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NASA NTRS2012

Algorithms for Determining Physical Responses of Structures Under Load

Ultra-efficient real-time structural monitoring algorithms have been developed to provide extensive information about the physical response of structures under load. These algorithms are driven by actual strain data to measure accurately local strains at multiple locations on the surface of a structure. Through a single point load calibration test, these structural strains are then used to calculate key physical properties of the structure at each measurement location. Such properties include the structure s flexural rigidity (the product of the structure's modulus of elasticity, and its moment of inertia) and the section modulus (the moment of inertia divided by the structure s half-depth). The resulting structural properties at each location can be used to determine the structure s bending moment, shear, and structural loads in real time while the structure is in service. The amount of structural information can be maximized through the use of highly multiplexed fiber Bragg grating technology using optical time domain reflectometry and optical frequency domain reflectometry, which can provide a local strain measurement every 10 mm on a single hair-sized optical fiber. Since local strain is used as input to the algorithms, this system serves multiple purposes of measuring strains and displacements, as well as determining structural bending moment, shear, and loads for assessing real-time structural health. The first step is to install a series of strain sensors on the structure s surface in such a way as to measure bending strains at desired locations. The next step is to perform a simple ground test calibration. For a beam of length l (see example), discretized into n sections and subjected to a tip load of P that places the beam in bending, the flexural rigidity of the beam can be experimentally determined at each measurement location x. The bending moment at each station can then be determined for any general set of loads applied during operation.

Richards, W. Lance↗

NASA NTRS2012

Transient Side Load Analysis of Out-of-Round Film-Cooled Nozzle Extensions

There was interest in understanding the impact of out-of-round nozzle extension on the nozzle side load during transient startup operations. The out-of-round nozzle extension could be the result of asymmetric internal stresses, deformation induced by previous tests, and asymmetric loads induced by hardware attached to the nozzle. The objective of this study was therefore to computationally investigate the effect of out-of-round nozzle extension on the nozzle side loads during an engine startup transient. The rocket engine studied encompasses a regeneratively cooled chamber and nozzle, along with a film cooled nozzle extension. The computational methodology is based on an unstructured-grid, pressure-based computational fluid dynamics formulation, and transient inlet boundary flow properties derived from an engine system simulation. Six three-dimensional cases were performed with the out-of-roundness achieved by three different degrees of ovalization, elongated on lateral y and z axes: one slightly out-of-round, one more out-of-round, and one significantly out-of-round. The results show that the separation line jump was the primary source of the peak side loads. Comparing to the peak side load of the perfectly round nozzle, the peak side loads increased for the slightly and more ovalized nozzle extensions, and either increased or decreased for the two significantly ovalized nozzle extensions. A theory based on the counteraction of the flow destabilizing effect of an exacerbated asymmetrical flow caused by a lower degree of ovalization, and the flow stabilizing effect of a more symmetrical flow, created also by ovalization, is presented to explain the observations obtained in this effort.

Wang, Ten-See↗

NASA NTRS2014

Assessing the Risk of Crew Injury Due to Dynamic Loads During Spaceflight

Spaceflight requires tremendous amounts of energy to achieve Earth orbit and to attain escape velocity for interplanetary missions. Although the majority of the energy is managed in such a way as to limit the accelerations on the crew, several mission phases may result in crew exposure to dynamic loads. In the automotive industry, risk of serious injury can be tolerated because the probability of a crash is remote each time a person enters a vehicle, resulting in a low total risk of injury. For spaceflight, the level of acceptable injury risk must be lower to achieve a low total risk of injury because the dynamic loads are expected on each flight. To mitigate the risk of injury due to dynamic loads, the NASA Human Research Program has developed a research plan to inform the knowledge gaps and develop relevant tools for assessing injury risk. The risk of injury due to dynamic loads can be further subdivided into extrinsic and intrinsic risk factors. Extrinsic risk factors include the vehicle dynamic profile, seat and restraint design, and spacesuit design. Human tolerance to loads varies considerably depending on the direction, amplitude, and rise-time of acceleration therefore the orientation of the body to the dynamic vector is critical to determining crew risk of injury. Although a particular vehicle dynamic profile may be safe for a particular design, the seat, restraint, and suit designs can affect the risk of injury due to localized effects. In addition, characteristics intrinsic to the crewmember may also contribute to the risk of injury, such as crewmember sex, age, anthropometry, and deconditioning due to spaceflight, and each astronaut may have a different risk profile because of these factors. The purpose of the research plan is to address any knowledge gaps in the risk factors to mitigate injury risk. Methods for assessing injury risk have been well documented in other analogous industries and include human volunteer testing, human exposure to dynamic environments, post-mortem human subject (PMHS) testing, animal testing, anthropomorphic test devices (ATD), dynamic models of the human, numerical models of ATDs, and numerical models of the human. Each has inherent strengths and limitations. For example, human volunteer testing is advantageous because a population can be selected that is similar to the astronaut corps; however, because of the inherent ethical limitations, only sub-injurious conditions can be tested. PMHSs can be tested in a variety of conditions including injurious levels, but the responses are not completely analogous to living human subjects. In addition, it is exceedingly difficult to select a PMHS population that is similar to the astronaut corps. ATDs are currently widely used in the automotive industry and military because they are highly repeatable and durable. Unfortunately, because they are mechanical models of the human body, the biofidelity of the responses are limited to dynamic conditions used to validate the ATD. Numerical models of the ATD, in addition to the strengths and limitations for ATDs, are easy to use for a variety of designs before a design is fabricated, but also have additional limitations for ATDs, are easy to use for a variety of designs before a design is fabricated, but also have additional uncertainty. Dynamic models are simple and easy to use, but do not account for localized effects of the seat and suit. Finally, numerical models of the human have the potential to have the most advantages; however, the current models are not validated for the conditions expected during spaceflight. To properly assess spaceflight conditions with numerical human models, human data would be needed to optimize the model responses for those conditions. Using the appropriate assessment method with the knowledge gained for each risk factor, an appropriate approach for mitigating the risk of injury due to dynamic loads can be developed ensuring crew safety in future NASA vehicles.

Somers, J. T.↗

NASA NTRS2014

Influence of Mixed Mode I-Mode II Loading on Fatigue Delamination Growth Characteristics of a Graphite Epoxy Tape Laminate

Mixed mode I-mode II interlaminar tests were conducted on IM7/8552 tape laminates using the mixed-mode bending test. Three mixed mode ratios, G(sub II)/G(sub T) = 0.2, 0.5, and 0.8, were considered. Tests were performed at all three mixed-mode ratios under quasi-static and cyclic loading conditions, where the former static tests were used to determine initial loading levels for the latter fatigue tests. Fatigue tests at each mixed-mode ratio were performed at four loading levels, Gmax, equal to 0.5G(sub c), 0.4G(sub c), 0.3G(sub c), and 0.2G(sub c), where G(sub c) is the interlaminar fracture toughness of the corresponding mixed-mode ratio at which a test was performed. All fatigue tests were performed using constant-amplitude load control and delamination growth was automatically documented using compliance solutions obtained from the corresponding quasi-static tests. Static fracture toughness data yielded a mixed-mode delamination criterion that exhibited monotonic increase in Gc with mixed-mode ratio, G(sub II)/G(sub T). Fatigue delamination onset parameters varied monotonically with G(sub II)/G(sub T), which was expected based on the fracture toughness data. Analysis of non-normalized data yielded a monotonic change in Paris law exponent with mode ratio. This was not the case when normalized data were analyzed. Fatigue data normalized by the static R-curve were most affected in specimens tested at G(sub II)/G(sub T)=0.2 (this process has little influence on the other data). In this case, the normalized data yielded a higher delamination growth rate compared to the raw data for a given loading level. Overall, fiber bridging appeared to be the dominant mechanism, affecting delamination growth rates in specimens tested at different load levels and differing mixed-mode ratios.

Ratcliffe, James G.↗

NASA NTRS2016

Moving Aerospace Structural Design Practice to a Load and Resistance Factor Approach

Aerospace structures are traditionally designed using the factor of safety (FOS) approach. The limit load on the structure is determined and the structure is then designed for FOS times the limit load - the ultimate load. Probabilistic approaches utilize distributions for loads and strengths. Failures are predicted to occur in the region of intersection of the two distributions. The load and resistance factor design (LRFD) approach judiciously combines these two approaches by intensive calibration studies on loads and strength to result in structures that are efficient and reliable. This paper discusses these three approaches.

Larsen, Curtis E.↗

NASA NTRS2018

Detection of Damage During Quasi-Static Loading of a Single Stringer Panel Using Passive Thermography and Acoustic Emission

The purpose of this paper is to describe the methodology of measuring damage onset and growth in a composite structure during quasi-static loading using passive thermography and acoustic emission. The early detection and measurement of damage progression is important to understand failure modes. A single stringer panel was subjected to quasi-static loading to induce deformation which resulted in the formation of damage between the stiffener flange and skin. The loading was stopped when damage growth was detected. Passive thermography and acoustic emission were used to detect damage in real-time as a function of the applied load. Of particular interest are the small transient thermographic signals resulting from damage formation which can be challenging to detect, as compared to the persistent passive thermography indications of cyclic fatigue loading. We describe a custom developed thermal inspection system for detection of composite damage during quasi-static loading. The thermal results are compared to a two-dimensional multi-layered thermal simulation based on the quadrupole method. Acoustic emission is used to further characterize the damage by comparing the acoustic emission events with the thermal imagery. Results are compared to ultrasonic measurements to document the damage through-the-thickness.

Zalameda, Joseph N.↗

NASA NTRS2018

Adaptive Load Control of Flexible Aircraft Wings Using Fiber Optic Sensing

Over the past century aircraft wing design has transformed from the morphing wing used on the Wright Flyer to rigid wings with little to no shape-tailoring abilities. Modern day wings are designed to fly at a single trim condition and optimized to have a maximum aerodynamic efficiency at only this condition. Shape morphing wings on the other hand have the potential to undergo geometric changes allowing them to adapt to their mission profiles. Several flight demonstrations have been conducted over the decades using morphing-wing technologies. Active wing-twist control was demonstrated on the X-53 Active Aeroelastic Wing (AAW) research project by utilizing multiple leading- and trailing-edge control surfaces. Passive morphing technology has been demonstrated on the Rockwell RPRV-870 Highly Maneuverable Aircraft Technology (HiMAT) aircraft. In the current study, the wings of a small unmanned aerial system (sUAS) were modified to have segmented control surfaces (SCS). The modifications include segmenting the original wing control surfaces (one flap and one aileron per wing) into 44 individual sections, each section having its own independent servo control motor. The wings were also instrumented with a network of over 1800 fiber-optic strain sensors (on four sensing fibers distributed over the top and bottom surfaces of the wing) monitoring the strain response of the wing to aerodynamic loading. The SCS positions were manipulated in real time to modify the spanwise lift distribution of the wings on the sUAS. The change in the structural response of the wings caused by load redistribution was quantified by measuring the bending strains on the upper and lower wing surfaces using an on-board compact fiber-optic strain sensing (cFOSS) system. A feedback controller was developed to control the SCS positions using strain-based shape estimations from the Displacement Transfer Function (DTF). Post-processing of the strain data allowed for the transverse displacement distributions and load distributions to be compared for the conventional and segmented control surface cases using displacements and loads algorithms developed by Richards and Ko at AFRC (refs. 5-10). While the current study focused on the shifting of the spanwise aerodynamic loads as quantified by displacements, future applications for loads or displacement control might include active gust alleviation and flutter suppression.

Pena, Francisco↗

NASA NTRSDate not supplied

Challenges Associated with In-Situ Calibration of Load Cells in Force Limited Vibration Testing

The difference in mounting configuration between flight and test can significantly impact the effectiveness of the test in environmental vibration testing. Many tests are performed with large electrodynamic shakers, which utilize interfaces that seek to replicate a fixed base, such as slip tables and head expanders. This fixed base configuration is rarely seen in flight configurations; rather a more realistic configuration would include a flexible mounting structure with its own compliance and dynamics. This causes significant over and under tests in various frequency bands depending on the differences between the test article and fixture dynamics. The traditional way of avoiding these high loads is to limit the acceleration responses at multiple locations on the test article. However, the effectiveness of this approach is highly dependent upon the validity of the test article’s analytical in order to derive accurate acceleration response limit specifications. Also this technique requires limiting the acceleration responses at many locations throughout the test article, which may not be implementable due to such things as access and cleanliness issues An improved environmental vibration testing technique known as force limiting incorporates measurements of the forces between the test article and shaker system interface and limiting them to a specification that more accurately replicates the interface impedance of the structure the test article will be mounted to in flight. In effect this transforms the high mechanical impedance at the test article to shaker interface to more closely match the mechanical impedance of the flight interface, which avoids producing the unrealistically high interface loads. Typically force gauges or load cells are used to measure these interface forces. However, utilizing load cells can present a multitude of challenges depending upon such things as their installation method, geometric layout, and test fixture setup. Regardless, it is extremely important to perform an in-situ calibration of the load cells prior to vibration testing at any significant levels. This paper will discuss the challenges associated with utilizing load cells during the NASA Evolutionary Xenon Thruster – Commercial (NEXT-C) gridded ion thruster proto-flight vibration test performed at the NASA Glenn Research Center’s Structural Dynamics Laboratory.

Kenneth J Pederson↗

NASA NTRSDate not supplied

Challenges Associated with In-Situ Calibration of Load Cells in Force Limited Vibration Testing

K. J. Pederson↗

NASA NTRSDate not supplied

Load Schedule Design Improvements for the Calibration of NASA's Semi-span Balances

Improvements of the load schedule design for the calibration of NASA's semi-span balances are discussed. These improvements became possible after a minor hardware change was made to Calspan's Large Load Rig. This change moves normal and axial force load points 12 inches closer to the balance moment center. Resulting data quality improvements are significant because Calspan's Large Load Rig cannot currently apply forces at the balance moment center of a semi-span balance. Now, the supported set of rolling and yawing moment arms is 28, 38, and 48 inches. Consequently, the new load schedule design better supports expected normal and axial force locations on semi-span models that are tested in the NASA Ames 11-ft Transonic Wind Tunnel. In addition, estimates of the primary sensitivities of the normal force, axial force, rolling moment, and yawing moment have become more accurate as the applied normal and axial forces have moved closer to the balance moment center. Fundamental moment arm range requirements and Calspan's hardware updates are discussed. In addition, calibration and check load analysis results of NASA's ARC30K semi-span balance are presented to illustrate benefits of the hardware changes.

wind tunnel test↗

NASA NTRSDate not supplied

Statistically Consistent Dispersions of Line Loads to Uncertain Integrated Forces and Moments

This work presents a method to generate line loads that are consistent with integrated force and moment constraints, while also following the statistical distribution underlying known line load data. The motivation for this method is to address a particular problem that can arise during the pre-flight analysis of an ascent vehicle. That problem is constructing separate databases, for line loads and integrated aerodynamic coefficients, that contain different data and thus different amounts of quantified uncertainty. A dispersed line load method is described and applied to a data set generated by 29 computational fluid dynamics simulations at various angles of attack and sideslip. The statistical similarity of the generated line loads to the known data is confirmed by using the Maximum Mean Discrepancy 2-sample test. The promising results from this novel method are compared against results from applying an existing line load adjustment procedure to the same data set. Finally, potential future avenues of research are briefly discussed.

Aaron C. Burkhead↗

NASA NTRS2024

Loads and Structural Dynamics Requirements for Spaceflight Hardware

The NASA Exploration Systems Development Mission Directorate requires Crewed Space Systems (CSS) to meet the intent of a set of Engineering Technical Authority (TA) documents called out in HEOMD-003, Crewed Deep Space Systems Human Rating Certification Requirements and Standards for NASA Missions. For the Loads and Dynamics technical discipline, the document invoked by the HEOMD-003 is JSC 65829, Loads and Structural Dynamics Requirements for Spaceflight Hardware. JSC 65829 was originally developed for the NASA Commercial Crew Program as an implementation of NASA STD-5002, Load Analyses of Spacecraft and Payloads, for that Program. Since that time, tailored alternatives to JSC 65829 have been produced for the Gateway, Human Landing System, and Extravehicular Activity and Human Surface Mobility Programs. Experience with those Programs has shown that the reduced set of less-prescriptive requirements in those tailored documents offers an advantage over the set of requirements in JSC 65829 Rev A and is a better fit for the paradigm of NASA procurement of commercially developed systems for crewed spaceflight. Revision B of JSC 65829 has been constructed to align with those tailored documents. The reduction in the number and specificity of requirements is balanced by a new requirement for hardware developers to create and provide a Loads Control Plan which describes how the approaches used to generate design-to loads and dynamic environments and substantiate dynamic model validity satisfy the requirements herein. The Plan will establish an agreement between the hardware developer and the TA for the loads and dynamics discipline and offer an opportunity for reengagement if the Plan changes during development.

Kenneth Schultz↗

DOE OSTI2020

Codisposal Waste Package Loading Options for DOE SNF and HLW PPT

The U.S. Department of Energy (DOE) is responsible for managing spent nuclear fuel (SNF) that is currently in, or will in the future come into, its possession. DOE must continue to safely store that SNF, transport it to an interim storage site or a repository, and dispose of it. These fuels come from a wide range of reactor types with various cladding materials and enrichments. Many of these reactors, now decommissioned, had unique design features, such as core configuration, fuel element and assembly geometry, moderator and coolant materials, operational characteristics, and neutron spatial and spectral properties, resulting in a large diversity of reactor and fuel designs. Because of the wide variety and conditions of SNF, a robust DOE Standard Canister was proposed that would confine radionuclides and preclude moderator. The DOE Standard Canister had four variations: 3.05-meter (10-foot) or 4.57-meter (15-foot) length, and 45.7-cm (18-inch) or 61.0-cm (24-inch) diameter. For ultimate disposal in the Yucca Mountain Repository, these canisters were to be grouped with 61.0-cm (24-inch) diameter high level waste (HLW) canisters in a 2.13-meter (84-inch) diameter co-disposal waste package. The smaller 45.7-cm (18-inch) diameter DOE Standard Canister could be placed in the middle of five HLW canisters. The larger 61.0-cm (24-inch) diameter DOE Standard Canister would take the place of one of the five HLW canisters on the outer ring in the co-disposal waste package. No DOE Standard Canisters have been loaded. A preliminary evaluation has estimated the number of elements of a fuel type that can fit into the different sizes of the DOE Standard Canister, but no definitive loading configuration has been selected. Changing the loading configuration could impact the number of loadable DOE Standard Canisters and the number of co-disposal waste packages needed for eventual disposition. This paper conveys the ranges of DOE Standard Canisters and HLW canisters that may be produced under certain conditions. It also examines the differences in the estimated number of co-disposal waste packages produced for eventual disposal when using different loading strategies in the DOE Standard Canister for Advanced Test Reactor (ATR), Peach Bottom, and High Flux Isotope Reactor (HFIR) SNF. Changing the loading configurations of ATR, Peach Bottom, and HFIR SNF slightly impacted the number of co-disposal waste packages that may be needed for ultimate disposal. The change in loading configuration was more impactful when a different canister was used, as opposed to varying the number of elements that could fit inside the same size canister. In one case, a reduction of co-disposal waste packages could be achieved by allowing mixing of short HLW canisters with long DOE Standard Canisters. The main conclusion from this analysis is that the ratio between HLW canisters and DOE Standard Canisters will drive the total number of co-disposal waste packages. If too many HLW canisters (i.e., more than five times the number of 18-inch DOE standard canisters) or DOE Standard Canisters are produced, some co-disposal waste packages may not have all positions filled. A co-disposal waste package may be filled with all HLW with no DOE Standard Canister, or a co-disposal waste package could be filled with a single DOE Standard Canister. A ratio that does not closely align to optimum could allow for the design of waste packages that hold just HLW canisters or just DOE SNF canisters.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2022

A Hybrid Reinforcement Learning-MPC Approach for Distribution System Critical Load Restoration: Preprint

This paper proposes a hybrid control approach for distribution system critical load restoration, combining deep reinforcement learning (RL) and model predictive control (MPC) aiming at maximizing total restored load following an extreme event. RL determines a policy for quantifying operating reserve requirements, thereby hedging against uncertainty, while MPC models grid operations incorporating RL policy actions, i.e., the reserve requirement, renewable (wind and solar) power predictions, and load demand forecasts. We formulate the reserve requirement determination problem as a sequential decision making problem based on the Markov Decision Process (MDP) and design an RL learning environment based on the OpenAI Gym framework and MPC. The RL agent reward and MPC objective function aim to maximize and monotonically increase total restored load and minimize load shedding and renewable power curtailment. The RL algorithm is trained off-line using historical forecast of renewable generation and load demand. The method is tested using a modified IEEE 13-bus distribution test feeder containing wind turbine, photovoltaic, microturbine and battery. Case studies demonstrated that the proposed method outperforms other operating reserve determination methods.

distribution system↗

DOE OSTI2022

A Hybrid Reinforcement Learning-MPC Approach for Distribution System Critical Load Restoration

This paper proposes a hybrid control approach for distribution system critical load restoration, combining deep reinforcement learning (RL) and model predictive control (MPC) aiming at maximizing total restored load following an extreme event. RL determines a policy for quantifying operating reserve requirements, thereby hedging against uncertainty, while MPC models grid operations incorporating RL policy actions (i.e., reserve requirements), renewable (wind and solar) power predictions, and load demand forecasts. We formulate the reserve requirement determination problem as a sequential decision-making problem based on the Markov Decision Process (MDP) and design an RL learning environment based on the OpenAI Gym framework and MPC simulation. The RL agent reward and MPC objective function aim to maximize and monotonically increase total restored load and minimize load shedding and renewable power curtailment. The RL algorithm is trained offline using a historical forecast of renewable generation and load demand. The method is tested using a modified IEEE 13-bus distribution test feeder containing wind turbine, photovoltaic, microturbine, and battery. Case studies demonstrated that the proposed method outperforms other policies with static operating reserves.

distribution system↗

DOE OSTI2022

Model predictive control for matrix converter operating in current control mode with load current estimation

A matrix converter system operating in current control mode is provided. The matrix converter includes a switching matrix coupled between a low voltage side and a high voltage side, wherein the matrix converter is coupled at its low voltage side to a generator for receiving an input power including an input current and transforming the input power into an output power at its high voltage side, wherein a load is coupled to the high voltage side. The control system includes a load observer and a model predictive controller (MPC). The load observer is configured to estimate a load current that flows to the load from the high voltage side of the matrix converter as a function of a switching state of the switching matrix, an output voltage output at the high voltage side of the matrix converter, and the input current. The MPC is configured to select the switching state of the switching matrix to meet one or more control objectives defined by minimization of a multi-objective function that tracks the output voltage and the input current using the estimated load current.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2022

A multi-level load shape clustering and disaggregation approach to characterize patterns of energy consumption behavior

This study presents representative electrical load shapes, disaggregated to the end-use level, for over 5000 customer clusters across California’s residential, commercial, industrial and agricultural sectors. We developed a novel, multi-level load shape clustering approach for residential and commercial sectors leveraging interval meter data for over 350,000 California utility customers collected as a part of the Phase 4 California Demand Response (DR) Potential Study. The clustering approach allowed us to identify typical consumption patterns and categorize customers based on their daily load shape displayed throughout the year. For example, we were able to identify customers with particular energy technologies such as electric vehicles and rooftop solar, as well as building occupancy types such as restaurants, grocery stores and even unoccupied buildings, based solely on whole-building interval data. We then combined the load shape-based clusters with other customer information including building type, climate, geographical area, total consumption and low-income status, to create a set of customer clusters based on both demographics and usage patterns. Total cluster electricity demand was then disaggregated into a wide variety of end-uses using weather normalization and other publicly available end-use load shape datasets. The resulting disaggregated cluster load shapes will be released in anonymized form as part of the Phase 4 DR Potential Study. They will have wide-ranging applications in energy research and policy analysis, including estimation of energy efficiency (EE) and DR potential on the end-use level, time-dependent valuation of EE savings, building stock modeling, and developing customer targeting strategies for EE and DR programs.

Murthy, Samanvitha↗

DOE OSTI2023

Pulsed source for driving non-linear current dependent loads

A pulsed current source comprises a power source, a discharge capacitor, and an inductive element. The discharge capacitor is selectively coupled to either of the power source or the inductive element. When coupled to the power source, the discharge capacitor is charged. The inductive element can be connected to a load. The load can have a current-dependent impedance. When the discharge capacitor is coupled to the inductive element, the discharge capacitor discharges through the inductive element and the load. The discharge capacitor and the inductive element are configured so that the current through the load exhibits a substantially linear rise in a linear operational region. The inductive element is configured to saturate during discharge of the capacitor through the load, so that the saturation of the inductive element causes the current through the load to continue to rise in a substantially linear fashion.

Curtis, III, William C.↗