Direct Variational and Weighted Residual Methods: Classical Trial Function Methods - Mechanics of Structures: Variational and Computational Methods - page 456

Thus

a 2 − b 2

ω = yz (10)

a 2

+

b 2

The δφ terms of (2) appear as

∂
∂

∂ω
∂

z 2

∂ω
∂

y 2 G φ dV

x δφ y +

+

z −

− m x δφ dx = 0 (11)

L

Rewrite this as

∂ω
∂

2 ∂ω

∂

2

∂
∂

z ∂ω
∂

y ∂ω
∂ x δφ

+ +

y − y z z

L A

z ∂ω
∂

z 2 dA G

y ∂ω
∂

y 2 φ

+

y − y +

+ dx − m x δφ dx = 0 (12)

L

The integral

∂ω
∂

2 ∂ω

∂

2 dA

z ∂ω
∂

y ∂ω
∂

+ +

y −

(13)

y z z

A

can be shown to be zero. To do so, use the displacement relationship 0

of Eq. (1.151), Green's integral theorem of Appendix II, and the surface condition of
Eq. (1.153). Introduce J of Eq. (1.154)

∂

2

ω/∂ y 2 + ∂

2

ω/∂ z 2 =

y y

z z

dA

− ∂ω
∂

+ ∂ω
∂

J = + (14)

z y

A

into (12), giving

d
dx δφ

GJ d

dx

dx − m x δφ dx = 0 (15)

L L

Integrate by parts and use appropriate boundary conditions to find

d dx

dx GJ d

dx

L δφ

=− m x δφ dx (16)

L

or

dx GJ d

d dx =−

m x (17)

This is the traditional equation of motion for torsion of a rod.
If ω of (10) is inserted in (14)

a 3 b 3 π

J = (18)

a 2 + b 2

The stresses become

2 zM t
ab 3

φ (

τ xy = G γ xy =− G z + ∂ω/∂ y ) =−

π

(19)

2 yM t
a 3 b

φ ( −

τ = γ =− + ∂ω/∂ ) =−

G G y z

xz xz

π

These example problems illustrate several interesting characteristics of the Ritz method,
especially that of its simplicity and preciseness. Also, when the trial functions span the