Hooke's Law Hooke's Law && Poisson's Ratio Poisson's Ratio

A lecture assembled for the course onStatics and Strength of Materials

by Jason E. Charalambides PhD, PE, M.ASCE, AIA, ENV_SP

Data composed exclusively by author(only for educational purposes)

2

Hooke's Law

4

Hooke's LawHooke's Law

Relation of Stress to Volume and Form: This brings to mind how the Classical Doric order and

the ancient architects gave so much emphasis on thevisual aspects of their designs where theyimplemented the effect of "entasis" on the columns;the bulging of the columns at mid to lower height.That accentuates a visual effect that can beanticipated when a form like the Doric column issubjected to extreme loads

By study of the example of the temple of Hera atPaestum, it is evident that the architect wanted toreveal through the architectural forms the stressesthat these columns were subjected to. The echinus ofthe capital is formed like a funnel, suggesting a fluidquality to the stresses that the column is receiving.

5

Hooke's LawHooke's Law

Relation of Stress to Volume and Form: Underneath that there is a form that would very

correctly translate to a hyperbolic curve, precisely theway it needed to be in order to efficiently distributethe stresses into the shaft. As the loads are addedby each drum, the girth of the column continues togrow in a this curvilinear fashion down to a pointclose to the column's mid-height where the growth ofthe girth becomes more linear, providing adequatearea to diminish the stress.

The ancient architect was certainly exaggerating thiseffect, and from the archaic and classical architecturewe see a more moderate statement in the Hellenisticera where this effect was minimized, giving more of amessage of lightness of forms.

6

Hooke's LawHooke's Law

Relation of Stress to Volume and Form: In the case of the temple we see the contraction in length

and expansion in girth that perfectly aligns with the logic ofHooke's theory and what we have addressed during lastsession with the necking effect of steel failing in tension. Inthis case, the element was expanding under compression

There is however a very strong connection to theunderstanding of material qualities represented by theancient architect and mason, that is taken to a level that laterperiods would not match.

Similar statements by the architect of the Baroque era, suchas the one we see on the entry to a house on Rue Nationalin Marseille, whilst beautiful and very gracefully executed,become more literal and direct, lacking the depth of what theancient classical orders were encompassing.

7

Poisson's Ratio

8

Poisson's RatioPoisson's Ratio

What is “ν”? Stone and marble may not be the materials that

come to mind when such effects are to bevisualized. The reason behind that is becausethese are brittle materials. Although neither of thetwo materials should be totally discredited for anylevel of elasticity, a much larger visual evidence willbe given from substances such as metals andalloys.

That aesthetic aspect has a very direct relation towhat is called the Poisson's ratio "ν".

where “ν” is Poisson's ratio and “ε” is thestrain either axial or lateral

ν=−εlateralε axial

9

Poisson's RatioPoisson's Ratio

“ν” is Poisson's Ratio! As seen in the formula, Poisson's ratio is the ratio of

lateral deformation that a material experiences as itcompensates to the axial deformation that occurs dueto the stress that is applied on it, e.g. a column bulgesoutwards as it is subjected to axial load and shortens.

Note the +ve sign on the numerator indicatingthat an elongation (being +ve) wouldcorrespond to a thinning effect (-ve) toproduce a +ve value to Poisson's ratio.

The ratio of most materials ranges between0.1 and 0.45, excluding special cases suchas Cork or special auxetic materials.

ν=−εlateralε axial

10

Poisson's RatioPoisson's Ratio

“ν” is Poisson's Ratio!

The table provides some typical values ofthe Poisson's ratio for a number ofmaterials:

Typical values of Poisson's ratioMaterial Poisson's ratio

Rubber ~ 0.5Gold 0.42Saturated clay 0.40–0.50Magnesium 0.35Titanium 0.34Copper 0.33Aluminum-alloy 0.33Clay 0.30–0.45Stainless Steel 0.30–0.31Steel 0.27–0.30Cast Iron 0.21–0.26Sand 0.20–0.45Concrete 0.2Glass 0.18–0.3Cork ~ 0

11

Poisson's RatioPoisson's Ratio

“ν” is Poisson's Ratio!

Poisson's ratio formula suggests alreadythat some strain will be exerted on anelement in the directions perpendicular tothe axis or the axes on which loadingoccurs.

This relation will have to incorporate stress“σ”, strain “ε”, Young's modulus of elasticity“Ε”, and Poisson's ratio “ν”.

The generalized formula for Hooke's law isthe following, adjusted for each axis of thethree dimensions

ε x=σ x−ν⋅(σ y+σ z)

E

ε y=σ y−ν⋅(σ z+σ x)

E

ε z=σ z−ν⋅(σ x+σ y)

E

13

Poisson's RatioPoisson's Ratio

In class example:

A cube of steel that has dimensions of 5” ·5” ·5” issubject to pressure of 80000psi on the z axis and50000psi on the y axis. No pressure is appliedalong the x axis.

The Modulus of Elasticity of Steel is 29000ksi andPoisson's ratio is 0.3. Determine the length of thiselement along the x axis at the time it remainsunder this set of stresses:

14

Poisson's RatioPoisson's Ratio

In class example:

The strain along the x axis can be calculated by the use of theabove formula:

So the change in length will be 0.0013 inches per inch oforiginal length. With 5 inches length → Δx of 5*0.0013=0.0067.Therefore the new length of the x axis will be:

ε x=σ x−ν (σ y+σ z)

E=0 psi−0.3⋅(50000 psi+80000 psi)

29000000 psi=0.0013in

Lx=Lx0+Δx=5.0067in

15

Poisson's RatioPoisson's Ratio

In class example:

i.e. the element will slightly elongate along the x axis. For thesake of argument the final lengths along the y and the z axiswill be 4.9955in and 4.9888in respectively.

With a modulus of elasticity as high as that of Steel, loads ofsuch magnitude cause very little deformation to be noticeableto the naked eye at these scales. In a scenario where 80ksi were applied on a steel column of 10', the deformation alongthe longitudinal axis would be of the magnitude of ⅓in .