Large axial forces - Poznań

POZNAŃ UNIVERSITY OF TECHNOLOGY
INSTITUTE OF STRUCTURAL ENGINEERING
DIVISION OF STRUCTURAL MECHANICS
EXERCISE NO. 3
INITIAL STABILITY AND STATIC ANALYSIS WITH LARGE AXIAL
FORCES FOR PLANE FRAMES – MATRIX METHOD
Name
Academic year
________________________________
________________________________
For the frame No. ___ :
1. Find the distribution of axial forces in the frame subjected to the reference loading (λ = 1) using any of the first-order
methods of static analysis.
2. Calculate the critical loading factor λ cr using the discretisation 1 beam – 1 element. Draw the buckling mode.
3. Calculate nodal displacements and internal forces taking into account the second-order effects for the loading corresponding
to ____ % of the calculated critical loading. Carry out 1 iteration.
Geometric stiffness matrices for beam elements
1. Clamped-clamped beam 2. Hinged-clamped beam
~
K
g
N
=
30l
⎡0
0 0 0 0 0 ⎤
⎢
⎥
⎢
0 36 3l
0 − 36 3l
⎥
⎢
2
2
0 3l
4l
0 − 3l
− l ⎥
⎢
⎥
⎢0
0 0 0 0 0 ⎥
⎢
⎥
⎢
0 − 36 − 3l
0 36 − 3l
⎥
⎢
2
2⎥
⎣0
3l
− l 0 − 3l
4l
⎦
~
K
g
N
=
30l
⎡0
0
⎢
⎢
0 36
⎢0
0
⎢
⎢0
0
⎢0
− 36
⎢
⎢⎣
0 6l
0
0
0
0
0
0
0 0 0 ⎤
0 − 36 6l
⎥
⎥
0 0 0 ⎥
⎥
0 0 0 ⎥
0 36 − 6l
⎥
⎥
2
0 − 6l
6l
⎥⎦
3. Clamped-hinged beam
~
K
g
⎡0
0 0 0 0
⎢
⎢
0 36 6l
0 − 36
⎢
2
N 0 6l
6l
0 − 6l
= ⎢
30l
⎢0
0 0 0 0
⎢
⎢
0 − 36 − 6l
0 36
⎢⎣
0 0 0 0 0
0 ⎤
0
⎥
⎥
0⎥
⎥
0 ⎥
0
⎥
⎥
0 ⎥⎦

1 2 3
2 2
2
2
1 1
1
1
k = ___ EA1
1
1
2
4 5 6
2 2
2
1
2
1
1
1
2
2
1
1
7 8 9
2 2
2
2
2
1
k = ___ EA1
1
1
2
1
1
10 11 12
2 2
2 2
2
1
1 1
1
1
1
1
13 14
2
1
1
2
1
2
2
1
Element cross-sections:
1 − ____________________
2 − ____________________