Cambridge Engineering Selector
CES 2009 Trainin Exercises
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Getting started with CES EduPack 2009 These exercises give an easy way to learn to use the CES EduPack software. The comprehensive Help file and CES. In depth within the software give more detailed guidance.
Thumbnail sketch of CES EduPack 2009 The CES EduPack software has three Levels of Database. Coverage 70 of the most widely used materials drawn from the classes: metals, polymers, ceramics, composites, foams and natural materials. 70 of the most widely used processes
Content A description, an image of the material in a familiar product, typical applications and limited data for mechanical, thermal and electrical properties, using rankings where appropriate.
Level 2
110 of the most widely used materials. 110 of the most commonly used processes
All the content of Level 1, supplemented by more extensive numerical data, design guidelines, ecological properties and technical notes.
Level 3
The core database contains more than 3,000 materials, including those in Levels 1 and 2. Specialist editions covering aerospace, polymers, architecture, bio-materials, and ecodesign are also available.
Extensive numerical data for all materials, allowing the full power of the CES selection system to be deployed.
Level 1
When the software opens you are asked to choose a Level. Chose Level 1 to start with.
At each Level there are a number of Data Tables. The most important are: Materials, Shaping Processes, Joining Processes, and Surface Treatments. Each of the three levels can be interrogated by • BROWSING: Exploring the database and retrieving records via a hierarchical index. • SEARCHING: Finding information via a full-text search of records. • SELECTION: Use of powerful selection engine to find records that meet an array of design criteria.
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Getting started with CES EduPack 2009 These exercises give an easy way to learn to use the CES EduPack software. The comprehensive Help file and CES. In depth within the software give more detailed guidance.
Thumbnail sketch of CES EduPack 2009 The CES EduPack software has three Levels of Database. Coverage 70 of the most widely used materials drawn from the classes: metals, polymers, ceramics, composites, foams and natural materials. 70 of the most widely used processes
Content A description, an image of the material in a familiar product, typical applications and limited data for mechanical, thermal and electrical properties, using rankings where appropriate.
Level 2
110 of the most widely used materials. 110 of the most commonly used processes
All the content of Level 1, supplemented by more extensive numerical data, design guidelines, ecological properties and technical notes.
Level 3
The core database contains more than 3,000 materials, including those in Levels 1 and 2. Specialist editions covering aerospace, polymers, architecture, bio-materials, and ecodesign are also available.
Extensive numerical data for all materials, allowing the full power of the CES selection system to be deployed.
Level 1
When the software opens you are asked to choose a Level. Chose Level 1 to start with.
At each Level there are a number of Data Tables. The most important are: Materials, Shaping Processes, Joining Processes, and Surface Treatments. Each of the three levels can be interrogated by • BROWSING: Exploring the database and retrieving records via a hierarchical index. • SEARCHING: Finding information via a full-text search of records. • SELECTION: Use of powerful selection engine to find records that meet an array of design criteria.
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BROWSING and SEARCHING The DEFAULT on loading CES EduPack Levels 1 & 2 is LEVEL 1, MATERIALS UNIVERSE Exercise 1. BROWSE materials • Find record for STAINLESS STEEL • Find record for CONCRETE • Find record for POLYPROPYLENE • Explore POLYPROPYLENE record at LEVEL 2 • Find PROCESSES that can shape POLYPROPYLENE using the LINK at the bottom of the record
Exercise 2. BROWSE processes Select LEVEL 2, ALL PROCESSES
• Find record for INJECTION MOLDING • Find record for LASER SURFACE HARDENING • Find record for FRICTION WELDING (METALS) • Find MATERIALS that can be DIE CAST, using the LINK at the bottom of the record for DIE CASTING
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Exercise 3. The SEARCH facility
• Find the material POLYACTIDE • Find materials for CUTTING TOOLS • Find the process RTM
PROPERTY CHARTS Exercise 4. Making PROPERTY CHARTS
• Make a BAR CHART of YOUNG’S MODULUS (E) (Set y-axis to Young’s modulus; leave x-axis at ) (Click on a few materials to label them; double-click to go to their record in the Data Table) • Make a BUBBLE CHART of YOUNG’S MODULUS (E) against DENSITY (ρ) (Set both x-axis and y-axis; the default is a log-log plot) (Materials can be labeled as before – click and drag to move the labels; use DEL to delete a label.)
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DELETE THE STAGE (Right click on stage and select “Delete”)
SELECTION using a LIMIT STAGE Exercise 5. Selection using a LIMIT stage
• Find materials with : MAX. SERVICE TEMPERATURE > 200 °C THERMAL CONDUCTIVITY > 25 W/mk ELECTRICAL CONDUCTOR = GOOD INSULATOR OR INSULATOR? (Enter the limits – minimum or maximum as appropriate – and click “Apply”) (Results at Level 1 or 2: aluminum nitride, alumina, silicon nitride) DELETE THE STAGE
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GRAPH SELECTION Exercise 6 Selection with a GRAPH stage
• Make a BAR CHART of Yield strength ( σ y ) (plotted on the y-axis). • Use a BOX SELECTION to find materials with high values of elastic limit (or strength). (Click the box icon, then click-drag-release to define the box) • Add, on the other axis, DENSITY ( ρ ) (Either: highlight Stage 1 in Selection Stages, and click Edit; or double-click the axis to edit) • Use a BOX SELECTION to find materials with high strength and low density. • Replace the BOX with a LINE SELECTION to find materials with high values of the “specific strength”, σ y / ρ. (Click the line icon, then enter slope required – 1 in this case – click the graph to position the line, click again to select the side required, i.e. above the line for high values of σ y /
ρ . Now click on the line and drag upwards, to refine the selection to just 3 materials). (Results at Level 1 or 2: CFRP (isotropic), Titanium alloys, Magnesium alloys) DELETE THE STAGE
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TREE SELECTION Exercise 7. Selection with a TREE Stage
• Find MATERIALS that can be MOLDED (In Tree Stage window, select ProcessUniverse, expand “Shaping” in the tree, select Molding, and click “Insert”, then OK) • DELETE THE STAGE • Find PROCESSES to join STEELS (First change Selection Data to select Processes: select LEVEL 2, JOINING PROCESSES.) (Then, in Tree Stage window, select MaterialUniverse, expand “Metals and alloys” in the tree, select Ferrous, and click “Insert”, then OK) DELETE THE STAGE
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GETTING IT ALL TOGETHER Exercise 8. Using ALL 3 STAGES together
Change Selection data to select materials: Select LEVEL 2, MATERIALS Find MATERIALS that are 3
• DENSITY < 2000 kg/m
• STRENGTH (Elastic limit) > 60 MPa • THERMAL CONDUCTIVITY < 10 W/mK (3 entries in a Limit Stage) • Can be THERMOFORMED (a Tree Stage: ProcessUniverse – Shaping - Molding) • Rank the results by PRICE (a Graph Stage: bar chart of Price) (On the final Graph Stage, click the “Intersect Stages” icon, like a small Venn diagram; materials failing one or more stages turn grey; label the remaining materials, which pass all stages. The RESULTS window shows the materials that pass all the stages.) (Results, cheapest first: PET, PMMA, Acetal (POM)) Exercise 9 Finding SUPPORTING INFORMATION (Requires Internet connection)
• With the PET record open, click on SEARCH WEB (CES translates the material ID to strings compatible with a group of high-quality material and process information sources and delivers the hits. Some of the sources are open access, others require a subscriber-based password. The ASM source is particularly recommended.) DELETE THE STAGE
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PROCESS SELECTION Exercise 10 Selecting PROCESSES Change Selection data to select processes:
Select LEVEL 2, SHAPING PROCESSES Find PRIMARY SHAPING PROCESSES to make a component with: • SHAPE = Dished sheet • MASS = 10 – 12 kg • SECTION THICKNESS = 4 mm • ECONOMIC BATCH SIZE > 1000 (3 entries in a Limit Stage) • Made of a THERMOPLASTIC, (a Tree Stage: MaterialUniverse – Polymer – Thermoplastics) (Result: manual compression molding, rotational molding, thermoforming)
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SAVING, COPYING, and REPORT WRITING Exercise 11. Saving Selection Stages as a PROJECT
• SAVE the project – exactly as if saving a file in Word (give it a filename and directory location; CES project files have the extension “.ces”). Exercise 12. COPYING CES OUTPUT into a Report Charts, Records and Results lists may be copied (CTRL-C) and pasted (CTRL-V) into Word.
• Display a chart, click on it, then COPY and PASTE it into a WORD document • Double click a selected material in the Results window to display its record, click on the record, then COPY and PASTE it. • Click on the Results window, then COPY and PASTE it. • Try editing the document (The records in Exercise 3 and the selection charts on Exercises 4 and 6 were made in this way.) (Warning: There is a problem with WORD 2000: the image in the record is not transferred with the text. The problem is overcome by copying the image and pasting it separately into the WORD document as a DEVICE INDEPENDENT BITMAP.)
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Toolbars in CES EduPack
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Training Exercises Using CES EduPack 2009 1. BROWSING materials and process records
Browsing lets you explore records, starting from the content list or “tree. Open the EduPack in Level 1. 1.1 Find the record for the thermoplastic polymer Polycarbonate, PC, a thermoplastic. What is it used to make? What, approximately, does it cost? Is it cheaper or more expensive than Polypropylene, PP? (Find the record for PP to decide.) 1.2 Find the record for the ferrous metal Stainless Steel. What is the value of its thermal conductivity? Is it a better or worse conductor than aluminum or copper? (All three are used to make cooking pans.) 1.3 Find the record for the Borosilicate Glass, commonly known as PYREX. What is the value of its “maximum service temperature” (the highest temperature at which it can used in a product)? What is Pyrex used for? 1.4 Find the records first for Titanium alloys and then for Aluminum alloys. Which has the higher tensile strength? Which has the lower density? 1.5 Find the records first for the composite CFRP (Carbon-fiber reinforced polymer). It is in the family HYBRID, under Composite Which has the higher tensile strength? Which has the lower density? What is CFRP used for? Is it denser or less dense than Magnesium? Click on the Process Universe link at the bottom of the CFRP record to find processes that can shape, join or finish CFRP. Can CFRP be shaped by Water-jet cutting? (Double click on any name in the list to see the record.)
Now switch from the Materials Universe to the Process Universe by changing the Browse table, using the dialog box just below the word Browse. 1.6 Find the Composite shaping record for Filament winding, a way of making high quality composite structures. What are its typical uses? 1.7 Find the shaping record for Injection molding, one of the used of all polymer shaping processes. Find materials that molded by clicking on the LINK button, labeled Materials bottom of the record. Can Polyethylene be injection molded? any name in the list to see the record.)
most commonly can be injection Universe, at the (Double click on
1.8 Find the shaping record for DIE CASTING, one of the most-used ways of shaping metals. What sort of products are made by die casting?
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2. SEARCHING
Searching lets you pull up any record that contains the word or word-string you enter in the search box. It is really useful when you only know the trade name of a material or process, or when you want to search for materials that are used to make a particular product . 2.1 Find the record for Plexiglas by searching. What is its proper name? Can it be injection molded? (Click on the Process Universe link at the bottom of the record to find out.) 2.2 What are spark plugs made of? Search on the name and find out. 2.3 Search on cutting tool to find materials that are used to make industrial cutting tools. You will find that some are metals, but others are ceramics – hard ceramics are good because they don’t wear, but they are expensive and hard to make. 2.4 Find what the process RTM is all about by searching on RTM. Since it is a process, not a material, you will have to change the table in which you search from Materials Universe to Process Universe, in the box immediately below the search box. 2.5 Find what the process SLS is all about by searching on SLS. Since it is a process, not a material, you will have to change the table in which you search from Materials Universe to Process Universe, in the box immediately below the search box. 3. SELECTING materials and process records There are three selection tools: GRAPH, LIMIT and TREE. We will start with the LIMIT STAGE tool. It lets you find materials or processes that meet requirement that you enter in a Limit Stage. To do this, set the system to Level 1 Materials (click on CHANGE, just below the word SETTINGS at the top of the selection window if necessary to allow you to change the settings). Then click on Select / New / Limit stage, as shown here.
3.1 Find materials that cost less that $1/kg and are good electrical conductors. Enter the upper limit on price and the constraint that the material must be a good conductor as shown above. Then click APPLY at the top of the screen. The materials that do not meet the constraints are deleted from the RESULTS window on the lower left, leaving those the do. 3.2 The property Fracture toughness is a measure of how well a material resists fracture. A brittle material like glass has a low value of fracture toughness – 1/2
around 1 in the units you will use (MPa.m ). Steel used for armour has a very high value – over 100, in the same units. Many engineers, when designing with
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Now we’ll do a GRAPH STAGE. It lets you plot properties and to select those that lie in a chosen part of the plot. Delete the limit stage (right-click on the stage name and select “delete”). Then open a graph stage: Select / New / Graph stage.
3.3 If you want to make a high quality cooking pan to go on the top of a gas stove, you need a material with a high thermal conductivity. The high conductivity is to spread the heat, preventing hot-spots where the flame hits the pan. The material must have enough elongation to be shaped to a pan (requiring o
elongation > 15%), and a maximum use temperature of at least 150 C. First make a limit stage and put these (lower) limits on elongation and maximum use temperature. Then make a graph with thermal conductivity on the Y-axis. To do this first click on Select, then on New and then on Graph stage, as in the diagram. Select the Y-axis tab and find Thermal conductivity in the Attribute list and double click to select it. When you click OK you get the graph shown above. To make the materials that fail the limits grey out, click on the little icon like two intersecting circles in the icon bar just above the graph. Use a box selection (the little box icon in the same icon bar) to select the materials with the highest thermal conductivities. Now we’ll do a TREE STAGE. A tree stage is essential if you want to find materials that can be processed in a particular way, or you want to limit the selection to just one class of material – just metals, for instance. To make a tree stage click Select / New / Tree stage. This gives a window in which you can select the part of the MaterialsUniverse or of the ProcessUniverse that you want to explore.
3.4 You want to make a casing for a mobile phone, exotic in color and design. It snaps onto the front of the phone, transforming it from a drab object to one of splendour. Research reveals that the shape is best made by Thermoforming (a very cheap process for shaping polymer sheet into dished and curved shapes) and that the decoration is best applied by In-mold decoration that can be done at the same time as the thermoforming. Find materials that can be processed in this way. 4
Exercises that don’t use indices. 4.1 Filament for a light bulb.
Background.
A headlight is an essential part of an automobile. Headlights differ in detail, but all have a bulb containing a filament enclosed in a transparent envelope. The filament is expose to harsh conditions: very high temperature, vibration and a risk of oxidation. The goal of the project is to use CES to select a material for the filament. Objective.
To select a material that meets the requirements for the filament.
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Requirements.
Must be an electrical conductor
Must resist oxidation
Must have the highest possible melting point.
Set the CES EduPack to Select / Level 2 / Materials. Use a Limit stage to apply the first two requirements, then a Graph stage of melting point to find the material with the highest value that also meets the first two requirements. Remember you can “grey-out” materials on the graph that do not meet the limit stage criteria by clicking the little icon like two intersecting circles in the row of icons along the top of the graph. 4. 2. Automotive headlight lens.
Background.
The lens of an automobile headlamp protects the bulb and reflector and focusses the light where it is most needed. The project is to use CES to select materials for the lens.
Objective.
To select materials that meets the requirements for the lens. Requirements.
Must be transparent with optical quality.
Must be able to be molded easily.
Must have very good resistance to fresh and salt water
Must have very good resistance to UV light
Good abrasion resistance, meaning a high hardness 18
Low cost
Set the CES EduPack to Select / Level 2 / Materials. Use a Limit stage to apply the first four requirements, selecting Optical properties to apply the first, Processability to apply the second, and Durabilit y to apply the third and the fourth (select both “good” and “very good” to avoid eliminating too many materials). Then make a Graph stage with Price on the X-axis and Hardness on the Y-axis to find the ones that are cheap and have high hardness. Remember you can “grey-out” materials on the graph that do not meet the limit stage criteria by clicking the little icon like two intersecting circles in the row of icons along the top of the graph. Project 3. Novel guitar case (guitar plus amplifier).
Background.
Guitars are delicate instruments. They need a case to protect them when moved, and if they are electric, they need an amplifier and speaker and they too have to be moved and protected. The mission is to simplify this protection problem by designing a case that will hold and protect both the guitar and the amplifier plus speaker using the case itself as the speaker cabinet and amplifier case. Objective.
To select materials and process method to make a case for guitar and amplifier.
Requirements.
• Must be tough – the rule of thumb here is that the fracture toughness should be 1/2
greater than 15 in the usual units (MPa.m ). • Must be moldable • Very good resistance to fresh and salt water • Must be light • Should not cost too much The procedure is the same as that for Project 1. Open the CES EduPack in Level 2. Click on Select. From the New Project box select Level 2, Materials. Use a Limit stage to apply the first 3 requirements, selecting Mechanical properties to apply the first, Processability to apply the second, and Durability to apply the third. Then make a graph stage with Price on the X-axis and Density on the Y-axis to find out which of the survivors is the cheapest, and which the lightest. Remember you can “grey-out” materials on the graph
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that do not meet the limit stage criteria by clicking the little icon like two intersecting circles in the row of icons along the top of the graph. Project 4. Design a CD case that doesn’t crack or scratch CDs
Background. The standard CD (“Jewel” case) cracks easily and, if broken, can scratch the CD. Jewel cases are made of injection moulded polystyrene, chosen because it is transparent, cheap and easy to mould. The project: redesign the case and choose a material for it. The redesign might be a minor refinement of the standard 3-part design, a new shape (circular instead of square?) or a single part moulding with a natural hinge linking lid to case.
Requirements:
• Optical properties : transparent or optically clear. • Fracture toughness better than polystyrene (get data for PS from its record). • Young’s modulus not too different from polystyrene (to make sure the case is stiff enough) • Able to be injection moulded. • Cost not more than twice that of polystyrene. Applying these using either the Level 1 or Level 2 database gives PMMA (acrylic) and PET as possible alternatives. Both are perfectly sensible choices. Project 5. Materials for knife-edges and pivots
Background. Precision instruments like clocks, watches, gyroscopes, and scientific equipment often contain moving parts located by knife-edges or pivots. The accuracy of location is limited by the deformation of the knife-edge or pivot and the mating surface. Elastic deformation is minimised by choosing materials with high Young’s modulus; plastic deformation is limited by choosing materials with high hardness. Requirements:
• Young’s modulus : as large as possible. • Hardness: as large as possible.
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The best way to tackle this using the Level 1 database is to make a graph stage of Young’s modulus and Hardness and pick the materials with high values of both. The very best are all ceramics: boron carbide, silicon carbide and tungsten carbide. If the selection box is relaxed so that the first metals appear, the selection picks up high carbon steel and low alloy steel. All are sensible choices: the ceramics when the ultimate precision is required, the steels when robust design able to deal with shock loading is needed. Project 6. Materials for heat sinks for a power electronics
Background. The power density of present day computer chips is such that removing the heat generated in them is a major consideration. The chip is attached to a heat sink that conducts the heat from the chip to a set of fins cooled by fan-driven air flow. The heat sink must conduct heat well, be able to operate continuously at 150 C and be electrically insulating.
Requirements:
• Electrical properties: good insulator • Maximum service temperature > 150 C (423 K) • As large a thermal conductivity as possible. Applying this using the Level 1 database, using a graph stage to plot thermal conductivity and selecting materials with the largest value gives aluminium nitride – the favored material for heat sinks. (Remember that you can hide materials on a graph stage that have failed previous limits by clicking on the two icons at the top of the graph.)
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Project 7. Materials for a fresh-water heat exchanger
Background. Heat exchangers, typically, consist of a set of tubes through which one fluid is pumped, immersed in a chamber through which the other fluid flows; heat passes from one fluid to the other. The material of the tubing must conduct heat well, have an maximum operating temperature above the operating temperature of the device, not corrode in the fluid, and – since the tubes have to be bent – have adequate ductility.
Requirements:
• Maximum service temperature > 150 C (423 K) • Elongation > 20% • Corrosion resistance in fresh water : very good • As large a thermal conductivity as possible. Applying this using the Level 2 database (necessary because Level 1 doesn’t have corrosion resistance), using a graph stage to plot thermal conductivity and selecting materials with the largest value gives copper alloys and aluminium alloys. Both are used for heat exchangers. E3. Exercises that make use of material selection charts The exercises in this section involve the simple use of the charts of Chapter 4 to find materials with give property profiles. They are answered by placing selection lines on the appropriate chart and reading off the materials that lie on the appropriate side of the line. It is a good idea to present the results as a table. All can be solved by using the printed charts. If the CES EDU Materials Selection software is available the same exercises can be solved by its use. This involves first creating the chart, then applying the appropriate box or line selection. The results, at Level 1 or 2, are the same as those read from the hard copy charts (which were made using the Level 2 database). The software offers links to processes, allows a wider search by using the Level 3 database, and gives access to supporting information via the “Search Web” function.
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5.1 A component is at present made from a brass, a copper alloy. Use the Young’s modulus – Density ( E-ρ) chart to suggest three other metals that, in the same shape, would be stiffer. “Stiffer”means a higher value of Young’s modulus. 5.2 Use the Young’s modulus – Density ( E-ρ) chart to identify materials with both a modulus E > 50 GPa and a density ρ < 2 Mg/m3. 5.3 Use the Young’s Modulus-Density ( E-ρ) chart to find (a) metals that are stiffer and less dense than steels and (b) materials (not just metals) that are both stiffer and less dense than steel. 5.4 Use the E-ρ chart to identify metals with both E > 100 GPa and E / ρ > 23 GPa/(Mg/m3). 5.5 Use the E-ρ chart to identify materials with both E >100 GPa and 1/3
1/ 3
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E / ρ > 3 (GPa) /(Mg/m ). Remember that, on taking logs, the index 1/ 3
M = E / ρ becomes Log ( E ) = 3 Log ( ρ ) + 3 Log (M ) and that this plots as a line
of slope 3 on the chart, passing through the point ρ = 1/3 = 0.33 at E = 1 in the units on the chart. 5.6 Use the E-ρ chart to establish whether woods have a higher specific stiffness E/ ρ than epoxies. 5.7 Do titanium alloys have a higher or lower specific strength (strength/density, σ f / ρ) than tungsten alloys? This is important when you want strength at low weight (landing gear of aircraft, mountain bikes). Use the σ f / ρ char t o decide. 5.8 Use the modulus-strength E −σ f chart to find materials that have E > 10 GPa and σ f ≥ 1000 MPa. 5.9 Are the fracture toughnesses, K 1c, of the common polymers polycarbonate, ABS, or polystyrene larger or smaller than the engineering ceramic alumina? Are their toughnesses G 1c = K 2 1c / E larger or smaller? The K1c –E chart, Figure 4.7, will help. 5.10
Use the fracture toughness-modulus chart to find materials that hav e a fracture
toughness K 1c greater than 100 MPa.m1/2 and a toughness G 1c = K 2 1c / E greater than 10 kJ/m3.
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5.11
Bell ring because they have a low loss (or damping) coefficient, η; a high
damping gives a dead sound. Use the Loss coefficient – Modulus (η − E) chart to identify material that should make good bells. 5.12 Use the Loss coefficient-Modulus (η − E) chart (Figure 4.9) to find metals with the highest possible damping. 5.13 The window through which the beam emerges from a high-powered laser must obviously be transparent to light. Even then, some of the energy of the beam is absorbed in the window and can cause it to heat and crack. This problem is minimized by choosing a window material with a high thermal conductivity λ (to conduct the heat away) and a low expansion coefficient α (to reduce thermal strains), that is, by seeking a window material with a high value of M = λ / α Use the α −λ chart to identify the best material for a ultra-high powered laser window. 5.14
Use the Thermal conductivity-Electrical resistivity (λ − ρ e ) chart to find three
materials with high thermal conductivity,λ , and high electrical resistivity, ρ e . 5.15 Use the Strength-Maximum service temperature (σ f −Tmax ) chart to find polymers that can be used above 2000 C. 5.16 Use the Friction coefficient chart, to find two materials with exceptionally low coefficient of friction. 5.17 Identical casings for a power tool could be die-cast in aluminum or molded in ABS or polyester GFRP. Use the appropriate production energy chart to decide which choice minimizes the material production energy.
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E5. Deriving and using material indices The exercises in this section give practice in deriving indices. (a) Start each by listing function, constraints, objectives and free variables; without having those straight, you will get in a mess. Then write down an equation for the objective. Consider whether it contains a freevariable other than material choice; if it does, identify the constraint thatlimits it, substitute, and read off the material index. (b) If the CES Edu software is available, use it to apply the constraints and rank the survivors using the index (start with the Level 2 database). Are the results sensible? If not, what constraint has been overlooked or incorrectly formulated?
6.1 Material indices for elastic beams with differing constraints.
Start each of the four parts of this problem by listing the function, the objective and the constraints. You will need the equations for the deflection of a cantilever beam with a square cross-section t x t. The two that matter are that for the deflection δ of a beam of length L under an end load F: 3 = F L / 3 E I and that for the deflection of a beam under a distributed load f per unit length: 4
= f L / 8 E I
Where I=t4/12. For a self-loaded beam f = Ag where is the density of the material of the beam, A its cross-sectional area and g the acceleration due to gravity. a) Show that the best material for a cantilever beam of given length L and given (i.e. fixed) square cross-section (t x t) that will deflect least under a given end load F is that with the largest value of the index M = E, where E is Young's modulus (neglect self-weight). b) Show that the best material choice for a cantilever beam of given length L and with a given section (t x t) that will deflect least under its own self-weight is that with the largest value of M = E/ρ, where ρ is the density. c) Show that the material index for the lightest cantilever beam of length L and square section (not given, i.e., the area is a free variable) that will not deflect by 2
more than under its own weight is M = E/ ρ . d) Show that the lightest cantilever beam of length L and square section (area free) that will not deflect by more than δ under an end load F is that made of the material with the largest value of M = E
1/2
/ ρ. (neglect self weight).
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Material index for a light, strong beam (Figure E5). In stiffness limited applications, it is elastic deflection that is the active constraint: it limits performance. In strength-limited applications, deflection is acceptable provided the component does not fail; strength is the active constraint. Derive the material index for selecting materials for a beam of length L, specified strength and minimum weight. For simplicity, assume the beam to have a solid square cross-section t x t. You will need the equation for the failure load of a beam (Appendix A, Section A4). It is
where ym is the distance between the neutral axis of the beam and its outer filament and I = t 4 / 12 = A2 / 12 is the second moment of the cross-section. The table itemizes the design requirements.
E5.4 Material index for a cheap, stiff column (Figure E6).
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In the last two exercises the objective has been that of minimizing weight. There are many others. In the selection of a material for a spring, the objective is that of maximizing the elastic energy it can store. In seeking materials for thermal-efficient insulation for a furnace, the best are those with the lowest thermal conductivity and heat capacity. And most common of all is the wish to minimize cost. So here is an example involving cost. Columns support compressive loads: the legs of a table; the pillars of the Parthenon. Derive the index for selecting materials for the cheapest cylindrical column of specified height, H, that will safely support a load F without buckling elastically. You will need the equation for the load Fcrit at which a slender column buckles. It is
where n is a constant that depends on the end constraints and I = π r 4 / 4 = A2 / 4π is the second moment of area of the column (see Appendix A for both). The table lists the requirements.
6.2 Springs for trucks. In vehicle suspension design it is desirable to minimize the mass of all components. You have been asked to select a material and dimensions for a light spring to replace the steel leaf-spring of an existing truck suspension. The existing leaf-spring is a beam, shown schematically in the figure. The new spring must have the same length L and stiffness S as the existing one, and must deflect through a maximum safe displacement δ max without failure. The width b and thickness t are free variables.
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Derive a material index for the selection of a material for this application. Note that this is a problem with two free variables: b and t; and there are two constraints, one on safe deflection δ max and the other on stiffness S. use the two constraints to fix free variables.
The table catalogs the requirements. Function Constraints
Objective Free variables
• Leaf spring for truck • Length L specified • Stiffness S specified • Maximum displacement δ max specified • Minimize the mass • Spring thickness t • Spring width b • Choice of material
You will need the equation for the mid-point deflection of an elastic beam of length L loaded in three-point bending by a central load F:
and that for the deflection at which failure occurs
where I is the second moment of area; for a beam of rectangular section, I = b t 3 / 12 and E and σ f are the modulus and failure stress of the material of the beam. (See Appendix
A.)
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7. Selecting processes The exercises of this section use the process selection charts of Chapters 7 and 8. They are useful in giving a feel for process attributes and the way in which process choice depends on material and the shape. Here the CES software offers greater advantages: what is cumbersome and of limited resolution with the charts is easy with the software, which offers much greater resolution. Each exercise has three parts, labeled (a), (b) and (c). The first involves translation. The second uses the selection charts of Chapter 7 (which you are free to copy) in the way that was illustrated in Chapter 8. The third involves the use of the CES software if available.
E6.3 Ceramic valves for taps (Figure E14). Few things are more irritating than a dripping tap. Taps drip because the rubber washer is worn or the brass seat is pitted by corrosion, or both. Ceramics wear well, and they have excellent corrosion resistance in both pure and salt water. Many household taps now use ceramic valves. The sketch shows how they work. A ceramic valve consists of two disks mounted one above the other, spring-loaded so that their faces are in contact. Each disk has a diameter of 20 mm, a thickness of 3 mm and weighs about 10 grams. In order to seal well, the mating surfaces of the two disks must be flat and smooth, requiring high levels of precision and surface finish; typically tolerance < 0.02 mm and surface roughness < 0.1 μm. The outer face of each has a slot that registers it, and allows the upper disc to be rotated through 90o (1/4 turn). In the “off” position the holes in the upper disc are blanked off by the solid part of the lower one; in the “on” position the holes are aligned. A production run of 105 –106 is envisaged.
(a) List the function and constraints, leave the objective blank and enter “Choice of process” for the free variable. (b) Use the charts of Chapter 7 to identify possible processes to make the casing. (c) Use the CES software to select materials for the casing.
7.1
Shaping plastic bottles. Plastic bottles are used to contain fluids as various as milk and engine oil. A typical polyethylene bottle weighs about 30 grams and has a
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wall thickness of about 0.8 mm. The shape is 3-D hollow. The batch size is large (1,000,000 bottles). What process could be used to make them?
a) List the function and constraints, leave the objective blank and enter “Choice of process” for the free variable. b) Use the charts of Chapter 7 to identify possible processes to make the casing. c) Use the CES software to select materials for the casing. 7.2 Car hood (bonnet). As weight-saving assumes greater importance in automobile design, the replacement of steel parts with polymer-composite substitutes becomes increasingly attractive. Weight can be saved by replacing a steel hood with one made from a thermosetting composites. The weight of the hood depends on the car model: a typical composite hood weighs is 8 - 10 kg. The shape is a dished-sheet and the requirements on tolerance and roughness are 0.2
mm and 2 μm, respectively. A production run of 100,000 is envisaged.
a) List the function and constraints, leave the objective blank and enter “Choice of process” for the free variable. b) Use the charts of Chapter 7 to identify possible processes to make the casing. c) Use the CES software to select materials for the casing.
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