Torsion members - The Behaviour and Design of Steel Structures to EC3 - page 438

(in which G is the shear modulus of elasticity) for the stress function θ (which
corresponds to the membrane displacement in Prandtl's analogy) defined by

,

τ xz = τ zx =− ∂θ/∂ y

(10.2)

τ xy = τ yx = ∂θ/∂ z

subject to the condition that

θ = constant (10.3)

aroundthesectionboundary.Theuniformtorque T t ,whichisthestaticequivalent
of the shear stresses τ xy , τ xz , is given by

T t = 2 θ d y d z . (10.4)

section

10.2.1.2 Solid cross-sections

Closed-form solutions of equations 10.1-10.3 are not generally available, except
for some very simple cross-sections. For a solid circular section of radius R , the
solution is
θ = G
2

d φ
d x ( R 2 − y 2 − z 2 ) .

(10.5)

Ifthisissubstitutedinequations10.2,thecircumferentialshearstress τ t ataradius

r = √ ( y 2 + z 2 ) is found to be
τ t = Gr d φ

d x .

(10.6)

The torque effect of these shear stresses is

R

T t = τ t ( 2 π r ) d r (10.7)

0

which can be expressed in the form
T t = GI t d φ
d x

(10.8)

where the torsion section constant I t is given by
I t = π R 4 / 2,

(10.9)

which can also be expressed as

A 4
4 π 2 ( I y + I z ) .

I t = (10.10)

The same result can be obtained by substituting equation 10.5 directly into
equation 10.4. Equation 10.6 indicates that the maximum shear stress occurs at