Materials and Shape. Part 1: Materials for efficient structure. A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka. Learning Objectives

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1 MME445: Lecture 27 Materials and Shape Part 1: Materials for efficient structure A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka Learning Objectives Knowledge & Understanding Understand the concept of shape and efficiency Skills & Abilities Ability to select efficient material-shape combinations Values & Attitudes Awareness of how material and shape interact Resources M F Ashby, Materials Selection in Mechanical Design, 4 th Ed., Ch. 09 1

2 Outline of today s lecture Introduction and synopsis Shape factors Limits to shape efficiency Exploring material shape combinations Introduction & Synopsis Design performance or function is determined by the combination of materials and process BUT, do not underestimate the impact of shape MATERIAL FUNCTION PROCESS SHAPE Shapes For Tension, Bending, Torsion, Buckling Shape Factors Performance Indices with Shape 2

3 Performance isn't just about materials - shape can also play an important role A material can be made stiffer and stronger when loaded in bending or twisting by shaping it into an I-beam or a hollow tube It can be made less stiff by flattening it into a flat leaf or winding it into a wire or helix Thinner shape helps dissipate heat; cellular shapes help conserve it Section shape becomes important when materials are loaded in bending, in torsion, or are used as slender columns. Shape can be optimized to maximize performance for a given loading condition Simple cross-sectional geometries are not always optimal Shaped sections can be made more efficient (i.e., uses less material) to carry load than a solid section. This lecture extends selection methods so as to include shape. This is not needed if we compared different materials with the same shape. But when two different materials are available, each with its own section shape, the more general problem arises: How to choose the best combination? from among the vast range of materials and the section shapes that are available or could potentially be made. A solid wood bicycle is certainly lighter for the same stiffness than a solid steel one, but is it lighter than one made of steel tubing? Might a magnesium I-section be lighter still? 3

To optimize both shape and material for a given loading condition, a shape factor is defined. A material can be thought of as having properties but no shape.

when shape is a variable, the shape factor appears in the expressions for the indices. they let you compare shaped materials and guide the choice of the best combination of material and shape.

matter area (A) and shape (J) both matter area (A) and shape (I min ) both matter The choice of section shape is linked to the mode of loading.

4 To optimize both shape and material for a given loading condition, a shape factor is defined. A material can be thought of as having properties but no shape. A structure is a material made into a shape (Figure 9.2). Shape factors are measures of the efficiency of material usage. when shape is a variable, the shape factor appears in the expressions for the indices. they let you compare shaped materials and guide the choice of the best combination of material and shape. Loading Conditions & Shape Different loading conditions are enhanced by maximizing different geometric properties shape not important, but the area (A) is area (A) and shape (I xx, I yy ) both matter area (A) and shape (J) both matter area (A) and shape (I min ) both matter The choice of section shape is linked to the mode of loading. I-sections or hollow boxes are better in bending than solid sections. Circular tubes are more efficient in torsion than either solid or I-sections. Tubes and box sections are good as columns, though I-sections are often used because their entire surface can be reached for painting and inspection, whereas the inner surface of tubes and boxes cannot 4

5 Certain materials can be made to certain shapes. What is the best material/shape combination (for each loading mode)? To characterize this we need a metric a way of measuring the structural efficiency of a section shape, independent of the material of which it is made. An obvious metric f (phi) is that given by the ratio of the stiffness or strength of the shaped section to that of a neutral reference shape a solid square section with the same cross-sectional area A can be taken as the reference shape 5

Properties of Shape Factor Shape factor is a dimensionless, pure number it characterizes the efficiency of a section shape in a given mode of loading Shape factor depends on

A shaped beam of shape factor for elastic bending, φ B e = 10, is 10 times stiffer than a solid square section beam of the same cross-sectional area increasing size at constant

6 Properties of Shape Factor Shape factor is a dimensionless, pure number it characterizes the efficiency of a section shape in a given mode of loading Shape factor depends on shape, not on scale But, what is the meaning of φ B e = 10? A shaped beam of shape factor for elastic bending, φ B e = 10, is 10 times stiffer than a solid square section beam of the same cross-sectional area increasing size at constant shape i.e., constant f φ B e = shape factor for elastic bending φ B f = shape factor for failure due to bending φ T e = shape factor for elastic torsion φ T f = shape factor for failure due to torsion They all equal to 1 for solid square section Shapes having values >1 are better, more efficient 6

efficiency It is achieved by spreading the material in a plane containing the axis of bending to form sheets or wires Shape factor for elastic bending Bending stiffness

7 But it is not always high stiffness that is wanted. Springs, suspensions, cables, and other structures that must flex yet have high tensile strength rely on having a low bending stiffness Then we want low shape efficiency It is achieved by spreading the material in a plane containing the axis of bending to form sheets or wires Shape factor for elastic bending Bending stiffness of a beam ; I 0 = second moment of area for a reference beam of square section b 0 = edge length of reference square section A = section area = b 0 2 Define shape factor as the ratio of the stiffness of the shaped beam to that of a solid rectangular section with the same cross-sectional area. 7

Define a standard reference section: a solid square bar Area A is constant A 0 = b 0 2 I 0 = A 0 2 12 b b neutral section shaped sections Then, comparing sections of same area (A = A 0 ), the shape

8 Define a standard reference section: a solid square bar Area A is constant A 0 = b 0 2 I 0 = A b b neutral section shaped sections Then, comparing sections of same area (A = A 0 ), the shape factor for elastic bending S = stiffness of cross-section under question S 0 = stiffness of reference solid cross-section Notice that shape factor is dimensionless. Effect of section shape on bending stiffness, EI Same area, but 2.5 times stiffer Same stiffness, but 4 times lighter 8

9 Example: Elastic shape factors of I-beam t = 1/8 b 0 h = 3 b 0 2t A 0 = b 0 2 Let, b 0 = 1 Then, A 0 = 1 For these dimensions, the shape increased stiffness over 13 times while using the same amount of material! A = 2 t (h + b) ; A = A 0 = 1 1 I = h 3 b t h = e 12 I f = = 13.5 B A 2 b = 1 Shape factor for bending failure/strength Failure of a beam in bending The stress s is largest at the point y m on the surface of the beam that lies furthest from the neutral axis: M = bending moment Z = section modulus = I/y m Define structure factor (a.k.a. strength efficiency) as the ratio of the failure moment of the shaped beam to that of reference beam: f B M f f = = M fo Z Z 0 The section modulus of the reference beam of square section with the same cross-sectional area, A The beam fails when the bending moment is large enough for σ to reach the failure stress of the material: Then the shape factor for bending failure M f = Zs f 9

10 Z = failure bending moment Q = failure twisting moment I = elastic bending moment K = elastic twisting moment 10

11 11

12 Given that, a column of length L, loaded in compression, buckles elastically when the load exceeds the Euler load 12

Limits to Shape Efficiency The conclusions so far: If you wish to make stiff, strong structures that are efficient (using as little material as possible), make the shape-efficiency factors as large

13 Limits to Shape Efficiency The conclusions so far: If you wish to make stiff, strong structures that are efficient (using as little material as possible), make the shape-efficiency factors as large as possible. It would seem, then, that the bigger the value of f the better. True, but there are limits. Two ways to examine these limits: 1 the empirical way of examining the shapes in which real materials are actually made, recording the limiting efficiency of available sections. 2 by the analysis of the mechanical stability of shaped sections. What values of f e do exist in reality? How much you can shape it 12 I e log( I) 2log(A) log e 2 A 12 Al max 44 Steel max 65 e 100 I e f slope 2; intercept = φ/12 Timber max 8 How big is the structure 13

14 Section Modulus (major), Z_max (m^3) Actual (empirical) values for f f? Section modulus Z f e-3 Z 6Z max 13 f log(z) 1.5log( A) log 3/ 2 Z A 6 o guide lines slope 1.5 Lines slope 1.5 Aluminium max 10 Steel Hot Rolled Steel Beam Timber max 3 1e-4 I 100 f 10 f 1e-5 1e-6 1e-7 Extruded Aluminium Timber, rectangular section Section area A 1 f 1e-4 1e Section Area, A (m^2) f 0.1 Practical (empirical) limits for f e and f f Material Max f e Max f f Steels Aluminium alloys GFRP and CFRP 39 9 Unreinforced polymers 12 5 Woods 8 3 Elastomers < 6 - Other materials can calculate 14

15 Theoretical limits for f e and f f φ B b 2.3 E σ y φ B f max. φ B b max. Material E s y f e Steels (65) Aluminium alloys (44) GFRP and CFRP (39) Unreinforced polymers (12) Theoretical Limit for f e Empirical Maximum fe What sets the upper limit on shape efficiency? And why does the limit depend on material? Limit set by : 1. The difficulty of making them a manufacturing constraint. Steels can be drawn to thin-walled tubes or I-sections with shape factor as high as 50 Wood cannot; plywood can be made into thin tubes or I-sections with shape factor as high as 5 only 2. Local buckling a more fundamental constraint on shape efficiency 15

$Examples of structures shaped to their limit Inefficient sections fail in a simple way - they yield, they fracture, or they suffer large-scale buckling.$

16 Examples of structures shaped to their limit Inefficient sections fail in a simple way - they yield, they fracture, or they suffer large-scale buckling. When efficient shapes can be fabricated, the limits of the efficiency are set by the competition between failure modes. Coke can. Dominant failure mode? Local buckling Egg shell? Local buckling / crushes locally Drinking straw/garden hose? Kink (form of local buckling) chessboard buckling Material limits to shape factors Tubes, axial loading Load factor = F/L 2 s y Yield Local buckling Global buckling Shape factor 16

17 Next Class MME445: Lecture 28 Materials and Shape Part 2: Materials indices with shape factor 17

Engineering Science OUTCOME 1 - TUTORIAL 4 COLUMNS

Engineering Science OUTCOME 1 - TUTORIAL 4 COLUMNS Unit 2: Unit code: QCF Level: Credit value: 15 Engineering Science L/601/10 OUTCOME 1 - TUTORIAL COLUMNS 1. Be able to determine the behavioural characteristics of elements of static engineering systems