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2,970 records · Page 37

H 2 O and CO 2 Sorption in Ion-Exchange Sorbents: Distinct Interactions in Amine Versus Quaternary Ammonium Materials

This article examines how water (H₂O) and carbon dioxide (CO₂) interact with two classes of ion-exchange sorbents — a primary amine sorbent and a quaternary ammonium (QA⁺) sorbent — using calorimetry, thermal gravimetric analysis, gas analysis, and molecular modeling. Here, the QA⁺ sorbent exhibits stronger binding to both H₂O and CO₂ but also shows thermal stability limitations. Mixed-gas experiments reveal that humidity strongly influences CO₂ uptake and that moisture-driven sorbent regeneration enables cyclic moisture swing CO₂ capture, with implications for low-energy CO₂ separation from dilute gas streams.

36 MATERIALS SCIENCE

Influential Factors for Liquid Acquisition Device Screen Selection for Cryogenic Propulsion Systems

This paper presents the influential factors which govern screen selection for liquid acquisition devices (LADs) operating in microgravity conditions for future in-space cryogenic propulsion engines and cryogenic propellant depots. Space flight requirements, which include mass flow rate, acceleration level and direction, and thermal environment, dictate screen selection for a particular mission. The five influential factors include bubble point pressure, flow-through-screen pressure drop, wicking rate, screen compliance, and material compatibility. Governing equations and analytical models for these parameters are developed from first principles. A comprehensive survey of the historical data on coarser LAD meshes over four decades of work is conducted, and liquid hydrogen data for finer Dutch Twill meshes (325 x 2300, 450 x 2750, 510 x 3600) from recently concluded experiments is also presented to validate analytical models. Each of these parameters is measurable from ground based tests, making it facile to predict flight system performance. Therefore analytical models in this paper will be valuable for future LAD designs for both cryogenic and storable propulsion systems. Additionally, analysis will be given on the impact of the factors on liquid hydrogen systems.

Fuel Depot

Unconventional solitonic high-temperature superfluorescence from perovskites

Fast thermal dephasing limits macroscopic quantum phenomena to cryogenic conditions and hinders their use at ambient temperatures. For electronic excitations in condensed media, dephasing is mediated by thermal lattice motion. Therefore, taming the lattice influence is essential for creating collective electronic quantum states at high temperatures. Although there are occasional reports of high-T c quantum effects across different platforms, it is unclear which lattice characteristics and electron–lattice interactions lead to macroscopically coherent electronic states in solids. Here we studied intensity fluctuations in the macroscopic polarization during the emergence of superfluorescence in a lead halide perovskite and showed that spontaneously synchronized polaronic lattice oscillations accompany collective electronic dipole emission. We further developed an effective field model and theoretically confirmed that exciton–lattice interactions lead to a new electronically and structurally entangled coherent extended solitonic state beyond a critical polaron density. The analysis shows a phase transition with two processes happening in tandem: incoherent disordered polaronic lattice deformations establish an order, while macroscopic quantum coherence among excitons simultaneously emerges. Recombination of excitons in this state culminates in superfluorescence at high temperatures. Our study establishes fundamental connections between the transient superfluorescence process observed after the impulsive excitation of perovskites and general equilibrium phase transitions achieved by thermal cooling. By identifying various electron–lattice interactions in the perovskite structure and their respective role in creating collectively coherent electronic effects in solids, our work provides unprecedented insight into the design and development of new materials that exhibit high-temperature macroscopic quantum phenomena.

36 MATERIALS SCIENCE

Battery Failure Databank

The Battery Failure Databank contains thermal runaway results gathered from nearly 300 small format fractional thermal runaway calorimetry (S-FTRC) experiments. A majority of these experiments were conducted at synchrotron facilities where high-speed x-ray videography was conducted of the cell while tested inside of the S-FTRC. The databank is a two-component system which consists of a Microsoft ExcelTM spreadsheet which provides S-FTRC results in tabular format and a radiographic video library containing the high-speed x-ray videos. Overall, the databank provides thermal results from S-FTRC experiments conducted on a mixture of commercially available lithium-ion (Li-ion) cells and specialized Li-ion test cells with varying cell format (18650, 21700, and D-cell) and trigger mechanism (heaters, heaters plus internal short circuiting device, and nail penetration). The radiography video component provides insight into the initiation and propagation of TR in the cells, in addition to consequences of TR measured by the FTRC. Fractions of mass ejected for the cell types are separated into regimes based on the different failure modes of the cells, such as purely venting, partial ejection, and total ejection. With knowledge of the rate and characterization of the internal degradation of the cells during TR, and the amounts of mass ejected and unrecovered, the extent of TR, and therefore the mitigation of TR, is revealed relative to cell types and failure modes.

Lithium-ion battery

First in operando transient grating spectroscopy measurements on tungsten during high flux plasma operation in PISCES-RF

This work details the construction and first in operando transient grating spectroscopy measurements conducted on PISCES-RF during plasma operation. A preliminary study on a tungsten sample with varied plasma species (Ar and D 2 ), ion flux (4 × 10 21 to 6 × 10 22 ion/m 2 /s), ion energy (5 to 80 eV), and surface temperature (20 to 200 °C) is presented. Prior to plasma exposure, the thermal diffusivity and surface acoustic wave (SAW) speed, (6.9 ± 0.2) × 10 −5 m 2 /s and (2.66 ± 0.02) km/s, were measured in situ with a thermal grating period of 12.5 μm. During Ar plasma exposure, these two quantities vary according to the sample surface temperature. At low flux and varied ion energy, D 2 plasma exposure also results in thermal diffusivity and SAW speed varying with surface temperature alone. At high flux, D 2 plasma exposure results in no convincing change to the thermal diffusivity, but the SAW speed is reduced. After returning to 20 °C, post plasma exposures, the thermal diffusivity and SAW speed almost recover to the initial value prior to exposure.

Surface acoustic wave SAW

A Multi-Tier Autonomous Aerial Architecture for Wildfire Detection, Characterization, and Communication in Infrastructure-Denied Environments

Wildfire response depends on how fast an ignition can be confirmed and located, especially in remote regions where ground-based communication and monitoring may be limited. Geostationary sensors provide frequent observations but at kilometer-scale resolution, which is too coarse to resolve small fires in remote terrain. Ground camera networks require sightlines and infrastructure that back-country areas lack. To address these limitations, this work proposes a Multi-Tier Autonomous Wildfire Intelligence System that combines wide-area monitoring with targeted, high-resolution sensing. A solar-powered high-altitude long endurance (HALE) platform operating at approximately 60,000 ft provides persistent wide-area thermal and optical surveillance, running onboard edge inference to screen candidate ignitions and reduce false positives and downlink bandwidth. When a candidate ignition is detected, low-altitude uncrewed aircraft systems (UAS) can be deployed to conduct localized observations, including high-resolution imaging and atmospheric measurements such as wind and plume observation. By combining persistent detection with local sensing, the proposed architecture is designed to provide first responders with timely, high-resolution information about fire location and behavior to aid in emergency decision making.

Wildfire management, UAS, drones

Ground Slope Effects on Lander Radiator Performance

Two sensitivity studies were performed to examine the drop in radiator heat rejection from slopes. Study #1 used a 5.5m tall reduced order asset to represent a small-lander/rover. Study #2 used a 20m asset represent a tall lander. The assets had body mounted horizontal, 45º inward-tilted, and vertical radiator orientations. The impact of slopes on different radiator orientations and on radiators pointing different directions (i.e., slope verses sun) was investigated. The studies were conducted at a polar location of -82.5ºS to capture the incident solar flux at the northmost, and therefore the hottest, range of potential Artemis landing sites (Mons Mouton Plateau).

Lunar Slope

Trajectory Engineering with Modular Patched Conics for Entry Systems and TPS (TEMPEST)

Brief Presenter Biography (35 word limit): Bohdan Wesely is an Aerospace Engineer in the Entry Systems and Technology Division at Ames. He has worked on a variety of projects for NASA including integrated TPS (thermal protection system) flight hardware deliveries and testing services for commercial partners. Introduction: TEMPEST is a new trajectory analysis framework that is designed to fill the gap between dedicated flight mechanics tools and aerothermal and TPS sizing tools. The project started as an SJSU master’s thesis and has since evolved into a general conceptual design tool capable of studying a wide variety of entry problems. Development is ongoing in the Entry Systems and Technology Division at NASA ARC. Why TEMPEST: Space missions involving entry into a planetary atmosphere involve a series of unique requirements across multiple disciplines. Whether it is traditional entry descent and landing (EDL), or aerocapture, the vehicle must navigate to its target landing location or orbit state, and the TPS must protect the payload during entry. The design process typically involves iterative handoffs between various flight mechanics, flow solver, and material response level tools. During the early conceptual phase, a wide variety of feasible trajectories are simulated in a Monte Carlo scenario which broadly satisfy the mission or landing requirements. Next, computational fluid dynamics (CFD), direct simulation Monte Carlo (DSMC), and other flow solver analyses are performed at various key trajectory points to generate an aero-database, heating and TPS design requirements also emerge at this stage. At this point, with updated aerodynamics from the various flow solvers, trajectories can be re-run, this in turn can change the required freestream conditions for the CFD tools, and as a project progresses, these analyses converge, and uncertainty is reduced. However, there is always a “hand-off” occurring between two inherently coupled phenomena. Analysis Description: One of the goals with TEMPEST is to use a variety of first principles estimation methods coupled with an atmosphere model to predict vehicle aerothermodynamics across the entire flight regime while propagating a 3 or 6 degree of freedom (DoF) trajectory. Aerodynamics methods include modified Newtonian, Maxwell and Cercignani- Lampis-Lord (CLL) for continuum, transitional, and free molecular flow regimes. Aerothermodynamics include boundary layer and reference enthalpy methods, and Mutation++ for non-equilibrium chemistry modeling. TEMPEST is also capable of stitching multiple trajectory segments together to study mission scenarios like multi-pass aerocapture and aero-gravity assists. Most of the program is implemented in MATLAB using modern system objects, it relies on several C++ shared libraries for supporting tools like Gmsh, the Global Reference Atmospheric Model (GRAM), and Mutation++. The various first principles aerothermal estimation methods are discretized across either a structured axisymmetric panel mesh or an unstructured tri-mesh generated from an open-source tool such as Gmsh, this allows solutions on the same mesh to be compared across tools such as CB-Aero. CFD Coupling. A physics-aware, gaussian process CFD anchoring scheme is proposed to adjust the various first principles methods as a CFD database is populated. One goal for this anchoring module is to inform the project where CFD should be run. Full knowledge of the entire trajectory, atmosphere, and aerothermodynamics allows for easier identification of high sensitivity areas and uncertainty quantification. While the first principles effects are well known and proven accurate in existing tools such as CB- Aero and Cart3D, a physics aware CFD anchoring scheme increases tool credibility across a project lifecycle. Material Response Modeling. Correct TPS sizing is critical for optimizing mass for science payloads and ensuring mission success. The process typically involves a thermal analysis along the trajectory with surface heating environments as a boundary condition. Several design constraints are maximum bondline temperature and maximum recession with various margining techniques. The material response tool FIAT, developed out of NASA Ames, is currently being integrated into the TEMPEST environment. TPS recession, shape change, mass loss, and mass property alteration are all factors that can perturb an entry trajectory. For missions like Mars 2020, recession was minimal and was safely handled separately as a post process. For missions such as Jupiter Galileo with a high TPS mass fraction or asteroid entries, recession plays a major role. The proposed fully coupled scheme is to use an epoch-based approach where the trajectory integration is halted after a recession threshold, the energy balance and FIAT are solved at each panel, the mesh, aerodynamics, and mass properties are updated, and the trajectory continues. Several computational tradeoffs have been made during the development of TEMPEST to limit the cost of a single trajectory and preserve its utility as a conceptual, rapid iteration tool. Conclusion: Development of TEMPEST is ongoing and the project is still in its infancy. This talk aims to showcase its unique capabilities to support future NASA entry systems missions.

Bohdan O Wesely

Tree tensor network hierarchical equations of motion based on time-dependent variational principle for efficient open quantum dynamics in structured thermal environments

In this work, we introduce an efficient method, TTN-HEOM, for exactly calculating the open quantum dynamics for driven quantum systems interacting with highly structured bosonic baths by combining the tree tensor network (TTN) decomposition scheme with the bexcitonic generalization of the numerically exact hierarchical equations of motion (HEOM). The method yields a series of quantum master equations for all core tensors in the TTN that efficiently and accurately capture the open quantum dynamics for non-Markovian environments to all orders in the system–bath interaction. These master equations are constructed based on the time-dependent Dirac–Frenkel variational principle, which isolates the optimal dynamics for the core tensors given the TTN ansatz. The dynamics converges to the HEOM when increasing the rank of the core tensors, a limit in which the TTN ansatz becomes exact. We introduce TENSO, tensor equations for non-Markovian structured open systems, as a general-purpose Python code to propagate the TTN-HEOM dynamics. We implement three general propagators for the coupled master equations: two fixed-rank methods that require a constant memory footprint during the dynamics and one adaptive-rank method with a variable memory footprint controlled by the target level of computational error. We exemplify the utility of these methods by simulating a two-level system coupled to a structured bath containing one Drude–Lorentz component and eight Brownian oscillators, which is beyond what can presently be computed using the standard HEOM. Our results show that the TTN-HEOM is capable of simulating both dephasing and relaxation dynamics of driven quantum systems interacting with structured baths, even those of chemical complexity, with an affordable computational cost.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH

High temperature resistant jet fuels

The thermal stability of aviation fuels is evaluated according to the ASTM D 2341 method. This characteristic is linked to the presence of generators of instability, mainly diolephins and N and S derivates. The objectives of the present work are as follows: to obtain, in the laboratory, kerosene which is thermally stable; the isolation and identification of generators of instability; and the regeneration of thermally unstable kerosenes.

Luis M. Peloche

Dronebase Photovoltaic (PV) Fleet Imagery Quantitative Evaluation (CRADA CRD-22-22941 Final Report)

Combine the Dronebase aerial imagery with corresponding sites in the NLR Photovoltaic (PV) Fleets database. By combining these two data sources in an aggregated, anonymized fashion, we can perform the following analyses: quantifying power loss due to outages caused by stuck trackers, string outages, and shading/snow, validate site metadata, including tilt and azimuth, and correlate.

14 SOLAR ENERGY

Decoupling Surface Topography from Gravitational Acceleration in Cryogenic Pool Boiling

Boiling heat transfer is governed by a complex interplay between surface conditions and gravitational acceleration. To isolate the sole effects of gravity, we investigated the pool boiling characteristics of liquid nitrogen on atomically smooth silicon dioxide (SiO 2 ) surfaces under terrestrial (1-g) and reduced gravity (0±0.02 g) conditions achieved via parabolic flight. Our results quantify a drastic reduction in the critical heat flux (CHF) in reduced gravity, decreasing from 16.15 W/cm 2 at 1-g to 5−6 W/cm 2 at μ-g due to the suppression of buoyancy. Conversely, we observed a distinct increase in the heat transfer coefficient (HTC) in the reduced gravity condition prior to CHF. By utilizing a surface with a maximum peak-to-valley height of ≈36.7 nm and low contact angle hysteresis (<10°), we confirm this HTC enhancement is an intrinsic response to the gravitational environment, decoupled from surface-defect-induced nucleation. These findings demonstrate that the influence of surface topography is significantly more prominent in reduced gravity than in terrestrial conditions, providing a critical baseline for rationalizing the design of cryogenic thermal management systems in space and quantum applications.

Liquid nitrogen

The Effect of Gravity on the Combustion of Bulk Metals

In recent years, metal combustion studies at the University of Colorado have focused on the effects of gravity (g) on the ignition and burning behavior of bulk metals. The impetus behind this effort is the understanding of the ignition conditions and flammability properties of structural metals found in oxygen (O,sub>2 ) systems for space applications. Since spacecraft are subjected to higher-than-1g loads during launch and reentry and to a zero-gravity environment while in orbit, the study of ignition and combustion of bulk metals at different gravitational accelerations is of great practical concern. From the scientific standpoint, studies conducted under low gravity conditions provide simplified boundary conditions, since buoyancy is removed, and make possible the identification of fundamental ignition and combustion mechanisms. This investigation is intended to provide experimental verification of the influence of natural convection on the burning behavior of metals. In addition, the study offers the first findings of the influence of gravity on ignition of bulk metals and on the combustion mechanism and structure of metal-oxygen, vapor-phase diffusion flames in a buoyancy-free environment. Titanium (Ti) and magnesium (Mg) metals were chosen because of their importance as elements of structural materials and their simple chemical composition-pure metals instead of multicomponent alloys to simplify chemical and spectroscopic analyses. In addition, these elements present the two different combustion modes observed in metals: heterogeneous or surface burning (for Ti) and homogeneous or gas-phase reaction (for Mg). Finally, Mg, Ti, and their oxides exhibit a wide range of thermophysical and chemical properties. Metal surface temperature profiles, critical and ignition temperatures, propagation rates, burning times, and spectroscopic measurements are obtained under normal and reduced gravity. Visual evidence of all phenomena is provided by high-speed photography.

Melvyn C Branch

Electron Inversion and Tunneling at Silicon Thermal Oxide Interfaces for Solar-Driven Molecular Catalysis to Syngas

Semiconductor photoelectrodes are regularly coupled to solid-state heterogeneous catalysts to perform solar-driven reduction of CO 2 . Less frequently, molecular catalysts are employed to better control the reactivity toward desired products, yet the development of robust semiconductor/molecule interfaces has proven challenging. Here, we demonstrate that a 2–3 nm thermal oxide layer on Si exhibits stability in aqueous solution, high photovoltage, and a photocurrent density of ∼10 mA/cm 2 for the solar-driven photoelectrochemical reduction of a homogeneous molecular catalyst, producing syngas with an ∼2:1 H 2 to CO ratio. Because of a low defect density, the oxide interface forms an electron inversion layer with metal-like electron density at cathodic potentials. This inversion layer facilitates electron transfer to redox-active molecules via tunneling even if the molecule’s reduction potential is beyond the semiconductor’s conduction band edge. Using an electrolyte solution composed of a homogeneous cobalt bis(terpyridine) catalyst in a water/organic solvent mixture, stable photoelectrochemistry was observed under 1-sun illumination, exhibiting an ∼30% Faradaic efficiency for CO that was similar to a glassy carbon electrode under comparable conditions. Furthermore, the results demonstrate that an ultrathin thermal oxide interface is a robust platform for development of aqueous-stable, molecule-driven photoelectrocatalysis.

Catalysts

Laser Beam Welding Advancements for In-Space Servicing, Assembly, and Manufacturing

NASA is currently working to develop in-space servicing, assembly, and manufacturing (ISAM) capabilities for low Earth orbit and the lunar surface. One crucial technology for this effort is laser beam welding. Laser systems can perform joining, cleaning, cutting, and repair activities, which will enable the construction of large in-space structures that could not fit on a single launch vehicle, such as trusses for solar panels, radiators, or communications infrastructure. Multiple projects studying laser welding for space applications are currently underway at NASA Marshall Space Flight Center. One of these, the DIsk-Shaped Configurable and Modular vAcuum uNit (DISCMAN), is a compact vacuum chamber designed to support parameter development for laser welding in microgravity. It contains a rotating platen with weld samples made from aluminum, steel, and titanium, a high-power infrared laser, and integrated pumps for pulling vacuum inside the sample cartridge. The DISCMAN payload is planned to launch to the International Space Station, where welds will be performed under sustained microgravity inside the Bishop Airlock. Another effort underway at Marshall is the Lunar Assembly and Servicing by Autonomous Robotics (LASAR) initiative. This project uses a space-rated robotic arm equipped with a laser weld head, wire feeder, and multiple cameras to perform welds in a thermal vacuum chamber simulating the lunar surface environment. Some of these welds are done on snowflake joints, which are specially designed to slot together to join segments of trussN structures, allowing for the construction of tall surface infrastructure. DISCMAN, LASAR, and other projects are being carried out to advance the technological maturity of in-space laser beam welding, collect data to inform computational models, and learn reliable processes for creating weld joints in space. This work supports NASA’s greater goals to expand humanity’s presence in low Earth orbit, establish a permanent moon base, and eventually send crewed missions to Mars and beyond.

Robotics

Laser Beam Welding Advancements for In-Space Servicing, Assembly, and Manufacturing

NASA is currently working to develop in-space servicing, assembly, and manufacturing (ISAM) capabilities for low Earth orbit and the lunar surface. One crucial technology for this effort is laser beam welding. Laser systems can perform joining, cleaning, cutting, and repair activities, which will enable the construction of large in-space structures that could not fit on a single launch vehicle, such as trusses for solar panels, radiators, or communications infrastructure. Multiple projects studying laser welding for space applications are currently underway at NASA Marshall Space Flight Center. One of these, the DIsk-Shaped Configurable and Modular vAcuum uNit (DISCMAN), is a compact vacuum chamber designed to support parameter development for laser welding in microgravity. It contains a rotating platen with weld samples made from aluminum, steel, and titanium, a high-power infrared laser, and integrated pumps for pulling vacuum inside the sample cartridge. The DISCMAN payload is planned to launch to the International Space Station, where welds will be performed under sustained microgravity inside the Bishop Airlock. Another effort underway at Marshall is the Lunar Assembly and Servicing by Autonomous Robotics (LASAR) initiative. This project uses a space-rated robotic arm equipped with a laser weld head, wire feeder, and multiple cameras to perform welds in a thermal vacuum chamber simulating the lunar surface environment. Some of these welds are done on snowflake joints, which are specially designed to slot together to join segments of trussN structures, allowing for the construction of tall surface infrastructure. DISCMAN, LASAR, and other projects are being carried out to advance the technological maturity of in-space laser beam welding, collect data to inform computational models, and learn reliable processes for creating weld joints in space. This work supports NASA’s greater goals to expand humanity’s presence in low Earth orbit, establish a permanent moon base, and eventually send crewed missions to Mars and beyond.

Manufacturing

Intermodal microwave-to-optical transduction using silicon-on-sapphire optomechanical ring resonator

Optomechanical and electro-optomechanical systems have emerged as one of the most promising approaches for quantum microwave-to-optical transduction to interconnect distributed quantum modalities for scaling the quantum systems. These systems use suspended structures to increase mode overlap and mitigate loss to achieve high efficiency. However, the suspended design’s poor heat dissipation under strong drive limits the ultimate efficiency. Here, we demonstrate an unsuspended optomechanical ring resonator (OMR) based on the silicon-on-sapphire (SOS) platform for microwave-to-optical frequency conversion. The OMR achieves a triply resonant optical-to-optical conversion with an enhanced coupling rateG b = 3.6 gigahertz per square-root milliwatt at a peak conversion efficiency of 1.2% with 3.6-milliwatt microwave drive power and a microwave-to-optical conversion efficiency of 1.5 × 10 −5 at 10-milliwatt optical drive power. Our results show that the unsuspended SOS platform, which mitigates the thermal effect and is compatible with superconducting qubits, is a promising platform for optomechanical circuitry and quantum transduction.

Science & Technology - Other Topics

Mechanics of Preloaded Bolt Tensile Loading With Focus on Load Introduction Factor

The bolt tensile and joint separation loads are directly influenced by the locations at which the external loads enter the clamped members of a preloaded bolted joint (PBJ) and the associated load-paths through the joint. This physical load introduction mechanism affecting the bolt tensile loading is typically represented in the bolt tensile load equation, in part, by a load introduction factor (LIF), which was shown by H.M. Lee of Marshall Spaceflight Center to be a natural product of the bolt tensile load equation using a linear spring stiffness model. This LIF, being a function of load-path stiffness, has subsequently been denoted as the stiffness-based LIF (SBLIF), providing a framework to calculate the LIF using whatever load-path stiffness approximations are appropriate. Expanding upon the work of Lee, it is shown that the SBLIF and the joint stiffness factor are functions of the stiffnesses of the same load-paths and regions within a PBJ, and thus they should not be treated as independent variables. Mathematical expressions for the SBLIF are presented. Comparisons are shown between the analytically calculated SBLIF, the analytically calculated geometric LIF (GLIF), which is a simple clamped-member thickness ratio, the experimentally derived LIF, and the LIF determined by finite element analysis (FEA). Using experiment and FEA as a benchmark, the SBLIF, using traditional load-path stiffness approximations, enables a more accurate prediction of bolt tensile loading than the GLIF, although it can be unconservative near joint separation. The GLIF generally attributes more of the externally applied tensile load to the bolt than does the SBLIF, potentially resulting in heavier and/or more costly bolted joints. Mathematical relationships between the SBLIF and the GLIF are developed. Supplemental material is provided in the appendixes where the historical practice of using the joint compressive stiffness in place of the joint tensile stiffness is evaluated. The appendixes include step-by-step examples demonstrating the calculation of the SBLIF using traditional stiffness approximations and conclude with the development of the joint diagram in terms of the SBLIF, culminating into formulas for the key features of a joint diagram, which is useful for programming.

Load Path