BE 5, Lecture 8Geometric Properties I

Part I: Beams

force

Types of beam:

‘cantilever’ ‘simple’

supported on one end supported onboth ends

3 point test:

Force deforms beam into arc of radius, R.

neutral plane

tension

R

ycompression

What is strain in section?

(y) = L/L = 2 (R+y) -2 (R) = y/R2 R

Thus, strain a distance y from neutral axis = y/R.If y is negative, strain changes sign:

-above neutral axis – compression-below neutral axis -- tension

What is stress in section?

(y) = E = E y/R

Very important principle in structural engineering

mg

tension

compression

What is stiffness of beam?

yb

y

dAyt

neutralplane

Calculate moment about neutral plane:

IR

EdAy

R

EM

dAyR

EM

dAyR

EdM

ER

y

t

b

t

b

y

y

y

y

2

2

2

dAyI 2Second moment of area:

beam equations:

yI

My

R

E

R

EIM ,,

Note:Larger for structures withstuff far away from neutralaxis.

What does beam equation tell us?

R R = infinite

EI

M

Rdeflectiong

1= flexural stiffness

Units: 2nd moment of area ~ Length4

flexural stiffness ~ force x Length2

neutral axis

What is second moment for common objects?

dAyI 2

44 RI

)( 444 io RRI

What if there is a cost to material?

Performance ~ ICost ~ Area

?A

I 4

2

2

441 R

R

R

A

I

4)(

)( 22

22

4441

i

i

i RR

RR

RR

A

I o

o

o

~10 million species

Animals with hollow exoskeletons are very successful:

How big do (did) they get?

What sets the limit on how big cylinders can be?

mg

compressiveload

L

n = 1, 2, or 4

1 2 4

Two important failure modes (for compression):

1) Euler buckling

2

2

L

EInFEuler

failure in tension

e.g. pencil or femur snapping

2) Brazier or ‘local’ buckling

ERRkF ioBrazier2)(

e.g. soda can0.5< k <0.8

Part II: Collagen

Most common protein in vertebrate body BY FAR!20% of a mouse by weight.

33% glycine, 20% hydroxyproline

Each tropo-collagen fiber held together by hydrogen bonds involving central glycines:

1 2 3 1

glycine

fiberwithin fiberconstruction:

Julian Voss-Andreae's sculpture Unraveling Collagen (2005)

Part III: Bone

1) When/How did it evolve?

ostracaderm

Dermal Bone=product of ectoderm

Endochondrial bone= product of mesoderm

2) What is bone?

Bone is composite of collagen and inorganicCa2+ salts (mostly hydroxyapatite)= structural composite (like fiberglass)

Two main organizations:1) dense cortical or compact bone2) trabecular bone

Bone is living tissue:• synthesized by osteoblasts• disolved by osteoclasts• maintained by osteocytes

Bone structure

How do bones articulate?

joint types

Linkage Systems

e.g. 4 bar system

Four bar system

4 bar system

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Torsional and Shear

E =

G =

E = Young’s modulus, = stress, = strain

G = Shear modulus, = shear stress, = shear strain

F

A

shear stress, = force/area

shear strain, = angular deflection

AreaL

L

Force

= force / cross sectional area = change in length / total length