Computer Aided Engineering Design

324 COMPUTER AIDED ENGINEERING DESIGN

ui vi θi uj vj θj

k =

00– 00

0 12 6 0 – 12 6

06 4 0–6 2

–00 00

0 – 12 – 6 0 12 – 6

06 2 0–6 4

32 32

22

32 3 2

22

AE

l

AE

l

EI

l

EI

l

EI

l

EI

l

EI

l

EI

l

EI

l

EI

l

AE

l

AE

l

EI

l

EI

l

EI

l

EI

l

EI

l

EI

l

EI

l

EI

l

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥⎥

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

u

u

i i i j j j

v

v

θ

θ

(11.9h)

11.5.1 Frame Elements and Transformations

For a frame element to be oriented at an angle θ in the x-y plane (Figure 11.10), the displacements
along the x and y global coordinate axes, namely, uix,uiy for node i and ujx,ujy for node j can be
extracted using the following transformation. Note that the rotations θi and θj remain invariant.

ue

ix

iy

i

jx

jy

j

i

i

i

j

j

u

u

u

u

u

u

= =

cos – sin 0 0 0 0

sin cos 0 0 0 0

001000

0 0 0 cos – sin 0

0 0 0 sin cos 0

000001

θ

θ

θθ

θθ

θθ

θθ

θ

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

v

v

θθj

T

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

=  u (11.9i)

The stiffness of the frame element for the displacements (and forces) along the x and y directions,
and rotations (moments) perpendicular to the plane containing the frame element is given similar to
the truss element as

ujy

uj

ujx

vj

θj

ξ

θ

uiy

ui

uix

vi

θi

y

x

Figure 11.10 A frame element oriented at an angle θθθθθ