parametric analysis is further conducted. Results reveals that the maximum interfacial stress appears at the end of bonding region so that BFRP debonding at the end is the main failure mode. The strain distribution of BFRP sheet obtained by multi-view DIC indicates that there is an effective bonding region. Besides, strain can be used as a variable to validate numerical model by analytical and experimental results. The validated numerical simulation could be conducted for the internal response study and application design of BFRP sheet reinforcement.

Nonlinear analytical solution for radiation stress of higher-order Stokes waves on a flat bottom

The radiation stress of water surface waves plays an important role in many aspects of marine hydrodynamics. The nonlinear effects of highly nonlinear waves on the radiation stress were investigated on a flat bottom under finite water depth and deep water conditions. Analytical solutions for radiation stress were derived based on different order nonlinear Stokes wave theory, respectively. The results indicate that the nonlinear radiation stresses are significantly greater than those based on the linear wave, and they increase as the nonlinearity strengthens. Through an inter-comparison of the expressions of nonlinear radiation stress of various orders of Stokes waves, it turned out that the results based on the second-order Stokes wave overestimated the radiation stress, while the expressions based on Fenton's fifth-order Stokes wave were more reasonable in reflecting the nonlinear effect of waves. In the case of extreme wave steepness, the fifth-order Stokes wave radiation stresses Sxx5 was approximately 17% larger than that of the linear wave. Finally, using the expressions of the fifth-order Stokes wave radiation stress, bound infragravity waves excited by the bichromatic wave groups formed by the superposition of two fifth-order Stokes waves were obtained.

Experimental and theoretical analysis of spalling in deep hard rock tunnels with different arch structures

To study the spalling characteristics and induction mechanism of tunnels with different arch structures, a series of true triaxial tests were conducted on three types of cubic granite specimens with a through-going arched hole and analytical stress solutions were obtained for the three arch sections. Test results show that the failure process of specimens includes four periods: a quiet period, a buckling deformation period, a gradual buckling and exfoliation period of rock fragments, and a V-shaped notch formation period. Specimens with different arch structures are listed in a descending order as the three-centered arch with f/B (arch rise/tunnel width) = 1/3 (II), three-centered arch with f/B = 1/4 (III), and semi-circular arch with f/B = 1/2 (I) according to the size of rock fragments. Before the spalling of arched tunnels, the acoustic emission (AE) count is low, while the AE count and cumulative AE count are high during spalling. Stress analysis results show that as the vertical stress rises, the arch feet of tunnels undergo initial failure at first due to stress concentration; spalling cracks appear in both side walls due to compressive stress concentration, which induce spalling failure; tensile cracks initiate in the vault and invert due to tensile stress concentration. The sequence of spalling in tunnels with different arch structures varies due to a change in section structure. Therefore, the quality of section structures can be evaluated according to the stress distribution characteristics of surrounding rocks. The vertical stress inducing the initial failure of tunnel surrounding rocks increases gradually in different arch structures from the II, III, to I; the failure severity of surrounding rocks decreases in different arch structures, also from II, III, to I. In addition, the stress analysis results also verify the accuracy of the test results. In practical engineering applications, to consider both the section utilization ratio and control of spalling in surrounding rocks, selecting tunnel sections with f/B = 1/4 is more economic and safer than that of f/B = 1/3.

Exact solution of magneto-elastoplastic problem of functionally graded cylinder subjected to internal pressure

With the development of composite material technology, the multi-field coupling analysis of functionally graded (FG) structures has attracted extensive attention. For FG cylindrical pressure vessels, due to the non-uniform distribution of stresses filed in the structure, plastic deformation occurs irreversibly under external load, and Lorentz's force under static magnetic field is also one of the key factors affecting the elastoplastic state. Assuming that the elastic modulus of FG cylinder changes as a power function along the thickness and Poisson's ratio is constant, the hollow cylinder under the combined action of applied magnetic field and internal pressure may have four deformation states containing pure elastic deformation, partial plastic deformation yielding from the inner surface, partial plastic deformation yielding from the outer surface, and full plasticity. In this paper, analytical solutions of stress and displacement fields of the FG cylinder under these four different elastoplastic states are given, and the critical load conditions corresponding to the transition between these four different states are proposed. The correctness of the present analytical solution in this paper is verified by comparing it with the existing results of related simplified problems. In addition, the analytical solution of residual stress distribution caused by unloading internal pressure is discussed for self-reinforcing processing. Based on the analytical solution in this paper, the influence of FG parameter, applied magnetic field and internal pressure on the elastoplastic state, plastic zone size, stresses and displacements distribution of FG cylinder is analyzed in detail. This study shows that for this magneto-elastoplastic problem of FG cylinder, plastic deformation may appear from the inner surface or outer surfaces. With the increase of FG parameter, the first yielding position of FG cylinder changes from the inner surface to the outer surface. In addition, the existence of an applied magnetic field reduces the critical internal pressure required for FG cylinders to yield, making the FG hollow cylinder more prone to plastic deformation.

Dynamic response of shallow-buried circular composite lining tunnel with imperfect interface under SV waves

In this paper, the dynamic response of lined shallow-buried composite lining tunnel with imperfect interface in semi-infinite elastic medium under plane SV waves is studied, and the spring model is introduced to describe the imperfect interface relationship among the surrounding rock, the isolation layer, the primary support layer and main support layer. By using the wave function expansion method, the analytical solutions of displacements and stresses are presented. The interacting effects of the interface properties and the buried depth under different dimensionless frequencies are discussed. The results show that the peak values of the dynamic stress in the linings gradually shift to the upper part of the tunnel with the increase of dimensionless frequency, and the maximum dynamic stress in the linings also tends to decrease. The dynamic stress around the main and primary support layers increases when the interfaces are weakly bonded, and the effect of transverse springs is greater than that of normal springs. The tunnel with diminutive buried depth has a larger dynamic stress under the low-frequency SV waves. Reducing the stiffness in the isolation layer will lead to the increase of dynamic stress in the surrounding rock, while the dynamic stress in the main and primary support will be weakened.

Consolidation of multilayered soil with fractional derivative viscoelasticity due to surface loading and internal pumping

Classical Terzaghi’s solution provides a useful tool to predict the one-dimensional consolidation of homogeneous elastic soil layer with fully drained/undrained boundary condition under instantaneous loading. Due to the sedimentation process and complex internal structure, natural soils usually exhibit multilayered inhomogeneity and viscoelastic behavior. In practical cases, the drainage capacity of the boundaries can change with time. Thus, these practical factors should be well in considered for accurate prediction of the consolidation. This paper extends the classical Terzaghi’s solution to multilayered viscoelastic soil with continuous drainage boundaries and subjected to time dependent loading. The fractional derivative Kelvin-Voigt model is used to describe the viscoelasticity of soil. Two loading cases namely surface loading and internal pumping are considered. The consolidation problem is solved by Laplace transform technique and a transfer matrix formulation. Analytical solutions for excess pore water pressure, effective stress and settlement are given. The present solution is general and can reduce to existing solutions available in the literatures. Numerical studies are conducted to investigate the effects of viscoelastic parameters, boundary drainage capacity and loading rate on the consolidation behavior. It is shown that the multilayered soil system with large viscosity coefficient and small fractional order can consolidate fast. For the consolidation due to internal pumping, the drainage parameters have no influence on the consolidation process.