BE 5, Lecture 8 Geometric Properties I

Part I: Beams force Types of beam: ‘cantilever’‘simple’ supported on one end supported on both ends 3 point test: Force deforms beam into arc of radius, R.

neutral plane tension R y compression  What is strain in section?  (y) =  L/L = 2  (R+y) -2  (R) = y / R 2  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? ybyb  y dA ytyt neutral plane Calculate moment about neutral plane: Second moment of area: beam equations: Note: Larger for structures with stuff far away from neutral axis.

What does beam equation tell us? R R = infinite = flexural stiffness Units: 2 nd moment of area ~ Length 4 flexural stiffness ~ force x Length 2 neutral axis

What is second moment for common objects? What if there is a cost to material? Performance ~ I Cost ~ Area

~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 compressive load L n = 1, 2, or Two important failure modes (for compression): 1) Euler buckling failure in tension e.g. pencil or femur snapping 2) Brazier or ‘local’ buckling e.g. soda can 0.5< 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: 1231 glycine

fiber within fiber construction:

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 inorganic Ca2+ salts (mostly hydroxyapatite) = structural composite (like fiberglass) Two main organizations: 1) dense cortical or compact bone 2) 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

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 Area LL L Force  = force / cross sectional area  = change in length / total length