Simple Schemes of Multi anchored Flexible Walls Dynamic Behavior

Transcription

1 6 th International Conference on Earthquake Geotechnical Engineering -4 Noveber 05 Christchurch, New Zealand Siple Schees of Multi anchored Flexible Walls Dynaic Behavior A. D. Garini ABSTRACT Siple schees of dynaic behavior of ulti anchored flexible retaining walls are exained in order to capture the typical frequencies involved during strong otion earthquakes. In particular ulti anchored retaining walls are sketched with claped ends, due to its own anchorages, three rigidity springs ( the wall s and the two anchored soil wedges, whose inertial effects are assued to be separated) and two asses as two earth wedges are involved. Introduction In an effort to better understand the actual behavior of a cantilever flexible wall to sustain a deep excavation during a strong otion earthquake and following the results of an iportant research reported in Al Atik and Sitar (00), Garini (04) hypothesized that the Mononobe Okabe seisic wedge should further be divided in two sub wedges that ove out of phase. In particular Garini found that the point of application of the seisic active thrust should indeed be deterined by this assuption, as the two wedges are individuated by the their own dynaic equilibriu. Indeed Al Atik and Sitar (00) state the following ain conclusions:. There sees to be no basis for the currently accepted position of the dynaic earth pressure force in dynaic L.E.A. at 0.6 to 0.67 H and, instead, the point of application should be at /3 H;. Maxiu dynaic earth pressures and axiu wall inertial forces do not tend to occur siultaneously. As a result, the current design ethods based on the Mononobe Okabe theory were found to significantly overestiate dynaic earth pressures and oents. 3. Seisic earth pressures on cantilever retaining walls can be neglected at accelerations below 0.4 g. Garini (04) gave an interpretation of the findings reported in Al Atik & Sitar (00) regarding the seisic active thrust on cantilever flexible walls resorting to correct rational echanics stateents i.e. the application of the seisic force in the centroid at /3H. In this paper I will focus on ulti anchored flexible retaining walls subjected to a strong otion earthquake and also in this case I assue that, due to the typical echanis of the thrust wedge that has an outer shallow/softer zone and an internal ore confined zone, it s possible to consider two independent sub wedges behind the ulti - anchored flexible wall. Consulting Engineer, Apeglio Diego Garini, Genova, Italy, adiego.garini@alice.it

2 Due to the typical constraints of ulti anchored flexible walls, this assuption leads to sketch a coupled oscillators echanical two asses and three rigidity springs syste. The Coupled Oscillators Mechanical Syste In the Coupled Oscillator echanical Syste we have the sketch reproduced in Figure. Figure. A double anchored flexible wall sketched with coupled oscillators with their own two asses and three rigidity springs representing the Mononobe Okabe seisic active wedge divided in two sub wedges, with claped ends in the wall and the tie back foundations. This syste has the following two natural frequencies (see Appendix): () where i (i=,) are the sub wedge i ass and for each half and double half sub wedge (for i =,,3), we have: i = E i A i /l i () while notice that for 3 = 0,we have the following: (3) It s clear fro physics that the coupled oscillators have one antisyetrical ode of vibration (out of phase) leaving unalterable the centroid position and one syetrical ode of vibration (in phase) leaving unalterable the relative distance between the two asses.

3 The out of phase ode, naely the antisyetrical ode of vibration, when excited by the strong otion, considering in this case, equation (3) valid for the wall ass and the entire Mononobe Okabe soil wedge ass, well explain the findings of Al Atik and Sitar (00), so this eans that the predoinant period fro 0. to 0.6 s ( ω = 5.3 rad/sec in average) in their work captures this typical frequency. Equation (3) can siulate the dynaic behavior of a cantilever flexible wall, either as above indicated for the work of Al Atik and Sitar (00) or for the soil sub wedges as reported in Garini(04). Equation () applies at each anchorage syste as the friction alongside the ass around anchorage cancel each other. More Accurate Coupled Oscillators Behavior So far I have assued that spring stiffness i, were behaving in a linear fashion with the sae value both in tension and copression. A ore precise law for soils ay be as that indicated in the hysteresis loop of Figure, where, during soil stress strain copression and extension along the anchorage line, we can distinguish between copression and tension stress respectively; so as in the sketches depicted in Figure 3 in general we will alternatively have, with obvious significance of the suffixes : T, T associated with 3C ; C, C associated with 3T ; C, T associated with 3C ; T, C associated with 3T ; And this eans that we have to consider a quadruple series of the two natural frequencies according to equation (). Figure. Hysteretic soil behaviour.

4 Figure 3. Quadruple schee to evaluate spring stiffness either in tension or in copression. Exaple Case We can consider the siplest case with the sae spring stiffness either in copression or in tension for a double anchorage at and 3 of depth, and an excavation height of 6 with the Young Modulus E = (0,3) Pa as in Al Atik and Sitar(00) and individuate the two sub wedges for exaple fro the second case described in Garini (04), where we get l subwedge =.9 and l wedge = 4.0. Considering the portions with A= of the sub wedges involved,we get the results showed in Table :

5 Table. Anchorage Coupled Oscillators natural frequencies Calculation. Anchorage (kn/) (kn/) 3 (kn/) (kg ) (kg ) ω (rad/s) ω (rad/s) So we have for shallow anchorages low natural frequencies naely high natural periods and vice versa for deeper anchorages and then corresponding probable resonance effects. It s iportant to notice that these results correspond with the work of Steedan and Zeng (990), where AE is shown as a function of the adiensional ratio H/TV S, which is the ratio of tie for a wave to travel the full excavation height to the period of lateral shacking. Indeed, for a given excavation depth H, the higher is the natural period and so the corresponding probable strong otion resonance effects period in a seisic aplification behavior, the lower the ratio H/TV S and the higher the coefficient AE naely the dynaic force on the wall and so the ass involved. Moreover this is particularly true for higher p.g.a., that is what I a considering due to the above said reported findings of Al Atik and Sitar (00) and Garini(04). Conclusions An interpretation of the dynaic behavior behind a ulti anchored flexible wall has been given having recourse to a siple coupled oscillators echanical syste, for which the natural frequencies have been recalculated even in the case of two hanging asses to siulate in this case a cantilever flexible wall. These considerations involve the iportance of lower natural frequencies in the shallower anchorages and vice versa for the deeper anchorages. Of course this eans that as an ordinary coupled oscillator we have one antisyetrical (out of phase) and one syetrical ode of vibration (in phase) so to rightly justify the dynaical behavior finding of Al Atik and Sitar (00), where it was argued a different oscillation between the wall and the soil, which in Garini(04), who assued a different oscillation between and outer soil wedge and an inner constrained sub wedge, found an interesting dynaic equilibriu explanation. In particular it is found a good agreeent of this echanical interpretation of the soil behavior behind the wall with the dependence of the seisic coefficient AE with the period of the seisic excitation as stated previously by Steedan and Zeng (990).

6 References Al Atik L., Sitar N. Seisic Earth Pressures on Cantilever Retaining Structures. Journal of Geotechnical and Geoenvironental Engineering 00; 36: Garini A.D. A New Approach for Seisic Active Thrusts in deep Excavations. Geotechnical Aspects of Underground Construction in soft Ground 04; Steedan R. S., Zeng X. The influence of phase on the calculation of pseudo-static earth pressure on a retaining wall. Géotechnique 990; 40: 03-. Appendix Fro the sketch in Figure we have: (4) (5) So if we write: (6) (7) (8) We find the natural frequencies given in Equation (), solving the followings: (9) 0 3 = ω ω (0) = ω ω ()