![𝑓𝑠0 𝑥 = [ 𝐸𝐼 𝑥 𝑢0
′′
𝑥 ]″
or alternatively
𝑓𝑠0 𝑥 = 𝑚 𝑥 𝜔2
𝑛 𝑢0 𝑥 = ෨
Γ𝐴 𝑚 𝑥 𝜓 𝑥
(11)
(12)
Thus, the shear and bending moment of height 𝑥 above the
base are:](https://image.slidesharecdn.com/lecture02-dvaspectraandgeneralizedsdofsystems-240520120526-66d2493a/85/Lecture-02-DVA-Spectra-and-Generalized-SDOF-Systems-pdf-23-320.jpg)
Lecture 02- DVA Spectra and Generalized SDOF Systems.pdf
- 1. CEE 424- Elementary Structural Dynamics Lecture 02- DVA Spectra and Generalized SDOF Systems Semester – January 2020 Dr.Tahir Mehmood
- 2. Combined D-V-A Spectrum: Each of the deformation, pseudo-velocity, and pseudo-acceleration response spectra for a given ground motion continue the same information-they are simply different ways of presenting the same information on structural response. 7.48= √7𝑥8
- 3. A single curve can simultaneously show three different quantities. • The peak deformation • The peak pseudo-velocity which is related to the peak strain energy • The peak pseudo-acceleration which is related to the peak value of equivalent static force (and base shear).
- 5. Example A 12-ft-long vertical cantilever, a 4-in.-nominal-diameter standard steel pipe, supports a 5200-lb weight attached at the tip as shown in Fig. E6.2. The properties of the pipe are: outside diameter, 𝑑𝑜= 4.500 in., inside diameter 𝑑𝑖= 4.026 in., thickness 𝑡 = 0.237 in., and second moment of cross-sectional area, 𝐼 = 7.23 in4, elastic modulus 𝐸 = 29,000 ksi, and weight = 10.79 lb/foot length. Determine the peak deformation and bending stress in the cantilever due to the El Centro ground motion. Assume that 𝜉 = 2%. Solution
- 7. The lateral stiffness of this SDF system is 𝑘 = 3𝐸𝐼 𝐿3 = 3 29 × 103 7.23 12 × 12 3 = 0.211 Τ 𝑘𝑖𝑝 𝑖𝑛. The total weight of the pipe is 10.79 × 12 = 129.5 𝑙𝑏, which may be neglected relative to the lumped weight of 5200 𝑙𝑏. Thus 𝑚 = 𝑤 𝑔 = 5.20 386 = 0.01347 𝑘𝑖𝑝 − Τ 𝑠𝑒𝑐2 𝑖𝑛.
- 8. From the response spectrum curve for 𝜉 = 2% (Fig. E6.2b), for 𝑇𝑛 = 1.59 sec, 𝐷 = 5.0 in. and 𝐴 = 0.20g. The peak deformation is 𝑢𝑜 = 𝐷 = 5.0 𝑖𝑛. The natural vibration frequency and period of the system are 𝜔𝑛 = 𝑘 𝑚 = 0.211 0.01374 = 3.958 Τ 𝑟𝑎𝑑 𝑠𝑒𝑐 𝑇𝑛 = 1.59 𝑠𝑒𝑐 w= 2𝜋/𝑇𝑛
- 9. Note: The unit of force is 𝑘𝑖𝑝, 1 kip = 1000 𝑙𝑏 The unit of mass is therefore the unit of force divided by the unit of acceleration. The unit of acceleration is 𝑖𝑛/sec2 1𝑔 = 9.81 𝑚/sec2 = 32.2 𝑓𝑡/ sec2 = 386 𝑖𝑛/ sec2 The unit of 𝐸 is the unit of force divided by the unit of area. The lateral stiffness 𝐾 in this case is determined from 𝐾 = 𝐹 𝛿
- 10. The peak value of the equivalent static force is 𝑓𝑆𝑜 = 𝐴 𝑔 𝑤 = 0.20 × 5.2 = 1.04 𝑘𝑖𝑝𝑠 The bending moment diagram is shown in Fig. E6.2d with the maximum moment at the base = 12.48 kip-ft. Points A and B shown in Fig. E6.2e are the locations of maximum bending stress: 𝜎𝑚𝑎𝑥 = 𝑀𝑐 𝐼 = (12.48 × 12)( Τ 4.5 2) 7.23 = 46.5 𝑘𝑠𝑖 As shown, 𝜎 = +46.5 ksi at 𝐴 and 𝜎 = −46.5 ksi at 𝐵, where + denotes tension. The algebraic signs of these stresses are irrelevant because the direction of the peak force is not known, as the pseudo-acceleration spectrum is, by definition, positive.
- 11. Response Spectrum Characteristics Combined D-V-A response spectrum (𝜉 = 0, 0.02, 0.05, 0.1) and peak values of ground acceleration, ground velocity, and ground displacement for El Centro ground motion
- 12. • For systems with very short period, the pseudo-acceleration 𝐴 for all damping values approach ሷ 𝑢𝑔0 and 𝐷 is very small. • For systems with very long period, 𝐷 for all the damping values approach 𝑢𝑔0 and 𝐴 is very small; thus the forces in the structures, which are related to the 𝑚𝐴, would be very small. • The reduction of response due to additional damping is different for the different spectral regions- greatest in the velocity-sensitive region. (The effectiveness of damping in reducing the structural response also depends on the ground motion characteristics). • For Mexico city 85 earthquake where ground motion is nearly harmonic over many cycles, the effect of damping would be large for a system near “resonance”. • For Park Filed 66 earth quake where ground motion is very short and shock like, the effect of damping would be small, as in the case of half cycle sine pulse excitation.
- 13. A very short period system is extremely stiff and rigid. Its deformation response to the ground motion is very small. So its mass move rigidly with the ground and its peak structural acceleration should be approximately ሷ 𝑢𝑔0. To drive the structural mass to move with acceleration of ሷ 𝑢𝑔0, it is necessary to have 𝑓𝑠0 ≈ 𝑚 ሷ 𝑢𝑔0, therefore, 𝐴 ≈ ሷ 𝑢𝑔0
- 14. A very long period system is extremely flexible. The mass would be expected to remain essentially stationary while the ground below moves. Thus 𝑢 𝑡 ≅ −𝑢𝑔 𝑡 that is 𝐷 ≅ 𝑢𝑔0
- 15. Generalized SDOF systems: Considering 3 examples of complex systems
- 16. If the deformation of complex systems can be (approximately) expressed as: 𝑢 𝑥, 𝑡 = 𝜓 𝑥 𝑧 𝑡 (1) where 𝜓 𝑥 : a shape function (dimensionless) or a mode shape 𝑧 𝑡 : a single generalized displacement or, for the case (c), 𝑢𝑖 𝑡 = 𝜓𝑖 𝑧 𝑡 (2) where 𝜓𝑖 : a shape factor (dimensionless) at the 𝑖𝑡ℎ story
- 17. Then it can be shown that the governing equation of motion of each of these complex systems is the form of 𝑚 ሷ 𝑧 𝑡 + ǁ 𝑐 ሷ 𝑧 𝑡 + ෨ 𝑘 𝑧 𝑡 = −෨ 𝐿 ሷ 𝑢𝑔 𝑡 (3) where 𝑚, ǁ 𝑐, ෨ 𝑘, ෨ 𝐿 are defined as the generalized mass, generalized damping, generalized stiffness and generalized force factor of the system, respectively.
- 18. ෨ 𝐿 = 𝑚𝑎 𝜓 𝑥𝑐 + 𝑚𝑏 𝜓 𝑥𝑏 For the system “a”: 𝑚 = 𝑚𝑎 𝜓2 𝑥𝑎 + 𝑚𝑏 𝜓2 𝑥𝑏 ǁ 𝑐 = 𝐶𝑒 𝜓2 𝑥𝑒 ෨ 𝑘 = 𝑘𝑐 𝜓2 𝑥𝑐 + 𝑘𝑑 𝜓2 𝑥𝑑 (4) For the system “b”: (5) ෨ 𝑘 = න 0 𝐿 𝐸𝐼 𝑥 𝜓′′ 𝑥 2 𝑑𝑥 𝑚 = න 0 𝐿 𝑚 𝑥 𝜓2 𝑥 𝑑𝑥 ෨ 𝐿 = න 0 𝐿 𝑚 𝑥 𝜓 𝑥 𝑑𝑥
- 19. For the system “c”: (6) 𝑚 = 𝑗=1 𝑁 𝑚𝑗 𝜓2 𝑗 (N=4 in this case) ෨ 𝑘 = 𝑗=1 𝑁 𝑘𝑗 (𝜓𝑗 − 𝜓𝑗−1)2 where ; ℎ: story height ෨ 𝐿 = 𝑗=1 𝑁 𝑚𝑗 𝜓𝑗 Therefore, Given the vibration shape and the distribution of mass and flexibility of a complex system, it is possible to evaluate all of these generalized properties of the system. 𝑘𝑗 = 𝑐𝑜𝑙𝑢𝑚𝑛 12𝐸𝐼 ℎ3
- 20. Dividing the Eq. (3) by 𝑚 gives: ሷ 𝑧 + 2 𝜉 𝜔𝑧 ሶ 𝑧 + 𝜔2 𝑛 𝑧 = −෨ Γ ሷ 𝑢𝑔 𝑡 (7) where ෨ Γ = ൗ ෨ 𝐿 𝑚 : (dimensionless factor) 𝜔𝑛 2 = ൗ ෨ 𝑘 𝑚 : the natural frequency of the generalized SDOF system 𝜉 = ൗ ǁ 𝑐 2෨ 𝑘 𝑚 : the (model) damping ratio (7′) The equation (7) says that the generalized displacement 𝑧 𝑡 of the generalized SDOF system due to ground motion 𝑢𝑔 𝑡 is identical to the displacement response 𝑢 𝑡 of a simple SDOF system (having the system 𝜔𝑛 and 𝜉) to ground motion ෨ Γ𝑢𝑔 𝑡 .
- 21. Suppose that the response spectrum of the ground motion 𝑢𝑔 𝑡 is available. It is then possible to estimate peak earthquake response of the generalized SDOF system:
- 22. Peak value of 𝑧 𝑡 = 𝑧0 = ෨ Γ 𝐷 = ෩ Γ 𝜔2 𝑛 𝐴 (8) where 𝐷 and 𝐴 are the deformation and pseudo-acceleration ordinates, respectively, of the spectrum at the modal period 𝑇𝑛 = Τ 2𝜋 𝜔𝑛 for the modal damping ratio 𝜉 Peak displacements 𝑢0 𝑥 = ෨ Γ 𝐷 𝜓 𝑥 𝑢𝑗0 = ෨ Γ 𝐷 𝜓𝑗 or (9) (10) Equivalent static forces (external static forces that would cause displacement 𝑢0 𝑥 for the system “b” can be derived from elementary beam theory as
- 23. 𝑓𝑠0 𝑥 = [ 𝐸𝐼 𝑥 𝑢0 ′′ 𝑥 ]″ or alternatively 𝑓𝑠0 𝑥 = 𝑚 𝑥 𝜔2 𝑛 𝑢0 𝑥 = ෨ Γ𝐴 𝑚 𝑥 𝜓 𝑥 (11) (12) Thus, the shear and bending moment of height 𝑥 above the base are:
- 24. 𝜐0 𝑥 = න 𝑥 𝐿 𝑓𝑠0 𝜂 𝑑𝜂 = ෨ Γ𝐴 න 𝑥 𝐿 𝑚 𝜂 𝜓 𝜂 𝑑𝜂 (13) 𝑀0 𝑥 = න 𝑥 𝐿 (𝜂 − 𝑥) 𝑓𝑠0 𝜂 𝑑𝜂 = ෨ Γ𝐴 න 𝑥 𝐿 (𝜂 − 𝑥) 𝑚 𝜂 𝜓 𝜂 𝑑𝜂 (14) The shear and bending moment at the base of the tower are 𝜐𝑏0 = 𝜐0 0 = ෨ 𝐿 ෨ Γ 𝐴 𝑀𝑏0 = 𝑀0 0 = ෨ 𝐿𝜃 ෨ Γ 𝐴 ෨ 𝐿𝜃 = න 0 𝐿 𝑥 𝑚 𝑥 𝜓 𝑥 𝑑𝑥 (15) (16) (17) where
- 25. For the case of “shear building” (system “c”) 𝑓𝑗0 = ෨ Γ𝐴 𝑚𝑗 𝜓𝑗 for j=1,2,3,……,N In this case N=4 (18)
- 26. The overturning moment 𝑀𝑖0 at the 𝑖𝑡ℎ story is : 𝑀𝑖0 = 𝑗=𝑖 𝑁 ( ℎ𝑗 − ℎ𝑖) 𝑓𝑗0 = ෨ Γ A 𝑗=𝑖 𝑁 ( ℎ𝑗 − ℎ𝑖) 𝑚𝑗 𝜓𝑗 The shear force 𝑉𝑖0 in the 𝑖𝑡ℎ story is: 𝑉𝑖0 = 𝑗=𝑖 𝑁 𝑓𝑗0 = ෨ Γ 𝐴 𝑗=𝑖 𝑁 𝑚𝑗 𝜓𝑗 (19) (20) where ℎ𝑖 is the height of the 𝑖𝑡ℎ floor above the base.
- 27. The shear and overturning moment at the base are 𝑉𝑏0 = 𝑗=𝑖 𝑁 𝑓𝑗0 = ෩ 𝐿 ෩ Γ 𝐴 (21) 𝑀𝑏0 = 𝑗=𝑖 𝑁 ℎ𝑗 𝑓𝑗0 = ෨ 𝐿𝜃 ෨ Γ 𝐴 where ෨ 𝐿𝜃 = 𝑗=𝑖 𝑁 ℎ𝑗 𝑚𝑗 𝜓𝑗 (Note here that the generalized factors ෨ 𝐿, ෨ 𝐿𝜃 and ෨ Γ depend only on the vibration shape and the mass distribution of the complex systems.) (22) (23)
- 28. Thank you