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The problem before us is to determine the total stresses in an axially loaded column of any degree of restraint which is also subject to transverse bending both from a uniformly distributed load and from concentrated loads.
An approach is presented for carrying out reliability-based design of a metallic, plate-like wing to meet strength and flutter requirements that are given in terms of risk/reliability. The design problem is to determine the thickness distribution such that wing weight is a minimum and the probability of failure is less than a specified value. Failure is assumed to occur if either the flutter speed is less than a specified allowable or the stress caused by a pressure loading is greater than a specified allowable. Four uncertain quantities are considered: wing thickness, calculated flutter speed, allowable stress, and magnitude of a uniform pressure load. The reliability-based design optimization approach described herein starts with a design obtained using conventional deterministic design optimization with margins on the allowables. Reliability is calculated using Monte Carlo simulation with response surfaces that provide values of stresses and flutter speed. During the reliability-based design optimization, the response surfaces and move limits are coordinated to ensure accuracy of the response surfaces. Studies carried out in the paper show the relationship between reliability and weight and indicate that, for the design problem considered, increases in reliability can be obtained with modest increases in weight.
Several aeroelastic stability analyses for a vertical fin similar to that of the National Aero-Space Plane are described. The objectives of the study were to design and obtain an experimental data base for a supersonic wind-tunnel model of the fin in order to examine the effects of thermal loading on the flutter characteristics. This paper describes the preliminary efforts to design the wind-tunnel model, including several of the geometric parameter variations that were analyzed. The dominant flutter mechanism involved a flap vibration mode and a fin bending mode. Variation of the thicknesses of flap and root flexures, used to attach the flap to the fin, and the fin to a support, significantly affected the flutter boundary. Uniform thermal loads, affecting only material properties, had little effect, as did the application of different uniform temperatures to each side of the fin. In contrast, the application of significant chord-wise thermal gradients induced stresses which reduced the flutter dynamic pressure by as much as 37 percent. For less extreme distributed loading, the low-aspect ratio fin was relatively unaffected.
Mode I displacement coefficients along the crack surface are presented for a radially cracked round compact specimen, treated as a plane elastostatic problem, subjected to two types of loading; a uniform tensile stress and a nominal bending stress distribution across the net section. By superposition the resultant displacement coefficient or the corresponding influence coefficient can be obtained for any practical load location. Load line displacements are presented for A/D ratios ranging from 0.40 to 0.95, where A is the crack length measured from the crack mouth to the crack tip and D is the specimen diameter. Through a linear extrapolation procedure crack mouth displacements are also obtained. Experimental evidence shows that the results are valid over the range of A/D ratios analyzed for a practical pin loaded round compact specimen.
Mode I displacement coefficients along the crack surface are presented for a radially cracked round compact specimen, treated as a plane elastostatic problem, subjected to two types of loading; a uniform tensile stress and a nominal bending stress distribution across the net section. By superposition the resultant displacement coefficient or the corresponding influence coefficient can be obtained for any practical load location. Load line displacements are presented for A/D ratios ranging from 0.40 to 0.95, where A is the crack length measured from the crack mouth to the crack tip and D is the specimen diameter. Through a linear extrapolation procedure crack mouth displacements are also obtained. Experimental evidence shows that the results of this study are valid over the range of A/D ratios analyzed for a practical pin loaded round compact specimen.
The elastostatic axisymmetric problem for a long thick-walled cylinder containing a ring-shaped internal or edge crack is considered. Using the standard transform technique the problem is formulated in terms of an integral equation which has a simple Cauchy kernel for the internal crack and a generalized Cauchy kernel for the edge crack as the dominant part. As examples the uniform axial load and the steady-state thermal stress problems have been solved and the related stress intensity factors have been calculated. Among other findings the results show that in the cylinder under uniform axial stress containing an internal crack the stress intensity factor at the inner tip is always greater than that at the outer tip for equal net ligament thicknesses and in the cylinder with an edge crack which is under a state of thermal stress the stress intensity factor is a decreasing function of the crack depth, tending to zero as the crack depth approaches the wall thickness.
The paper considers the elastostatic axisymmetric problem for a long thick-walled cylinder containing a ring-shaped internal or edge crack. Using the standard transform technique the problem is formulated in terms of an integral equation which has a simple Cauchy kernel for the internal crack and a generalized Cauchy kernel for the edge crack as the dominant part. As examples the uniform axial load and the steady-state thermal stress problems have been solved and the related stress-intensity factors have been calculated. Among other findings the results show that in the cylinder under uniform axial stress containing an internal crack the stress-intensity factor at the inner tip is always greater than that at the outer tip for equal net ligament thicknesses and in the cylinder with an edge crack which is under a state of thermal stress the stress-intensity factor is a decreasing function of the crack depth, tending to zero as the crack depth approaches the wall thickness.
Here, this study evaluates the energy efficiency of multi-chiller systems in large office buildings, focusing on their optimization across various climate zones as defined by ASHRAE. Using EnergyPlus for simulations, the research examines five different load distribution algorithms in multi-chiller systems that range from one to ten chillers, aiming to understand their effectiveness in 15 distinct climate zones. The primary objectives of the study include identifying the energy efficiency of multi-chiller systems in each climate zone, determining the appropriate number of chillers for each zone, and evaluating the performance of the load distribution algorithms. Based on the U.S. Department of Energy’s commercial building model, the results suggest that multi-chiller systems can significantly reduce cooling energy consumption in various climates. Among the algorithms evaluated, the Sequential Uniform Part Load Ratio (SUPLR) algorithm demonstrates notable efficiency, especially in the 4A climate zone (Baltimore), where it achieves substantial energy savings. Applying the SUPLR algorithm in a multi-chiller setup with four chillers in this zone leads to an estimated 24.5 % reduction in energy usage, equivalent to 183 MW annually. The research indicates that a range of 3 to 5 chillers is typically optimal for most climate zones. In-depth analysis in the 4A climate zone highlights the importance of minimizing operation hours at low Part Load Ratios (PLR) to ensure that chillers operate at a high Coefficient of Performance (COP). This strategy underscores the potential of well-designed multi-chiller systems to reduce cooling energy demand, particularly in climates with transitional seasons. This study provides an overview of the energy-saving potential of multi-chiller systems, applicable across a variety of climatic scenarios.
The results of an experimental and analytical study of the effects of circular holes on the postbuckling behavior of graphite-epoxy cylindrical panels loaded in axial compression are presented. The STAGSC-1 general shell analysis computer code is used to determine the buckling and postbuckling response of the panels. The loaded, curved ends of the specimens were clamped by fixtures and the unloaded, straight edges were simply supported by knife-edge restraints. The panels are loaded by uniform end shortening to several times the end shortening at buckling. The unstable equilibrium path of the postbuckling response is obtained analytically by using a method based on controlling an equilibrium-path-arc-length parameter instead of the traditional load parameter. The effects of hole diameter, panel radius, and panel thickness on postbuckling response are considered in the study. Experimental results are compared with the analytical results and the failure characteristics of the graphite-epoxy panels are described.
Interrupted flow, impingement cooling, and axial power distribution are employed to enhance the heat-transfer configuration of a solid-core nuclear thermal rocket engine. Impingement cooling is introduced to increase the local heat-transfer coefficients between the reactor material and the coolants. Increased fuel loading is used at the inlet end of the reactor to enhance heat-transfer capability where the temperature differences are the greatest. A thermal-hydraulics computer program for an unfueled NERVA reactor core is employed to analyze the proposed configuration with attention given to uniform fuel loading, number of channels through the impingement wafers, fuel-element length, mass-flow rate, and wafer gap. The impingement wafer concept (IWC) is shown to have heat-transfer characteristics that are better than those of the NERVA-derived reactor at 2500 K. The IWC concept is argued to be an effective heat-transfer configuration for solid-core nuclear thermal rocket engines.
The out-of-plane fracture of a 2-D carbon-carbon composite was observed and characterized to gain an understanding of the factors influencing the stress distribution in such a laminate. Finite element analyses of a two-ply carbon-carbon composite under in-plane, out-of-plane, and thermal loading were performed. Under in-plane loading all components of stress were strong functions of geometry. Additionally, large thermal stresses were predicted. Out-of-plane tensile tests revealed that failure was interlaminar, and that cracks propagated along the fiber-matrix interface. An elasticity solution was utilized to analyze an orthotropic fiber in an isotropic matrix under uniform thermal load. The analysis reveals that the stress distributions in a transversely orthotropic fiber are radically different than those predicted assuming the fiber to be transversely isotropic.
An optimization study was performed to develop a minimum weight spreader bar to allow two helicopters to lift the same payload. With this arrangement, the maximum payload that can be lifted is almost doubled without the expense of designing and building a new helicopter. The concept has had some limited use by civil helicopter operators using small helicopters and has been demonstrated in large scale by two CH-54's which successfully lifted a total load of 20 ton. To this point, rather heavy available beams or tower structures have been used for the spreader bar. Since the weight of the bar not only detracts from payload but also adds to the logistics problem, there are more than the usual incentives to minimize weight. Since the design requirement is for classic beam column with uniform side loads resulting from bar weight and aerodynamic drag, the design problem is particularly amenable to optimization. A study has been performed at Sikorsky to establish the minimum weight for a spreader bar sized to carry a load equal to the capacity of two Army BLACK HAWK helicopters. Toward this end, a computer program was written to analyze the spreader bar deflections and stresses and coupled to the NASA developed CONMIN optimization routines.
No rational column formula has yet been developed which gives results which are sufficiently precise for the design of airplane members, and consequently it is necessary to fall back upon experimental testing. In order to derive the maximum benefit from experiments, however, it is necessary that the experiments be guided by theory. It is the object here to modify existing formulae that may be obtained with a minimum number of tests. A discussion is given that is limited to the case of a simple column supported at both ends and subjected to uniformly distributed loads perpendicular to its axis and to end loads either axially or eccentrically applied. Discussed here are forces in the bracing of latticed columns, the application of theory to practical columns, loads in lattice members, and the strength of individual lattices.
Static and cyclic load tests were conducted to determine the static and fatigue strength of the RIS tile/SIP thermal protection system used on the orbiter of the space shuttle. The material systems investigated include the densified and undensified LI-900 tile system on the .40 cm thick SIP and the densified and undensified LI-2200 tile system on the .23 cm (.090 inch) thick SIP. The tests were conducted at room temperature with a fully reversed uniform cyclic loading at 1 Hertz. Cyclic loading causes a relatively large reduction in the stress level that each of the SIP/tile systems can withstand for a small number of cycles. For example, the average static strength of the .40 cm thick SIP/LI-900 tile system is reduced from 86 kPa to 62 kPa for a thousand cycles. Although the .23 cm thick SIP/LI-2200 tile system has a higher static strength, similar reductions in the fatigue strength are noted. Densifying the faying surface of the RSI tile changes the failure mode from the SIP/tile interface to the parent RSI or the SIP and thus greatly increases the static strength of the system. Fatigue failure for the densified tile system, however, occurs due to complete separation or excessive elongation of the SIP and the fatigue strength is only slightly greater than that for the undensified tile system.
The objective of this two-year project was to develop and deliver to the NASA-Glenn Research Center a two-dimensional higher-order theory, and related computer codes, for the analysis and design of cylindrical functionally graded materials/structural components for use in advanced aircraft engines (e.g., combustor linings, rotor disks, heat shields, brisk blades). To satisfy this objective, two-dimensional version of the higher-order theory, HOTCFGM-2D, and four computer codes based on this theory, for the analysis and design of structural components functionally graded in the radial and circumferential directions were developed in the cylindrical coordinate system r-Theta-z. This version of the higher-order theory is a significant generalization of the one-dimensional theory, HOTCFGM-1D, developed during the FY97 for the analysis and design of cylindrical structural components with radially graded microstructures. The generalized theory is applicable to thin multi-phased composite shells/cylinders subjected to steady-state thermomechanical, transient thermal and inertial loading applied uniformly along the axial direction such that the overall deformation is characterized by a constant average axial strain. The reinforcement phases are uniformly distributed in the axial direction, and arbitrarily distributed in the radial and circumferential direction, thereby allowing functional grading of the internal reinforcement in the r-Theta plane. The four computer codes fgmc3dq.cylindrical.f, fgmp3dq.cylindrical.f, fgmgvips3dq.cylindrical.f, and fgmc3dq.cylindrical.transient.f are research-oriented codes for investigating the effect of functionally graded architectures, as well as the properties of the multi-phase reinforcement, in thin shells subjected to thermomechanical and inertial loading, on the internal temperature, stress and (inelastic) strain fields. The reinforcement distribution in the radial and circumferential directions is specified by the user. The thermal and inelastic properties of the individual phases can vary with temperature. The inelastic phases are presently modeled by the power-law creep model generalized to multi-directional loading (within fgmc3dq.cylindrical.f and fgmc3dq.cylindrical.transient.f for steady-state and transient thermal loading, respectively), and incremental plasticity and GVIPS unified viscoplasticity theories (within the steady-state loading versions fgmp3dq.cylindrical.f and fgmgvips3dq.cylindrical.f).
The effect of load eccentricity and substructure deformation on the ultimate strength and stress displacement properties of the shuttle orbiter thermal protection system (TPS) was determined. The LI-900 Reusable Surface Insulation (RSI) tiles mounted on the .41 cm thick Strain Isolator Pad (SIP) were investigated. Substructure deformations reduce the ultimate strength of the SIP/tile TPS and increase the scatter in the ultimate strength data. Substructure deformations that occur unsymmetric to the tile can cause the tile to rotate when subjected to a uniform applied load. Load eccentricity reduces SIP/tile TPS ultimate strength and causes tile rotation.
A three-dimensional finite element program called VISCOPAC was developed and used to conduct a micromechanics analysis of titanium metal matrix composites. The VISCOPAC program uses a modified Eisenberg-Yen thermo-viscoplastic constitutive model to predict matrix behavior under thermomechanical fatigue loading. The analysis incorporated temperature-dependent elastic properties in the fiber and temperature-dependent viscoplastic properties in the matrix. The material model was described and the necessary material constants were determined experimentally. Fiber-matrix interfacial behavior was analyzed using a discrete fiber-matrix model. The thermal residual stresses due to the fabrication cycle were predicted with a failed interface, The failed interface resulted in lower thermal residual stresses in the matrix and fiber. Stresses due to a uniform transverse load were calculated at two temperatures, room temperature and an elevated temperature of 650 C. At both temperatures, a large stress concentration was calculated when the interface had failed. The results indicate the importance of accuracy accounting for fiber-matrix interface failure and the need for a micromechanics-based analytical technique to understand and predict the behavior of titanium metal matrix composites.
The problem of buckling of clamped shallow spherical shells has recently been considered in several theoretical investigations. Buck-ling loads under uniform external pressure were obtained in these investigations which show a surprisingly good agreement with each other, but show a marked disagreement with available experimental values. In all previous studies it has been assumed that the shell deformations are rotationally symmetric. In this paper, the buckling problem is re-examined by introducing asymmetric modes of deformation. The approach is to superimpose small asymmetric deflections on finite axisymmetric deflections, and to show that the symmetric states of deformation are unstable over certain ranges of load and geometry parameter. Numerical results are obtained by means of a digital computer and are compared with previous theoretical and experimental results.