Beams - Mechanics of Structures: Variational and Computational Methods - page 732

corresponds to Bernoulli's hypothesis of plane cross-sections remaining plane and orthog-
onal to the axis during bending. For such cases, i.e., when shear deformation is not taken
into account,

θ y =− w θ z = v (12.12)

Torsion: Wagner's Hypothesis
Wagner's hypothesis is the assumption that a cross-sectional area retains its shape during
torsion, i.e., it remains undistorted. This hypothesis is equivalent to specifying that warping
torsion generated shear strains can be neglected, i.e.,

γ tS = φ x + ψ = ψ =− φ x 0or (12.13)

Internal Virtual Work
The underlined terms of Eq. (12.10) contain the strains

(γ γ γ )

that are zero for
the kinematic assumptions just discussed. If these terms are left out and the definitions
=

y , z , and tS

= φ x are introduced in Eq. (12.10), the internal
virtual work will contain five terms rather than eight. Thus,

u , = θ z , = θ y , κ ω = ψ , and

κ κ γ

z y tP

u + δv − δw −

ω δφ x +

δφ x ) − δ W i =

x (

N δ M z M y M M tP dx (12.14)

Introduction of the Material Law
Return to the internal virtual work of Eq. (12.8), and introduce the material law in order to
obtain an expression in terms of strains. Thus, substitution of Hooke's law into Eq. (12.8)
gives

V (

− δ W i = E x δ x + G γ xy δγ xy + G γ xz δγ xz ) dV (12.15)

Neglect the shear terms related to γ tS , and introduce the strain displacement

relations of Eq. (12.7) into Eq. (12.15). This leads to

γ y , γ z , and

E

− δ W i = ( − y κ z + z κ y + ωκ ω )δ( − y κ z + z κ y + ωκ ω )

x A

G z

tP z

G

tP tP dA dx

+ ∂ω
∂

+ ∂ω
∂

+ ∂ω
∂

+ ∂ω
∂

+ γ δγ + − y γ − y δγ

tP

y y z z

For constant E and G over the cross-section, this expression can be written as

dA ydA zdA dA − κ z + κ y +

A ω

κ ω

− δ W i = E δ

A A A

x

A I y I z I

ω

ydA yy dA yz dA dA − κ z + κ y + y ω κ ω

− E δκ

z

A A A A

I y I yy I yz I y ω