Part V: Geometric Stiffness of Frames

Structural Design Lab.(Prof. Ho-Kyung Kim) Dept. of Civil & Environmental Eng.

Seoul National University

457.649 Advanced Structural Analysis

Part V:

Geometric Stiffness of Frames

▶

Geometric effects:

§

Initial imperfections

§

P-Δ effect

§

P-δ effect

▶

Material effects:

§

Plastic deformation of steel structures

§

Cracking or creep of reinforced concrete structures

§

Inelastic interaction of axial force, bending, shear and torsion

Source of Nonlinearity

P-δ and P-Δ Effects

Moment Amplification or Second-Order Analysis for P-δ

( C P

^m

P

_e

)

M

M = -

0

1

max

Moment Amplification or Second-Order Analysis for P-Δ

2

2

1 1

1

^nt

e

B P

P

= ³

- å

å

Types of Analysis

▶

Linear Elastic Analysis

▶

Incremental form with tangential stiffness matrix

▶

Second-order Elastic Analysis

▶

First-order Inelastic Analysis

▶

Second-order Inelastic Analysis

▶

Eigenvalue Problem

Matrix Representation

[ ] K

e

{ } { } D = P

[ ] K

t

{ } { } d D = dP

{ } { }

e g

é + ù D =

ë K K û d dP

[ K

e

+ K

m

] { } { } d D = dP

{ } { }

e g m

é + + ù D =

ë K K K û d dP

{ } { }

e

l ˆ

_g

0

é + ù D =

ë K K û l 1 { } D

^f

= - é ë K

^ef

ù û ë

^-1

é K ˆ

^gf

ù û { } D

^f

Large Displacement

▶

e.g. for axial force element,

§

▶

for pure torsion element,

§

▶

for flexural element,

§

Displacement in terms of nodal displacements

[ ] { } D

= D

=

D +

+ D +

+ D +

D

= D

å

=

N N

N N

N N

n i

i i

n n i

i

1

2 2 1

1

! !

2 2 1

1

u N u

N

u = +

2 2 1

1 x x

x

N q N q

q = +

2 4 1

3 2

2 1

1

v N v N

z

N

z

N

v = + + q + q

Shape Functions for Axial Member

x a a dx u

e

_x

= du Þ =

₁

+

₂

( )

_{1 2}

2 1

2 2 1

1

1 1

u u

L u u x

L x

u N u

N u

x x + -

=

÷ + ø ç ö

è æ -

=

+

=

Shape Functions for Flexural Member

1 1 2 2 3 1 4 2

2 3

1

2 2

2 3

3

2 4

1 3 2

1

3 2

z z

v N v N v N N

x x

N L L

N x x

L

x x

N L L

x x

N x

L L

q q

= + + +

æ ö æ ö

= - ç ÷ è ø + ç ÷ è ø

æ ö

= ç è - ÷ ø

æ ö æ ö

= ç ÷ è ø - ç ÷ è ø é æ ö ù

= ê ê ë ç ÷ è ø - ú ú û

그림

7.3 삽입

그림

7.5 삽입

Element Stiffness Matrix

= 0 -

= W

_ext

W

_int

W d d d

[ ][ ] { } ( ) d vol

W

int

= ò

vol

d e E e d

[ ] d Δ { } F

d

d = å D =

= n i

i i

ext

F

W

1

[ ] { } [ ][ ] { } ( ) [ ] { }

vol

d d vol d

é ¢ ¢ ù =

ë ò Δ N E N Δ û Δ F

[ ] k = é ë ò

_vol

{ } N ¢ [ ][ ] E N ¢ d vol ( ) ù û

Geometric Stiffness Matrices: Axial Force Member

Geometric Stiffness Matrices: Combined Bending and Axial Force

2 3

Combined Torsion and axial force

Three Dimensional Geometric Nonlinear Analysis

Three Dimensional Geometric Nonlinear Analysis