Eas207-Formula Sheet -final-exam PDF

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EAS 207 Formula Sheet Chapters 5 and 6 (Trusses and Frames) Static Equilibrium Equations: Vector form: FR = ƩF = 0 and MR = Ʃ (r x F) + Ʃ C (Couple moments) = 0 Scalar form: ƩFx = 0, ƩFy = 0, ƩFz = 0, ƩMx = 0, ƩMy = 0 and ƩMz = 0

Chapter 7 (Beams and Cables) Beams:

V2 – V1 = ʃ 𝑤𝑑𝑥

dV/dx = w

M2 – M1 = ʃ 𝑉𝑑𝑥

dM/dx = V

Cables:

dy 1  dx FH

dy

1

 w( x) dx; dx  F  w(s )ds

T = FH / cosθ

Tmax = FH / cosθmax

H

Cable supporting uniformly distributed load ‘w’ along a horizontal plane: y = ax2/2

a= w/To = w/FH

T = To √1 + 𝑎2 𝑥 2

S = 0.5 [ x √1 + 𝑎2 𝑥 2 + (1/a) ln{ ax + √1 + 𝑎2 𝑥 2 } Cables supporting its own self weight (uniformly distributed): y = (1/2a) { 𝑒 𝑎𝑥 + 𝑒 −𝑎𝑥 – 2}

S = (1/2a) { 𝑒 𝑎𝑥 + 𝑒 −𝑎𝑥 }

T = To √1 + 0.25 ∗ (𝑒𝑎𝑥 − 𝑒−𝑎𝑥)2

Chapter 8 (Friction) Sliding Friction:

f ≤ μs N

fmax = μs N

φs = tan-1(μs)

fdyn = μk N

φk = tan-1(μk)

Belt-Drum Friction: T1/T2 = exp(μsβ) , where T1 > T2 Dynamic case:

Torque = ( T1 – T2 ) x Radius

T1/T2 = exp(μkβ) , where T1 > T2

Screws: Pitch= p = 2πr tanθ Journal Bearings:

Thrust Bearings:

M = F r tan(φs + θ)

or

M = P r sinφs

M = (2μsF / 3cosα) {(R13 – R23)/ (R12 – R22)}

Clutches and Simple Thrust Bearing:

M = (2/3) μs F r

M = F r tan(φs - θ)

Chapter 9 (Centroids and Fluid Statics)

Theorems of Pappus and Guldinus 1) Surface Area of Revolution:

A=θȳL

2) Volume of Revolution:

V=θȳA

Where, ȳ = Centroid (from axis of revolution) and θ = angle of revolution in radians Fluid Statics Fluid pressure = ϒ z Where, ϒ = unit weight of the fluid = ρg ; z = depth below the fluid surface

Chapter 10 (Moments of Inertia)

Area Moment of Inertia or 2nd Moment of Area I x   y 2 dA;

Transformation Equations:

Principal Axes Orientation:

I y   x2 dA; A

_

I  I  Ad 2 I xy   xydA; A

I xy  I x' y'  Ad x d y J O  I x  I y  Iu  I v

Principal Values:

Mass Moment of Inertia

I   r 2 dm   r 2 dV ; I  I G  md 2 m

V...