Segment Stiffness

Three-Dimensional Columns

5.3 Stiffness Formulation .1 Fiber Stiffness

5.3.2 Segment Stiffness

The segment stiffness for 2D elements was formulated in section 3.3.2. The 3D formulation is similar, using separate interpolation of displacements and rotations.

As with the 2D segment, shearing is elastic and considered separately.

The relation between fibers and segments is shown in figures 5.18 and 5.19. Each fiber i has its centroid located a distance hi from the neutral axis in the X{ direction and bi in the X~ direction. The transformation from segment displacements to fiber displacements is easily determined by individually displacing the segment degrees of freedom. Axial, shearing, and rotational displacements at the right end of the segment are shown for both X{-X~ plane and X~-X~ plane displacements in figure 5.20. The resulting relation between segment displacements Ail⁸ and individual fiber displacements

Auf

^is

(5.29)

_ 2 X'

~x·

- 2 X'

~x·1

1

,-

~.

_I•

'---

1 ^l

I l l I i I I ' I

- - - - ~~I I I I I

I I

L_j

X' ₃

t

_X'1

Figure 5.20 Relation between fiber and segment displacements.

where the transformation from segment to fiber displacements is given by

0 1 0 -hi -bi 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0

(5.30) R?F=

~

0 0 0 0 0 0 1 0 -hi -bi 0 0 0 0 0 ^-1 0 0 0 0

0 0 0 0 0 0 0 1 0 0

I

•

I

_~x·1

^{_ 2 X'}

X' ₃

tx·

_ 2

The contribution of a fiber's forces

pf

to the segment forces is denoted by

pf

and their relation is (5.31)

Using the fiber equation (5.28), and substituting the displacement transformation (5.29) and the force transformation (5.31) in incremental form, gives

(5.32)

Summing the contributions from each fiber gives (5.33)

where the tangent segment stiffness due to fibers is summed for the total number of fibers, nF:

np

(5.34) Kf'F

=

L([RfF]T[K~][RfF]).

i=l

The shear stiffness is determined using mid-span sampling as in the 2D case:

A1 0 0 ^A1 0 ^A1 0 0 ^A1 0

11 2 -11 ₂

0 0 0 0 0 0 0 0 0

A3 0 ^A3 0 0 ^Ag 0 &

lg 2 -13 2

A1l1 0 ^A1 0 0 ^A1h 0

- 4 - - 2 ^{- 4 -}

A3lg

0 0 ^_A3 0 ^A3l3

(5.35) Kf'SH

=

G ^{- 4 - 2 - 4 -}

A1 0 0 ^A1 0

11 ^{- 2}

0 0 0 0

sym 13 A3 0 ^{- 2 A3}

A1h 0

- 4 - A3lg

4

The projected lengths are used in the formulation since the local axes are in the projected planes (figure 5.1). Theses projected lengths are:

(5.36) (5.37)

The shear area A1 is dtw for the strong axis of a wide flange section or 5/6(2btt) for the minor axis. For a box column, A1 is 2dtw for the strong axis or 2btt for the weak axis. The shear area A3 corresponds to the direction not being used for the X1 axis. For example, if the weak axis is aligned with the X 1 axis, then strong axis properties should be used for the

x3

axis.

Local segment forces can be determined from fiber stresses O'i, geometric prop-

erties and end rotations:

(5.38) (5.39) (5.40) (5.41)

(5.42)

(5.43)

(5.44)

i=1

Q1

=

2GA1(0u 1

+

012- 2a1) Q3

=

2GA3(031 1

+

^032-2a3)

Np 1

Mu = -

L

^uiAihi

⁺

^2Q1h

i=1

Np 1

M12

= L

^O'iAihi

+

2Qlh i=1

Np 1

M31 = -

L

^O'iAibi

+

2Q3l3 i=l

Np 1

M32

= L

^O'iAibi

⁺

^2Q3h·

i=1

The strains developed in the fibers are determined from the segment end displace- ments in each plane

(5.45)

Refer to figures 5.21 and 5.22 to see the positive sense of all segment local properties in the Xi-X~ and X~-X~ planes, respectively. Additionally, these figures define the end angles, On, 012, On and 012, in terms of the segment rotations,

u4, ug, u5

^and

u

10 , respectively.

The global end forces of the segment are determined from the transformation

(5.46) ^A

>T

P10

X'

t

²

u~ +u~

'Yt\

_{_ u i} ^AS

End 1

• - X ' I

End2 ^{u6 AS}

~AS

-

A u 9

u7

Axes Dofs

M11~

p

--

-

Ql

~M12

Forces

()(,1

~----

"'S

912= u9

Angles, Fibers Figure 5.21 Positive directions of various segment parameters, Xf-X~ plane.

where

(5.47)

and Tis the transformation from equation (5.1). The displacement transformation is u.s

=

T sus. Refer to figures 5.21 and 5.22 for the positive sense of the segment end forces and to figures 5.1 and 5.2 for local and global parameters. The global segment stiffness matrix can be determined from the segment transformation (5.48)

resulting in (5.49)

X'

l

^{2 '}

• ____,._X 3

End2 ^{Ug AS}

---

A ~AS uio u7

Axes Dofs

p M 3 I r K

Q3

---

~M32

Forces

as

~- - - - -

"'S 832= U10

Angles, Fibers Figure 5.22 Positive directions of various segment parameters, X~-X~ plane.

5.3.3 Column Stiffness

Now that the global segment relations have been determined, the segment stiffnesses are summed to formulate the member stiffness. Dropping superscripts denoting 3D columns, the member stiffness is

(5.50)

and the member equation for all interior and exterior dofs is

(5.51) K~u = f -p = ~p,

where f are the applied forces and p are the member stiffness forces. The stiffness matrix can be partitioned as in the case of beam-columns from section 3.3.3:

(5.52)

The member incremental displacements are solved with individual member itera- tions using the exterior dof displacement increments recovered from the frame as input, as discussed for the 2D members. The same derivation (section 3.3.3) pro- vides the equation for a 3D column (denoted CC) in terms of its end (exterior) degrees of freedom

(5.53) (5.54) (5.55) (5.56) (5.57) (5.58)

Kcc ~ucc =~Pee, where

~Pee = fcc _ Pee fcc= fE

P00

=

^{{PE -} KEIK]}PI}

u00

=

^uE and

Kcc

= [

KEE- KEIK]}KIE

J .

The column end forces p⁰⁰ are applied to the edges of the panel zones and assem- bled to the frame nodal forces that have applied forces f already assembled. The forces f 00 never actually have to be determined since the columns are not solved individually for the condensed exterior-only system. The tangent stiffness matrix K⁰⁰ is assembled to the frame stiffness matrix. Equation (5.53) gives the column end displacements and forces in column global coordinates.

The column output forces (see figures 5.21 and 5.22 for segments, which are

analogous) in local coordinates are calculated using the updated geometry as follows:

(5.59) (5.60) (5.61) (5.62) (5.63) (5.64) (5.65)

Ns

p

= 2_

'"'p.S N L..t ^t

s

i=l

Mn

= Mfi,

^segment 1 M12

=

M~, segment Ns

Q1

= L~

^(Mn

⁺

^M12)

SlllO<l

M31

= M£,

segment 1 M32

=

M~, segment N

s

1

Q3

= -

L~ (M31

+

^M32).

Slll0<3

These forces are for output purposes only.

The column end forces (equation 5.56) act at the end of the column which is located at the edge of a panel zone. These forces must be transformed to frame nodal forces that are applied at the center of the panel zone. The panel zone depth d~ and widths dJ.i and d3i, and the beam and column rotations

ot,

^(}~i' OJ.i and 03i at each end ( i or j) of the columns are used to determine the frame nodal loading from the columns.