同济大学 双语 钢结构4-Compression

Mechanical imperfection
residual stress
Flexural equilibrium equation
EIy '' Ny 0
EIy '' Ny Ne0
elastic
N ym eo /(sec 1) 2 NE
Elastic-plastic
Compression members
Refer to §5.1
introduction
Compression members in structures - truss members - bracing - pinned columns Structural shapes commonly used in compression members
BASIC PRINCIPLES OF STEEL STRUCTURES
Prepared by: Dr. Feng ZHOU
Room: Rui’an Building 207A ( welcome to walk-in)
Phone: 15800660849
BASIC PRINCIPLES OF STEEL STRUCTURES
Effective length
Effective length factor for different end constraint
- End constraint: bending and torsional deformation
- Constraint types: free end, pinned, fixed end - Effective length factor: Table 5-4 in page 101
N-y
Compression members
Effects of initial imperfections
Geometrical imperfection
initial out-of-straightness (crookedness), initial eccentricity
Mechanical imperfection
It － torsional constant
2 2 r02 (Ix Iy) / A x 0 y0
Relate to torsion problem
A － cross-section area
y0
Centroid o
x0
where a shear force can be applied without inducing any torsion
residual stress
Ideal stress-strain curve and buckling curve
stress-strain curve with residual stress and corresponding buckling curve
Compression members
- doubly symmetric - singly symmetric - unsymmetric
Compression members
possible failure modes
• Material yield, squash load
• Member (overall) buckling
• Plate (local) buckling
A=0:
Compression members
column theory
Pcr l n EI
Pcr
l / n 2
2 EI
•
values of n larger than 1 are not possible unless the compression member is physically restrained from deflecting at the points where the reversal of curvature would occur.
u = 1.0
u = 2.0
u = 0.5
u = 0.7
How about the critical stress? E N E / A 2 E / 2

ul i
Compression members
More on effective length
Minor axis buckling
members
Local buckling of plate element Laced and battened compression members Rigidity of compression members Design of axially loaded compression members
N u An f y
Design equation
N Nu / R An f y / R An f d
N fd An
Compression members
member (overall) buckling
Compression members
plate (local) buckling
Topic 4 Compression members
Compression members
Outlines
Introduction Resistance of cross-section Overall stability of uniform (solid-web) compression
N
•
Pcr
2 EI
l
2
2 EA Pcr 2
“Euler load”
N N N
where, E — modulus of elasticity of the material I — moment of inertia of the cross-section area with respect to the minor principle axis l — length between points of supported
EI ω IV GI t '' Nx0v '' Ny 0u '' ( Nr02 R ) '' 0
where：Ix、Iy － moment of inertia with respect to x, y axis
Iw － warping constant N
Y (v) Shear center S(x0, y0) X (u)
Compression members
Refer to §5.3, §5.5
basic concept of stability
After temporary transverse load P is removed
N
N
△
＜ Ncr ，δ＝ 0，equilibrium state is stable N ＝ Ncr ，δ kept small, critical state ＞ Ncr ，δ increases，equilibrium state is unstable
Compression members
Effects of initial imperfections
Geometrical imperfection
initial out-of-straightness (crookedness), initial eccentricity
Mechanical imperfection
Compression members
equilibrium equations of elastic buckling for ideal members
Z（rotation）
N
EI x v IV Nv '' Nx0 '' 0
EI yu IV Nu'' Ny0 '' 0
δ
P
N
N
stable
critical
unstable
Critical buckling load —the load that is just large enough to maintain the deflected shape when the temporary transverse load is removed.
Equilibrium equation
EIy Ny 0
''
N N y A cos x B sin x EI EI
Pcr l n EI
At x=0, y=0:
At x=l, y=0:
N B sin EI l 0
or N (1 yom / ym ) N E
N
N－y
N ult N E
Compression members
Effects of initial imperfections
Geometrical imperfection
initial out-of-straightness (crookedness), initial eccentricity
Ib＝∞ Ib＝∞
Major axis buckling
u=1.0
支撑
u=0.5
Compression members
Member (overall) bucking modes
(1) Flexural buckling (2) Torsion buckling