Matrix Methods of Analysis - Structural and Stress Analysis - page 559

y, v

y, v

x, w

j

i

x, w

F IGURE 17.5 Local and
global coordinate systems for
a member in a space truss

z, u

z, u

6 so that [ K ij ] of Eq. (17.15) must be expanded to the same order to
allow for this. Hence

×

of the order 6

w i v i u i w j v j u j

100
000 000
000 000
−

100 −

AE
L

[ K ij ] = (17.24)

100 100
000 000
000 000

InFig. 17.5 themember ij is of length L , cross-sectional area A andmodulus of elasticity
E . Global and local coordinate systems are designated as for the two-dimensional case.
Further, we suppose that

¯

θ xx = angle between x and x

θ xy = angle between x and

¯ y

.

¯

θ zy = angle between z and y

.

Therefore, nodal forces referred to the two systems of axes are related as follows

!

F x = F x cos θ xx + F y cos θ xy + F z cos θ xz

(17.25)

F y = F x cos θ yx + F y cos θ yy + F z cos θ yz

"

F z = F x cos θ zx + F y cos θ zy + F z cos θ zz

Writing

!

λ = cos θ xx λ = cos θ xy λ = cos θ xz

¯ x ¯ y ¯ z

µ x = cos θ yx µ y = cos θ yy µ z = cos θ yz (17.26)

"

ν x = cos θ zx ν y = cos θ zy ν z = cos θ zz