Influence of elastic support on the bending of parallelogram plates

Summary The influence of elastic support on the centre deflections and maximum centre and edge moments in clamped parallelogram shaped plates is examined. A polynomial series is assumed for the deflection function, and by applying Galerkin's process, an approximate solution to the governing differential equation is obtained. Convergence of the results were verified. Results for various skew angles and aspect ratios are presented graphically.Nomenclature 2a, 2b lengths of the sides of the parallelogram - a _mn undetermined parameter in series representing the deflection W - c = cosine - D flexural rigidity of plate =Eh ³/[12(1– ²)] - E Young's modulus - h thickness of plate - k, l, m, n, M, N positive integers - p aspect ratio = b/a - q intensity of uniform load - s = sine - u, v oblique co-ordinates - W lateral deflection - x, y Cartesian co-ordinates - reaction of foundation per unit area for unit deflection - adjustable constant, zero or one - outwardly drawn normal - , dimensionless oblique co-ordinates equal to v/b and u/a respectively - skew angle - Poisson's ratio     

Adding a Structural Context to the Deprotometalation and Trans‐Metal Trapping Chemistry of Phenyl‐Substituted Benzotriazole

Organometallic bases are becoming increasingly complex, because mixing components can lead to bases superior to single‐component bases. To better understand this superiority, it is useful to study metalated intermediate structures prior to quenching. This study is on 1‐phenyl‐1H‐benzotriazole, which was previously deprotonated by an in situ ZnCl₂ ? TMEDA/LiTMP (TMEDA=N,N,N′,N′‐tetramethylethylenediamine; TMP=2,2,6,6‐tetramethylpiperidide) mixture and then iodinated. Herein, reaction with LiTMP exposes the deficiency of the single‐component base as the crystalline product obtained was [{4‐R‐1‐(2‐lithiophenyl)‐1H‐benzotriazole ? 3THF}₂], [R=2‐C₆H₄(Ph)NLi], in which ring opening of benzotriazole and N₂ extrusion had occurred. Supporting lithiation by adding iBu₂Al(TMP) induces trans‐metal trapping, in which C?Li bonds transform into C?Al bonds to stabilise the metalated intermediate. X‐ray diffraction studies revealed homodimeric [(4‐R′‐1‐phenyl‐1H‐benzotriazole)₂], [R′=(iBu)₂Al(μ‐TMP)Li], and its heterodimeric isomer [(4‐R′‐1‐phenyl‐1H‐benzotriazole){2‐R′‐1‐phenyl‐1H‐benzotriazole}], whose structure and slow conformational dynamics were probed by solution NMR spectroscopy.     

A machine for static and dynamic triaxial testing

A machine has been developed for studying the static and dynamic triaxial constitutive behavior of large specimens of geologic and construction materials. Test specimens can also contain a cylindrical tunnel cavity to permit study of tunnel-reinforcement structures and rock-structure interaction. The specimens are 0.3 m in diameter and 0.3 to 0.45 m high; the model tunnels can be up to 50 mm in diameter. Static and dynamic triaxial loads can be applied with maximum pressures of 200 MPa in static tests and 100 MPa in dynamic tests. Dynamic loading can also be superimposed on a static preload as large as 20 MPa. To facilitate study of tunnel reinforcement, the tunnel is maintained at ambient pressure, with access at both ends for instrumentation and photography. Example results show the influence on tunnel deformation of loading rate as well as the presence of joints and their orientation. For a given allowable tunnel closure, substantially greater pressures can be sustained under dynamic loading than under static loading, and substantially greater pressures can be sustained by an intact specimen than by a jointed specimen.     

Aerosol formation in an isothermal stagnation flow

An experiment has been performed in a laminar stagnation point flow in which two non-premixed reactants produced an aerosol of sub-micron particles. The reactants were NH₃ and HCl. The rate of mixing of the reactants was determined by the velocity gradient or strain rate of the flow; the response of the aerosol dynamics to the flow field was measured with a laser light scattering technique. Laser Doppler Spectroscopy was used to measure the particle size. It was found that the particle size was independent of the strain rate of the flow. On the other hand, the particle number density decreased as the strain rate increased. It is argued that the intensity of light scattered from the aerosol is, therefore, a measure of the amount of product of the relatively slow NH₃-HCl reaction.     

Analysis of a method and a system to determine heat transfer and properties of fluids in the absence of temperature measurements and a modified Fourier Equation

The technique to determine by capacitance measurements heat transfer, thermal transport and dielectric properties of fluids introduced recently is now analyzed for a simple system of spherical geometry. The temperature distribution under programmed heat input to a fluid annulus between solid walls is computed by finite difference method for the determination of the capacitance time function of the arrangement. A system of heavy wall structure and heated long enough will produce a capacitance-time curve which is a function of thermal conductivity only. Thermal diffusivity is of influence in thin wall systems. The capacitance change of a heavy wall arrangement is related to the thermal conductivity of the test fluid by a modified Fourier equation. This equation describes the heat flow through the fluid layer but includes the thermal expansion of the solid walls. The change of geometry with T is therefore accounted for. For other multicomposite structures the Fourier equation must be further modified by including the thermal expansion of all materials of the structure and possibly also their compressibilities.

---

Zusammenfassung Die kürzlich eingeführte Methode der Bestimmung von Wärmeübergang, thermischen Transport und dielektrischen Größen mittels Kapazitäts-Zeit-Messung wird analysiert für ein einfaches kugeliges System. Die Temperaturverteilung in der Flüssigkeit im Kugelspalt zwischen zwei festen Körpern wird für konstante Wärmezufuhr von außen mittels der Differenzmethode bestimmt und daraus die Kapazitäts-Zeit-Funktion ermittelt. Es wird gezeigt, daß die Kapazitäts-Zeit-Kurve nur eine Funktion der Wärmeleitzahl ist für den Fall dickwandiger Anordnungen. Für dünnwandige Systeme wird sie auch abhängig von der Temperaturleitzahl. Es wird eine modifizierte Fourier-Gleichung eingeführt, die den Wärmetransport durch die Flüssigkeit beschreibt, dabei aber die Änderung der Geometrie der Schicht berücksichtigt, die sich wegen der thermischen Ausdehnung der festen Wände bei der Einstellung der Temperaturdifferenz ergibt. Für andere mehrschichtige Körper muß die Fourier-Gleichung weiterhin modifiziert werden durch Berücksichtigung der thermischen Ausdehnungskoeffizienten aller beteiligten Materialien und möglicherweise auch ihrer Kompressibilitäten.

---

Nomenclature A average cross-sectional area of fluid layer - A coefficient matrix - B matrix defined by Eq. (20) - B₀ geometric constant of fluid layer (A/L) at reference temperature - C capacitance of arrangement - C_i, C_r capacitance of layer of fluid i and reference fluid at temperature T - capacitances at reference temperature - C_H, c_l specific heats of outer and inner wall - FA...FE constants defined in Eqs. (13 ... 17) - L thickness of fluid layer - M_H, M_L mass of outer and inner wall - P power input to the system - R constant defined by Eq. (24) - T temperature - T_ref reference temperature - T (O, t), T (L, t) temperatures of outer and inner wall at time t - T _i ⁿ , T _i+0 ^n+m temperatures at location i and time n (m=number of t's; 0=number of x's) - T temperature difference across fluid layer - T apparent temperature difference - t_h, T_l temperature increases of outer and inner wall - T_max temperature change of system from one to another thermal equilibrium condition a thermal diffusivity - k, k_i, k_r thermal qonductivity of fluids and of fluid i and reference fluid - q heat flow through fluid layer - ^rh,^rl inner radius of outer wall and outer radius of inner wall - ^rOH,^rOL radii at reference temperature - t time - t time interval - x coordinate - ¯x vector of unknown T_i ⁿ⁺¹ - x length interval Greek symbols linear thermal expansion coefficient - _H, _L linear thermal expansion coefficient of materials of outer and inner wall - dielectric constant - _i, _ref dielectric constant of fluid i and reference fluid - ₀ permittivity of free space - multiplyer of conduction Eq. (7) in finite difference form - time needed to establish quasi-steady state conditions in the system heated by a constant power input In honor of Prof. Dr. E. Schmidt to his 80th Birthday.     

The Effects of a Mathematics Infusion Curriculum on Middle School Student Mathematics Achievement

Increasing mathematical competencies of American students has been a focus for educators, researchers, and policy makers alike. One purported approach to increase student learning is through connecting mathematics and science curricula. Yet there is a lack of research examining the impact of making these connections. The Mathematics Infusion into Science Project, funded by the National Science Foundation, developed a middle school mathematics‐infused science curriculum. Twenty teachers utilized this curriculum with over 1,200 students. The current research evaluated the effects of this curriculum on students' mathematics learning and compared effects to students who did not receive the curriculum. Students who were taught the infusion curriculum showed a significant increase in mathematical content scores when compared with the control students.     

The ferroelectric phase of CdTiO3: A powder neutron diffraction study

The synthesis of bulk samples of polycrystalline CdTiO₃ in both the rhombohedral ilmenite and orthorhombic perovskite forms is described and the structures of these have been refined using powder neutron diffraction data. This involved the preparation of samples enriched in ¹¹⁴Cd. Cooling perovskite-type CdTiO₃ to 4 K induces a ferroelectric phase transition, with the neutron data suggesting the low temperature structure is in Pna2₁. Mode analysis shows the polar mode to be dominant at low temperatures. The ilmenite-structure of CdTiO₃ is compared with that of ZnTiO₃. The refined scattering length of the ¹¹⁴Cd is estimated to be 5.56 fm. Attempts to dope CdTiO₃ with Ca and Sr are described.     

Shedding new light on ZnCl2-mediated addition reactions of Grignard reagents to ketones: structural authentication of key intermediates and diffusion-ordered NMR studies

Building on recent advances in synthesis showing that the addition of inorganic salts to Grignard reagents can greatly enhance their performance in alkylation reactions to ketones, this study explores the reactions of EtMgCl with benzophenone in the presence of stoichiometric or catalytic amounts of ZnCl(2) with the aim of furthering the understanding of the role and constitution of the organometallic species involved in these transformations. Investigations into the metathesis reactions of three molar equivalents of EtMgCl with ZnCl(2) led to the isolation and characterisation (X-ray crystallography and (1)H and (13)C NMR spectroscopy) of novel magnesium "zinc-rich" zincate [{(THF)(6)Mg(2)Cl(3)}(+){Zn(2)Et(5)}(-)] (1), whose complicated constitution in THF solutions was assessed by variable-temperature (1)H DOSY NMR studies. Compound 1 reacted with one equivalent of benzophenone to yield magnesium magnesiate [{(THF)(6)Mg(2)Cl(3)}(+){Mg(2)(OC(Et)Ph(2))(2)Cl(3)(THF)}(-)] (3), whose structure was determined by X-ray crystallography. (1)H NMR monitoring of this reaction showed two equivalents of ZnEt(2) formed as a co-product, which together with the "magnesium only constitution" of 3 provides experimental insights into how zinc can be efficiently recycled in these reactions, and therefore used catalytically. The chemoselectivity of this reaction can be rationalised in terms of the synergic effect of magnesium and zinc and contrasts with the results obtained when benzophenone was allowed to react with EtMgCl in the absence of ZnCl(2), where the reduction of the ketone takes place preferentially. The reduction product [{(THF)(5)Mg(3)Cl(4){OC(H)Ph(CF(3))}(2)] (4) obtained from the reaction of EtMgCl with 2,2,2-trifluoroacetophenone was established by X-ray crystallography and multinuclear ((1)H, (13)C and (19)F) NMR spectroscopy. Compounds 3 and 4 exhibit new structural motifs in magnesium chemistry having MgCl(2) integrated within their constitution, which highlights the new role of this inorganic salt in providing structural support for the newly generated alkoxide ligand.