SE104 6 StrengthLimitedDesign - University of...

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


Structural behaviors

Materialbehaviors

Figure 4.19Atomic bond stiffness, Sb

Young’s modulus

Sb

Bond length at eq’m (atomic size)

Bond stiffness

Stiffness of structure

……

Stiffness

2


Structural behaviors

MaterialbehaviorsCritical resolved

shear stress (CRSS)

Yield strengthy = 3.1y

Figure 6.1

Maximum loadingFmax = yA

(tension/compression)Mmax = yZe (bending)Tmax = crK/R (torsion)……

Strength

Yielding of Ties and Columns


A tie rod is loaded in tension while a columnis loaded in compression

The state of stress within them is uniform andthe material remains elastic at stresses below

the yield strength

Fmax = yA

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Figure 7.1

Beam in Bending

If σmax exceeds σy then thecomponent is no

longer elastic

Mmax = yZe

In bending, the maximum longitudinal stress on a

beam or panel is

ym – distance from the neutral axisZe – elastic section modulus

Figure 7.2

At a moment well beyond the yield strength, a component will collapse by rotating about itsplastic hinge


Maximum moment

FL

FL/4

FL2/8

Fmax = Mmax/Ze

= FL/Ze

The most efficient design: removing all the ‘redundant” materials: Ze = bh2/6 (L-x)

x

Fmax = yZe/L

Fmax = 4yZe/L

Fmax = 8yZe/L2

Maximum Allowed Force

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Figure 7.3


Ze and Zp are functions of shape thathelp define the failure of a beam in bending –in either case, the moment required is Zσy

Cross-sectional Area and Second Moments for Four Shapes


Figure 5.2

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Figure 7.5

Spinning Disks (Flywheels)


Spinning disks store kinetic energy U – centrifugalforces generate a radial tensile stress in the disk

The disk yields when σmax

exceeds σy, and this definesthe maximum angular velocityand limits the potential for energy storage

Figure 7.4

Yielding of Shafts and Helical Springs


For a shaft loaded in torsion,failure occurs when the maximum shear stress exceeds the shear strength of the material

T – applied torqueR – radius of shaftK – polar second moment of area

Helical springs are a specialcase of torsional loading

S – spring stiffnessn – number of turns of wired – diameter of wireG – shear modulus of wireR – radiusF – applied axial forceu – extension of spring

Critical force to causethe onset of plasticity

Tmax = crK/R (cr~1/2y: shear strength)

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Helical Springs

u

F

Stored Energy U = ½Fu= ½F2/S= ½Su2

Design a helical spring that can store as much energy as possible without plastic damage, when it is subjected to a given force, F0. The wire diameter, d, and the coil length, n, are specified.

Objective: U = F02/2S Constraints: n, d, F0 = (/32)d3y/R

Free variables: Choice of material, R

U = [32(/32)3](d5/F0)(y3/G)

We need to maximize y3/G or y

3/E (EG); or to minimize E/y3

http://www.dikshasprings.com


Figure 7.9

E/y3=constant

3

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Helical Springs

u

F

Stored Energy U = ½Fu= ½F2/S= ½Su2

Design a helical spring that can store as much energy as possible without plastic damage, when it works at a given displacement, u0. The wire diameter, d, and the coil length, n, are specified.

Objective: U = Su02/2 = Gd4u0

2/(128nR3) Constraints: n, d, u0 = … Free variables: Choice of material, R

u0 = Fcrit/S = 2ynR2/(Gd) – Constraint function R = [Gdu0/(2yn)]1/2

U = […](y3/G)1/2 The same performance index: max(y

3/E) or min(E/y3)

http://www.dikshasprings.com

Elastic Hinge


Table 7.5

Stress caused by bending

Minimum radius

Best material is one that canbe bent to the smallest radius

Thickness

Radius

Strain caused by bending

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Figure 7.7

Stress Concentrations


Holes, slots, threads, and changes in section concentratestress locally – yielding will start at these places but theinitial yielding is not usually catastrophic

Stress concentration factor

σnom – nominal loadσmax – maximum local stressα – loading factorc – characteristic dimension ρsc – radius of curvature

Material Indices: Strong, Light Tie-Rod


Table 7.1

ObjectiveMinimize mass

ConstraintsLength is specifiedMust not fail under F

Free VariablesMaterial choiceSection area (eliminate)

Maximize

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Figure 7.8

Strength - Density


Strong, Light Panel


Table 7.2

Same procedure

• Set up an equation for mass• Find an expression for themaximum stress

• Use this express to eliminate thefree variable h

Maximize

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Light, Strong Beam


Table 7.3

For a beam, we are free to changethe shape as well as the dimensions

of the cross-section

For a square cross-section:

SE104 6 StrengthLimitedDesign - University of...

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