midterm 1 design of mech. comp.

Load

Any force or moment (torque) pushing‚ pulling‚ bending‚ or twisting a part. Loads are the INPUT: Load + Geometry + Material → stress‚ strain‚ deformation → does it work or fail?

Stress (concept)

How hard the material is being loaded INSIDE‚ per unit of area. Same force on a skinny rod vs. a thick rod = more stress in the skinny one. Think: how ‘crowded’ the force is. Internal quantity‚ units Pa (N/m²) or psi.

Stress (formula)

stress = force / area → σ = F/A (normal) or τ = F/A (shear). Units: Pa = N/m². 1 MPa = 10⁶ Pa = 1 N/mm².

Normal stress

Stress acting PERPENDICULAR to the cut surface — pulling it apart (tension‚ +) or pushing it together (compression‚ −). Symbol σ.

Shear stress

Stress acting PARALLEL to the cut surface — layers trying to slide past each other‚ like scissors. Symbol τ.

Tension vs. compression

Tension = being stretched/pulled (positive normal stress). Compression = being squished/pushed (negative normal stress).

Engineering stress vs. true stress

Engineering stress uses the ORIGINAL area (what we normally use). True stress uses the CURRENT (deformed) area.

Deformation

How much a part actually changes size or shape — an absolute amount (e.g.‚ stretched 2 mm). Not normalized. Often written u or δ. Also called deflection.

Strain (concept)

Deformation relative to original size — a percentage-style stretch. 2 mm stretch on a 1000 mm rod = 0.002 strain. Unitless (or mm/mm). Measurable (strain gauges).

Strain (formula)

ε = ΔL / L (change in length ÷ original length). As a derivative: ε = du/dx.

Normal strain

Strain from a change in LENGTH (stretching or shrinking). Symbol ε.

Shear strain

Strain from a change in ANGLE — a square getting skewed into a diamond. Symbol γ (gamma). Measured in radians.

Hooke’s law (1D)

σ = Eε. Stress = stiffness × strain. Only valid in the elastic (linear) region for simple uniaxial loading.

Elastic

Deformation that fully springs back when the load is removed (like a rubber band).

Plastic

Permanent deformation — the part stays bent after you unload it (like a bent paperclip). Starts at yield.

Yield strength

Stress at which the material starts to permanently deform. Used for FOS with DUCTILE materials. Symbol S_y.

Ultimate strength

Maximum stress the material can take before breaking. Used for FOS with BRITTLE materials. Symbol S_u or S_ut.

Strength vs. stress

STRESS is what the load is doing to the part. STRENGTH is what the material can handle (a material property). Design goal: stress < strength.

Ductile material

Stretches/bends a lot before breaking (steel‚ aluminum). Failure = yielding‚ so compare to S_y.

Brittle material

Breaks with little warning or stretching (glass‚ cast iron‚ ceramic). Failure = fracture‚ so compare to S_u.

Factor of safety (FOS/n)

How many times stronger the part is than it needs to be. FOS = strength / actual max stress. Ductile: S_y/σ_max. Brittle: S_u/σ_max. FOS > 1 means it survives.

Stiffness vs. strength

Stiffness (E) = how much it resists deforming. Strength (S_y‚ S_u) = how much stress before it yields/breaks. A material can be stiff but weak (glass) or flexible but tough.

Static

System doesn’t move. ΣF = 0 and ΣM = 0.

Quasi-static

System moves‚ but slowly enough to ignore acceleration/inertia. Still use ΣF = 0‚ ΣM = 0.

Dynamic

Accelerates enough that inertia matters. ΣF = ma and ΣM = Iα.

Isotropic

Same properties in every direction (most metals). Opposite: anisotropic (wood‚ composites).

Homogeneous

Same material properties at every point throughout the part.

Stress concentration

Local spike in stress near holes‚ notches‚ sharp corners‚ or damage. Factor K_t multiplies the nominal stress.

F

Force. Units N (newtons) or lbf.

A

Cross-sectional area — area of the cut face the force acts on. Units m² or mm².

L

Length (original‚ unloaded length of a part).

ΔL

Change in length (Δ = ‘change in’).

u/v/w

Displacements of the material in the x‚ y‚ z directions.

E

Young’s modulus / modulus of elasticity — material STIFFNESS in tension/compression. Steel ≈ 200 GPa‚ aluminum ≈ 70 GPa. Units Pa.

G

Shear modulus — stiffness in shear. τ = Gγ.

M

Moment. In beams‚ the internal BENDING moment at a section. Units N·m.

V

Internal SHEAR force in a beam at a section. Units N.

w (beam loading)

Distributed load — force spread over a length‚ e.g.‚ 1800 N/m.

T

Torque — twisting moment on a shaft. Units N·m.

I

Area moment of inertia (second moment of area) — how well a cross-section’s SHAPE resists bending. Bigger = stiffer/stronger in bending. Units m⁴ or mm⁴. Rectangle: bh³/12. Solid circle: πr⁴/4 = πd⁴/64.

J

Polar moment of inertia — resistance to TWISTING. Solid circle: πr⁴/2 = πd⁴/32.

y (bending)

Distance from the neutral axis to the point you’re checking.

c

Distance from the neutral axis to the OUTERMOST fiber (where bending stress is max). For a circle‚ c = r.

Q

First moment of area — used for transverse shear stress in beams (τ = VQ/It).

t or b (shear formula)

Width of the cross-section at the location where you’re computing shear stress.

P

Axial (along the length) load‚ usually compressive in buckling problems.

P_cr

Critical buckling load — the load where a column suddenly bows sideways.

L_e

Effective (equivalent) length of a column — actual length adjusted for end conditions. Fixed-fixed: L_e = 0.65L (AISC). Pinned-pinned: L_e = L.

S_y

Yield strength (S = strength).

S_u / S_ut

Ultimate (tensile) strength.

S_cr

Critical buckling stress = P_cr / A.

n (buckling)

Mode number in Euler’s formula. Use n = 1 for the critical (lowest) load. (Elsewhere n can mean factor of safety — check context!)

K_t

Stress concentration factor.

m/a

Mass and acceleration (ΣF = ma).

σ (sigma/lowercase)

NORMAL stress. σ = F/A.

Σ (Sigma/capital)

‘Sum of’ — NOT stress! ΣF = 0 means all forces add to zero.

τ (tau)

SHEAR stress.

ε (epsilon)

NORMAL strain (stretch ÷ original length).

γ (gamma)

SHEAR strain (change in angle‚ radians).

δ (delta/lowercase)

Deflection/deformation — how far something moved. Also used for elongation.

Δ (Delta/capital)

‘Change in’ — ΔL = change in length.

ν (nu)

Poisson’s ratio — when you stretch something‚ how much it gets thinner sideways. Most metals ≈ 0.3. Looks like a ‘v’!

ρ (rho)

In THIS course (buckling): radius of gyration‚ from I = Aρ². (In other classes ρ often means density — careful!)

θ (theta)

Angle — e.g.‚ angle of twist in a shaft or slope of a beam.

α (alpha)

Angular acceleration (ΣM = Iα). Sometimes coefficient of thermal expansion.

π (pi)

3.14159… shows up in circle areas‚ I‚ J‚ and Euler buckling.

ω (omega)

Angular velocity (rotation speed‚ rad/s) — shows up with shafts/gears.

∂ (partial)

Partial derivative — ε_x = ∂u/∂x means strain in x is how fast x-displacement changes along x.

σ = F/A

Axial normal stress = force ÷ area.

τ = F/A

Average (direct) shear stress = force ÷ area being sheared.

σ = Mc/I (or My/I)

Bending stress in a beam. Max at the outer fiber (y = c)‚ zero at the neutral axis.

τ = VQ/(It)

Transverse shear stress in a beam. Max at the neutral axis (opposite of bending stress!).

Max transverse shear/rectangle

τ_max = 3V/(2A).

Max transverse shear/solid circle

τ_max = 4V/(3A).

τ = Tr/J

Torsional shear stress in a round shaft‚ max at the outer surface.

P_cr = n²π²EI / L_e²

Euler buckling load. Use n = 1. Use the SMALLEST I of the cross-section.

I = Aρ²

Defines radius of gyration ρ = √(I/A).

S_cr = π²E / (L_e/ρ)²

Euler critical buckling STRESS.

S_cr = S_y − (S_y² / 4π²E)(L_e/ρ)²

Johnson parabola — critical buckling stress for intermediate (not super slender) columns.

Beam

Long member loaded sideways (perpendicular to its length) so it bends.

Shear-moment diagram

Graphs of internal shear V and bending moment M along a beam. Used to find where M_max (→ max bending stress) and V_max (→ max shear stress) occur.

Neutral axis

Line through the cross-section’s centroid where bending stress is zero — one side is in tension‚ the other in compression.

Centroid

Geometric center of a cross-section’s area.

Outer fiber

The material farthest from the neutral axis — where bending stress is largest.

Simple support

Support that holds a beam up but lets it rotate (pin or roller). Moment = 0 there.

Cantilever

Beam fixed at one end‚ free at the other (diving board). Max moment at the wall.

Fixed support

Prevents both movement and rotation‚ has a reaction force AND reaction moment.

Beam sign convention

Internal bending moment is positive when the beam bends into a ‘smile’ (concave up) — different from the right-hand-rule convention.

Right-hand rule

Curl fingers in the direction of rotation‚ thumb points along the moment vector. Counterclockwise = positive (out of page).

Buckling

Sudden sideways bowing of a column under compression — failure from instability‚ not from the material breaking. Can happen at stresses below yield.

Column

Long‚ slender member loaded in axial compression.

Radius of gyration (ρ)

√(I/A). Describes how spread out a cross-section’s area is. Bigger ρ = more resistant to buckling.

Slenderness ratio

L_e/ρ. Big = long & skinny (Euler). Medium = Johnson parabola. Less than ~10 = short column‚ just check S_y.

Euler vs. Johnson

Euler: long/slender columns (L_e/ρ above the tangent point √(2π²E/S_y)). Johnson: intermediate columns below it. Euler overpredicts strength for stubby columns.

Tangent point (Euler/Johnson)

L_e/ρ = √(2π²E/S_y)‚ where S_cr = S_y/2. Above → Euler‚ below → Johnson.

Shell buckling

Thin-walled tubes wrinkling locally while the axis stays straight. Hard to predict‚ designers use knockdown factors