seegongsik

mechanics of materials · how things carry load

The limits of materials, seen through stress

Through stress, not force: see when a member stretches, twists, and breaks.

15 / 15

01Stress Is Internal Force per Unit Area

At an imaginary cut, the internal force each patch of area carries, σ = P / A

#elevator cable#bolt#column

02Strain Is the Ratio of Stretch

Stretch over original length, ε = ΔL / L, dimensionless; shear is an angle γ

#rubber band#bungee cord#strain gauge

03In the Elastic Range, σ Is Proportional to ε

The straight part of the curve, σ = E·ε, with slope E as stiffness

#spring scale#steel beam#guitar string

04Shear Slides Things Along a Face

Force parallel to a face, τ = V/A, shear strain γ, Hooke as τ = G·γ

#scissors#rivet#hole punch

05Stretch One Way and It Shrinks the Other

Ratio of lateral to axial strain ν, volume change ε(1−2ν), G = E/(2(1+ν))

#rubber eraser#cork stopper#wine bottle

06Pull an Axial Member and It Stretches by δ=PL/AE

Whole-member stretch δ = PL/AE, summed over segments, stiffness k = AE/L

#suspension cable#tie rod#steel tower

07Twist a Shaft and the Shear Grows with Radius

Shear τ = Tρ/J max at surface, twist φ = TL/GJ, hollow-shaft efficiency

#drive shaft#drill bit#screwdriver

08Bending Makes Stress Linear in Depth, σ=My/I

Zero at neutral axis, linear in depth, σ = My/I, second moment I, modulus S = I/c

#bridge girder#bookshelf#diving board

09How V and M Change Along a Beam

V and M diagrams from reactions and cuts, dM/dx = V, max M is the critical point

#balcony#crane boom#footbridge

10Shear in a Beam Section Peaks at the Neutral Axis

Transverse shear τ = VQ/It, Q the first moment, parabolic, max at neutral axis, I-beam web

#I-beam web#glued plywood#nailed beam

11Deflection Is the Curvature M/EI Integrated Twice

Curvature = M/EI, integrate EI·y″ = M twice, boundary conditions, superposition

#sagging shelf#diving board#floor joist

12Stress at a Point Depends on the Angle You View It

Rotate the element, σ′·τ′ as sine·cosine of 2θ, principal planes (τ=0), max shear at 45°

#pressure vessel#welded joint#aircraft fuselage

13The Normal Stress on the Zero-Shear Face Is the Principal Stress

σ1,2 = (σx+σy)/2 ± R, angle tan2θp, max shear τ_max = R, brittle vs ductile failure

#cracked concrete#snapped chalk#pipeline

14A Stress State Is a Circle, Rotation Is Rotation on It

A point’s stress as a circle in σ-τ, ends = principal, top = max shear, θ rotation = 2θ

#geologic fault#bridge bearing#tunnel lining

15A Slender Column Bends Before It Crushes

Critical load P_cr = π²EI/(KL)², slenderness λ = KL/r, end factor K, buckling as a limit of stiffness and shape

#building column#scaffold pipe#bicycle spoke

Was this helpful? Support seegongsik