2.7 - Materials selection and shape. Outline. The shape factor, and shape limits. Material indices that include shape

Transcription

1 - shy Method.7 - Materials selection and shape Outline Shape efficiency The shape factor, and shape limits Material indices that include shape Graphical ways of dealing with shape Resources: M.. shy, Materials Selection in Mechanical Design Butterworth Heinemann, 1999 Chapter 7 W. C. Young, R. G. Budynas, Roark s ormulas for Stress and Strain 7 th ed, McGraw-Hill, 00 The Camridge Material Selector (CES) software -- Granta Design, Camridge (

2 Structural components Moments of area They depend on shapes d Moment of area aout axis x I xx y d Moment of area aout axis y I yy x d Polar moment of area J r d

3 Modes of loading: xial loading σ δ ε Stress Strain rom Hooke s aw (linearly elastic material): σ σ Eε ε E E rom the definition of strain: δ ε δ E σ S δ E Stiffness Modes of loading: Bending Pure Bending: Prismatic memers sujected to couples acting in the longitudinal plane crossing one of the principal inertia axes fter deformation, the length of the neutral surface remains. t other sections: x z δ ' ε ε ε z max z ( ρ y) θ ( ρ y) δ yθ y ρθ ρ c c ρ ρ ε y ε c max θ ρθ yθ max Strain (varies linearly)

4 Modes of loading: Bending or a linearly elastic material: y σ z Eεz Eεmax c y σmax Stress c (varies linearly) σ z σ M M (c y max ) Z I /c max xx Ixx Z Bending strength modulus c σ z M I /y xx I XX moment of area aout the ending axis Modes of loading: Bending z δ S δ CEI xx 3 Stiffness d y EI M( z) xx dz I XX moment of area aout the ending axis C constant (depending on the loading conditions)

5 Modes of loading: Torsion T Torsion: Prismatic memers sujected to twisting couples or torques Consider an interior section of the shaft. s a torsional load is applied, the shear strain is equal to angle of twist. ) ) ρφ ' γ ρφ γ c ρ γ max φ γ γmax c Shear strain ( twist angle and radius) Modes of loading: Torsion T or a linearly elastic material: ρ τ Gγ Gγ max c ρ τ max Shear stress c (varies linearly) T T τ max (c ρ max ) Q K/c K Q c Twisting strength modulus τ T K/ρ K torsional moment of area

6 Modes of loading: Torsion T T KG γ τ Tρ S T Stiffness φ φ ρ ρ G ρg K Cross-sections of noncircular (non-axisymmetric) shafts are distorted when sujected to torsion. Cross-sections for hollow and solid circular shafts remain plain and undistorted ecause a circular shaft is axisymmetric. K J for circular sections only T T τ max (c ρ max ) Q J/c τ T J/ρ J polar moment of area T JG S T Stiffness φ Modes of loading: Buckling Buckling: Prismatic memers sujected to compression in unstale equilirium In the design of columns, cross-sectional area is selected such that - allowale stress is not exceeded σ σ y - deformation falls within specifications δ δ E lim fter these design calculations, may discover that the column is unstale under loading and that it suddenly uckles.

7 Modes of loading: Buckling Consider ideal model with two rods and torsional spring. fter a small perturation k ( θ ) restoring moment sin θ θ destailizing moment Column is stale (tends to return to aligned orientation) if θ < k θ < cr ( ) 4k Modes of loading: Buckling The critical loading is calculated from Euler s formula cr π EImin Stress corresponding to critical loading σ σ cr cr π ( r) E π E λ ( r Imin/ ( λ /r inertia radius ) slenderness )

8 Modes of loading: Buckling cr π EI e min e Equivalent length (length of free inflexion, distance etween two susequent inflexion points) Shape efficiency Shape cross section formed to a tues I-sections hollow ox-section sandwich panels Efficient use least material for given stiffness or strength Shapes to which a material can e formed are limited y the material itself (processaility and mechanical ehaviour) Goals: - quantify the efficiency of shape - understand the limits to shape - develop methods for co-selecting material and shape Certain materials can e made to certain shapes: what is the est comination?

9 Shape and mode of loading When materials are loaded in ending, in torsion, or are used as columns, section shape ecomes important rea matters, not shape rea and shape (I XX ) matter rea and shape (J, K) matter rea and shape (I min ) matter Shape and mode of loading unction Tie-rod rea Ojective Constraints Minimise mass m: m ρ Stiffness of the tie S: E S rea matters, not shape m mass area length ρ density S stiffness E Youngs Modulus

10 Shape and mode of loading unction Tie-rod rea Ojective Constraints Minimise mass m: m ρ Must not fail under load : / < σ y m mass area length ρ density σ y yield strength rea matters, not shape Shape and mode of loading unction Beam (solid square section). Ojective Constraint Minimise mass, m, where: m ρ ρ Stiffness of the eam S: CEI S 3 rea and shape matter m mass area length ρ density edge length S stiffness I second moment of area E Youngs Modulus I is the second moment of area: 4 I 1

11 Shape and mode of loading unction Beam (solid square section). Ojective Minimise mass, m, where: m ρ ρ Constraint Must not fail under load σ y M M / > Z I m mass area length ρ density edge length I second moment of area σ y yield strength rea and shape matter I is the second moment of area: 4 I 1 Shape and mode of loading

12 Definition of Shape actor Bending has its est shape: eams with hollow-ox or I-sections are etter than solid sections of the same cross-sectional area Torsion too has its est shape: circular tues are etter than either solid sections or I-sections of the same cross-sectional area To characterize this we need a metric - the shape factor a way of measuring the structural efficiency of a section shape - specific for each mode of loading - independent of the material of which the component is made - dimensionless (regardless of shape scale) We define shape factor the ratio of the stiffness (or strength) of the shaped section to the stiffness (or strength) of a reference shape, with the same cross-sectional area (and thus the same mass per unit length) Shape efficiency: Bending stiffness Take ratio of ending stiffness S of shaped section to that (S o ) of a neutral reference section of the same cross-section area Define a standard reference section: a solid square with area (alternatively: solid circular section) Second moment of area is I; stiffness scales as EI (S CEI/ 3 ) CEI S 3

13 Shape efficiency: Bending stiffness Take ratio of ending stiffness S of shaped section to that (S o ) of a neutral reference section of the same cross-section area Define a standard reference section: a solid square with area (alternatively: solid circular section) Second moment of area is I; stiffness scales as EI (S CEI/ 3 ) 4 Io 1 1 rea is constant rea rea and modulus E unchanged Define shape factor for elastic ending, measuring efficiency, as φ e B S S o EI EI o I 1 Shape efficiency: Bending stiffness

14 Properties of Shape actor The shape factor is dimensionless -- a pure numer It characterizes shape Increasing size at constant shape Each of these is roughly times stiffer in ending than a solid square section of the same cross-sectional area Taulation of Shape actors φ e B S S o EI EI o I 1 (standard reference section: solid square section) 4 o I 1 1

15 Taulation of Shape actors φ e B S S o EI EI o I 4π (standard reference section: solid circular section) 4 Io π r 4 4π Shape efficiency: Bending strength Take ratio of ending strength (failure moment) M f of shaped section to that (M f,o ) of a reference section (solid square) of the same cross-section area Z Section modulus for ending is Z; strength (M f ) scales as σ y Z (M f σ y Z) I Z y max o rea 6 3/ rea is constant rea and yield strength σ y unchanged Define shape factor for failure in ending, measuring efficiency, as φ f B M M f fo σ σ y y Z Z o Z 6 3/

16 Shape efficiency: Bending strength Shape efficiency: Twisting stiffness Take ratio of twisting stiffness S T of shaped section to that (S T,o ) of a reference section (solid square) of the same cross-section area T KG S T θ Torsional moment of area is K ( J for circular sections); stiffness scales as KG 3 h o 1 0,58 0,14 K 3 h h rea rea and modulus G unchanged Define shape factor for elastic twisting, measuring efficiency, as φ e T S S T T,o KG K G o K 7,14

17 Shape efficiency: Twisting stiffness Shape efficiency: Twisting strength Take ratio of twisting strength (failure torque) T f of shaped section to that (T f,o ) of a reference section (solid square) of the same cross-section area Section modulus for twisting is Q; strength (T f ) scales as τ Q (T f τ Q) Q J r max (for circular sections only) Q 3 o h 3h + 1,8 4,8 h rea 3/ 4,8 rea and strength τ unchanged Define shape factor for failure in twisting, measuring efficiency, as φ f T Tf T f,o τ Q τ Q o Q 4,8 3/

18 Shape efficiency: Twisting strength Shape efficiency: Resistance to uckling Take ratio of critical load (Euler load) cr of shaped section to that ( cr,o ) of a reference section (solid square) of the same cross-section area EI min Critical load ( cr ) scales as ( cr π EI min / e ) e The shape factor is the same as that for elastic ending ( φ B), with I replaced y I min Imin, o Io 1 rea 4 1 rea and modulus E unchanged Define shape factor for resistance to uckling, measuring efficiency, as φ Bck cr cr,o EI EI min min,o I 1 min

19 Taulation of Shape actors imits for Shape actors If you wish to make stiff, strong structures that are efficient (using as little material as possile) then make shapes with shape factors as large as possile Two types of limit for shape factors - manufacturing constraints (processaility of materials) - mechanical staility of shaped sections

20 imits for Shape actors If you wish to make stiff, strong structures that are efficient (using as little material as possile) then make shapes with shape factors as large as possile Two types of limit for shape factors - manufacturing constraints (processaility of materials) - mechanical staility of shaped sections In seeking greater efficiency, a shape is chosen that raises the load required for the simple failure modes (yield, fracture). But in doing so, the structure is pushed nearer the load at which new failure modes ecome dominant. ocal uckling Modulus Theoretical limit: φ e B E.3 σ y f φ B φ e B Yield strength What values of φ Be exist in reality? e I φ B 1 log I φ B ( ) log( ) + log 1 Slope e z x CEI S xx 3 I xx > I xx > I xx x

21 What values of φ Bf exist in reality? f Z φ B 6 3/ log 3 φ B ( Z) log( ) + log 6 Slope 3/ f z x I Z y xx max I xx > I xx > I xx x What values of φ Be exist in reality? Data for structural steel, 6061 aluminium, pultruded GRP and wood φ ϕ e 100 Be ϕ e Second Second Moment moment of rea of (major), area, I_max I (m (m^4) 4 ) e-003 1e-004 1e-005 1e-006 1e-007 1e-008 1e-009 1e-010 1e-011 1I ϕ ϕ log( ) log() log e e I + 1 Pultruded GRP I-section Pultruded GRP Channel Pultruded GRP ngle Steel tue Extruded l-channel Extruded l I-section Extruded l-tue Extruded l-angle Extruded l -angle Steel Universal Beam Pultruded GRP tue Glulam rectangular Steel tue Softwood rectangular 1e-005 1e-004 1e Section rea, (m^) ) ϕ e 1 Slope

22 Indices that include shape unction Beam (shaped section) rea Ojective Constraint ϕ Minimise mass, m, where: m ρ Bending stiffness of the eam S: 3 CEI S S I 3 CE I is the second moment of area: e I 1I 1 ϕe Comining the equations gives: 1/ m mass area length ρ density edge length S stiffness I second moment of area E Youngs Modulus 1 S m C 5 1/ ρ ( ) 1/ ϕee Chose materials with smallest ρ ( ) 1/ ϕee Selecting material-shape cominations Materials for stiff, shaped eams of minimum weight ixed shape (ϕ e fixed): choose materials with low ρ 1/ E Shape ϕ e a variale: choose materials and shapes with low ρ ( ϕ ) 1/ e E Material ρ, Mg/m 3 E, GPa ϕ e,max 1/ /E 100 Steel T4 l GRP Wood (oak) ρ ρ/ ( ϕ ) 1/ e E,max Commentary: ixed shape (up to ϕ e 8): wood is est Maximum shape (ϕ e ϕ e,max ): l-alloy is est Steel recovers some performance through high ϕ e,max

23 Selecting material-shape cominations log e I φ B 1 φ B ( I) log ( ) + log 1 e Selecting material-shape cominations S E I log ( E) - log ( I) + log ( ) S

24 Selecting material-shape cominations m/ ρ log ( ) - log ( ρ) + log ( m/ ) Selecting material-shape cominations Required section stiffness: EI 10 6 N.m Shape factor: φ B e 10

25 Selecting material-shape cominations Required section stiffness: EI 10 6 N.m Shape factor: φ Be 10 Selecting material-shape cominations Required section stiffness: EI 10 6 N.m Shape factor: φ B e

26 Selecting material-shape cominations Required section stiffness: EI 106 N.m Shape factor: φbe 30 Selecting material-shape cominations Required section strength: σyz > Vmin

27 Selecting material-shape cominations Required section strength: σ y Z > V min Selecting material-shape cominations Selection with fixed shape

28 Selecting material-shape cominations Selection with variale shape Shape S Selecting material-shape cominations four 4 S When the groups are separale, the optimum choice of material and shape ecomes independent of the detail of the design. It is the same for all geometries G and all values of functional requirements. The performance for all and G is maximized y maximizing f 3 (M) and f 4 (S).

29 Selecting material-shape cominations four 4 S In theory f 4 (S) is independent of the material (shape factors depend on shape only). In reality the shape factors depend on material (ecause of constraints from material-process-shape relations, and limits from processaility and mechanical ehaviour of material which form the shape), therefore f 3 (M). f 4 (S) constitutes the new performance index. Shaped material can e considered as a new material with modified (improved) properties. Shape on selection charts Note that ρ ρ/ ϕ 1/ ( ϕ E) ( E/ ϕ ) 1/ ( E ) 1/ e e * e ρ* New material with ρ* ρ/ ϕe E* E / ϕe Young s modulus (GPa) Young's Modulus (typical) (GPa) Rigid Polymer oams l: ϕ e 1 Bamoo Wood Plywood Cork PP Silicon Caride lumina Boron Caride Silicon l alloys Mg alloys CRP GRP EV Titanium Steels Concrete PET PVC PUR PE PTE Silicone Tungsten Carides Nickel alloys Copper alloys Zinc alloys ead alloys ρ C E 1 / 1e-003 lexile Polymer oams Polyisoprene Butyl Ruer Polyurethane Neoprene 1e Density (typical) (Mg/m^3) Density (Mg/m 3 )

30 Shape on selection charts Note that ρ ρ/ ϕ 1/ ( ϕ E) ( E/ ϕ ) 1/ ( E ) 1/ e e * e ρ* New material with ρ* ρ/ ϕe E* E / ϕe Young s modulus (GPa) Young's Modulus (typical) (GPa) E l / l: ϕ e 44 Rigid Polymer oams l: ϕ e 1 Bamoo Wood Plywood Cork PP Silicon Caride lumina Boron Caride Silicon l alloys Mg alloys CRP GRP EV Titanium Steels Concrete PET PVC PUR PE PTE Silicone Tungsten Carides Nickel alloys Copper alloys Zinc alloys ead alloys ρ C E 1 / 1e-003 lexile Polymer oams Polyisoprene Butyl Ruer Polyurethane Neoprene 1e-004 ρ l / Density (typical) (Mg/m^3) Density (Mg/m 3 ) Data organisation: Structural sections Standard prismatic sections Kingdom amily Material and Memer ttriutes Structural sections ngles Channels I-sections Rectangular T-sections Tues Extruded l alloy Pultruded GRP Structural steel Softwood Material properties ρ, E, σ y Dimensions... Section properties: I, Z, K, Q... Structural properties: EI, σ y Z, GK... record

31 Part of a record for a structural section Pultruded GRP Vinyl Ester (44 x 3.18) Material properties Price GBP/kg Density Mg/m^3 Young's Modulus17-18 GPa Yield Strength MPa Dimensions Diameter, B m Thickness, t.54e e-3 m Section properties Section rea, 3.3e e-004 m^ Second Moment of rea (maj.), I_max 7.11e e-007 m^4 Second Moment of rea (min.), I_min 7.11e e-007 m^4 Section Modulus (major), Z_max 3.3e e-006 m^3 Section Modulus (minor), Z_min 3.3e e-006 m^3 Etc. Structural properties Mass per unit length, m/l kg/m Bending Stiffness (major), E.I_max N.m^ Bending Stiffness (minor), E.I_min N.m^ ailure Moment (major), Y. Z_max N.m ailure Moment (minor), Y. Z_min N.m Etc. Example: Selection of a eam Specification B x D unction Constraint Beam Required stiffness: EI max > 10 5 N.m Required strength: σ y Z > 10 3 N.m Dimension B < 100 mm D < 00 mm m a mass/unit length C a cost/unit length D eam depth B width I second moment of area E Young s modulus Z section modulus σ y yield strength Ojectives (a) ind lightest eam () ind cheapest eam

32 pplying constraints with a limit stage Dimensions Minimum Maximum Depth D 00 Width B 100 mm mm Section attriutes Bending Stiffness E.I 100,000 N.m ailure Moment Y. Z 1000 N.m Optimisation: Minimising mass/length 1e+009 Bending Bending Stiffness Stiffness (major), E.I_max (Nm (N.m^) ) 1e+008 1e+007 1e Selection ox E.I max 10 5 Nm Extruded l I-section Extruded l-tue Steel Universal Beam Steel Rect.Hollow Steel tue Extruded l ngle Steel Equal ngle 10 Pultruded GRP tue Bending Stiffness EI vs. mass per unit length Mass per unit length, m/l (kg/m) Mass per unit length (kg/m)

33 Results: Selection of a eam OUTPUT: ojective minimum weight Extruded aluminium ox section, YS 55 MPa (15 x 56 x 3.0 mm) Extruded aluminium ox section, YS 55 MPa (135 x 35 x 4.0 mm) Extruded aluminium ox section, YS 55 MPa (15 x 44 x 3. mm) Extruded aluminium ox section, YS 55 MPa (15 x 64 x 3. mm) OUTPUT: ojective minimum cost Sawn softwood, rectangular section (150 x 36) Sawn softwood, rectangular section (150 x 38) Sawn softwood, rectangular section (175 x 3) Sawn softwood, rectangular section (00 x ) The main points When materials carry ending, torsion or axial compression, the section shape ecomes important. The shape efficiency quantify the amount of material needed to carry the load. It is measured y the shape factor, φ. If two materials have the same shape, the standard indices for 1/ ending (eg ρ /E ) guide the choice. If materials can e made -- or are availale -- in different shapes, then indices which include the shape (eg ρ /( φe) 1/ ) guide the choice. The CES Structural Sections dataase allows standard sections to e explored and selected.