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Concept: Beam


Classic beam theory is frequently used in biomechanics to model the stress behaviour of vertebrate long bones, particularly when creating intraspecific scaling models. Although methodologically straightforward, classic beam theory requires complex irregular bones to be approximated as slender beams, and the errors associated with simplifying complex organic structures to such an extent are unknown. Alternative approaches, such as finite element analysis (FEA), while much more time-consuming to perform, require no such assumptions. This study compares the results obtained using classic beam theory with those from FEA to quantify the beam theory errors and to provide recommendations about when a full FEA is essential for reasonable biomechanical predictions. High-resolution computed tomographic scans of eight vertebrate long bones were used to calculate diaphyseal stress owing to various loading regimes. Under compression, FEA values of minimum principal stress (σ(min)) were on average 142 per cent (±28% s.e.) larger than those predicted by beam theory, with deviation between the two models correlated to shaft curvature (two-tailed p = 0.03, r(2) = 0.56). Under bending, FEA values of maximum principal stress (σ(max)) and beam theory values differed on average by 12 per cent (±4% s.e.), with deviation between the models significantly correlated to cross-sectional asymmetry at midshaft (two-tailed p = 0.02, r(2) = 0.62). In torsion, assuming maximum stress values occurred at the location of minimum cortical thickness brought beam theory and FEA values closest in line, and in this case FEA values of τ(torsion) were on average 14 per cent (±5% s.e.) higher than beam theory. Therefore, FEA is the preferred modelling solution when estimates of absolute diaphyseal stress are required, although values calculated by beam theory for bending may be acceptable in some situations.

Concepts: Interval finite element, Finite element method, Model, Biomechanics, Solid mechanics, Beam, Torsion, Structural analysis


The mathematical modelling of column buckling or beam bending under an axial or transverse load is well established. However, the existent models generally assume a high degree of symmetry in the structure of the column and minor longitudinal and transverse displacements. The situation when the column is made of several components with different mechanical properties asymmetrically distributed in the transverse section, semi-rigid, and subjected to multiple axial loads with significant longitudinal and transverse displacements through compression and bending has not been well characterised. A more comprehensive theoretical model allowing for these possibilities and assuming a circular arc contour for the bend is developed, and used to establish the bending axes, balance between compression and bending, and equivalent stiffness of the column. In certain situations, such as with pull cable catheters commonly used for minimally invasive surgical procedures, the compression loads are applied via cables running through channels inside a semi-rigid column. The model predicts the mathematical relationships between the radius of curvature of the bend and the tension in and normal force exerted by such cables. Conjugate extension with reciprocal compression-bending is a special structural arrangement for a semi-rigid column such that extension of one segment is linked to compression-bending of another by inextensible cables running between them. Leads are cords containing insulated electrical conductor coil and cables between the heart muscle and cardiac implantable electronic devices. Leads can behave like pull cable catheters through differential component pulling, providing a possible mechanism for inside-out abrasion and conductor cable externalisation. Certain design features may predispose to this mode of structural failure.

Concepts: Force, Structural engineering, Beam, Column, Bending, Buckling, Skin effect, Truss


Microcantilever biosensors in the static operation mode translate molecular recognition into a surface stress signal. Surface stress is derived from the nanomechanical cantilever bending by applying Stoney’s equation, derived more than 100 years ago. This equation ignores the clamping effect on the cantilever deformation, which induces significant errors in the quantification of the biosensing response. This leads to discrepancies in the surface stress induced by biomolecular interactions in measurements with cantilevers with different sizes and geometries. So far, more accurate solutions have been precluded by the formidable complexity of the theoretical problem that involves solving the two-dimensional biharmonic equation. In this paper, we present an accurate and simple analytical expression to quantify the response of microcantilever biosensors. The equation exhibits an excellent agreement with finite element simulations and DNA immobilization experiments on gold-coated microcantilevers.

Concepts: Linear elasticity, Cantilever, Beam, Cantilever bridge, Fallingwater


Flexoelectricity allows a dielectric material to polarize in response to a mechanical bending moment and, conversely, to bend in response to an electric field. Compared with piezoelectricity, flexoelectricity is a weak effect of little practical significance in bulk materials. However, the roles can be reversed at the nanoscale. Here, we demonstrate that flexoelectricity is a viable route to lead-free microelectromechanical and nanoelectromechanical systems. Specifically, we have fabricated a silicon-compatible thin-film cantilever actuator with a single flexoelectrically active layer of strontium titanate with a figure of merit (curvature divided by electric field) of 3.33 MV(-1), comparable to that of state-of-the-art piezoelectric bimorph cantilevers.

Concepts: Nanotechnology, Materials science, Atomic force microscopy, Microelectromechanical systems, Piezoelectricity, Beam, Second moment of area, Nanoelectromechanical systems


Engineered cartilage is a promising option for auricular reconstruction. We have previously demonstrated that a titanium wire framework within a composite collagen ear-shaped scaffold helped to maintain the gross dimensions of the engineered ear after implantation, resisting the deformation forces encountered during neocartilage maturation and wound healing. The ear geometry was redesigned to achieve a more accurate aesthetic result when implanted subcutaneously in a nude rat model. A non-invasive method was developed to assess size and shape changes of the engineered ear in three dimensions. Computer models of the titanium framework were obtained from CT scans before and after implantation. Several parameters were measured including the overall length, width and depth, the minimum intrahelical distance and overall curvature values for each beam section within the framework. Local curvature values were measured to gain understanding of the bending forces experienced by the framework structure in situ. Length and width changed by less than 2%, whereas the depth decreased by approximately 8% and the minimum intrahelical distance changed by approximately 12%. Overall curvature changes identified regions most susceptible to deformation. Eighty-nine per cent of local curvature measurements experienced a bending moment less than 50 µN-m owing to deformation forces during implantation. These quantitative shape analysis results have identified opportunities to improve shape fidelity of engineered ear constructs.

Concepts: Dimension, Euclidean space, Depth perception, Units of measurement, Height, Length, Distance, Beam


We explore and exploit diffraction effects that have been previously neglected when modelling optical measurement techniques for the bending of micro-mechanical transducers such as cantilevers for atomic force microscopy. The illumination of a cantilever edge causes an asymmetric diffraction pattern at the photo-detector affecting the calibration of the measured signal in the popular optical beam deflection technique (OBDT). The conditions that avoid such detection artefacts conflict with the use of smaller cantilevers. Embracing diffraction patterns as data yields a potent detection technique that decouples tilt and curvature and simultaneously relaxes the requirements on the illumination alignment and detector position through a measurable which is invariant to translation and rotation. We show analytical results, numerical simulations and physiologically relevant experimental data demonstrating the utility of the diffraction patterns. We offer experimental design guidelines and quantify possible sources of systematic error in OBDT. We demonstrate a new nanometre resolution detection method that can replace OBDT, where diffraction effects from finite sized or patterned cantilevers are exploited. Such effects are readily generalized to cantilever arrays, and allow transmission detection of mechanical curvature, enabling instrumentation with simpler geometry. We highlight the comparative advantages over OBDT by detecting molecular activity of antibiotic Vancomycin.

Concepts: Diffraction, Optics, Measurement, Wavelength, Pattern, Atomic force microscopy, Cantilever, Beam


In this paper, we present a technique for increasing the strength of thermoplastic fused deposition manufactured printed parts while retaining the benefits of the process such as ease, speed of implementation, and complex part geometries. By carefully placing voids in the printed parts and filling them with high-strength resins, we can improve the overall part strength and stiffness by up to 45% and 25%, respectively. We discuss the process parameters necessary to use this strengthening technique and the theoretically possible strength improvements to bending beam members. We then show three-point bend testing data comparing solid printed ABS samples with those strengthened through the fill compositing process, as well as examples of 3D printed parts used in real-world applications.

Concepts: Elasticity, Solid mechanics, Beam, Second moment of area, Bending, Statics, Buckling, Truss


Enhancing the short-term force precision of atomic force microscopy (AFM) while maintaining excellent long-term force stability would result in improved performance across multiple AFM modalities, including single molecule force spectroscopy (SMFS). SMFS is a powerful method to probe the nanometer-scale dynamics and energetics of biomolecules (DNA, RNA, and proteins). The folding and unfolding rates of such macromolecules are sensitive to sub-pN changes in force. Recently, we demonstrated sub-pN stability over a broad bandwidth (Δf = 0.01-16 Hz) by removing the gold coating from a 100 μm long cantilever. However, this stability came at the cost of increased short-term force noise, decreased temporal response, and poor sensitivity. Here, we avoided these compromises while retaining excellent force stability by modifying a short (L = 40 μm) cantilever with a focused ion beam. Our process led to a ~10-fold reduction in both a cantilever’s stiffness and its hydrodynamic drag near a surface. We also preserved the benefits of a highly reflective cantilever while mitigating gold-coating induced long-term drift. As a result, we extended AFM’s sub-pN bandwidth by a factor of ~50 to span five decades of bandwidth (Δf ≈ 0.01-1,000 Hz). Measurements of mechanically stretching individual proteins showed improved force precision coupled with state-of-the-art force stability and no significant loss in temporal resolution compared to the stiffer, unmodified cantilever. Finally, these cantilevers were robust and were reused for SFMS over multiple days. Hence, we expect these responsive, yet stable, cantilevers to broadly benefit diverse AFM-based studies.

Concepts: Molecule, Atom, Biochemistry, Atomic force microscopy, Drag, Cantilever, Beam, Force spectroscopy


With the aim of providing reliable benchmark values, we have measured the Soret, thermodiffusion and molecular diffusion coefficients for the ternary mixture formed by 1,2,3,4-tetrahydronaphthalene, isobutylbenzene and n-dodecane for a mass fraction of 0.8-0.1-0.1 and at a temperature of 25°C. The experimental techniques used by the six participating laboratories are Optical Digital Interferometry, Taylor Dispersion technique, Open Ended Capillary, Optical Beam Deflection, Thermogravitational technique and Sliding Symmetric Tubes technique in ground conditions and Selectable Optical Diagnostic Instrument (SODI) in microgravity conditions. The measurements obtained in the SODI installation have been analyzed independently by four laboratories. Benchmark values are proposed for the thermodiffusion and Soret coefficients and for the eigenvalues of the diffusion matrix in ground conditions, and for Soret coefficients in microgravity conditions.

Concepts: Fundamental physics concepts, Optical fiber, Measurement, Chemistry, Molecular diffusion, Diffusion, Beam, Deflection


In this study, we investigate the performance of the Gunma University Heavy Ion Medical Center’s ion computed tomography (CT) system, which measures the residual range of a carbon-ion beam using a fluoroscopy screen, a charge-coupled-device camera (CCD), and a moving wedge absorber and collects CT reconstruction images from each projection angle. Each 2D image was obtained by changing the polymethyl methacrylate (PMMA) thickness, such that all images for one projection could be expressed as the depth distribution in PMMA. The residual range as a function of PMMA depth was related to the range in water through a calibration factor, which was determined by comparing the PMMA-equivalent thickness (PET) measured by the ion CT system to the water-equivalent thickness (WET) measured by a water column. Aluminium, graphite, PMMA, and five biological phantoms were placed in a sample holder, and the residual range for each was quantified simultaneously. A novel method of CT reconstruction to correct for the angular deflection of incident carbon ions in the heterogeneous region utilising the Bragg peak reduction (BPR) is also introduced in this paper, and its performance is compared with other methods present in the literature such as the decomposition and differential methods. Stopping power ratio (SPR) values derived with the BPR method from carbon-ion CT images matched closely with the true WEL values obtained from the validation slab experiment.

Concepts: Particle physics, Hydrogen, Medical imaging, Carbon, Ions, Beam, Bragg peak, Stopping power