Transcript Design of Timber Bending Members

Design of Bending Members in Steel
Steel wide
flange
beams in
an office
building
Composite Steel-Concrete Girders
An example of
curved steel
beams
German Historical Museum
Steel Girders for Bridge Decks
Sea to sky highway, Squamish
Cantilevered
arms for steel
pole
What can go wrong ?
STEEL BEAMS:
• Bending failure
• Lateral torsional buckling
• Shear failure
• Bearing failure (web crippling)
• Excessive deflections
Bending Strength
Linear elastic stresses
 max
y
M
Design Equation:
My M


I
S
M   max S
for rectangular sections
bh2
S
6
M r   Fb S
Where Fb is the characteristic bending strength
For steel this is Fb = Fy
For timber it is Fb = fb (KDKHKSbKT)
Plastic moment capacity of steel beams
Fy
Fy
Ac
C
My
Mp
a
T
Yield moment
M y  Fy S
Plastic moment
Mp
At
 Ca  Ta
 Fy AC a  Fy AT a
 Fy Z
So, when do we use
the one or the other
??
Which is the definition of the plastic section
modulus Z
Z can be found by halving the cross-sectional
area and multiplying the distance between the
centroids of the two areas with one of the areas
This is also called the first moment of area
Steel beam design equation
For laterally supported
beams (no lateral
torsional buckling)
Mr =  Fy Z
for class 1 and 2
sections
Mr =  Fy S
for class 3 sections
where  = 0.9
Steel cross-section classes
Class 4 Real thin plate sections
Mr < My
(Use Cold
Will buckle before reaching Fy at
Formed Section
Code S136)
extreme fibres
Class 3 Fairly thin (slender) flanges and web
Mr = My
Will not buckle until reaching Fy in
extreme fibres
Class 2 Stocky plate sections
Mr = Mp
Will not buckle until at least the
plastic moment capacity is reached
Class 1 Very stockyplate sections
Can be bent beyond Mp and can
therefore be used for plastic
analysis
Mr = Mp
Load deflection curves for Class 1
to Class 4 sections
Local buckling of the
compression flange
Local torsional buckling of the
compression flange
Local web buckling
Lateral torsional buckling
Elastic buckling:
Mu = ωπ / Le √(GJ EIy ) + (π/L)2EIy ECw
Moment gradient factor
Torsional stiffness
Lateral bending stiffness
Warping stiffness
y
y
Δx
x
Δy
x
Le
θ
x
y
y
Mr / 
Mmax
Moment resistance of laterally
unsupported steel beams
Mmax = My for class 3
or Mp for class 1 and 2
0.67Mmax
1.15 Mmax [1- (0.28Mmax/Mu)]
Mu
Le
Shear stress in a beam
A
y
d
N.A.
b
τ
y
A
τmax
b=w
d
N.A.
= V(0.5A)(d/4)
(bd3/12)b
=1.5 V/A
VAy VQ


Ib
Ib
τ
τmax
≈ V/Aw
=V/wd
Shear design of a steel I-beam
Vr = φ Aw 0.66 Fy
for h/w ≤ 1018/√Fy = 54.4 for 350W steel
d
h
w
This is the case for all rolled shapes
For welded plate girders when
h/w ≥ 1018/√Fy
the shear stress is reduced to
account for buckling of the web
(see clause 13.4.1.1)
Aw = d.w for rolled shapes and h.w for welded girders
Bearing
failures in a
steel beam
N+10t
w
N+4t
k
For end reactions
Br  bi w  N  4t  Fy , or ,1.45bi w2 Fy E
For interior reactions
Br  bi w  N  10t  Fy , or , 0.60be w2 Fy E
N
Deflections
• A serviceability criterion
– Avoid damage to cladding etc. (Δ ≤ L/180)
– Avoid vibrations (Δ ≤ L/360)
– Aesthetics (Δ ≤ L/240)
• Use unfactored loads
• Typically not part of the code
Δ