Download Effective static load distributions

Wind loading and structural response
Lecture 13 Dr. J.D. Holmes
Effective static loading
distributions
Effective static loading distributions
• Static load distributions which give correct peak load effects under
fluctuating wind loading
Separately calculate e.s.l.d  s for :
•
mean component
•
background component
•
resonant components
• Generally e.s.l.d. s depend on load effect (e.g. bending moment, shear)
Effective static loading distributions
Influence coefficient :
Value of a load effect as a unit load is moved around a structure :
Effective static loading distributions
Influence coefficient :
Value of a load effect as a unit load is moved around a structure :
Effective static loading distributions
Influence coefficient :
Value of a load effect as a unit load is moved around a structure :
Effective static loading distributions
Influence coefficient :
Value of a load effect as a unit load is moved around a structure :
Effective static loading distributions
Influence coefficient :
Value of a load effect as a unit load is moved around a structure :
Effective static loading distributions
Influence coefficient :
Value of a load effect as a unit load is moved around a structure :
Influence line - Central B.M.
0.2
0.1
0
-0.1
0
0.2
0.4
0.6
position/total beam length
0.8
1
Effective static loading distributions
Influence coefficient :
Value of a load effect as a unit load is moved around a structure :
Influence line - Central B.M.
0.2
0.1
0
-0.1
0
0.2
0.4
0.6
position/total beam length
0.8
1
Effective static loading distributions
Influence coefficient :
Value of a load effect as a unit load is moved around a structure :
Influence line - Central B.M.
0.2
0.1
0
-0.1
0
0.2
0.4
0.6
position/total beam length
0.8
1
Effective static loading distributions
Influence coefficient :
Value of a load effect as a unit load is moved around a structure :
Influence line - Central B.M.
0.2
0.1
0
-0.1
0
0.2
0.4
0.6
position/total beam length
0.8
1
Effective static loading distributions
Influence coefficient :
Value of a load effect as a unit load is moved around a structure :
Influence line - Central B.M.
0.2
0.1
0
-0.1
0
0.2
0.4
0.6
position/total beam length
0.8
1
Effective static loading distributions
Influence coefficient :
Value of a load effect as a unit load is moved around a structure :
Influence line - Central B.M.
0.2
Ir(z)
0.1
z
0
-0.1
0
0.2
0.4
0.6
0.8
position/total beam length
L
For a distributed load p(z) , r =  p(z) I r (z) dz
0
1
Effective static loading distributions
Mean component
 p(z) = [0.5 aUh2] Cp
on a tower :  f (z) = [0.5 a  U(z)2] Cd
b(z) (per unit height)
Effective static loading distributions
Background (quasi-static) component
(Kasperski 1992)
f B (z)  g .ρ pr (z) .σ (z)
B
p
pr(z) : correlation coefficient between the fluctuating
load effect, and the fluctuating pressure at position, z
Effective static loading distributions
Background (quasi-static) component
Consider a load effect r with influence line Ir(z):
L
Instantaneous value of r : r(t) =
 p(z, t)
0
p(z,t) = fluctuating pressure at z
L is length of the structure
L
Mean value of r :
r   p(z) I r (z) dz
0
I r (z) dz
Effective static loading distributions
Background (quasi-static) component
Standard deviation of r : σ r,B
(background) (Lecture 9)
L
 
0
Expected maximum value of r :



0 p (z1 )p (z 2 ) Ir (z1 ) Ir (z 2 ) dz1dz 2 
L
1/ 2
r̂  r  g Bσ r,B
Distribution for maximum response : pB(z) = gB pr (z) p (z)
L
L
ρ pr (z) 
p(z, t)  p(z1 , t) I r (z1 ) dz1
0
σ p (z) σ r,B


p(z, t) p(z1 , t) I r (z1 ) dz1
0
σ p (z) σ r,B
Effective static loading distributions
Background (quasi-static) component
L
r̂   p(z) r̂ I r (z) dz
Check :
L
0
L


  p(z)  p B (z)  I r (z) dz   p(z)  g Bρ pr (z) σ p (z) I r (z) dz
0
0
L
L
  p(z) I r (z) dz  g B
0
 r  gB
r̂  r  g Bσ r,B

0


  p(z, t) p(z1 , t) I r (z1 ) dz1 I r (z)dz


0

σ r,B
σ r,B
L
2
σ r,B
Effective static loading distributions
Background (quasi-static) component
Discrete form of pr :

ρ r,pi   p i (t) p k (t )I k
 σ σ 
pi
r
k
This form is useful when using using wind-tunnel data obtained
from area-averaging over discrete measurement panels
Standard deviation of load effect :
σ 2    p (t) p (t) I I
r
i
k
i k
i
k
Effective static loading distributions
Example (pitched free roof) :
(Appendix F in book)
2
1
h
22.5
Effective static loading distributions
Wind-tunnel test results :
Correlation
coefficient =
-0.17
+0.46, (0.35)
0.60,
(0.20)
-
mean,std.dev. Cp’s
+0.03, (-1.90)
peak Cp’s
+2.53, (-0.65)
Effective static loading distributions
Mean drag force :
Influence coefficients :
Panel 1 : +h
Panel 2 : -h
Mean drag force :D = (0.46) qh (+h) + (-0.60) qh (-h)
= 1.06 qh (h)
qh is the reference mean dynamic pressure at roof height
 1
2
  ρa U h 
 2

Effective static loading distributions
Standard deviation of drag force :
σ D    pi (t) p k (t) Ii I k
2
i 1,2 k 1,2
D = qh [(0.35)2 (+h)2+ (0.20)2(-h)2 + 2(-0.17).(0.35)
(0.20)(+h)(-h)]1/2
= 0.432 qh h
qh is the reference mean dynamic pressure at roof height
 1
2
  ρa U h 
 2

Peak drag force : D̂ = 1.06 qh h + 4 0.432 qh h
= 2.79 qh h
assuming a peak factor g of 4
Effective static loading distributions
Effective pressures for maximum drag force :
Covariance between p1(t) and drag D(t) : 
 p (t)p (t )I 
k 1,2
1
k
k
[ p1 2 .(h) + p1 p2 (-h)] = qh2 h [(0.35)2 - (-0.17)(0.35)(0.20)]
= (0.134) qh2 h
Correlation coefficient :
ρ p1,D 
 p (t)p
k 1,2
1
2
k
(t )I k
0.134q h h

(0.35q h )(0.432q h h)
 σ σ 
p1 D
= 0.886
Pressure on panel 1 when D is maximum :
p1 D̂ = qh [Cp1 + g p1,D Cp1] = qh [(0.46) + 4 (0.886) (0.35)] = 1.70 qh
Effective static loading distributions
Effective pressures for maximum drag force :
Covariance between p2(t) and drag D(t) : 
 p (t) p (t )I 
k 1,2
2
k
k
[p2 p1.(h) + p2 2 (-h)] = qh2 h [ (-0.17)(0.20)(0.35)- (0.20)2 )]
= -(0.052) qh2 h
Correlation coefficient :
ρ p2,D 
 p (t)p
k 1,2
2
2
k
(t )I k
- 0.052q h h

(0.20q h )(0.432q h h)
 σ
σD 
p2
= -0.602
Pressure on panel 2 when D is maximum :
p 2 D̂ = qh [Cp2 + g p2,D Cp2] = qh [(-0.60) + 4 (-0.602) (0.20)] = -1.08 qh
Effective static loading distributions
Effective pressures for maximum drag force :
Pressure coefficients corresponding to maximum drag :
C 
p1 D̂
C 
 +1.70
p2 D̂
-1.08
 -1.08
+1.70
Check : maximum drag force : D̂ = (1.70) qh (+h) + (-1.08) qh (-h)
= 2.78 qh(h) (previously 2.79 qh(h) )
Effective static loading distributions
Effective pressures for maximum lift force :
Pressure coefficients corresponding to maximum uplift force:
C 
p1 L̂
C 
 -0.73
2
p2 L̂
-0.90
-0.73
1
 -0.90
Effective static loading distributions
Effective pressures for minimum lift force :
Pressure coefficients corresponding to minimum uplift force:
(maximum down force)
C 

p1 L
C 
 +1.65
2

p2 L
-0.30
+1.65
1
 -0.30
Effective static loading distributions
Resonant load distribution :
fR (z) = gR m(z) (2 n1)2 a'2
1 (z)
gR is peak factor for resonant response
m(z) is mass per unit length
n1 is first mode natural frequency
a'2 (=a) is the standard deviation of the modal coordinate
1 (z) is the mode shape for the first mode of vibration
where, x(z,t) = j aj (t) j (z) (modal analysis)
Effective static loading distributions
Combined load distribution :
fc (z) = f (z) +Wback fB(z) + Wres(z) fR(z)
Wback and Wres are weighting factors
Wback 
g
Check :
g Bσ r,B
2
B
σ r,B  g R σ r,R
2
2

Wres 
2 1/2
L
L
0
0

g
g R σ r,R
2
B
σ r,B  g R σ r,R
2
2

2 1/2

r̂   f c (z) I r (z) dz   f (z)  Wback f B (z)  Wres f R (z) .I r (z) dz
 r  g B σ r,B  g R σ r,R
2
2
2
2
(correct expression)
Effective static loading distributions
Example :
Effective static load distributions for end reaction and bending
moment on an arched roof (no resonant contribution):
45
-
+
C
Cp =0.5
R
Extreme load distribution for the support reaction, R
Extreme load distribution for the bending moment at C
Gust pressure envelope
Effective static loading distributions
Example :
Effective static load distributions for base bending
moment on a tower :
160
140 Res onant
C o m b in e d
Height (m )
120
100
B a c kg r o u n d
80
Me a n
60
40
20
0
0 .0
0 .2
0 .4
0 .6
0 .8
E ffe c ti ve p r e s s u r e ( k P a )
1 .0
End of Lecture 13
John Holmes
225-405-3789 [email protected]