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Forumula sheet for 2018...

Description

Formula Sheet Normal stress

σ avg = P / A σ avg = normal stress P = internal resultant normal force A = area normal to the force at that section Shear stress

τ avg = V / A τavg = average shear stress V = shear force A = area tangent to the force at that section Factor of safety F .S . = σ fail / σ allow F .S . = τ fail /τ allow F .S . = Pfail / Pallow

Strain

ε = δ /L ε = strain δ = change in length L= original length Hooke’s law

σ = Eε E = Young’s modulus Strain energy density ∆U

1

∆V

2

U = − = −σε U = strain energy density Poisson’s ratio ε lat

υ=− −

εlong

Shear modulus of elasticity τ

G=− γ

γ = shear strain Also, E

G= −

2 (1+υ )

Elastic deformation of axially loaded member

δ = PL / AE

δ = displacement of one point on a bar relative another point L= original distance between the points P= internal force A= cross-sectional area E= modulus of elasticity of the material Thermal Stress

δT = α∆ TL α = linear coefficient of thermal expansion ΔT =algebraic change in temperature of the member L= original length of the member δT = algebraic change in length of the member Castigliano’s 2nd Theorem applied to trusses

 ∂N  L ∆ = ∑N −  −  ∂ P  AE Δ = joint displacement of the truss P = external force of variable magnitude applied to the truss joint in the direction of Δ N = internal axial force in the member caused by both the force P and the load on the truss L = length of the member A = cross-sectional area of a member E = modulus of elasticity of the material. Torsion formulas of a circular shaft Shear stress Tρ

τ=− J

τ = shear stress in the shaft T = the resultant torque acting at the cross section ρ = radial position of the element J = polar moment of inertia of the cross-sectional area

Angle of twist TL

ϕ=−

JG

φ = angle of twist of one end of the shaft with respect to the other end, measured in radians T = internal torque J = the shaft’s polar moment of inertia. G = shear modulus of elasticity for the material. Bending formulas for beams and shafts dV

− = − w(x )

dX

dM

− =V

dX

W = distribute loading on the shaft V = shear force M = bending moment MY

σ =− − I

σ = normal stress in the member M = resultant internal moment, determined from method of sections and equations of equilibrium, and computed about the neutral axis of the cross section I = moment of inertia of the cross-sectional area computed about the neutral axis Y = intermediate distance from the neutral axis to the point where the bending stress is to be determined Composite Beams E1

n=−

E2

n = transformation factor E1 = modulus of elasticity of material 1 E2 = modulus of elasticity of material 2 Transverse Shear VQ

τ=− It

τ = shear stress in the member at the point located a distance Y ′ from the neutral axis. This stress is assumed to be constant and therefore averaged across the width t of the member V = internal resultant shear force, determined from the method of sections and equations of equilibrium I = moment of inertia of the entire cross- sectional area computed about the neutral axis

t = the width of the member’s cross-sectional area. Measured at the point where τ is to be determined

Q = ∫ A′YdA′ = Y ′A′ A’ = is the top (or bottom) portion of the member’s cross-sectional area, defined from the section where t is measured. Y’= is the distance to the centroid of A′ , measured from the neutral axis Thin-walled pressure vessels Cylindrical Vessels Pr

σ1 = − t

Pr

σ2 = −

2t

σ1, σ2 = the normal stress in the hoop and loongtudinal directions respectively. Each is assumed to be constant throughout the wall of the cylinder, and each subjests the material to tension. P = the internal pressure developed by the contained gas or fluid. r = inner radius of the cylinder. t = thickness of the wall r / t ≥ 10 Spherical vessels Pr

σ1 =σ 2 = −

2t

Stress Transformation General equations of plane stress transformation

σ x' = σ y' =

σx + σ y 2 σx +σy

τ x 'y ' = −

2

+ −

σ x −σ y 2

σx −σy 2 σ x −σ y 2

cos 2θ + τ xy sin 2θ cos 2θ + τ xy sin 2θ

sin 2θ + τ xy cos 2θ

In-plane principle stresses

σ 1,2 =

σx + σy 2

σ x −σ y ±  2 

2

  + τ xy 2 where σ 1 > σ 2 

σ1,2 = principle stresses (maximum and minimum in-plane stresses) The planes of maximum and minimum normal stresses

tan 2θ p =

τ xy (σ x − σ y ) / 2

θp = orientation of the planes of maximum and minimum normal stress Maximum Shear Stress in-plane 2

σ x − σ y   + τ xy 2 τ max in -plane =  2   − (σ x − σ y ) / 2 tan 2θ s = τ xy θs = orientation of the planes of maximum shear stress

σ avg = Mohr’s Circle – Plane Stress 2

 σ x −σ y  2  + τ xy   2

τ max in - plane = 

σ x −σ y 2

Deflection of Beams and Shafts Slope and Displacement by Integration

d4 y EI 4 = −w (x ) dx d3 y EI 3 = V (x ) dx d2 y EI 2 = M ( x) dx dy θ= dx y = elastic curve deflection w(x) = load on the beam V(x) = shear force in the beam M(x) = the internal moment in the beam θ = slope of the elastic curve of the beam E= the material’s modulus of elasticity I = beam’s moment of inertia computed about the neutral axis Macaulay Functions

x −a

n

0 = n ( x − a) n ≥a

for x < a for x ≥ a

Slope and displacement by the moment-area method THEOREM 1 Angle between the tangents at any 2 points on the elastic curve equals the area under the M/EI diagram between these two points. THEOREM 2 Tangent at A on the elastic curve with respect to the tangent extended from B equals to the moment of the area under M/EI diagram. This moment is computed about A where the vertical deviation is to be determined.

Buckling of Columns Pcr = σ cr =

π 2 EI L2 π2E

( L / r )2

Pcr = maximum axial load σcr = critical stress E = modulus of elasticity for the material I = least moment of inertia for the column’s cross-sectional area L = unsupported length of the column r = smallest radius of gyration of the column L/r = slenderness ratio Columns Having Various Types of Supports

Pcr =

σ cr = Le = KL K = dimensionless coefficient Le= effective length factor

π 2 EI

(KL )2 π 2E

(KL / r )2

Centroid of an area ∫ A xdA

∫ A ydA

x= −

y= −

∑ xA x= − ∑A

∑ yA y= − ∑A

∫ A dA

∫ A dA

Composite area

Moment of inertia for an area

I x = ∫ A y2 dA

I y = ∫ A x 2dA

Polar moment of inertia

I 0 = ∫ A r 2dA = I x + I y...