Beams - Mechanics of Structures: Variational and Computational Methods - page 733

zdA zy dA zz dA dA − κ z + κ y + z ω κ ω

+ E δκ

y

A A A A

I z I zy I zz = I I z ω

dA ydA zdA dA

A ω

−

A ω

κ z +

A ω

κ y +

A ωω

κ ω

+ E δκ ω

I I I I

ω ω y ω z ωω

z 2 2 dA

G

+ ∂ω
∂

+ ∂ω
∂

+ − y γ tP δγ tP

+ dx . (12.16)

y z

A

J t

This provides definitions of the cross-sectional characteristics for arbitrary coordinate axes
y and z . Thus

dA [ L 2 ] ydA [ L 3 ] zdA [ L 3 ] A = I y = I z =

A A A

dA [ L 4 ] yz dA [ L 4 ] dA [ L 5 ] I

ω =

A ω

I yz = I y ω = y ω

A A

(12.17)

dA [ L 5 ] y 2 dA [ L 4 ] z 2 dA [ L 4 ] I z ω = z ω I yy = I zz =

A A A

z 2 2 dA [ L 4 ]

+ ∂ω
∂

+ ∂ω
∂

2 dA [ L 6 ]

I

ωω =

A ω

J t = + − y

y z

A

where the dimensions are in brackets [ ], with L indicating length. These cross-sectional
properties are displayed in Fig. 12.6a [Bornscheuer, 1952].
Special cross-sectional axes are of interest. If the cross-sectional coordinates y, z are the
centroidal axes y C ,z C , the first moments of the areas are zero, i.e.,

I y = 0 y = y − y C y C = I y / A

(12.18a)

I z = 0 z = z − z C z C = I z / A .

The corresponding ω coordinate is ω C , and Eq. (12.18a) is supplemented with the condition

I

ω =

0 ω = ω − ω C ω C = I

ω /

A . (12.18b)

These relations substituted into Eq. (12.17) lead to

I y I z
A

I y I

A

[ L 4 ] [ L 5 ]

I y z = I yz − I y ω = I y ω −

I z I A I y I y

A

[ L 5 ] [ L 4 ]

I z ω = I z ω − I y y = I yy − (12.18c)

I z I z
A

I I

A ω

[ L 4 ] [ L 6 ]

I z z = I zz − I

ω ω =

I

ωω −