Buckling of Beam-Columns – I -...
28
Buckling of Beam-Columns – I Prof. Tzuyang Yu Structural Engineering Research Group (SERG) Department of Civil and Environmental Engineering University of Massachusetts Lowell Lowell, Massachusetts CIVE.5120 Structural Stability (3-0-3) 02/07/17
Transcript of Buckling of Beam-Columns – I -...
Buckling of Beam-Columns – I
Prof. Tzuyang Yu Structural Engineering Research Group (SERG)
Department of Civil and Environmental Engineering University of Massachusetts Lowell
Lowell, Massachusetts
CIVE.5120 Structural Stability (3-0-3) 02/07/17
2
Beam-Columns – I
(Source: Structural Stability Research Council)
3
Beam-Columns – I
Buckling of a steel beam-column
4
Beam-Columns – I
Buckling of a steel beam-column
5
Outline
• Beam-columns with various loading conditions – Uniformly distributed lateral load, w(x) – A concentrated lateral load, Q(x=a) – End moments, MA and MB
• single curvature • double curvature
• Summary
6
Beam-Columns – I
• Uniformly distributed lateral load: w(x) – 2nd-order D.E. approach – Governing equation:
– Solution: y = yc + yp
7
Beam-Columns – I
– Maximum deflection: ymax
– Primary and secondary effects:
– Critical load:
8
Beam-Columns – I
– Interactions among P, w(x), and ymax:
• At constant P, w(x) and ymax are proportional to each other.
• At constant w(x), P and ymax are NOT proportional to each other.
9
Beam-Columns – I
– Maximum internal bending moment: Mmax
– Primary and secondary effects:
10
Beam-Columns – I
• Theoretical and design values: – Maximum deflection: ymax
– Maximum internal bending moment: Mmax
11
Beam-Columns – I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60
70
80
90
100
P/Pe
Ratio
of l
ater
al d
efle
ctio
n, y
/yo
ytheoreticalydesign
12
Beam-Columns – I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60
70
80
90
100
P/Pe
Ratio
of b
endi
ng m
omen
t, M
max
/Mo
MtheoreticalMdesign
13
Beam-Columns – I
• Note: Taylor series expansion
14
Beam-Columns – I
• A concentrated lateral load: Q(x=a) – 2nd-order D.E. approach – Governing equation:
– Solution: y = yc + yp
15
Beam-Columns – I
– Maximum deflection: ymax
– Primary and secondary effects:
– Critical load, Pcr
16
Beam-Columns – I
– Maximum internal bending moment: Mmax
– Primary and secondary effects:
17
Beam-Columns – I
• Theoretical and design values: – Maximum deflection: ymax
– Maximum internal bending moment: Mmax
18
Beam-Columns – I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60
70
80
90
100
P/Pe
Ratio
of l
ater
al d
efle
ctio
n, y
max
/yo
ytheoreticalydesign
19
Beam-Columns – I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60
70
80
90
P/Pe
Ratio
of b
endi
ng m
omen
t, M
max
/Mo
MtheoreticalMdesign
20
Beam-Columns – I
• End moments: MA and MB – 2nd-order D.E. approach – Governing equation:
– Solution: y = yc + yp
21
Beam-Columns – I
– Maximum deflection: ymax
– Primary and secondary effects:
– Critical load, Pcr
22
Beam-Columns – I
– Maximum internal bending moment: Mmax
– Primary and secondary effects:
23
Beam-Columns – I
• Theoretical and design values: – Maximum deflection: ymax
– Maximum internal bending moment: Mmax
24
Beam-Columns – I
• Concept of equivalent moment (Equivalent moment factor, Cm)
25
Beam-Columns – I
• Failure modes – Beam-column joint failed in shear
(Source: Toshihiro MIKI, Daido Inst. Tech., Japan)
26
Beam-Columns – I
• Failure modes – Beam-column joint failed in bending
(Source: Toshihiro MIKI, Daido Inst. Tech., Japan)
27
Beam-Columns – I
• Failure modes – Beam-column joint failed in torsion
(Source: Toshihiro MIKI, Daido Inst. Tech., Japan)
28
Summary
• In reality, all members in a frame are beam-columns.
• When analyzing beam-columns, the primary effect (bending moment and deflection) is due to bending moment and the secondary effect (bending moment and deflection) is to due to axial force; however, the secondary bending moment and deflection can be more significant than the primary ones.
• At constant P, w(x) and ymax are proportional to each other; however, at constant w(x), P and ymax are NOT proportional to each other.
• For beam-columns subjected to end moments, : Double curvature;
: Single curvature
0A
B
MM
>
0A
B
MM
<
