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Probabilistic Risk Assessment Procedures Guide for NASA Managers and Practitioners (Second Edition)

Probabilistic Risk Assessment (PRA) is a comprehensive, structured, and logical analysis method aimed at identifying and assessing risks in complex technological systems for the purpose of cost-effectively improving their safety and performance. NASA's objective is to better understand and effectively manage risk, and thus more effectively ensure mission and programmatic success, and to achieve and maintain high safety standards at NASA. NASA intends to use risk assessment in its programs and projects to support optimal management decision making for the improvement of safety and program performance. In addition to using quantitative/probabilistic risk assessment to improve safety and enhance the safety decision process, NASA has incorporated quantitative risk assessment into its system safety assessment process, which until now has relied primarily on a qualitative representation of risk. Also, NASA has recently adopted the Risk-Informed Decision Making (RIDM) process [1-1] as a valuable addition to supplement existing deterministic and experience-based engineering methods and tools. Over the years, NASA has been a leader in most of the technologies it has employed in its programs. One would think that PRA should be no exception. In fact, it would be natural for NASA to be a leader in PRA because, as a technology pioneer, NASA uses risk assessment and management implicitly or explicitly on a daily basis. NASA has probabilistic safety requirements (thresholds and goals) for crew transportation system missions to the International Space Station (ISS) [1-2]. NASA intends to have probabilistic requirements for any new human spaceflight transportation system acquisition. Methods to perform risk and reliability assessment in the early 1960s originated in U.S. aerospace and missile programs. Fault tree analysis (FTA) is an example. It would have been a reasonable extrapolation to expect that NASA would also become the world leader in the application of PRA. That was, however, not to happen. Early in the Apollo program, estimates of the probability for a successful roundtrip human mission to the moon yielded disappointingly low (and suspect) values and NASA became discouraged from further performing quantitative risk analyses until some two decades later when the methods were more refined, rigorous, and repeatable. Instead, NASA decided to rely primarily on the Hazard Analysis (HA) and Failure Modes and Effects Analysis (FMEA) methods for system safety assessment.

Stamatelatos,Michael↗

Advanced Concept Studies for Supersonic Commercial Transports Entering Service in the 2018-2020 Period Phase 2

Lockheed Martin Aeronautics Company (LM), working in conjunction with General Electric Global Research (GE GR) and Stanford University, executed a 19 month program responsive to the NASA sponsored "N+2 Supersonic Validation: Advanced Concept Studies for Supersonic Commercial Transports Entering Service in the 2018-2020 Period" contract. The key technical objective of this effort was to validate integrated airframe and propulsion technologies and design methodologies necessary to realize a supersonic vehicle capable of meeting the N+2 environmental and performance goals. The N+2 program is aligned with NASA's Supersonic Project and is focused on providing system level solutions capable of overcoming the efficiency, environmental, and performance barriers to practical supersonic flight. The N+2 environmental and performance goals are outlined in the technical paper, AIAA-2014-2138 (Ref. 1) along with the validated N+2 Phase 2 results. Our Phase 2 efforts built upon our Phase 1 studies (Ref. 2) and successfully demonstrated the ability to design and test realistic configurations capable of shaped sonic booms over the width of the sonic boom carpet. Developing a shaped boom configuration capable of meeting the N+2 shaped boom targets is a key goal for the N+2 program. During the LM Phase 1 effort, LM successfully designed and tested a shaped boom trijet configuration (1021) capable of achieving 85 PLdB under track (forward and aft shock) and up to 28 deg off-track at Mach 1.6. In Phase 2 we developed a refined configuration (1044-2) that extended the under 85 PLdB sonic boom level over the entire carpet of 52 deg off-track at a cruise Mach number of 1.7. Further, the loudness level of the configuration throughout operational conditions calculates to an average of 79 PLdB. These calculations rely on propagation employing Burger's (sBOOM) rounding methodology, and there are indications that the configuration average loudness would actually be 75 PLdB. We also added significant fidelity to the design of the configuration in this phase by performing a low speed wind tunnel test at our LTWT facility in Palmdale, by more complete modelling of propulsion effects in our sonic boom analysis, and by refining our configuration packaging and performance assessments. Working with General Electric, LM performed an assessment of the impact of inlet and nozzle effects on the sonic boom signature of the LM N+2 configurations. Our results indicate that inlet/exhaust streamtube boundary conditions are adequate for conceptual design studies, but realistic propulsion modeling at similar stream-tube conditions does have a small but measurable impact on the sonic boom signature. Previous supersonic transport studies have identified aeroelastic effects as one of the major challenges associated with the long, slender vehicles particularly common with shaped boom aircraft (Ref. 3). Under the Phase 2 effort, we have developed a detailed structural analysis model to evaluate the impact of flexibility and structural considerations on the feasibility of future quiet supersonic transports. We looked in particular at dynamic structural modes and flutter as a failure that must be avoided. We found that for our N+2 design in particular, adequate flutter margin existed. Our flutter margin is large enough to cover uncertainties like large increases in engine weight and the margin is relatively easy to increase with additional stiffening mass. The lack of major aeroelastic problems probably derives somewhat from an early design bias. While shaped boom aircraft require long length, they are not required to be thin. We intentionally developed our structural depths to avoid major flexibility problems. So at the end of Phase 2, we have validated that aeroelastic problems are not necessarily endemic to shaped boom designs. Experimental validation of sonic boom design and analysis techniques was the primary objective of the N+2 Supersonic Validations contract; and in this Phase, LM participated in four high speed wind tunnel tests. The first so-called Parametric Test in the Ames 9x7 tunnel did an exhaustive look at variation effects of the parameters: humidity, total pressure, sample time, spatial averaging distance and number of measurement locations, and more. From the results we learned to obtain data faster and more accurately, and made test condition tolerances easy to meet (eliminating earlier 60 percent wasted time when condition tolerances could not be held). The next two tests used different tunnels. The Ames 11 ft tunnel was used to test lower Mach numbers of 1.2 and 1.4. There were several difficulties using this tunnel for the first time for sonic boom including having to shift the measurement Mach numbers to 1.15 and 1.3 to avoid flow problems. It is believed that the 11 ft could be used successfully to measure sonic boom but there are likely to be a number of test condition restrictions. The Glenn 8x6 ft tunnel was used next and the tunnel has a number of desirable features for sonic boom measurement. While the Ames 9x7 can only test Mach 1.55 to 2.55 and the 11 ft can only test Mach 1.3 and lower, the Glenn 8x6 can test continuously from Mach 0.3 to 2.0. Unfortunately test measurement accuracy was compromised by a reference pressure drift. Post-test analysis revealed that the drift occurred when Mach number drifted slightly. Test measurements indicated that if Mach number drift is eliminated, results from the 8x6 would be more accurate, especially at longer distances, than results from the 9x7. The fourth test in the 9x7, called LM4, used everything we learned to comprehensively and accurately measure our new 1044-02 configuration with a full-carpet shaped signature design. Productivity was 8 times greater than our Phase 1 LM3 test. Measurement accuracy and repeatability was excellent out to 42 in. However, measurements at greater distances require the rail in the aft position and become substantially less accurate. Further signature processing or measurement improvements are needed for beyond near-field signature validation.

Morgenstern, John↗

Dynamic Positioning System (DPS) Risk Analysis Using Probabilistic Risk Assessment (PRA)

The National Aeronautics and Space Administration (NASA) Safety & Mission Assurance (S&MA) directorate at the Johnson Space Center (JSC) has applied its knowledge and experience with Probabilistic Risk Assessment (PRA) to projects in industries ranging from spacecraft to nuclear power plants. PRA is a comprehensive and structured process for analyzing risk in complex engineered systems and/or processes. The PRA process enables the user to identify potential risk contributors such as, hardware and software failure, human error, and external events. Recent developments in the oil and gas industry have presented opportunities for NASA to lend their PRA expertise to both ongoing and developmental projects within the industry. This paper provides an overview of the PRA process and demonstrates how this process was applied in estimating the probability that a Mobile Offshore Drilling Unit (MODU) operating in the Gulf of Mexico and equipped with a generically configured Dynamic Positioning System (DPS) loses location and needs to initiate an emergency disconnect. The PRA described in this paper is intended to be generic such that the vessel meets the general requirements of an International Maritime Organization (IMO) Maritime Safety Committee (MSC)/Circ. 645 Class 3 dynamically positioned vessel. The results of this analysis are not intended to be applied to any specific drilling vessel, although provisions were made to allow the analysis to be configured to a specific vessel if required.

Thigpen, Eric B.↗

Dynamic Positioning System (DPS) Risk Analysis Using Probabilistic Risk Assessment (PRA)

The National Aeronautics and Space Administration (NASA) Safety & Mission Assurance (S&MA) directorate at the Johnson Space Center (JSC) has applied its knowledge and experience with Probabilistic Risk Assessment (PRA) to projects in industries ranging from spacecraft to nuclear power plants. PRA is a comprehensive and structured process for analyzing risk in complex engineered systems and/or processes. The PRA process enables the user to identify potential risk contributors such as, hardware and software failure, human error, and external events. Recent developments in the oil and gas industry have presented opportunities for NASA to lend their PRA expertise to both ongoing and developmental projects within the industry. This paper provides an overview of the PRA process and demonstrates how this process was applied in estimating the probability that a Mobile Offshore Drilling Unit (MODU) operating in the Gulf of Mexico and equipped with a generically configured Dynamic Positioning System (DPS) loses location and needs to initiate an emergency disconnect. The PRA described in this paper is intended to be generic such that the vessel meets the general requirements of an International Maritime Organization (IMO) Maritime Safety Committee (MSC)/Circ. 645 Class 3 dynamically positioned vessel. The results of this analysis are not intended to be applied to any specific drilling vessel, although provisions were made to allow the analysis to be configured to a specific vessel if required.

Thigpen, Eric B.↗

Component level modeling of materials degradation for insights into operational flexibility of Existing Coal Power Plants

Increasingly, coal-fired power plants are required to balance power grids by compensating for the variable electricity supply from renewable energy sources. Fossil-fueled power plants, originally designed to be base loaded, will increasingly need to operate on a load following or cyclic basis. This demanding requirement for operational flexibility needs insights into accelerated material degradation arising due to the harsh operating conditions (e.g., fatigue, early oxide exfoliation due to stresses) along with current damage mechanisms (fireside corrosion, creep and erosion) observed in service. Our research objective is to develop component level modeling toolkit for materials-based degradation for two key mechanisms that can accelerate with cyclic operations. In more detail, this includes the fireside corrosion/steam oxidation/erosion/creep/fatigue of superheaters/reheaters and steam pipework and also the water droplet erosion/ fatigue of last stage steam turbine blades degradation mechanisms, that demand routine and sometimes unplanned maintenance and repair. The innovation is in developing a computational fluid dynamics/finite element (CFD/FE) modeling toolkit for the component level models of the boilers and low-pressure steam turbines in coal power plants that can tackle multidisciplinary failure mechanisms occurring concurrently for extreme environment materials. Lifetime assessment in such environments also needs to account for the unit-specific analyses, operational history and fuel feedstock; this can only be obtained by destructive analysis of components. This, in turn, enables validation of the model toolkits utilizing service feedback data, improving the probability of time/temperature dependent life prediction.

20 FOSSIL-FUELED POWER PLANTS↗

Optimization of laminated stacking sequence for buckling load maximization by genetic algorithm

The use of a genetic algorithm to optimize the stacking sequence of a composite laminate for buckling load maximization is studied. Various genetic parameters including the population size, the probability of mutation, and the probability of crossover are optimized by numerical experiments. A new genetic operator - permutation - is proposed and shown to be effective in reducing the cost of the genetic search. Results are obtained for a graphite-epoxy plate, first when only the buckling load is considered, and then when constraints on ply contiguity and strain failure are added. The influence on the genetic search of the penalty parameter enforcing the contiguity constraint is studied. The advantage of the genetic algorithm in producing several near-optimal designs is discussed.

Le Riche, Rodolphe↗

Common Cause Case Study: An Estimated Probability of Four Solid Rocket Booster Hold-down Post Stud Hang-ups

Until Solid Rocket Motor ignition, the Space Shuttle is mated to the Mobil Launch Platform in part via eight (8) Solid Rocket Booster (SRB) hold-down bolts. The bolts are fractured using redundant pyrotechnics, and are designed to drop through a hold-down post on the Mobile Launch Platform before the Space Shuttle begins movement. The Space Shuttle program has experienced numerous failures where a bolt has "hung-up." That is, it did not clear the hold-down post before liftoff and was caught by the SRBs. This places an additional structural load on the vehicle that was not included in the original certification requirements. The Space Shuttle is currently being certified to withstand the loads induced by up to three (3) of eight (8) SRB hold-down post studs experiencing a "hang-up." The results af loads analyses performed for four (4) stud-hang ups indicate that the internal vehicle loads exceed current structural certification limits at several locations. To determine the risk to the vehicle from four (4) stud hang-ups, the likelihood of the scenario occurring must first be evaluated. Prior to the analysis discussed in this paper, the likelihood of occurrence had been estimated assuming that the stud hang-ups were completely independent events. That is, it was assumed that no common causes or factors existed between the individual stud hang-up events. A review of the data associated with the hang-up events, showed that a common factor (timing skew) was present. This paper summarizes a revised likelihood evaluation performed for the four (4) stud hang-ups case considering that there are common factors associated with the stud hang-ups. The results show that explicitly (i.e. not using standard common cause methodologies such as beta factor or Multiple Greek Letter modeling) taking into account the common factor of timing skew results in an increase in the estimated likelihood of four (4) stud hang-ups of an order of magnitude over the independent failure case.

Cross, Robert↗

Common Cause Case Study: An Estimated Probability of Four Solid Rocket Booster Hold-Down Post Stud Hang-ups

Until Solid Rocket Motor ignition, the Space Shuttle is mated to the Mobil Launch Platform in part via eight (8) Solid Rocket Booster (SRB) hold-down bolts. The bolts are fractured using redundant pyrotechnics, and are designed to drop through a hold-down post on the Mobile Launch Platform before the Space Shuttle begins movement. The Space Shuttle program has experienced numerous failures where a bolt has hung up. That is, it did not clear the hold-down post before liftoff and was caught by the SRBs. This places an additional structural load on the vehicle that was not included in the original certification requirements. The Space Shuttle is currently being certified to withstand the loads induced by up to three (3) of eight (8) SRB hold-down experiencing a "hang-up". The results of loads analyses performed for (4) stud hang-ups indicate that the internal vehicle loads exceed current structural certification limits at several locations. To determine the risk to the vehicle from four (4) stud hang-ups, the likelihood of the scenario occurring must first be evaluated. Prior to the analysis discussed in this paper, the likelihood of occurrence had been estimated assuming that the stud hang-ups were completely independent events. That is, it was assumed that no common causes or factors existed between the individual stud hang-up events. A review of the data associated with the hang-up events, showed that a common factor (timing skew) was present. This paper summarizes a revised likelihood evaluation performed for the four (4) stud hang-ups case considering that there are common factors associated with the stud hang-ups. The results show that explicitly (i.e. not using standard common cause methodologies such as beta factor or Multiple Greek Letter modeling) taking into account the common factor of timing skew results in an increase in the estimated likelihood of four (4) stud hang-ups of an order of magnitude over the independent failure case.

Cross, Robert↗

An evaluation of the use of new Doppler methods for detecting longitudinal function abnormalities in a pacing-induced heart failure model

BACKGROUND: Doppler tissue echocardiography and color M-mode Doppler flow propagation velocity have proven useful in evaluating cross-sections of patients with left ventricular (LV) dysfunction, but experience with serial changes is limited. Purpose and methods: We tested their use by evaluating the temporal changes of LV function in a pacing-induced congestive heart failure model. Rapid ventricular pacing was initiated and maintained in 20 dogs for 4 weeks. Echocardiography was performed at baseline and weekly during brief pacing cessation. RESULTS: With rapid pacing, LV volume significantly increased and ejection fraction (57%-28%), stroke volume (37-18 mL), and mitral annulus systolic velocity (16.1-6.6 cm/s) by Doppler tissue echocardiography significantly decreased, with ejection fraction and mitral annulus systolic velocity closely correlated (r = 0.706, P <.0001). In contrast to the mitral inflow velocities, mitral annulus early diastolic velocity decreased steadily (12.3-7.3 cm/s) resulting in a dramatic decrease in mitral annulus early/late (1.22-0.57) diastolic velocity with no tendency toward pseudonormalization. The color M-mode Doppler flow propagation velocity also showed significant steady decrease (57-24 cm/s) throughout the pacing period. Multiple regression analysis chose mitral annulus systolic velocity (r = 0.895, P <.0001) and propagation velocity (r = 0.782, P <.0001) for the most important factor predicting LV systolic and diastolic function, respectively. CONCLUSIONS: Doppler tissue echocardiography and color M-mode Doppler flow could evaluate the serial deterioration in LV dysfunction throughout the pacing period. These were more useful in quantifying progressive LV dysfunction than conventional ehocardiographic techniques, and were probably relatively independent of preload. These techniques could be suitable for longitudinal evaluation in addition to the cross-sectional study.

Evaluation Studies↗

Materials and Processes Engineering's Role in the Reconstruction and Failure Analysis of The Space Shuttle Columbia

Conclusions: The hot gases, having flooded the wing interior, quickly heated the upper and lower wing surfaces allowing the aluminum honeycomb facesheets and the wing tiles to debond. The thin-wall aluminum truss tubes would soon collapse and the aerodynamic and structural integrity of the left wing would be effectively destroyed. The forensic evidence is consistent with the observed External Tank foam impact 81 seconds into launch. This is the most probable cause of the damage to the Reinforced Carbon-Carbon. (RCC) leading edge.

Russell, Richard↗

Estimating Orion Heat Shield Failure Due To Ablator Cracking During The EFT-1 Mission

The Orion EFT-1 heatshield suffered from two major certification challenges: First, the mechanical properties used in design were not evident in the flight hardware and second, the flight article itself cracked during fabrication. The combination of these events motivated the Orion Program to pursue an engineering-level Probabilistic Risk Assessment (PRA) as part of heatshield certification rationale. The PRA provided loss of Mission (LOM) likelihoods considering the probability of a crack occurring during the mission and the likelihood of subsequent structure over-temperature. The methods and input data for the PRA are presented along with a discussion of the test data used to anchor the results. The Orion program accepted an EFT-1 Loss of Vehicle (LOV) risk of 1-in-160,000 due to in-mission Avcoat cracking based on the results of this analysis. Conservatisms in the result, along with future considerations for Exploration Missions (EM) are also addressed.

Avcoat↗

Flexible tactile sensing arrays for robotics: Architectural robustness and yield considerations

No tactile sensor array developed to date fills all, nor even most, of the requirements of: (1) many sensor elements to minimize tactile imaging time, (2) close spacing of small sensing elements for fine spatial discrimination, (3) ability to be molded to a convex fingertip surface for taction on concave surfaces, (4) ability to sense both force and temperature in order to fully mimic biological touch, (5) tolerance to overscale forces without damage, and (6) graceful failure in use, i.e., functional survival of most of the array when part of it is damaged, well. Several investigators have recently developed tactile sensing arrays suitable for imaging shapes on hard flat surfaces, but not well suited for taction on curved surfaces. Others have developed single tactile sensors suitable for incorporation into arrays, which will probably prove uneconomical when large numbers of sensors are needed. Still others have developed flexible capacitive arrays suitable for use on curved surfaces but incapable of obtaining temperature or proximity data. A technology is being developed herein suitable to the achievement of all of the above goals, and to incrementally demonstrate useful steps toward those goals.

Barth, Phillip W.↗

Changes in Climate and Its Effect on Timing of Snowmelt and Intensity-Duration-Frequency Curves

Snow is a critical water resource for much of the U.S. and failure to account for changes in climate could deleteriously impact military assets. In this study, we produced historical and future snow trends through modeling at three military sites (in Washington, Colorado, and North Dakota) and the Western U.S. For selected rivers, we performed seasonal trend analysis of discharge extremes. We calculated flood frequency curves and estimated the probability of occurrence of future annual maximum daily rainfall depths. Additionally, we generated intensity-duration-frequency curves (IDF) to find rainfall intensities at several return levels. Generally, our results showed a decreasing trend in historical and future snow duration, rain-on-snow events, and snowmelt runoff. This decreasing trend in snowpack could reduce water resources. A statistically significant increase in maximum streamflow for most rivers at the Washington and North Dakota sites occurred for several months of the year. In Colorado, only a few months indicated such an increase. Future IDF curves for Colorado and North Dakota indicated a slight increase in rainfall intensity whereas the Washington site had about a twofold increase. This increase in rainfall intensity could result in major flood events, demonstrating the importance of accounting for climate changes in infrastructure planning.

58 GEOSCIENCES↗

Guidelines for Proof Test Analysis

These guidelines integrate state-of-the-art elastic-plastic fracture mechanics (EPFM) and proof test implementation issues into a comprehensive proof test analysis procedure in the form of a road map which identifies the types of data, fracture mechanics based parameters, and calculations needed to perform flaw screening and minimum proof load analyses of fracture critical components. Worked examples are presented to illustrate the application of the road map to proof test analysis. The state-of-the art fracture technology employed in these guidelines is based on the EPFM parameter, J, and a pictorial representation of a J fracture analysis, called the failure assessment diagram (FAD) approach. The recommended fracture technology is validated using finite element J results, and laboratory and hardware fracture test results on the nickel-based superalloy Inconel 718, the aluminum alloy 2024-T3511, and ferritic pressure vessel steels. In all cases the laboratory specimens and hardware failed by ductile mechanisms. Advanced proof test analyses involving probability analysis and multiple-cycle proof testing (MCPT) are addressed. Finally, recommendations are provided on how to account for the effects of the proof test overload on subsequent service fatigue and fracture behaviors.

Chell, G. G.↗

Limitations of Reliability for Long-Endurance Human Spaceflight

Long-endurance human spaceflight - such as missions to Mars or its moons - will present a never-before-seen maintenance logistics challenge. Crews will be in space for longer and be farther way from Earth than ever before. Resupply and abort options will be heavily constrained, and will have timescales much longer than current and past experience. Spare parts and/or redundant systems will have to be included to reduce risk. However, the high cost of transportation means that this risk reduction must be achieved while also minimizing mass. The concept of increasing system and component reliability is commonly discussed as a means to reduce risk and mass by reducing the probability that components will fail during a mission. While increased reliability can reduce maintenance logistics mass requirements, the rate of mass reduction decreases over time. In addition, reliability growth requires increased test time and cost. This paper assesses trends in test time requirements, cost, and maintenance logistics mass savings as a function of increase in Mean Time Between Failures (MTBF) for some or all of the components in a system. In general, reliability growth results in superlinear growth in test time requirements, exponential growth in cost, and sublinear benefits (in terms of logistics mass saved). These trends indicate that it is unlikely that reliability growth alone will be a cost-effective approach to maintenance logistics mass reduction and risk mitigation for long-endurance missions. This paper discusses these trends as well as other options to reduce logistics mass such as direct reduction of part mass, commonality, or In-Space Manufacturing (ISM). Overall, it is likely that some combination of all available options - including reliability growth - will be required to reduce mass and mitigate risk for future deep space missions.

Owens, Andrew C.↗

Minimum Detectable Quantity Calculation for Radiation Portal Monitors

The minimum detectable quantity of a radiation portal monitor is the smallest amount of radioactive material that can be detected passing through the monitor (with a specified detection probability). The minimum detectable quantity is a function of many factors, including isotope, velocity, distance between pillars, height of portal and vehicle, source distribution and position, background radiation, background suppression, detector volume and positions, detector efficiency, decision metrics and algorithms, the presence of NORM (naturally occurring radioactive material), and alarm thresholds. Experimentally testing a radiation portal monitor to failure for all combinations of these factors is extremely time-consuming and often cost prohibitive. This document outlines a straightforward method for quickly estimating the minimum detectable quantity for moving sources over a large variety of conditions with a minimal number of static measurements. The document also describes a proof of concept software application that the International Atomic Energy Agency has designed based on the described method that can be used by Member States to estimate a monitor’s minimum detectable quantity.

Blessinger, Christopher S.↗

Do asteroids have satellites?

A substantial body of indirect evidence suggests that some asteroids have satellites, although none has been detected unambiguously. Collisions between asteroids provide physically plausible mechanisms for the production of binaries, but these operate with low probability; only a small minority of asteroids are likely to have satellites. The abundance of binary asteorids can constrain the collisional history of the entire belt population. The allowed angular momentum of binaries and their rate of tidal evolution limit separations to no more than a few tens of the primary's radii. Their expected properties are consistent with failure to detect them by current imaging techniques.

Weidenschilling, Stuart J.↗

Factors controlling the structures of magma chambers in basaltic volcanoes

The depths, vertical extents, and lateral extents of magma chambers and their formation are discussed. The depth to the center of a magma chamber is most probably determined by the density structure of the lithosphere; this process is explained. It is commonly assumed that magma chambers grow until the stress on the roof, floor, and side-wall boundaries exceed the strength of the wall rocks. Attempts to grow further lead to dike propagation events which reduce the stresses below the critical values of rock failure. The tensile or compressive failure of the walls is discussed with respect to magma migration. The later growth of magma chambers is accomplished by lateral dike injection into the country rocks. The factors controlling the patterns of growth and cooling of such dikes are briefly mentioned.

Wilson, L.↗