Beams - Mechanics of Structures: Variational and Computational Methods - page 730

12.1.3 Strain-Displacement Relations

Substitution of the displacements of Eq. (12.4) into the linear strain-displacement relations
for the continuum [Chapter 1, Eq. (1.18)] leads to

Continuum
Representation of a Bar

Bar variables

x = ∂

u x
∂

u θ z + θ y + ωψ

x =

− y z

= − y κ + z κ + ωκ ω .

z y

(12.7a)

Extension Curvatures

= ∂

u y
∂ x + ∂

u x
∂

φ x + ∂ω
∂ y = v − θ

γ − z

y ψ

xy z

z

+ ∂ω
∂

φ x + ∂ω

= v − θ z −

y γ tS y ∂

z

+ ∂ω
∂

γ tP + ∂ω
∂

= γ y −

y γ tS

(12.7b)

y

Shear Shear Strain from Warping

Force Primary Torsion

Torsion

γ xz = ∂

u z
∂ x + ∂

u x
∂

φ x + ∂ω

z = w + θ y +

y

z ψ. ∂

+ ∂ω
∂

+ ∂ω
∂

= γ − − y γ

z γ

z tP tS

(12.7c)

z

Shear Shear Strain from Warping

Force Primary Torsion

Torsion

u , κ z = θ z , κ y = θ y ,

where the strains and curvatures of the bar axis are defined as = κ ω =

ψ , ψ = γ tS − φ x , γ y = v − θ z , γ tP = φ x , and γ z = w + θ y . The prime indicates the partial

derivative with respect to x .

12.1.4 Internal Virtual Work
From Eq. (12.1), the internal virtual work is given by

− δ W i =

V (σ

δ + τ δγ + τ δγ ) dV . (12.8)

x x xy xy xz xz

Substitution of the strain expressions of Eq. (12.7) into Eq. (12.8) leads to

xy

∂ω
∂

φ x u − θ z + θ y + ωψ ) + τ δ(v − θ

− δ W i = σ δ( y z ) + δ

y ψ −

z

x z

x A

xz

∂ω
∂

φ x dA dx δ(w + θ

+ τ ) + δ

z ψ +

y

y