1. Effective static loading distributions Wind loading and structural response Lecture 13 Dr. J.D. Holmes

2. 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)

3. Effective static loading distributions Influence coefficient : Value of a load effect as a unit load is moved around a structure :

4. Effective static loading distributions Influence coefficient : Value of a load effect as a unit load is moved around a structure :

5. Effective static loading distributions Influence coefficient : Value of a load effect as a unit load is moved around a structure :

6. Effective static loading distributions Influence coefficient : Value of a load effect as a unit load is moved around a structure :

7. Effective static loading distributions Influence coefficient : Value of a load effect as a unit load is moved around a structure :

8. Effective static loading distributions Influence coefficient : Value of a load effect as a unit load is moved around a structure :

9. Effective static loading distributions Influence coefficient : Value of a load effect as a unit load is moved around a structure :

10. Effective static loading distributions Influence coefficient : Value of a load effect as a unit load is moved around a structure :

11. Effective static loading distributions Influence coefficient : Value of a load effect as a unit load is moved around a structure :

12. Effective static loading distributions Influence coefficient : Value of a load effect as a unit load is moved around a structure :

13. Effective static loading distributions Influence coefficient : Value of a load effect as a unit load is moved around a structure : I r (z) z For a distributed load p(z), r =

14. Effective static loading distributions Mean component  p(z) = [0.5  a  U h 2 ]  C p on a tower :  f (z) = [0.5  a  U(z) 2 ] C d b(z) (per unit height)

15. Effective static loading distributions Background (quasi-static) component (Kasperski 1992)  pr (z) : correlation coefficient between the fluctuating load effect, and the fluctuating pressure at position, z

16. Effective static loading distributions Background (quasi-static) component Consider a load effect r with influence line I r (z): Instantaneous value of r : r(t) = p(z,t) = fluctuating pressure at z L is length of the structure Mean value of r :

17. Effective static loading distributions Background (quasi-static) component Standard deviation of r : (background) (Lecture 9) Expected maximum value of r : Distribution for maximum response : p B (z) = g B  pr (z)  p (z)

18. Effective static loading distributions Background (quasi-static) component Check :

19. Effective static loading distributions Background (quasi-static) component Discrete form of  pr : This form is useful when using using wind-tunnel data obtained from area-averaging over discrete measurement panels Standard deviation of load effect :

20. Effective static loading distributions Example (pitched free roof) : (Appendix F in book) 12 22.5  h

21. Effective static loading distributions +2.53, (-0.65) +0.03, (-1.90) peak C p’s +0.46, (0.35) - 0.60, (0.20) Correlation coefficient = -0.17 mean,std.dev. C p’s Wind-tunnel test results :

22. Effective static loading distributions Mean drag force : Influence coefficients : Panel 1 : +h Panel 2 : -h Mean drag force :  D = (0.46) q h (+h) + (-0.60) q h (-h) = 1.06 q h (h) q h is the reference mean dynamic pressure at roof height

23. Effective static loading distributions Standard deviation of drag force :  D = q h [(0.35) 2 (+h) 2 + (0.20) 2 (-h) 2 + 2(-0.17).(0.35) (0.20)(+h)(-h)] 1/2 = 0.432 q h h q h is the reference mean dynamic pressure at roof height Peak drag force : = 1.06 q h h + 4  0.432 q h h = 2.79 q h h assuming a peak factor g of 4

24. Effective static loading distributions Effective pressures for maximum drag force : Covariance between p 1 (t) and drag D(t) : Correlation coefficient : = (0.134) q h 2 h [. (h) + (-h)] = q h 2 h [(0.35) 2 - (-0.17)(0.35)(0.20)] = 0.886 = q h [  C p1 + g  p1,D  Cp1 ] = q h [(0.46) + 4 (0.886) (0.35)] = 1.70 q h Pressure on panel 1 when D is maximum :

25. Effective static loading distributions Effective pressures for maximum drag force : Covariance between p 2 (t) and drag D(t) : Correlation coefficient : = -(0.052) q h 2 h [. (h) + (-h)] = q h 2 h [ (-0.17)(0.20)(0.35)- (0.20) 2 )] = -0.602 = q h [  C p2 + g  p2,D  Cp2 ] = q h [(-0.60) + 4 (-0.602) (0.20)] = -1.08 q h Pressure on panel 2 when D is maximum :

26. Effective static loading distributions Effective pressures for maximum drag force : Pressure coefficients corresponding to maximum drag : +1.70 -1.08 +1.70 -1.08 Check : maximum drag force : = (1.70) q h (+h) + (-1.08) q h (-h) = 2.78 q h (h) (previously 2.79 q h (h) )

27. Effective static loading distributions Effective pressures for maximum lift force : Pressure coefficients corresponding to maximum uplift force: -0.73 -0.90 -0.73 -0.90 1 2

28. Effective static loading distributions Effective pressures for minimum lift force : Pressure coefficients corresponding to minimum uplift force: (maximum down force) +1.65 -0.30 +1.65 -0.30 1 2

29. Effective static loading distributions Resonant load distribution : f R (z) = g R m(z) (2  n 1 ) 2  1 (z) g R is peak factor for resonant response m(z) is mass per unit length n 1 is first mode natural frequency (=  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 a j (t)  j (z) (modal analysis)

30. Effective static loading distributions Combined load distribution : W back and W res are weighting factors Check : (correct expression)

31. Effective static loading distributions Example : Effective static load distributions for end reaction and bending moment on an arched roof (no resonant contribution): Extreme load distribution for the support reaction, R Extreme load distribution for the bending moment at C Gust pressure envelope C =0.5 p C R 45 + -

32. Effective static loading distributions Example : Effective static load distributions for base bending moment on a tower :

33. End of Lecture 13 John Holmes 225-405-3789 JHolmes@lsu.edu