Statics and Strength of Materials Formula Sheet

Statics and Strength of Materials Formula Sheet

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  Statics and Strengthof Materials Formula Sheet12/12/94 — A. Ruina Not given here are the conditions under which the formulae are accurate or useful. Basic Statics Free Body Diagram The FBD is a picture of any system for which you would like to apply mechanics equations and of all the external forces and torques which act on the system. Action & Reaction If A feels force FFFFFFFFFF and couple MMMMMMMMMM from B. then B feels force −FFFFFFFFFF and couple −MMMMMMMMMM from A. (With FFFFFFFFFF and −FFFFFFFFFF acting on the same line of action.) Force and Moment Balance These equations apply to every FBD in equilibrium: F orce Balance All external forces FFFFFFFFFF = 0000000000 Moment Balance about pt C All external torques MMMMMMMMMM/C = 0000000000 • The torque MMMMMMMMMM/C of a force depends on the reference point C. But, for a body in equilibrium, and for any point C, the sum of all the torques relative to point C must add to zero ). • Dotting the force balance equation with a unit vector gives a scalar equation, e.g. { FFFFFFFFFF} · iiiiiiiiii = 0 ⇒ Fx = 0. • Dotting the moment balance equation with a unit vector gives a scalar equation, e.g. { MMMMMMMMMM/C }·λλλλλλλλλλ = 0 ⇒ net moment about axis in direction λλλλλλλλλλ through C = 0. Some Statics Facts and Definitions • The moment of a force is unchanged if the force is slid along its line of action. • For many purposes the words ‘moment’, ‘torque’, and ‘couple’ have the same meaning. • Two-force body. If a body in equilibrium has only two forces acting on it then the two forces must be equal and opposite and have a common line of action. • Three-force body. If a body in equilibrium has only three forces acting on it then the three forces must be coplanar and have lines of action that intersect at one point. • truss: A collection of weightless two-force bodies connected with hinges (2D) or ball and socket joints (3D). • Method of joints. Draw free body diagrams of each of the joints in a truss. • Method of sections. Draw free body diagrams of various regions of a truss. Try to make the FBD cuts for the sections go through only three bars with unknown forces (2D). • Caution: Machine and frame components are often not two-force bodies. • Hydrostatics: p = ρgh, F = p dA Cross Section Geometry thin-wall annulus annulus Definition Composite (circle: c1 = 0) (approx) rectangle A = dA Ai π(c2 2 − c2 1 ) 2πct bh J = ρ2 dA π 2 (c4 2 − c4 1 ) 2πc3t I = y2 dA (Ii + d2 i Ai) π 4 (c4 2 − c4 1 ) πc3t bh3/12 ¯y = y dA dA yiAi Ai center center center Q = y dA = A ¯y A i ¯y i b( h2 4 −y2) 2 Stress, strain, and Hooke’s Law Stress Strain Hooke’s Law Normal: σ = P⊥/A = δ/L0 = L−L0 L0 σ = E [ = σ/E + α∆T ] tran = −ν long Shear: τ = P /A γ = change of formerly right angle τ = Gγ 2G = E 1+ν Stress and deformation of some things Equilibrium Geometry Results Tension P = σA = δ/L δ = P L AE [δ = P L AE + αL∆T ] Torsion T = ρτ dA γ = ρφ/L φ = T L JG τ = T ρ J Bending M = − yσ dA = −y/ρ = −yκ u = M EI and Shear in dM dx = V , dV dx = −w u = d2 dx2 u = 1 ρ = κ σ = −My I Beams V = τ dA τ = V Q It τt∆x = ∆MQ/I Pressure pAgas = σAsolid σ = pr 2t (sphere) Vessels σl = pr 2t (cylinder) σc = pr t (cylinder) Buckling Critical buckling load = Pcrit = π2EI L2 eff . pinned-pinned clamped-free clamped-clamped clamped-pinned Leff = L Leff = 2L Leff = L/2 Leff = .7L Mohr’s Circle Rotating the surface of interest an angle θ in physical space corresponds to a rotation of 2θ on the Mohr’s circle in the same direction. C = σ1+σ2 2 = σx + σy 2 R = σ1−σ2 2 = (σx − C)2 + τ2 xy = σx − σy 2 2 + τ2 xy tan 2θ = τ σ−C = 2τ σx − σy Miscellaneous • Power in a shaft: P = T ω. • Saint Venant’s Principle: Far from the region of loading, the stresses in a structure would only change slightly if a load system were replaced with any other load system having the same net force and moment.