Modes of Loading - mmrl.ucsd.edummrl.ucsd.edu/Courses/SE104/SE104_4_StiffnessLimitedDesign.pdf · 9...

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Materials: engineering, science, processing and design, 2nd edition Copyright (c)2010 Michael Ashby, Hugh Shercliff, David Cebon


Modes of Loading

(1) – tension (a)

(2) – compression (b)

(3) – bending (c)

(4) – torsion (d)

and

combinations of them (e)

Figure 4.2

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Standard Solution to Elastic Problems


Figure 5.1

Three common modes of loading:

(a)– tie with a circular cross-sectionloaded in tension

(b) – beam with rectangular cross-section loaded in bending

(c) – shaft of circular cross-sectionloaded in torsion

Elastic Extension or Compression


Relation between load, deflection and stiffness

Shape of cross-section does not matter becausethe stress is uniform across the section

Stiffness of material (a free variable – choice of material: a parameter that we need to find out)

Stiffness of structure (a functional requirement)

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Elastic Bending of Beams


Curvature of initial straight axis………………………

Stress cause by bending moment in a beam with a given Young’s modulus………………

Second moment of inertia…………………………….

u: displacement parallel to y-axis M: bending moment y: distance from neutral axis

Cross-sectional Area and Second Moments for Four Shapes


Figure 5.2

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Figure 5.3

Stiffness of a beam undertransverse loading

C1 is the only value that depends on the distribution of the load

= FL3/C1EI

Buckling of Columns and Plates


If sufficiently slender, an elastic column or plate, loaded in compression, fails by elastic buckling at a critical load

Figure 5.5

Value of n depends on the endconstraints of the beam

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Torsion of Shafts


Figure 5.4

τ: shear stressr: radial distance from axis of symmetryK: resistance to twisting (torsional

equivalent of I)G: shear modulus = (3/8)E when = 1/3T/θ: torsional rigity

L

GKTST

Shear stress

= TL/GK

Vibrating of Beams and Plates


Figure 5.6

An undamped system vibrating at its natural frequency can be expressedas the simple problem of a mass, m, attached

to a spring of stiffness, k

Lowest natural frequencyof system

Natural frequency based on spring stiffness, k, of a beam in bending

mo is the mass per unit length or (area x density);therefore, frequencies scalewith (E/ρ)1/2

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Material Indices for Elastic Design


Figure 3.6

Minimizing Weight

• Length, Lo, is specified

• Cross-sectional area must be A0

• Must carry a tensile force F0 without extending elastically by more than δ0 Stiffness of the structure must be at least S* = F0/δ0

• Must have some toughness

• Should be as lightweight as possible


A light, stiff tie-rod:

Figure 5.7aDesign Requirements

Constraints

Objective

Free Variable: Choice of Materials In this course, choice of material (, E, y…) is always a free variable

Uncoupled: Can be used to screen out irrelevant choices, later on

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Compare to: No free variable –might not have any solution!

Constraints: S*, L0, A, must be 6061 Al alloy…Objective: Mass (m)Free variables: N/A

Or: Only one free variable – group all the materials parameters in a single item, based on which materials candidates can be ranked

Constraints: S*, L0, A… Eliminate irrelevant choicesObjective: Mass (m) Rank relevant candidatesFree variables: Choice of material (toughness, mass density…)

Complicated case: There are 2 or more free variables affecting both constraints and objective Must derive the performance index (discussed later),


Objective function: equation that describes the quantity

to be maximized or minimized

The goal is to minimize the value of the objective functionwithin the given constraints

Constraint: Section area A must be sufficient to provide a stiffness of S*

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If there were only one free variables (an “easy” problem):Constraints: S* (stiffness of rod), L0, A0… Eliminate irrelevant choicesObjective: Mass (m) Rank relevant candidatesFree variables: Choice of material

Figure 5.10

1) Use constraints to screen: E* = S*L0/A0

(e.g. 10 GPa)

2) Use objective to rank

One Free Variable

Optimum choice: natural material (wood)

Minimizing Weight

• Length, Lo, is specified

• Cross-sectional area must be A0

• Must carry a tensile force F0 without extending elastically by more than δ0 Stiffness must be at least S* = F0/δ0

• Must have some toughness

• Should be as lightweight as possible


A light, stiff tie-rod:

Figure 5.7aDesign Requirements

Constraints

Objective

Free Variable: Choice of MaterialsCross-sectional area, A

Free variables affecting constraints or objective are listed here

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Eliminate the free variable A to obtain:

Objective functionConstraint

S* and Lo are both specified; the lightest tie that will provide a stiffness S* is that made of the material

with the smallest value of ρ/E

There must be at least one free variable (degree of freedom); otherwise the design is impossible.

Two free variables; one of them should be “eliminated”

Only one free variable (choice of material) left, grouped in a single term (perfor-mance index)

Two Free Variables With unknown A, constraint on E cannot be derived from S*

High values of Mt are the best choice; the function E/ρ is called the specific stiffness

Performance Index for material selection


Eliminate the free variable A to obtain:

Objective functionConstraint

There must be at least one free variable (degree of freedom); otherwise the design is impossible.

Two free variables; one of them should be “eliminated”

Only one free variable (choice of material) left, grouped in a single term (performance index)

Two Free Variables With unknown A, constraint on E cannot be derived from S*

The smaller the mass density (), the larger the performance index, Mt (the smaller the total mass, m)

The larger the Young’s modulus (E), the smaller the cross-sectional area (A) needs to be to reach the required S*, resulting in a smaller total mass, m

Performance Index for material selection

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Figure 5.10

Mt = E/

Constant line for Mt or log(Mt) is:

log(E) – log() = log(Mt0)orlog(E) = log() + log(Mt0)ory = x + c in the left figure

In the family of parallel lines of y = x+c, the higher the line, the larger the value of Mt0.

But it cannot exceed the area of bubbles.

Higher

Lower

Highest

Lowest

Two Free Variables

Optimum choice: Ceramics (e.g. alumina)


If: When the force is Fmax, the elongation must be close to L0;A must be in the range of A0A; L is a free variable

Objective function Constraint

The material performance index becomesMt = (E)

Mt collectively sets the criterion of material selection

Two Free Variables

A smaller E leads to a shorter rod (and thus, a lower mass), to achieve S*

A smaller leads to a lower mass, m

=

m = A2(E)/S*

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Two Free Variables

Figure 5.10

Mt = E

Constant line for Mt or log(Mt) is:

log(E) + log() = log(Mt0)orlog(E) = -log() + log(Mt0)ory = -x + c in the left figure

In the family of parallel lines of y = -x+c, the higher the line, the larger the value of Mt0.

But it cannot exceed the area of bubbles.

Lowest

Highest

Optimum choice: foams (e.g. PU foam)

Minimizing Weight: A light, stiff panel

• Length L and width b are specified

• Thickness h is free

• Loaded by bending by a central load F

• Stiffness constraint requires it must not deflect more

than δ under the load

• Objective it to make the panel as light as possible


Figure 5.7b

Design Requirements

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Table 5.2

Objective function:

Stiffness Constraint:

Second moment of inertiafor rectangular section:

Material Index: Light, Stiff Panel


Stiffness S*, length L, and width b arespecified; thickness h is free

Eliminating h from the objective function gives:

Unspecified variables leaves ρ/E1/3 which would be minimized to minimize mass; as before the function will be inverted so a

maximum value will be sought

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Figure 5.10

Ranking: indices on charts

Selection lines are used based on the material indices

All materials that lie on the selectionline perform equally well; those

that lie above the line perform better


Figure 5.11

For the selection of a light, stiff panel, the material index is E1/3/ρ

A material with M = 3 givesa panel that has one-tenththe weight of one with M = 0.3

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Minimizing WeightA light, stiff beam


Figure 5.7c

Table 5.3


Objective function:

Stiffness constraint and second moment of inertia:

Eliminating the free variable A from the objective functionand identifying the unspecified variables

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Figure 5.10

Ranking: indices on charts

Selection lines are used based on the material indices

All materials that lie on the selectionline perform equally well; those

that lie above the line perform better

Shape Factor


Figure 5.8

By reshaping the cross-section of a beam, it is possible to increase I – thus increasing

stiffness – without increasing the total area

The ratio of I for the shaped section to that for a solid square section

with the same area is defined asthe shape factor Φ

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Shaping is also used to make structures lighter; it is a way to achieve the same stiffness

with less material

Table 5.4

Shaping could increase the value of Iby 64X if steel was used in the constructionof a beam with a square cross-section; if woodwas used, the maximum increase would be 9X

Case Studies: Lever for corkscrew


Figure 5.12

Table 5.5

Lever is loaded in bending and acts as a light, stiff beam

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Figure 5.14

Based on Figure 5.14,the materials most

suited for a stiff, lightcorkscrew lever are

ceramics, composites,and woods

Other considerationssuch as cost will helpnarrow the selection

further

Selection of corkscrew lever by plottingmaterial index

E1/2/ = const log(E1/2/) = const logE = 2log + const

E1/2 = const log(E1/2) = const logE = -2log + const

Minimizing Materials Cost


Material indices change as the objective changes; the objective function for minimizing cost of the tie rod, panel, or beam would be:

With A (cross-sectional area) and L (length) being specified, the goal of a material selection would be to minimize Cmρ or

maximize 1/ Cmρ It is equivalent to using a weight factor of Cm

for the volume density () in the analysis of light, stiff beam.

Materials cost per unit massMaterials cost Mass

CV, materials cost per unit volume

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Plotting Limits and Indices on Charts


Figure 5.9

Screening: attribute limits on charts

Constraints can be plotted as horizontal or vertical lines on material property charts

Constraints in Figure 5.9:

E > 10 GPa

Relative Cost < 3

All materials in the searchregion meet both constraints

Case Studies: Structural materials


Figure 5.15

A stiff beam with minimum cost is required

Table 5.6

The analysis process is similar with that of a lightweight, stiff beam, except that the density is weighted by the materials cost per mass (Cm)

~ Cost per unit volume, CV

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Materials: engineering, science, processing and design, 2nd edition Copyright (c)2010 Michael Ashby, Hugh Shercliff, David CebonFigure 5.16

Selection of stiff floor beam

Concrete, stone, and brick

have poor tensile strength

Wood, steel, and reinforced

concrete have strength

in tension and compression

Steel is also has a higher

shape factor allowing

for greater freedom of

the form of the building

Selection of stiff bar stool shaft

What is the best material for the shaft in a bar stool?

Constraint: Cannot deform too much under external loading:

A. Tension?

B. Compression?

C. Bending?

D. Torsion?

Objective: minimize or maximize cost?

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Bendy Design: Part stiff, Part-Flexible Structures


Figure 5.17

Figure 5.18

Components can be designed to be strong but not stiff

Elastic bending locates and guidesthe plunger as well as provides the

restoring forced provided by the spring

Rotational hinges is replacedby elastic connecting strip


Figure 5.19

Various section shapes, eachallowing one or more degrees

of freedom, while retaining stiffness and strength in

the other directions

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