ROADMAP

Mechanics Of Materials - Gabriel Thompson

Putting It Together!

Basics of deformation

Loading Diagram VS FBD

Basic Equations

Tension (+)

Stress Strain Diagrams help understand material behavior

Moh'rs Circle

σavg = F/A

Consider a bird sitting on a beam

Like taffy, deforming by normal stress changes all dimensions

A Loading Diagram considers the beam supports and bird weight

ε= δ/L

E= σ/ε

ν = εlat/εlong

δ=NL/AE

Compression (-)

Stress strain curves relate stress and deformation until fracture

A normal force will either compress, or pull on a material

Becomes

A FBD removes supports and places forces

Normal Stress

Deformation by shear will slant a shape

τ=VQ/It Q = A'y

τavg=V/A

Mohr's Circle allows for transformation of axies. Tilting the axis by θ results in a change in orientation of 2θ on the circle, allowing easy calculation for normal and shear stress in new axies

γ=θ-θ'

Now Consider the beam heating up

Maximum Shear Stress occurs at neutral axis

As the beam heats up, it expands according to δ=α*ΔT. α varies by material

G=τ/γ

A' is the partial area from the closest extreme fiber to the neutral axis.

Shear forces come in sets of four for static equilibrium. No shear stress occurs at extreme fibers

Becomes

Mohr's Circle also allows calculation of maximum shear and normal stress for a loaded structure

If constrained, the beam will be under compression due to it's expansion

Shear Stress

+ = :)

Moments Cause Tension and Compression

For Torsion, think Tornado

M = Fd

The way a beam is supported changes how easy it is to bend

Moments bend a material into either :) or :(, or a combination

the larger the k, the easier to bend

Solid Cylinder: J = π/2 * r^4 Hollow Cylinder: J = π/2(R^4-r^4) τmax = Tr/J

k = 2

k = 0.5

k = 0.7

k = 1

Wind (stresses) are largest at the outside of the tornado, and none at the center

σmax = My/I I = ΣI + ΣAd

Pcr is the critical buckling load. Often, a factor of safety is included, which is FS = σ/Pcr. This gives leeway on the amount of stress a beam can take before buckling

Becomes

- = :(

Pcr=π^2AE/(KL/r)^2 r = sqrt(I/A)

Torsion max at extreme fibers

Y is distance from extreme fiber to neutral axis

Moment/torsion