End Constraints Research Articles

A non-linear model for the dynamics of open cross-section thin-walled beams—Part II: forced motion

The discrete equations developed in Part I are here used to analyze the non-linear dynamics of an inextensional shear indeformable beam with given end constraints. The model takes into account the non-linear effects of warping and of torsional elongation. Non-linear 3D oscillations of a beam with a cross-section having one symmetry axis is examined. Only terms of higher magnitude are retained in the equations, which exhibit quadratic, cubic and combination resonances. A harmonic load acting in the direction of the symmetry axis and in resonance with the corresponding natural frequency, is considered. Steady-state solutions and their stability are studied; in particular the effects of non-linear warping and of torsional elongation on the response are highlighted.

FREE VIBRATIONS OF SHEAR DEFORMABLE CIRCULAR CURVED BEAMS RESTING ON ELASTIC FOUNDATIONS

The governing differential equations for the out-of-plane, free vibration of circular curved beams resting on elastic foundations are derived and solved numerically. The formulation takes into consideration the effects of rotary inertia and transverse shear deformation. The lowest three natural frequencies are calculated for beams with hinged–hinged, hinged-clamped, and clamped–clamped end constraints. The effects of various system parameters as well as rotary inertia and shear deformation on the natural frequencies are investigated.

Effect of Temperature on the Tensile Strength and Failure Modes of Angle Ply Aramid Fibre (KRP) Tubes Under Hoop Loading

A comprehensive study was undertaken to characterise Kevlar reinforced plastic (KRP) angle ply filament wound tubes at different temperatures. Quasi-static burst tests were performed on tubes of 25°, 55° and 75° winding angle. The tubes were burst under internal radial pressure with minimum end constraints. An experimental rig and two conditioning tanks were designed and built to test the specimens at three temperatures; −46°C (low temperature) and +20°C (room temperature) and +70°C (high temperature). For each test the internal pressure and the strains in both circumferential and longitudinal directions were recorded on suitable digital processing equipment.

Finite element calculated uniaxial apparent stiffness is a consistent predictor of uniaxial apparent strength in human vertebral cancellous bone tested with different boundary conditions

Strong correspondence between the uniaxial apparent strength and stiffness of cancellous bone allows the use of stiffness as a predictor of bone strength. Measured values of mechanical properties in cancellous bone can be different between experiments due to different experimental conditions. In the current study, bone volume fraction, experimentally determined and finite element (FE) predicted stiffness were examined as predictors of cancellous bone ultimate strength in two different groups each of which was tested using a different end constraint. It is demonstrated that, although always significant, the relationships of strength with bone volume fraction and experimentally determined stiffness are different between test groups. Apparent stiffness, estimated by FE modeling, predicts the ultimate strength of human cancellous bone consistently for all examined experimental protocols.

Influence of end constraints on the collapse of axially impacted frusta

The energy absorption performance of right circular frusta subjected to dynamic axial load is studied and compared with the results of quasi-static tests. Frusta of different geometric ratios and end constraints were axially crushed using a drop hammer at initial velocities in the range of 2–5 m s. The effect of heat treatment on the collapse behaviour and energy absorption is also investigated. The experimental observations indicate that the effects of end constraints and heat treatment on energy absorption were qualitatively similar to those observed under quasi-static testing. Due to inertia effects, the absolute values of the energy absorbed by similar frusta were higher under dynamic loads than under quasi-static loads. It has been established that constraining the frusta enhances their energy absorption capacity under static and dynamic loading particularly at the top (smaller diameter). The optimum geometric parameters for maximum energy absorption performance are identified when residual stresses and strain hardening characteristics, arising from spinning the frusta, were removed.

Uniaxial crushing of constrained tubes

By changing the end constraints, the behaviour of mild steel and aluminium alloy tubes of relatively low D/ t ratio, subjected to an axial crushing load, is studied. Many combinations of end constraints were produced by radially constraining one or both ends of the loaded tube outwards, inwards or in both directions. Partially constrained mild steel and aluminium tubes are found to collapse into either a pure concertina mode or a pure diamond mode, depending on the ratio D/ t and the material characteristics. Mixed concertina-diamond modes would generally result by radially constraining the tube at both ends. Constraining the tube radially outwards at one end does not significantly affect the load-displacement characteristics and the mode of collapse. However, constraining the tube radially inwards produces a different mode than that observed under outward, inward or combined outward/inward constraint. New modes of collapse have been observed under different combinations of outward and inward constraints at both ends. Fully constraining the tubes in the radial direction results in the removal of the initial overshoot in the load-displacement curve, which is responsible for the overestimation of the mean crushing load and the energy absorption capacity of tubular elements. The mode of collapse and the energy absorbed could be controlled with the proper choice of the end constraints.

Temperature and Rate Effects on GRP Tubes Under Tensile Hoop Loading

A comprehensive study was undertaken to characterise glass fibre reinforced plastic (GRP) tubes at different temperatures and strain rates. The tests were performed on tubes of 25°, 55° and 75° winding angle. The tubes were burst under internal radial pressure with minimum end constraints. Two separate rigs were used, one for the static and the other for the dynamic tests. The tests were carried out at three temperatures; −46°C (low temperature), +20°C (room temperature) and +70°C (high temperature). For each test the internal pressure and the strains in both circumferential and longitudinal directions were recorded on suitable digital processing equipment. For a particular batch of tubes tested at three different temperatures, there is in general a decrease in hoop strength with increasing temperature during quasi-static tests. The use of a non-structural liner during such tests led to an increase in ultimate hoop strain of 55° tubes, especially at high temperature. The corresponding increase in ultimate hoop strain was markedly less in the case of 75° and almost negligible in the case of 25° tubes. Testing the tubes at high strain rates resulted in substantial increases in burst strength and ultimate hoop strain as compared with the quasi-static and low strain rate values. The mode of failure of 75° tube is a catastrophic fibre breakage under all test conditions. The mode of failure of 55° tube is a combination of weeping and fibre failure. The 25° tubes are characterised by matrix failure, which is very severe at high strain rates.

Mode of collapse and energy absorption characteristics of constrained frusta under axial impact loading

The energy absorption performance of right circular frusta subjected to dynamic axial load is studied and compared with the results of quasi-static tests. Frusta of different geometric ratios and end constraints were axially crushed using a drop hammer at initial velocities in the range of 2–5 m/s. The effect of heat treatment on the collapse behaviour and energy absorption is also investigated. The experimental observations indicate that the effects of the end constraints and heat treatment on the energy absorption were qualitatively similar to those observed under quasi-static testing. Due to inertia effects, the absolute values of the energy absorbed by similar frusta were higher under dynamic loads than under quasi-static loads. It has been established that constraining the frusta enhances their energy absorption capacity under static and dynamic loading particularly at the top (smaller diameter). The optimum geometric parameters for maximum energy absorption performance are identified when residual stresses and strain hardening characteristics, arising from spinning the frusta, were removed.

An Experimental Investigation On The AxialCrush Of A Stainless Steel Box Component

Axial crush can provide controlled and reliable energy absorption for components requiring structural crashworthiness. An experimental investigation using thin-walled, commercial 304 stainless steel box components examined the effect of alloy composition and microstructure on quasi-static (room temperature) axial crush response. The experimental program consisted of two phases. The objective of the first phase was to develop experimental methods that would generate reproducible axial crush response for a given material. End constraints (removable, grooved caps) and collapse initiators (shallow groove patterns on specimen sidewalls) were used to ensure a specific crush mode and collapse location. A progressive axial crush study was performed in which nine specimens were compressed to record a deformation sequence during a fold formation cycle. Good agreement was obtained for crush characteristic values. The percent difference (average value basis) was less than 3% for maximum loads, 6% for minimum loads, and 3% for energy absorption. In the second phase, the same experimental methodology was used to investigate the effect of alloy composition and microstructure. A higher concentration of carbon and smaller grains resulted in an 18% increase in energy absorption in a secondary fold cycle. Overall, results showed that if an axial crush component's structural engineering response is controlled, material behavior can be isolated and then, alloy composition and microstructure can be modified to enhance energy absorption performance. Structures under Shock & Impact VI, C.A. Brebbia & N. Jones (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-820-1

Numerical studies of the influence of microstructure on rock failure in uniaxial compression — Part II: constraint, slenderness and size effect

Abstract Numerical simulations of uniaxial compression have been conducted to evaluate the effects of the loading system and specimen geometry on the deformation and failure behavior of brittle and heterogeneous rock. This was done using the Rock Failure Process Analysis program (RFPA 2D ). Numerical model specimens with different Young’s modulus ratios of platen to specimen ( Ep / Es =0, 0.1, 1, 2, and 10), different slenderness in terms of height to width ratios (H/W=0.5, 0.67, 1, 1.5, and 3), and different sizes (H×W=30×20, 100×67, 120×80, 150×100, and 190×127 mm) have been numerically analyzed. The numerical simulations not only qualitatively reproduce the experimentally observed pre- and post-peak failure phenomena of the loaded specimens, but also provide a quantitative evaluation of the influence of the parameters studied on the complete stress–strain curves and the strength characteristics. The results presented here indicate that a more ductile response is simulated and the peak strength increases when the end constraint increases. The crack patterns show that almost-vertical splitting cracks develop in specimens loaded with softer loading platens, and the well-known hour-glass failure mode develops in specimens loaded with stiffer loading platens. The peak stress sustained by a specimen decreases with increasing slenderness of the specimen. The pre-peak portion of the stress–strain curves shows no significant dependence on slenderness; however, the post-peak curves are highly dependent on the ratio of specimen height to width. As far as the effect of size is concerned, numerical results reveal a strength reduction with an increase in specimen size, whereas the change of the specimen size does not obviously change the failure patterns.

Free vibrations of non-circular arches with non-uniform cross-section

The differential equations governing free, in-plane vibrations of linearly elastic non-circular arches with non-uniform cross-section are derived and solved numerically to obtain frequencies and mode shapes. Numerical results are presented for the quadratic, parabolic, catenary and elliptic arches with hinged–hinged, hinged–clamped, and clamped–clamped end constraints. Three general taper types for a rectangular section are considered. Experimental measurements of frequencies and their corresponding mode shapes agree closely with those predicted by theory.

Scaling of honeycomb compressive yield stresses

In this study, the compressive yield stresses of phenolic-dipped Nomex ® rings were scaled and correlated to honeycomb core. A honeycomb-scaling factor and geometric end constraint factor were found to relate the rings and core through the relative yield stresses and their physical dimensions and properties of the honeycomb materials. The compressive properties of the Nomex rings were also investigated using both a model and a commercial phenolic honeycomb dip resin. The Nomex rings manufactured from the model resin were found to have higher compressive properties. These properties were attributed to the higher fracture toughness of the resin, and both resins were found to accurately scale from rings to core with the honeycomb-scaling factor. Ultimately, this research can be used to innovate and efficiently characterize the compressive properties of different honeycomb webs and dip resins without having to make the significant investment of time and resources associated with large scale honeycomb testing.

Feasibility and stability of constrained finite receding horizon control

Issues of feasibility and stability are considered for a finite horizon formulation of receding horizon control for linear systems under mixed linear state and control constraints. We prove that given any compact set of initial conditions that is feasible for the infinite horizon problem, there exists a finite horizon length above which a receding horizon policy will provide both feasibility and stability, even when no end or stability constraint is imposed. Finally, computations for determining a sufficient horizon length are carried out on a simple open-loop stable example under control saturation constraints.

A new approach to stability analysis for constrained finite receding horizon control without end constraints

We present a new approach to the stability analysis of finite receding horizon control applied to constrained linear systems. By relating the final predicted state to the current state through a bound on the terminal cost, it is shown that knowledge of upper and lower bounds for the finite horizon costs is sufficient to determine the stability of a receding horizon controller. This analysis is valid for receding horizon schemes with arbitrary positive-definite terminal weights and does not rely on the use of stabilizing constraints. The result is a computable test for stability, and two simple examples are used to illustrate its application.

Interplay of factors influencing collapse modes in axially crushed tubes

The effects of various schemes of end constraints, heat-treatment, filler density and impact velocity on the behaviour of axially crushed tubes of different L/D and D/t ratios are investigated. It is evident that these factors, individually or collectively, influence the collapse mechanism and hence determine the collapse mode. It follows that the total energy absorption capacity. The concertina mode represents the low energy mode whereas the multi-lobe mode represents the high-Icnergy mode. The former is attributed to instability and/or geometric and loading imperfections. The latter mode, however, occurs as a result of stiffness enhancement.

Lateral Load Capacity and Passive Resistance of Full-Scale Pile Group and Cap

Static and dynamic (statnamic) lateral load tests were performed on a full-scale 3 × 3 pile group driven in saturated low-plasticity silts and clays. The 324-mm outside diameter steel pipe piles were attached to a reinforced concrete pile cap (2.74 m square in plan and 1.21 m high), which created an essentially fixed-head end constraint. A gravel backfill was compacted in place on the back side of the cap. Lateral resistance was therefore provided by pile-soil-pile interaction as well as by base friction and passive pressure on the cap. In this case, passive resistance contributed about 40 percent of the measured static capacity. The measured resistance was compared with that computed by several techniques. The log-spiral method provided the best agreement with measured resistance. Estimates of passive pressure computed using the Rankine or GROUP p-y curve methods significantly underestimated the resistance, whereas the Coulomb method overestimated resistance. The wall movement required to fully mobilize passive resistance in the dense gravel backfill was approximately 0.06 times the wall height, which is in good agreement with design recommendations. The p-multipliers developed for the free-head pile group provided reasonable estimates of the pile-soil-pile resistance for the fixed-head pile group. Default p-multipliers in the program GROUP led to a 35 percent overestimate of pile capacity. Overall dynamic resistance was typically 100 to 125 percent higher than static; however, dynamic passive pressure resistance was over 200 percent higher than static.

Buckling of a Thin Face Layer on Winkler Foundation with Debonds

The buckling behavior of a face layer debonded locally from the core of a sandwich panel is analyzed by considering a Euler beam on Winkler foundation with debonds subjected to in-plane compression. Exact closed form solutions of the buckling load and mode shape are obtained, and corresponding numerical results are given to illustrate the solution. Results indicate that the wrinkling wavelength of the perfectly bonded face layer can be used as an appropriate characteristic length for normalizing the debond length of a relatively long face layer. The effects of length and location of debonds and the end constraints of the face layer on the load carrying capacity are discussed. Interactive effects due to two debonds and the overlapping of debond faces are also studied. A master curve based on a classical solution is developed by employing the new normalization of the debond length. A modification to the usual Winkler foundation constants is made for an isotropic core, and it agrees very well with published results of a finite element analysis.

B2 stress index for elbow analysis

The purpose of this study is to evaluate the effect of end constraint, geometric dimension and internal pressure on the B 2 stress indices, and thus to determine the conservativeness of the code equation for B 2 stress indices. For a series of stainless steel elbow-straight pipe connections of different sizes and schedules, with flanges attached at different locations and with different levels of applied internal pressure, nonlinear finite element analyses were performed considering both material and geometrical nonlinearities. Load-deflection curves were obtained, and from these curves, collapse limit loads were determined. B 2 stress indices were computed using the equation B 2= S y/( M cl/ Z). The B 2 stress indices of elbows with flanges attached at different locations were compared. Also compared were B 2 stress indices of elbows with different levels of internal pressure. Finally the B 2 stress indices from the finite element analysis are compared with the B 2 stress indices computed from code equation. Regression analysis is used to determine revised numerical values for the code equation for B 2. A new, less conservative equation is proposed to be used to compute B 2 stress indices for elbow component.

Free vibrations of tapered piles embedded partially in Winkler type foundations

The numerical methods for calculating the natural frequencies and the mode shapes of the tapered piles embedded partially in the Winkler type foundations are developed. The ordinary differential equation governing the free vibrations of such piles and the boundary conditions are derived, in which three types of linear tapered piles the with rectangular cross-sections i.e. breadth, depth and square tapers are considered. This equation subjected to the boundary conditions is solved numerically. The Runge-Kutta method and the determinant search method combined with Regula-Falsi method are used for integrating the differential equation and determining the eigenvalues(natural frequencies), respectively. In the numerical examples, the two lowest natural frequencies for six kind of end constraints are presented as the functions of four non-dimensional system parameters: the embedded length parameter, the foundation parameter, the section ratio, and the load parameter. The effects of taper types on the natural frequencies are reported. In addition, the typical mode shapes of free vibrations of piles are presented in the figure.

On one-dimensional random fields with fixed end values

For a one-dimensional, uni-variate random field with deterministic fixed end values, expressions are derived for the conditional mean, variance, and covariance functions in terms of given mean, variance, and correlation functions for an unrestricted, variance-homogeneous Gaussian random field. Also, a relation is derived between the conditional random field and the underlying unrestricted random field. This relation is useful for simulation purposes. Further, expressions are derived for the coefficients in a series expansion for the conditional random field. The present results are obtained from known general formulas for conditional Gaussian distributions, conditional estimation, and series expansion. An earlier alternate approach to enforcing end conditions is also examined. An example is given to illustrate the effect of conditioning a random field by zero end constraints. The present results have direct application to the representation of random imperfections in probabilistic stability analysis of columns and arches.