The research team set out to answer a deceptively simple question: what happens to the oil films at all four supports of a multi-bearing rotor when a load is applied at just one end? The answer required solving a coupled problem that most earlier studies avoided. Previous work on misaligned journal bearings, stretching back to Sun and Gui’s foundational lubrication analyses in 2004 and confirmed experimentally in 2005, established that shaft deformation distorts the convergent wedge of oil, thins the minimum film thickness, and reshapes the pressure field. But those studies largely examined single bearings or prescribed the misalignment in advance. In a real multi-support system, the four journal attitudes are not independent quantities. The deformation of the shared shaft and the hydrodynamic reaction forces generated at all supports collectively determine the global equilibrium, so strengthening one oil film can weaken, shift, or redistribute the constraints at its neighbors.

To capture this system-level coupling, the researchers modeled a three-disk, four-support bearing-rotor system resembling a camshaft under asymmetric loading. The shaft was represented with a Timoshenko beam formulation, which accounts for both bending and shear deformation, while each bearing’s lubricant was described by the Reynolds equation for a finite-length journal bearing under laminar, incompressible, isoviscous assumptions. Crucially, the eccentricity and inclination of all four journals were solved simultaneously through a fluid-structure equilibrium iteration: the shaft deformation determines each journal’s displacement and tilt, the resulting oil-film pressures generate reaction forces, and those forces feed back into the global structural balance until the two fields become mutually consistent. The team also introduced a compact descriptor, a flexible-deformation index denoted M_r, that combines the journal’s eccentricity ratio with a dimensionless measure of its inclination, allowing the coupled eccentricity-inclination state at each support to be summarized in a single number.

The numerical campaign was ambitious. The team built an offline database spanning end loads from 0 to 30 newtons and speeds from 1000 to 8000 rpm, sampling 211,001 load-speed states, of which 200,304, or 94.93 percent, returned valid directly converged fluid-structure solutions. At each converged state, the full oil-film stiffness and damping matrices were extracted by differential perturbation of the journal position and velocity, then stored for interpolation into a finite-element rotordynamic model. Validation was layered: the predicted maximum oil-film pressures under journal inclination agreed closely with the established results of Sun and Gui and of Lv and colleagues, with deviations of roughly 9 to 11 percent only at the most extreme inclination of 0.010 degrees, where the exact peak is known to be sensitive to cavitation treatment and thermal effects. The dynamic coefficients matched the classic calculations of Lund and Orcutt, and the computed bending mode shapes near 53 and 89 hertz agreed well with an independent ANSYS three-dimensional finite-element model.

The headline finding concerns how violently the support conditions redistribute under end loading. At rest, the journal attitudes are nearly symmetric, with the outer pair of bearings mirroring each other. Apply a modest end load, and that symmetry collapses. Within the 0 to 25 newton range, the flexible-deformation index at the loaded-end bearing, labeled J4, surges by 266.5 to 402.2 percent, while the inner supports J1 and J3 actually decrease and J2 barely moves. At 25 newtons, the index at J4 reaches 4.19 to 6.02 times that of J1 and 1.76 to 2.33 times that of J3. The physical picture is one of concentration: shaft flexibility funnels the eccentricity-inclination coupling into the loaded-end bearing while the inner supports respond mainly through load redistribution.

The consequences for the oil films themselves are even more striking. At 3000 rpm, the non-dimensional stiffness trace of the loaded-end support increases by 4298.0 percent and its damping trace by 293.6 percent. At 2884 rpm, the direct-stiffness trace of J4 climbs from 2.66 million to 125 million newtons per meter, roughly 47 times its initial value. Meanwhile the neighboring support J3 weakens markedly, its stiffness and damping traces falling by 62.4 and 36.8 percent, and J1 and J2 change only modestly. The cross-coupled stiffness term at the loaded end even reverses sign near a deformation index of 0.96, signaling a strongly asymmetric pressure field at high tilt. In short, the four-bearing boundary evolves from a comparatively shared support distribution into a strongly nonuniform, loaded-end-dominated one. The authors are careful to note that the exact extreme pressure peaks and coefficient amplitudes near the limiting thin-film state remain model-dependent, since the formulation is isothermal with a non-mass-conserving half-Sommerfeld cavitation treatment, but the direction and ordering of the redistribution proved robust across sensitivity checks.

Here is where the story takes its most counterintuitive turn. One might expect that a support suddenly 47 times stiffer would dramatically shift the rotor’s critical speeds, the rotation rates at which synchronous vibration resonates. It barely does. By continuously tracking the synchronous modal branches with a mass-weighted complex modal assurance criterion, which prevents closely spaced modes from being confused with one another, the team found that the two low-order forward and backward whirl branches near 2884 and 2895 rpm shift by only minus 0.00037 percent and plus 0.00039 percent respectively, essentially nothing. The higher-order branches H1 and H2 move a bit more, by plus 0.0140 and plus 0.1477 percent, and an additional high-order synchronous intersection appears at 5843.2 rpm in the highly deformed state. Yet even these larger shifts are tiny compared with the local stiffness explosion, and the global mode shapes themselves remain almost unchanged.

The resolution of this paradox lies in modal participation. The bearing locations are simply not very important players in the low-order bending modes: the shaft flexes between the supports in ways that barely engage the oil films, so even a massive local stiffness change barely registers. In the higher-order modes, and especially H2, the bearings matter far more. The generalized bearing-stiffness participation and the bearing-location displacement participation for H2 reach 1.449 percent and 0.0568 percent in the low-deformation state, approximately 23.6 and 22.1 times the corresponding forward-whirl values, and its critical-speed sensitivity rises in the same order. The support redistribution acts as a filter: the same local hydrodynamic change is transmitted into the global dynamics only in proportion to how strongly each modal branch involves the bearing locations. This explains why the first bending mode is reshaped more clearly in character than the main low-order critical speeds are shifted in frequency.

For engineers designing camshafts, compressors, and any flexible rotor carried on multiple oil-film bearings, the message is practical and profound. Local bearing coefficients measured or assumed in isolation cannot predict system behavior, because the four supports form a single coupled hydrodynamic network whose state emerges from shared shaft deformation. A designer who hardens one bearing without accounting for the induced weakening of its neighbors may misjudge both the lubrication regime and the vibration margin. The study’s framework, linking shaft flexibility, journal attitude redistribution, oil-film coefficient evolution, and branch-dependent modal sensitivity in one coherent chain, extends misaligned-bearing research from local lubrication analysis to the evolution of dynamic boundaries in whole multi-support systems. The authors note that conditions involving mixed lubrication, mass-conserving cavitation dynamics, or large-amplitude transient squeeze would require extended formulations, but within the converged hydrodynamic regime they examined, the mechanism is clear: the global dynamic consequence of local oil-film strengthening is governed not by the magnitude of the local change alone, but by how the redistributed support couples into each mode of the spinning shaft.

Subject of Research: Hydrodynamic support redistribution and modal response in flexible multi-support bearing-rotor systems

Article Title: Shaft-flexibility-induced hydrodynamic support redistribution and modal response in a multi-support bearing-rotor system

Article References: Zhang, Y., Li, B., Xu, W., Guo, H., & Yuan, Y. (2026). Shaft-flexibility-induced hydrodynamic support redistribution and modal response in a multi-support bearing-rotor system. Mechanical Sciences, 17(2), 857-871. https://doi.org/10.5194/ms-17-857-2026

Image Credits: AI Generated

DOI: 10.5194/ms-17-857-2026

Keywords: journal bearings, rotor dynamics, shaft flexibility, hydrodynamic lubrication, journal misalignment, oil-film stiffness, critical speeds, modal analysis, fluid-structure interaction, Timoshenko beam, Reynolds equation, camshaft

The challenge begins with the ground itself. Deep urban excavations rarely pass through uniform soil. Instead, a wall might thread through soft clay sandwiched between dense sands and stiff silts, and at each interface the earth pressure pushing on the structure can change abruptly. Soft layers tend to let the wall bulge outward and concentrate bending moments, while hard layers push back with strong constraint reactions. Traditional design models, which typically assume a smooth, linearly distributed pressure, simply cannot capture these jumps, and the discrepancies between calculated and observed deformation grow most pronounced precisely where projects are deepest and most demanding.

A second blind spot concerns the walls themselves. To resist enormous pressures at depth, engineers build ever thicker diaphragm walls, and thick structures do not behave the way classical beam theory assumes. Conventional models idealise the retaining wall as an Euler-Bernoulli beam, which accounts only for bending-induced rotation and ignores shear deformation, the internal sliding of one cross-section past another. For slender walls in shallow pits this simplification is harmless, but for the massive walls of deep excavations it introduces real error. Earlier work on shear effects had focused on buckling and boundary conditions rather than on the coupled bending-shear response of braced retaining structures in layered ground.

The research team, led by Jun Guan, Zejia Wu and Yanbin Fu, built a model that tackles both problems at once. Above the excavation level, the wall is loaded by an active earth pressure that depends on the wall’s own displacement, so the pressure redistributes as the wall moves, a feedback loop that conventional approaches freeze out. Below the excavation level, the embedded portion of the wall is treated as a Timoshenko beam resting on a Winkler elastic foundation, with soil resistance described by the widely used m-method, in which the ground’s reaction stiffness grows linearly with depth. Internal struts are represented as elastic supports whose reaction depends on their pre-applied force and stiffness, and the wall is sliced into computational segments at every strut location, soil-layer interface and current excavation level so that each segment sits within a single soil layer with a definite set of parameters.

Assembling these ingredients yields a fourth-order nonhomogeneous linear differential equation with variable coefficients for the wall’s horizontal displacement, an equation far too unwieldy for closed-form treatment by standard techniques. Rather than resorting to finite element modelling, which demands fresh meshing and re-analysis for every new project and excavation stage, the researchers turned to a power series method. They expressed the deflection of each wall segment as an infinite series, substituted it into the governing equation, and matched coefficients of like powers of the depth coordinate to derive a constructive analytical solution. Boundary conditions at the wall ends, together with deformation compatibility and force equilibrium between adjacent segments, pin down the unknown coefficients, and the same procedure is applied below the excavation level with the appropriate foundation reaction terms.

A crucial theoretical step was proving that the resulting series actually converges. Using the ratio test, the team showed that the limit of the ratio of successive terms falls below one, guaranteeing convergence, and they demonstrated that a ten-term expansion already satisfies engineering accuracy requirements. This matters practically: once the governing equations and solution are established, engineers can analyse different excavation stages, stratigraphic conditions and boundary conditions simply by updating input parameters, with no repeated mesh generation or staged numerical modelling, making the method especially convenient for the multi-scenario and parametric studies that dominate real design workflows.

To test the model, the researchers turned to an ultra-deep excavation in Hangzhou, a rectangular pit 22.40 metres long and 10.80 metres wide, retained by a cast-in-place diaphragm wall 800 millimetres thick and embedded to a depth of 60.37 metres, with its toe socketed into weathered rock. Excavation proceeded in eleven stages down to a final depth of 46.2 metres, threading through eleven distinct strata ranging from fill and sandy silts to a very soft muddy clay, rounded gravel and argillaceous siltstone. The team validated their predictions against measured wall displacements during the final four excavation stages, when the wall was working hardest.

The results were striking. In stage 8, the measured maximum wall displacement was 43.14 millimetres; the new model predicted 45.76 millimetres, an error of 6.07 percent, while a conventional Euler-Bernoulli benchmark predicted 40.60 millimetres, off by 12.7 percent. In stage 9 the new model erred by only 4.67 percent against the benchmark’s 11.9 percent; in stage 10 by 5.02 percent against 12.9 percent; and in stage 11 by 4.03 percent against 11.4 percent. Even more impressive was the prediction of where the maximum displacement occurs, the depth that most concerns designers deciding where to place struts. The new model located the critical depth within two metres of the measured value in every stage, and in stage 10 it was off by just 0.07 metres, whereas the benchmark consistently placed the bulge several metres too deep.

The parametric studies added further insight. Comparing simulations at excavation depths from 31 to 37 metres, the team found that shear deformation matters most in the segment between the lowest strut and the excavation surface, where it can account for up to 9.6 percent of total displacement when the ratio of that segment’s length to the wall thickness falls between roughly 0.85 and 4.17. Outside that range the shear effect fades to near insignificance, meaning the classical model remains a useful simplification for overall trends but must be supplemented in the critical zone. A separate comparison showed why accounting for layered ground is essential: when the model was reduced to assume homogeneous, linearly distributed earth pressure, it placed the maximum displacement depth 10 percent away from the field measurement, while the full heterogeneous model was within 2 percent. A scenario in which the properties of the soft sixth soil layer were modified shifted the maximum displacement magnitude by 12.13 percent and moved its depth upward by 8.03 percent, underscoring how sensitive deep walls are to local stratigraphic changes.

The authors are careful to note the model’s scope. It addresses monotonic inward wall movement during normal staged excavation and does not yet explicitly capture reversed wall movement, unloading-reloading behaviour, soil-structure separation or the progressive mobilisation of ultimate passive resistance, and the independent effects of individual soil parameters such as unit weight, cohesion and friction angle call for further controlled-variable study. Even so, the combination of displacement-dependent earth pressure, layered-strata handling and Timoshenko shear coupling, delivered as a fast, convergent analytical formula rather than a bespoke numerical model, offers deep-urban projects something they have lacked: a rigorous first-pass tool that tells engineers not just how far a wall will move, but exactly where it will move most, and therefore where to concentrate the bracing that keeps a city’s underground ambitions safe.

Subject of Research: Analytical prediction of lateral displacement of retaining structures in deep urban excavations

Article Title: Analytical formulation for lateral displacement of retaining structures in deep underground space and its application

Article References: Guan, J., Wu, Z., Zhang, B., Gu, J., Chen, F., Yelv, G., Zhou, Y., & Fu, Y. (2026). Analytical formulation for lateral displacement of retaining structures in deep underground space and its application. Results in Engineering, 32, Article 113205. https://doi.org/10.1016/j.rineng.2026.113205

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113205

Keywords: deep excavation, retaining structures, Timoshenko beam, earth pressure, soil-structure interaction, diaphragm wall, power series solution, shear deformation, underground space, geotechnical engineering, Hangzhou case study, deformation prediction