/Optimization for Mass Properties Engineers Training Primer Course Objective: This class is a one-day session that covers structural design and analysis considerations and their on Mass Properties. This course is for working engineers outside of the or Loads organization to provide insight into the decision tools and processes affecting structural integrity. Specific emphasis will be given to identify and explain the typical methods utilized by Structural Analyst and how these can sometimes adversely affect the optimization of the structure. Illustration of several accepted practices and their impact will be covered in the case studies.

Course Outline: (8 hours) Morning (8am – 11:30am) Overview of course objectives Basic Airframe Configuration Issues Common Structural Design Features Understanding Design Drivers and the Mass Properties Impact Strength Designed Stiffness Designed Designed Damage Tolerant Safe Life

Afternoon (1:00pm – 5pm) Common practices Hidden Gotcha’s Case Studies / Examples Aero Commander Aloha Incident 767-300 Freighter Cargo Barrier Shear and Moment Optimization

This Primer covers the following Topics

Basic Strength and Material Review Metals properties Composites properties

Failure Theories and Analysis Static Loading (Brittle & Ductile Materials) Alternating Stresses Cumulative Fatigue

Loads Review Ground Air Ground Cycle for Aircraft Cycle Counting methods

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This document is intended as a quick refresher for topics that the student is already familiar with either through their education or professional background. This is not intended to be a comprehensive treatise or textbook on structural analysis. The basic knowledge presented here is the upon which the lecture material will expand upon. The intent is to re-familiarize the students with the underlying principals of Structural Analysis, Strength of Materials and Failure criteria development.

The actual class will start with the presumption that the student is familiar with these concepts. That does not imply that they will be discussed directly during the class. The intent of providing this information is to give everyone common basis to understand the concepts discussed in the class. These will serve as the basis for explanations on how current structure may not be designed to these criterion and what that means with respect to structural sizing. Much of the in class discussion will address how to identify what criteria is being used to establish the structural requirements.

Basic Strength and Material Review

The fundamental basis of all structural analysis is an understanding the mechanical properties of the material being used. In fact, degree or level of understanding by the designer and analyst often determines how aggressive or conservative a design approach is taken. The old stress adage, “When in doubt, make it stout… out of things we know about.” still holds true to most structural engineers. This can lead to non-optimized designs or even worse, unsafe designs due to a lack of understanding of the mechanical properties and failure mechanisms of the materials being used.

Today the basic materials used in aerospace or transportation are metals and composites. There are even more exotic product forms out there, CMC’s, nano structures, advanced crystalline forms etc. However the distinct differences in these two basic product forms will be sufficient to illustrate the importance of understanding the material properties and their influence on design.

The Basics

The materials in use today are often characterized by several basic parameters in common usage. Some differences occur using composite laminates, or hybrid structures. The most common of these is the conventional Stress-Strain Diagram. Data collected from tensile or compressive tests can be used to determine the nominal stress in the part from:

P σ = A0

Where A0 is the original cross-sectional area of the part and P is the applied load.

Copyrighted SAWE 2004 Page 2 of 18 Structural Analysis/Optimization for Mass Properties Engineers Training Primer Nominal strain for a part can also be determined from strain gage reading taken during testing from the following:

δ ε = L0

Where L0 is the original length of the part and δ is the change in length.

The corresponding values for stress and strain can then be plotted as a graph, with stress on the ordinate and strain on the abscissa. This plot is known as a stress-strain diagram. This provides a quick means of obtaining data about a material’s strength properties independent of the actual part configuration. A typical stress strain curve is shown below.

As can be seen, a significant portion of the stress-strain diagram shows a linear relationship between stress and strain. This is often referred to as the linear-elastic portion of the stress-strain curve. This fact was discovered in 1676 and became known as Hooke’s Law, after Robert Hooke. This can be expressed as:

Copyrighted SAWE 2004 Page 3 of 18 Structural Analysis/Optimization for Mass Properties Engineers Training Primer σ = Eε Where E represents Young’s modulus for the material.

This equation represents a very good approximation of the initial straight lined portion of the stress-strain diagram. In addition, Young’s modulus then represents the slope of this line. The modulus of or Young’s modulus is one of the most important mechanical properties for material characterization. However, E can only be used if a material has linear-elastic behavior over the entire loading spectrum.

Metals properties Metals used in these applications are typically selected for a specific set of properties. Key properties, depending on application are: resistance to flow or creep at high , Stiffness, Tensile and Compressive Strength, durability, toughness and resistance to fracture. Many of these properties can be found in the military handbook MIL-HDBK-5 or the FAA Metallic Material Properties Development and Standardization (MMPDS). Typical product forms include; sheet and plate, extrusions, forgings, and castings.

For many of these materials, more than one set of design allowables are common. The type of allowable or its level of statistical certainty is known as its Basis. The general categories for allowables are A, B, or S, depending on how the allowables were established.

∞ A basis – The allowable value above which at least 99% of the population values will fall with a 95% confidence level. ∞ B basis– The allowable value above which at least 90% of the population values will fall with a 95% confidence level. ∞ S basis – The minimum guaranteed value from the governing material specification with no statistically defined confidence level.

A basis allowables are required for single load path structures where the failure of that load path would result in the loss of structural integrity for a vehicle. B basis allowables are more commonly applied to redundant load path or “fail safe” structure. S basis allowables are very conservative and are typically applied only in applications of new material and/or product form, where the required testing is not yet complete and the perceived mechanical property benefits outweigh the use of more common materials.

Most metals exhibit elastic behavior under relatively low loadings. The level of loading is material dependant. Elastic behavior means that the specimen returns to its original shape or length when the applied load is removed. This elastic behavior occurs when the strains in the specimen are with in the first portion of the previous chart. This portion of the curve is actually a straight line. In other words the stresses are proportional to the strains. The upper stress limit of this linear relationship is defined as the proportional limit, typically designated as σpl. As the stresses exceed the proportional limit, most materials will respond elastically, however, the return curve tends to bend and flatten out such that a greater amount of strain corresponds to an increment of stress. This behavior

Copyrighted SAWE 2004 Page 4 of 18 Structural Analysis/Optimization for Mass Properties Engineers Training Primer continues until the stress reaches the elastic limit of the material. After this point on the curve, permanent occurs.

This increase in stress above the elastic limit will result in the breakdown of the material causing the permanent deformation. This is known as the point or stress for the material, σY. In contrast to elastic loading, any load that causes yielding of the material permanently changes the properties of that material. Some materials have two distinct yield points, an upper yield point that occurs first, followed by a sudden decrease in load carrying capability until the lower yield point is reached. If load equal to this lower load point is applied, the material will continue to deform without any increase in load. At this time a material is often referred to as being perfectly plastic.

Strain hardening is the next portion of the diagram. After a material has finished yielding, additional load can be applied to the specimen. This results in the rising portion of the curve that becomes flatter until it reaches its apex, known as the ultimate stress, σu. This rise is referred to as strain hardening. This type of hardening can increase the apparent elastic limit of a material by trading off some of the materials ductility.

The last part of the curve is correlated with the “necking” behavior of the part. After reaching the ultimate stress, the cross-sectional area of the part begins to decrease in a local area of the specimen. This is due to the slip planes formed with the material and the actual strains produced are due to the shear stresses. This results in a “neck” or reduction of cross-section in a local area as the specimen elongates. Since the cross-sectional area is decreasing, the load carrying capability of the structure continues to decline until the specimen fractures. This point is of academic interest only and is not directly correlated with the fracture toughness or durability of the material.

Composites properties

The use of composites in vehicle structures has grown over the years. This growth is due, in part, to the perceived benefits of unique properties of these materials. It is the ability to tailor the mechanical properties of the final part to match the expected loading conditions that allows the manufacture of structure that is lighter than equivalent metal structure. A key difference is that metallic parts result from processes that remove material to achieve a defined shape, and the majority of these processes do not change the material properties. (E.g. drilling, machining, etc.) In addition, those processes that do change both the shape and the properties are well understood on an empirical basis, from years of use and familiarity. Composites parts are the result of additive processes and can even be considered an amalgam of distinctly different structural components with unique properties.

The biggest problem with composites (from a Materials viewpoint) is that it is difficult to assign mechanical properties to the constituent parts of the composite part. This is a part of the basic material attributes associated with composites. These attributes are:

Copyrighted SAWE 2004 Page 5 of 18 Structural Analysis/Optimization for Mass Properties Engineers Training Primer Material Inhomogeneity – That is the materials differ in fiber material, diameter and architecture, as well as resin material and content. Each of these characteristics directly affects the final material properties, structural response and failure mode.

Process sensitivity – Most notably composites are sensitive to lay-up, and cure variation. Typically it is the resin-dominated properties that are most process sensitive, though fiber dominated properties may also be affected due to improper lay-up practices.

Anisotropy – The key attribute of composite materials is the ability to develop structures with a high degree of anisotropy. On the lamina level, this requires longitudinal, transverse and shear stiffness properties to characterize the material. On the laminate level, the ply orientations and their combinations drive the final mechanical properties.

Deriving Composite Allowables

The industry has two general approaches to analyzing composite strength. Both of these approaches are based on laminated plate theory for stiffness calculations based on ply moduli values. The approaches calculate the ply level strains at a given point in the structure using the applied loads. Failure criteria are then applied to each ply of the laminate. The principle differences in the two methods are in the failure theories applied and the resulting test requirements for substantiation.

The ply or lamina failure theory uses allowables for the individual plies of the material. These allowables are based on unidirectional tests or computed from cross-ply tests. These values are then used as inputs for a lamina failure theory model with empirically defined correction factors to account for structural load paths not observed in the testing. The advantage of this approach is that the ply level data can be rapidly and cheaply established with test coupons. The disadvantage is that current lamina based failure theories tend not to correlate well with actual structural applications. Therefore excessively conservative allowables must often be utilized or expensive full-scale validations test must accompany this approach.

The laminate failure theory uses allowables and design criteria established from tests on representative laminate components. The ply level data is utilized to establish the moduli and the remaining allowables are based on linearized laminate failure strains. These are used in a maximum strain failure criteria that is evaluated on a ply-by-ply basis within the laminate. The variation in stacking sequence, and processing changes can then be included in the testing to allow for the derivation of statically based allowables. This method does not require additional correlation factors, but requires larger and more test coupons to represent usage in the actual structure.

Because of these limitations in establishing composite allowables, a large number of composite structures are point designed. This is essential a certification by test approach

Copyrighted SAWE 2004 Page 6 of 18 Structural Analysis/Optimization for Mass Properties Engineers Training Primer when the final verification of the structure is a proof test of the complete structure. This reduces some of the need for statistically verified allowables and a well documented understanding of all of the failure modes. This has the affect of reducing the costs for a specific program, but often at the price of increasing the risk and weight of the structure.

Failure Theories and Analysis

Static Loading (Brittle & Ductile Materials) Structures can fail under many different loading conditions. Materials identified as brittle, such as cast iron, can fail by sudden fracturing, with very little yielding prior to failure. The most common approach to designing around this failure mechanism is to ensure that the maximum applied load results in a stress that is significantly less than the structures ultimate strength. This can be accommodated by additional factors for castings or by adjusting the “basis” of the allowable. However, for brittle materials, failure occurs when the applied stress is greater than the Ultimate Strength Allowable:

σ > Ftu

Most current failure criteria for brittle material are based on empirical (test derived) modifications to the Coulomb-Mohr theory. The Coulomb-Mohr theory is a conservative representation of failure for combinations of tensile and compressive tensile stresses. Typically this is shown graphically as below.

This Failure Criterion is:

(FS)σ 1 (FS)σ 2 + ≥ 1 Ftu Fcu

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The regions are then validated by experimental data that addresses the specific application

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Ductile materials, unlike brittle materials, fail by yielding, not by fracture. All design criterion for ductile materials are intended to keep the maximum stress below the yield strength of the material. The simplest theory in this area is the maximum shear stress theory, which as the name implies, uses shear stress to indicate yielding. The key feature of this theory is the assumption that for most materials, the yield strength in shear is half of the tensile yield strength.

For simple biaxial loading:

σ 1 −σ 2 τ 12 = 2 Therefore:

Fyt τ max > Fys = 2

From this the combinations of allowable principal stresses for a given maximum shear stress can be depicted as shown below.

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However, the most common failure theory is the von Mises theory. This theory is commonly used to predict tensile and shear failure in metallic parts. von Mises stress distributions are commonly depicted in most commercial Finite Element Codes. The von Mises stress or effective stress is calculated from principal stresses.

2 2 2 σ ' = (1/ 2)[(σ 1 −σ 2) + (σ 2 −σ 3) + (σ 3 −σ1) ]

This then simplifies for biaxial loading to:

2 2 σ ' = σ 1 +σ 2 −σ 1σ 2

Failure is predicted as:

σ ' > Fyt

Alternating Stresses

Copyrighted SAWE 2004 Page 10 of 18 Structural Analysis/Optimization for Mass Properties Engineers Training Primer So far we have been concerned with single load applications failure criteria. However, most engineering applications subject materials to some form of alternating stresses. These can be typified as – Tension Loading, Tension-Compression loading or Compression-Compression loading. For parts undergoing alternating loading over time, failure cannot be determined by comparing the stresses with the yield strength. The average stress and the effect of the stress reversal must be considered. The diagram below shows how a typical Tension-Compression loading spectrum can be illustrated.

From the diagram the mean stress is defined as:

σ +σ σ = max min m 2

The range stress is:

σ r = σ max −σ min

and the alternating stress is half of the range stress:

σ alt = 1/ 2σ r = 1/ 2(σ max −σ min )

Another key characteristic for evaluating the alternating stresses is the ration of the minimum stress to the maximum stress. This is termed the R-ration and is defined as:

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σ R = min σ max

R>0 Tension-Tension or Compression-Compression R=0 Tension or Compression only -1

To date there is no strict consensus on establishing failure criteria for design. The typical methods, Soderberg diagrams, equivalent stress, Smith diagrams for cast materials and Goodman diagrams form the basis of most engineering criteria, but they are typically modified by empirical data from relevant tests that result in proprietary criteria and methodology within different companies.

Cumulative Fatigue

One of the most common structural applications in the transportation industry is a part that undergoes a variety of cyclical loadings. Often the part is subject to a wide variety of maximum and minimum stresses for a series of cycles. Miner’s Rule is almost universally used to provide a reasonable approximation for cumulative damage. Miner’s Rule states that the total damage accumulated is equal to the summation of the ratios of fatigue damage accumulated at each corresponding stress level:

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n ∑ i ≥ C ≅ 1 Ni

Where ni, is the number of cycles accumulated at stress level δi And Ni is the number of cycles to failure at stress level δi

Or expanded to show that cumulative damage, D, is:

n n n D = ∑( 1 + 2 + .... + i ) ≤ 1 N1 N2 Nii

However, cyclical failure and cumulative damage is very material, load application, load sequencing and initial condition sensitive. It is not unusual to see scatter factors of 4 or 6 in test data. Because of this values for C can vary significantly. Fatigue or durability methods are straight forward in theory, but vary greatly in implementation across companies, academia and various government agencies.

Structural Load Environment

Air Vehicle or Transportation Operational Loads

Any platform used for transportation, or a specific mission will be exposed to a unique operational environment. This environment includes both environmental factors like, heat, humidity, pressure, and corrosion products and operational loads. Operational loads for most vehicles or aerospace platforms include, inertia loads, air/hydrodynamic loads, Cabin pressure, and point loads.

Inertia loads are those loads due to the gross accelerations of the vehicle. For aircraft, the vertical loads are most significant on the horizontal aero surfaces like, wings, horizontal stabilizers, canards and external engine pylon structures. These loads are typically caused by gusts, pilot induced maneuvers, landing, runway rollout, taxing, etc.

Air loads are caused by dynamic pressure on the vehicle surfaces. These can be the intended design loads for airfoils and the like, or the unintended loading of any external surface exposed to the flow. This is most significant on wings and control surfaces such as flaps, spoilers and rudders. The transient nature of this type of load in high performance vehicles can lead to very intensive structural reactions of short duration, often known as buffet.

Copyrighted SAWE 2004 Page 13 of 18 Structural Analysis/Optimization for Mass Properties Engineers Training Primer The differential pressure required for mission performance causes pressurization loads. For aircraft this is a positive cabin pressure required to sustain life for high altitude flight. This is the most significant loading for transport fuselage structure. For other vehicles this can be for tankerage of volatiles, or maintaining hull integrity under external pressure loads.

Point loads are characterized by the application of an external load acting at a discrete location on the structure. Examples of this loading type include, snubbing, jacking, and towing.

For air vehicle operations, the Ground-Air-Ground or GAG cycle is often used to illustrate the operational environment. This cycle is most relevant for structure that is dominated by the pressurization cycle. Due to the nature of commercial transport aircraft missions, it has also been successfully used to represent other structures as well. The illustration below shows the typical loading events that occur during a commercial jet transport operation.

These events can then be represented as a stress spectrum for a given location on the airframe as shown:

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Once this stress spectrum has been established, the means to evaluate how the structure reacts to these loadings is required. It is far too cumbersome, even with today’s computing capabilities, to actually evaluate the structural reaction for every loading event. Therefore, a way to reduce these random cycles into a nominal spectrum for analysis is required and is shown in the section on cycle counting.

In evaluating the stress spectrum for structural analysis, some knowledge of the frequency and probability of any single loading event is required. The loading environment is random. This is addressed in the aerospace industry by establishing design criteria for different loading events. Criterion are established to satisfy structural integrity requirements for Ultimate, Limit and Fatigue loadings.

Ultimate loads are taken to be a once in a lifetime or fleet lifetime event. Typically criteria allow for the deformation or yielding of the structure as long as the load carrying capability is not lost. An ultimate load event will result in the inspection and potential repair or removal of affected structural components. Ultimate loads are typically taken to be 1.5 times the limit load. This is a historically established value that has some readily apparent correlation with the ratio between the proportional limit and rupture levels of early steel and aluminum alloys.

Limit loads are expected to be loading events that can occur once in a vehicles lifetime. These are addressed in the basic fatigue and stress analysis during the design. The analysis typically requires metallic materials to stay in the linear-elastic range of their stress-strain diagram with no elongation or deformation occurring under this load. This is typical of events that can readily occur, like landing on a rough runway, heavy maneuvering or flying through turbulent air.

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Fatigue loads are the typical operational loads that occur during ground and flight operations. Due to the life requirements for today’s structural applications, these loads can be significantly less than ultimate or limit loads. For some metals, most notably steels, the goal is often to get the stresses below the material’s endurance limit to ensure long life. For composite materials, fatigue loads are typically not a major driver and some other criteria tend to establish sizing requirements.

Cycle counting methods

The cyclical loading of structure in transport applications is often random. Typically large excursions in amplitude occur less frequently and stresses or loads that occur more often ten to be lower in magnitude. This understanding does little to help us translate these events into some assumed loading profile for analysis. There are several methods listed in the ASTM to address how to turn these random events into a load cycle profile suitable for analysis. The most common of these methods are rainflow counting, simple range counting, peak counting and level crossing counts. Among these methods, rainflow counting is the most commonly accepted in the transportation industry.

Simple range counting is the easiest method to apply. In this method, the range between adjacent load reversals is counted as half a cycle and both peak-to-valley and valley-to- peak ranges are used as shown below:

In the peak counting method, the highest peak is paired with the lowest valley and then removed from the spectrum. This is repeated for each subsequent maximum and minimum until the highest peak is no longer higher than the lowest valley.

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For level crossing counts, the stress or load range time history is divided into a number of load or stress increments. A count is then made at each level every time a positively or negatively sloped portion of the load or stress history crosses that level. The highest positive crossing is then paired with the lowest negative crossing to form a cycle. The crossing counts at each level between the highest and lowest levels are then reduced by one. This is repeated until the highest positive crossing value is no longer higher than the lowest negative crossing value.

Rainflow counting, as stated previously, is the most common approach to reducing measured or theoretical loading cycles for analysis. There are two ways to approach this method. The first approach also describes how the method got its name. Visualize the loading spectrum rotated 90 degrees clockwise with water running down the slopes. The flow starts at the top and flows downhill. The flow also is assumed to start at each unused peak or valley, and stop where there is a peak higher than the starting peak or a valley deeper than the starting valley. The Flow also stops where there is a previously established flow. After this visualization, Peak/Valley pairs are established and counted as shown below.

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Another means of establishing the same result is to identify the smallest peak to valley range, add it to the cycle count and remove it from the original spectrum. Then this is repeated for the next smallest peak to valley range and repeated until no cycles remain. This method is illustrated below and results in the same cycle count as the traditional rainflow count.

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