PSAD Merged

Resultant of a Force System: Components

R_x = ΣFcosα

R_y = ΣFsinα

α is angle wrt x-axis

Resultant of a Force System: Magnitude and Direction

R = √(R_x² + R_y²)

Θ = tan^-1(R_y / R_x)

α is angle wrt x-axis

Moment about a Point

M = Fd

Centroid and Effective Forces

Concentrated Load (Point): F = P

Uniformly Distributed Load: F = wL @ L/2

Uniformly Varying Load (Triangular): F = wL/2 @ L/3

Squared Property of Parabola (Parabolic Cable)

d₁² / h₁ = d₂² / h₂

d = distance from lowest point to support

h = height of supports from lowest point

Parabolic Cable: Minimum and Maximum Tension

T_min = (w d_max² / 2 h_max)

T_max = √((T_min)² + (w d_max)²)

Length of Catenary Cable

S = (T₀ / W₀) sinh(W₀ x / T₀)

Catenary Cable: Tension at any Point

T = T₀ cosh(W₀ x / T₀)

Catenary Cable: Vertical Distance of Lowest Point to any Point of the Cable

y = (T₀ / W₀) [cosh(W₀ x / T₀) - 1]

Friction Force

F_f = µ_s N (static friction)

F_f = µ_k N (dynamic friction)

Angle of Friction

tanΘ = F / N

when Θ = α, tanα = µ = F / N

Belt Friction

T₂ / T₁ = e^µß

T₂ > T₁

ß = subtended angle in radians

Rectangle: Area, Centroid, Moment of Inertia

A = bh

Centroid = (b/2, h/2)

I_x = bh³ / 12

I_y = hb³ / 12

Triangle: Area, Centroid, Moment of Inertia

A = bh / 2

Centroid = (b/3, h/3)

I_x = bh³ / 36

I_y = hb³ / 36

Circle: Area, Centroid, Moment of Inertia

A = πr² = πd² / 4

Centroid = (d/2, d/2)

I_x = πd⁴ / 64

I_y = πd⁴ / 64

Quarter Circle / Spandrel

A = πr² / 4

Centroid = (4r/3π, 4r/3π)

I_x = 0.055r⁴

I_y = 0.055r⁴

Rectilinear Motion: Uniform Motion

s = vt

a = 0

Rectilinear Motion: Uniformly Accelerated Motion

a = constant

v = v₀ ± at

s = v₀ t ± at² / 2

v² = v₀² ± 2a (x - x₀)

Rectilinear Motion: Instantaneous vs. Average

Instantaneous Velocity: v = dx / dt

Instantaneous Acceleration: a = dv / dt = d²x / dt²

Average Velocity: v_ave = (x - x₀) / (t - t₀)

Average Acceleration: a_ave = (v - v₀) / (t - t₀)

Rectilinear Motion: Free Fall

v₂ = v₁ ± gt

v₂² = v₁² ± 2gy

y = v₁t ± gt² / 2

t = √(2y / g)

y_max = v₁² / 2g

(+) - upwards

(-) - downwards

Projectile Motion

v_0x = v₀ cosΘ

v_0y = v₀ sinΘ

x = v_0x t

t = v₀ sinΘ / g + √(2h_max / g)

h_max = y_max + h

y_max = v₀²sin²Θ / 2g

Projectile Motion: General Formula for height of projectile

y = xtanΘ - gx² / (2v₀²cos²Θ)

Rotational Motion

ω = dΘ / dt

α = dω / dt

s = rΘ

v = rω

Rotational Motion: Acceleration Components

a_T = αr

a_N = ω²r = v² / r

a = √(a_T² + a_N²)

Work

W = F d

Power

P = W / t

Gravitational Potential Energy

PE = m g h

Spring Potential Energy

PE = kx² / 2

Kinetic Energy

KE = mv² / 2

Conservation of Mechanical Energy

PE₁ + KE₁ = PE₂ + KE₂

For PE = KE: m g h = mv² / 2

v = √2gh

Linear Momentum and Impulse

p = m × v

Conservation: mv₁ = mv₂

J = F × t

Coefficient of Resitution

e = - (v_B2 - v_A2) / (v_B1 - v_A1)

Particle Kinetics: Centripetal Force

F_cf = mv² / r

Particle Kinetics: Banked Curves

Friction is Considered: tan (Θ + Φ) = v² / gR

Car is Slipping: tan (Θ - Φ) = v² / gR

Design Angle of Banking: tan (Θ) = v² / gR

Allowable vs. Ultimate

Allowable = Ultimate / Factor of Safety

Axial Stress

σ = P / A

(P = axial load, A = cross-sectional area where axial load is applied)

Axial Strain

ε = δ / L = σ / E

(δ = deflection/change in length, L = original length, E = modulus of elasticity)

Modulus of Elasticity

E = σ / ε

Axial Deformation/Deflection

δ = PL / AE = σL / E

Thermal Deformation

δ_T = αLΔT

(α = Coefficient of Thermal Expansion)

Shear Stress (General)

τ = V / A_s

(V = Shear Force, A_s = sheared area)

Shear Strain

γ = δ_s / L

Modulus of Rigidity

G = δ_s / γ

Shear Deformation

δ_s = VL / A_sG

Bearing Stress

σ_b = P_b / A_b

Cylindrical Thin-Walled Pressure Vessels

Tangential/Hoop: σ_T = pD / 2t

Longitudinal/Axial: σ_L = pD / 4t

(p = internal pressure, D = diameter of vessel, t = thickness)

Spherical Thin-Walled Pressure Vessels

σ = pD / 4t

(p = internal pressure, D = diameter of vessel, t = thickness)

Torsional Shear Stress

τ = Tr / J

(T = torque, r = radius of curvature, J = polar moment of inertia)

Torsion: Angle of Twist

θ = TL / JG

(T = torque, L = length, J = polar moment of inertia, G = modulus of rigidity)

Polar Moment of Inertia

J = I_x + I_y

For circular solid shafts: J = πD⁴ / 32

For circular hollow shafts: J = π(D_o⁴ - D_i⁴)

Helical Springs: Approximate Maximum Shear Stress

S_s = 16PR/πd³ * (1 + d/4R)

(P = load applied, R = spring radius, d = diameter)

Helical Spring: AM Wahl’s Formula

S_s = 16PR/πd³ * ((4m - 1) / (4m+4) + 0.615/m)

m = 2R/d

(4m - 1) / (4m+4) = Wahl factor

Helical Springs: Deformation

δ = 64PR³n / Gd⁴

(n = number of turns, G = modulus of rigidity, R = spring radius, P = load applied, d = diameter of wire)

Flexure: Bending Stress

f_b = My / I

(M = maximum moment, y = distance of point from NA, I = moment of inertia)

Flexural Shear Stress

f_v = VQ / Ib

(V = maximum shear force, Q = A’y’, I = moment of inertia, b = width at section)

Note: A’ = area above/below NA, y’ = distance from centroid of A’ to NA

Shear Flow

q = VQ / I = f_v b

(V = maximum shear force, Q = A’y’, I = moment of inertia)

Shear Connector

s = RI / VQ = R/q

(s = pitch of shear connector, R = total resistance of connectors within pitch length)

Flexure for Curved Beams

f_b = Ey / ρ

(E = modulus of elasticity, y = point from NA, ρ = radius of curvature)

Flexure: Non-homogeneous beams

n = E_s / E_b

b → nb

f_transformed = nMc / I

Combined Stresses

σ = ± P/A ± Mc/I

τ = V/A

τ = VQ/Ib + Tr/J

Mohr’s Circle: Sign Convention

Normal Stress: (+) Tension, (-) Compression

Shear Stress: (+) Clockwise, (-) Counterclockwise

Static Determinacy of Beams (Planar)

DI = R - 3 - C

R - external reactions, C - releases

Static Determinacy of Beams (3D)

DI = R - 6 - C

R - external reactions, C - releases

Static Determinacy of Frames (Planar)

DI = R + 3M - 3J - C

R - external reactions, M - members, J - joints, C - releases

Static Determinacy of Frames (3D)

DI = R + 6M - 6J - C

R - external reactions, M - members, J - joints, C - releases

Static Determinacy of Trusses (Planar)

DI = R + M - 3J

R - external reactions, M - members, J - joints

Static Determinacy of Trusses (3D)

DI = R + M - 2J

R - external reactions, M - members, J - joints

Equations of Conditions (C) for Internal Hinge

M = 0; 2 internal reactions (Fx, Fy)

Equations of Conditions (C) for Internal Roller

Fx = 0, M = 0; 1 internal reaction (Fy)

Equations of Conditions (C) for Internal Fixed

3 internal reactions (Fx, Fy, M)

Equations of Conditions (C) for Internal Slider

Fy = 0; 2 internal reactions (Fx, M)

DIM: Moment Equation

EI y” = M

E = modulus of elasticity, I = moment of inertia

DIM: Slope Equation

EI y’ = ∫ M dx + C₁

E = modulus of elasticity, I = moment of inertia

DIM: Beam Deflection Equation

EI y’ = ∫ ∫ (M dx) dx + C₁x + C₂

E = modulus of elasticity, I = moment of inertia

Area-Moment for Cantilever → Moment at Free End

Area = bh (see figure for components)

Area-Moment for Cantilever → Point Load

Area = bh/2 (see figure for components)

Area-Moment for Cantilever → Uniformly Distributed Load

Area = bh/3 (see figure for components)

Area-Moment for Cantilever → Uniformly Varying Load (max at fixed end)

Area = bh/4 (see figure for components)

Area-Moment Method: Theorem 1

θ_AB = (Area)_AB / EI

Area-Moment Method: Theorem 2

t_B/A = (Area)_AB / EI * x̄_B

Area-Moment Method: Rotation at A

y’_A = θ_A = t_B/A / L = (t_C/A + δ_C) / x_C

Area-Moment Method: Rotation at B

y’_B = θ_B = t_A/B / L = (t_C/B + δ_C) / (L - x_C)

Superposition Method - Simply Supported Beam: Concentrated Load at Midspan

Superposition Method - Simply Supported Beam: Concentrated Load, General

Superposition Method - Simple Beam: Uniform Load over Entire Span

Superposition Method - Simple Beam: Uniformly Varying Load over Entire Span with Zero load at one end

Superposition Method - Simple Beam: Symmetrical Triangular Load over Entire Span

Superposition Method - Simple Beam: Moment Load at Right End

Superposition Method - Cantilever Beam: Point Load at Free End

Superposition Method - Cantilever Beam: Point Load, General

Superposition Method - Cantilever Beam: Uniform Load over Entire Span

Superposition Method - Cantilever Beam: Uniformly Varying Load over Entire Span with Zero Load at Free End

Three-Moment Equation: General

Three-Moment Equation: Constant E and I

M₁L₁ + 2M₂ (L₁ + L₂) + M₃L₂ + 6A₁a₁ / L₁ + 6A₂b₂ / L₂ = 6EI (h₁ / L₁ + h₃ / L₂)

Values of 6Aa/L and 6Ab/L for Point Load, General

6Aa/L = Pa (L² - a²) / L

6Ab/L = Pb (L² - b²) / L

Values of 6Aa/L and 6Ab/L for Point Load at Midspan

6Aa/L = 3PL² / 8

6Ab/L = 3PL² / 8

Values of 6Aa/L and 6Ab/L for Uniform Load

6Aa/L = w₀L³ / 4

6Ab/L = w₀L³ / 4

Values of 6Aa/L and 6Ab/L for Triangular Load, 0 at left end

6Aa/L = 8w₀L³ / 60

6Ab/L = 7w₀L³ / 60

Values of 6Aa/L and 6Ab/L for Symmetric Triangular Load

6Aa/L = 5w₀L³ / 32

6Ab/L = 5w₀L³ / 32

Values of 6Aa/L and 6Ab/L for Moment

6Aa/L = -M/L × (3a² - L²)

6Ab/L = M/L × (3a² - L²)