# Mechanical design

Some hopefully helpful formulas and thoughts.

We typically design for stiffness, ie small deflection (delta below)… but we also want to check that we are not going to deform or break the relevant material.  For alloys, plastic deformation starts when you get to the yield strength of a material; we use that number. You have to load it further to break it; this is the tensile strength, which we’ll avoid mentioning from now on, though it may be important to you if you’re designing a balloon gondola for chute shock. Brittle materials like carbon fiber don’t have a yield point, so for those we give the tensile strength.

By the way… make sure you work entirely in one unit system, eg english or SI.

Materials properties: (ballpark)

Steel:  yield strength typically 150-400 MPa, E = 200GPa (29 Mpsi)

Aluminum:  yield strength \sim 100MPa, E = 70GPa (10Mpsi)

Carbon fiber: tensile strength \sim 2.5-7GPa, E = 200-600GPa.
http://blog.velocite-bikes.com/carbon-fiber/#sthash.WANEvkUY.dpbs

Bending

Some good formulas for bending beams:  http://www.atcpublications.com/Sample_pages_from_FDG.pdf

Tube bending (fixed at one end, load P applied perpendicularly at far end):

E = elastic modulus,
L = length,
I = cross sectional moment of inertia,
Z = section modulus.
A = pi*(r1^2 – r0^2)
I = (pi/4)*(r1^4 – r0^4)
Z = I/r1

Deflection: delta = (P*L^3)/(3*E*I) .
Obviously this scales as L^3.
For fixed wall thickness it scales as 1/r^3 .
For fixed cross sectional area (and thickness), which is a more relevant constraint for thermal isolation applications, it scales as 1/r^2.

Maximum stress  sigma = P*L/Z.
This scales as 1/r^2 for fixed wall thickness, but for fixed cross sectional area (and thickness) it scales as 1/r^2.

Circular plate:
formulas from http://www.roymech.co.uk/Useful_Tables/Mechanics/Plates.html

t = thickness
p = distributed load pressure, in N/m^2
E = Young’s modulus
nu = poisson’s ratio (0.3 for steel)
D = flexural rigidity = (E*t^3)/(12*(1-nu^2))

Fixed at edges, uniformly distributed load:

sigma_max = (3*p*r^2)/(4*t^2)   [at center!]
delta_max = (p*r^4)/(64*D)

Simply supported at edges, uniformly distributed load:

sigma_max = (3*(3+nu)*p*r^2)/(8*t^2)          [at edges!]
delta_max = ( (5+nu)*p*r^4)/(64*(1+nu)*D)