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[Sharon Harris]

Although the moment M ( x ) \displaystyle M(x) and displacement y \displaystyle y generally result from external loads and may vary along the length of the beam or rod, the flexural rigidity (defined as E I \displaystyle EI ) is a property of the beam itself and is generally constant for prismatic members. However, in cases of non-prismatic members, such as the case of the tapered beams or columns or notched stair stringers, the flexural rigidity will vary along the length of the beam as well. The flexural rigidity, moment, and transverse displacement are related by the following equation along the length of the rod, x \displaystyle x :

In the study of geology, lithospheric flexure affects the thin lithospheric plates covering the surface of the Earth when a load or force is applied to them. On a geological timescale, the lithosphere behaves elastically (in first approach) and can therefore bend under loading by mountain chains, volcanoes and other heavy objects. Isostatic depression caused by the weight of ice sheets during the last glacial period is an example of the effects of such loading.

Flexural rigidity of a plate has units of Pam3, i.e. one dimension of length less than the same property for the rod, as it refers to the moment per unit length per unit of curvature, and not the total moment. I is termed as moment of inertia. J is denoted as 2nd moment of inertia/polar moment of inertia.

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The biomechanics of embryonic notochords are studied using an elastic membrane model. An initial study varying internal pressure and stiffness ratio determines tension and geometric ratios as a function of internal pressure, membrane stiffness ratio, and cell packing pattern. A subsequent three-point bending study determines flexural rigidity as a function of internal pressure, configuration, and orientation. Flexural rigidity is found to be independent of membrane stiffness ratio. Controlling for number and volume of cells and their internal pressure, the eccentric staircase pattern of cell packing has more than double the flexural rigidity of the radially symmetric bamboo pattern. Moreover, the eccentric staircase pattern is found to be more than twice as stiff in lateral bending than in dorsoventral bending. This suggests a mechanical advantage to the eccentric WT staircase pattern of the embryonic notochord, over patterns with round cross-section.

The flexural rigidity of single microtubules is measured using optical tweezers. Two new methods are presented. In both the optical forces of the laser trap are used to directly manipulate microtubules grown off the ends of Chlamydomonas axonemes. The shapes of the microtubules are observed by video microscopy as the hydrodynamic forces of viscous flow counteract the elastic restoring forces when the microtubules are moved actively relative to the surrounding buffer medium. To determine the flexural rigidity, the bending of a microtubule is analyzed under a given velocity distribution along its length. Microtubules incubated with taxol after polymerization are measured to be more flexible than those without taxol added. On the other hand, MAPs are shown to increase microtubule stiffness.

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Wedge water entry can be used to study slamming on high-speed craft, seaplane landing, and biological flows such as birds diving into the water. The focus of this paper is on experimental measurements and simulations for the primary application of slamming of small craft, where the wedge boundary conditions are closed on all edges. The simulation is comprised of a one-way coupled nonlinear hydrodynamic code developed for water entry with an Euler-Bernoulli beam finite element solver. In this paper, the dynamic surface pressures on the bottom of the wedge are measured along with structural deflection, water contact line (or spray root), and rigid body motions of the vertically constrained wedge. Five different panel types with large variations in flexural rigidity were tested at impact speeds ranging from 1.2 m/s to 3.0 m/s. Comparisons of one set of experiments are compared with the simulation predictions. For the experimental results, it was found that the nondimensionalized spray root position versus time is self-similar despite the three orders of magnitude difference in the panel flexural rigidity values. When the time for the pressure wave to reach each pressure sensor on the bottom of the panel is normalized and compared with the hydroelasticity factor (the ratio of the structural period to the wetting time for the panel), the nondimensional times of arrival are approximately constant for each nondimensional location along the panel.

Schematic of the experimental setup showing the wedge model above the water surface, the rig above the tank, the positions of the cameras above and below the tank, as well as the pressure sensor locations.

Schematic of the 1D structural finite element model used to represent the behavior of the centerline of the wedge panel. The model is discretized using two-node beam elements with 2 degrees of freedom at each node.

Summary of all experimental conditions for (a) keel submergence, (b) keel velocity, and (c) vertical acceleration results for all conditions from keel impact to chine wetting time. Each curve represents an average of three repeated runs. The color scale goes from dark to light with increasing impact velocity, W0, that is tabulated in Table 2.

(a) Single frame from the bottom view camera. The red line denotes the spray root [49]. (b) The results from the spray root taken from the high-speed videos. The red dashed line indicates the location, at the centerline of the wedge, that the spray root position time histories are reported in Fig. 10.

Spray root position time history from the bottom view camera with the spray root location taken at the midspan of the panel for D22=46.4 Pa m3 (composite panel A) with impact speed indicated in the legend.

Summary of kinematics for all flexible aluminum numerical and experimental results. (a) Keel submergence, (b) keel velocity, and (c) vertical acceleration from keel impact to chine wetting time. The color scale goes from dark to light with increasing impact velocity, W0, which is tabulated in Table 2.

Model predictions of hydrodynamic pressure and structural deformation compared with experimental pressure sensor and S-DIC measurements. (g) Distributions shown correspond to the time of arrival of the peak of the pressure wave at sensor P7. The color gradient becomes lighter as the impact velocity increases.

Summary of nondimensional spray root time histories in the experimental program. Position is normalized by the length of plate the spray root must travel, and the time is normalized by the wetting time. The color gradient becomes lighter as the impact velocity increases.

Summary of nondimensional spray root velocities in the experimental program. Velocity is normalized by the impact velocity, W0, and the time is normalized by the wetting time. The color gradient becomes lighter as the impact velocity increases.

Summary of spray root position time histories for the flexible aluminum panel for the experiment and simulation at all impact velocities. The color gradient becomes lighter as the impact velocity increases.

Nondimensional time of pressure peak arrival (horizontal axis) at each nondimensional pressure sensor location (vertical axis). The position of the pressure sensor is normalized by the plate length, and the time is normalized by the wetting time. The color gradient becomes lighter as the impact velocity increases.

Comparison of the experimental and numerical time of pressure peak arrival (horizontal axis) at each pressure sensor location (vertical axis) for flexible aluminum panel. The color gradient becomes lighter as the impact velocity increases. The inset shows the correlation between time of arrival between the simulation prediction and experimental measurement.

Comparison of the experimental and numerical nondimensional time of pressure peak arrival (horizontal axis) at each nondimensional pressure sensor location (vertical axis) for flexible aluminum panels. The color gradient becomes lighter as the impact velocity increases. The position of the pressure sensor is normalized by the plate length, and the time is normalized by the wetting time. The inset shows the correlation between time of arrival between the simulation prediction and experimental measurement.

Microtubules (MTs) are highly dynamic tubular cytoskeleton filaments that are essential for cellular morphology and intracellular transport. In vivo, the flexural rigidity of MTs can be dynamically regulated depending on their intracellular function. In the in vitro reconstructed MT-motor system, flexural rigidity affects MT gliding behaviors and trajectories. Despite the importance of flexural rigidity for both biological functions and in vitro applications, there is no clear interpretation of the regulation of MT flexural rigidity, and the results of many studies are contradictory. These discrepancies impede our understanding of the regulation of MT flexural rigidity, thereby challenging its precise manipulation.

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