Dynamic Earth Pressure Problems and Retaining Walls 1/15 Behavior of Retaining Walls During Earthquakes - Permanent displacement = cc ' 2 2 due to one cycle of ground motion
2/15 Hence, questions are : 1 What is the change in earth pressure? 2 Where is its point of application? 3 How much displacement of the wall occurs? (What is the permissible displacement?) Modification of Coulomb`s Theory - Inertia forces on the trial wedge in horizontal and vertical directions = Wa g and Wa 1 / g respectively 1 h / v ( a : horizontal acceleration, a : vertical acceleration) h v - During the worst conditions for wall stability Wa g acts toward the wall, 1 h / Wa 1 v / g acts downward (or upward)
3/15 - a g h α h =, av α v = g horizontal, vertical seismic coefficients - The forces acting on the wedge abc 1 : Weight of the wedge abc 1, W 1, acting at its CG. Earth pressure P 1 inclined at an angle δ to normal to the wall Soil reaction R 1 inclined at an angle φ to normal to the face bc1 Horizontal inertia force Wα 1 h Vertical inertia force Wα 1 v - Constant force polygon to obtain P ( ) 1 = P = P +Δ P total a d - The maximum value of P total is determined by considering other trial surfaces - Analytically, Ptotal 2 2 cos ( φ ψ α)(1 ± αv) = γ H 2 1 1 1 2 cosψ cos αcos( δ + α + ψ) sin( φ+ δ)sin( φ i ψ) 2 {1 + [ ] } cos( α i) cos( δ + α + ψ) 2
Modified Culmann`s construction 4/15 - bs at an angle φ ψ with the horizontal line - bl = bl - use W ( W (1 ) 1α + W1 + α ) h v Point of Application of the total ( P ) a dyn - at midheight of the wall - at the upper third point of the wall Displacement analysis - no information on permissible displacement of retaining wall 1. Draw a dimensional sketch 2. Draw a line bs ' and bl (= bl ') 3. Intercept bd1, equal to the wt. of wedge abc 1, to a convenient scale along 4. Draw a line de 1 1 parallel to bl ( = bl ' ) 5. Trace a Culmann s line 6. Draw a line parallel to bs ' and tangential to the Culmann`s line. 7. Find de P a obtained bs '
5/15 Earthquake Loads on Footings (Pseudostatic method) Effect of moment only - Eccentricity M e = Q v Qv M : moment( = a l ) g Q v : central vertical static load due to the shaking of structure i.e., horizontal acceleration force (lumped mass at CG) B - For e < 6 Q e qmax = (1+ 6 ) A B B - For e > 6 Q 4B qmax = ( ) A 3B 6e - Effective width, B ' = B 2e Effect of trust only - Inclination Pp + F Pa FS = Q H F : Frictional Resistance 1 2 - Pa = kaγ h 2 1 2 - Pp = kpγ H, take ( 1 ~ 1 2 2 3 ) of Pp
At the present state of practice for earthquake loading 6/15 - Bearing capacity & settlement may be calculated by the pseudostatic method - Interconnecting beams (Tie beams) be provided to tie the foundations. It should be capable of carrying, in tension or compression, a force equal to A a /4of the largest footing or column load ( A a =effective peak acceleration). ATC(1978) Tie beams - transmits the unbalanced shear and moment - reduces the differential settlement - PC pile no use(depends on the building class & earthquake intensity)
Dynamic Analysis for Pile Foundations 7/15 - Evaluate the free-vibration characteristics of pile-soil system Calculate the spectral displacement ( y g ) Maximum bending moment ( M ) Computation of soil reaction ( Px ( ) ) max - Preliminaries - Soil properties are known ( kn, h...) - Pile characteristics are known (size, EI, length, type) - Lateral load-deflection of the pile under static conditions - nondimensional frequency factor ( F SL1 ) (based on parametric study) (k varies linearly with depth) F = w W SL1 n1 2 g nh T T =, 5 EI n h w n1 : the first natural angular frequency (rad/sec) W / g : the lumped mass at the pile top k : the soil modulus T : the relative stiffness factor
Soil Dynamics week # 12 8/15 - the depth coefficient(z) Z= z (if Z max < 2, rigid body / if Z max > 5, long pile) T - the max. depth coefficient ( Z max ) Z max = D T
Soil Dynamics week # 12 9/15 - max. bending moment coefficients (Fig 7.24)
Soil Dynamics week # 12 10/15 Step-by-step procedures ① Estimate the dynamic soil modulus k (or nh ) (modified value from a static lateral load test based on the engineering judgement) ② Compute the relative stiffness factor, T ③ Calculate the maximum depth factor Z max for a pile ; ( Z max should be > 5) ④ Read the frequency factor FSL1 for the Z max and the pile end condition from Fig 7.22
Soil Dynamics week # 12 11/15 ⑤ Estimate the dead load on the pile top mt = W / g ⑥ Determine the natural frequency wn1 wn1 = FSL1 W gnht 2 ⑦ Compute the time period Tn1 = 2π / wn1 ⑧ From Fig 7.25 determine the spectral displacement S d for assumed damping (for pile-soil system 5% damping may be assumed) maximum displacement of the pile head
9 Estimate the maximum bending moment in the pile section 12/15 M = B n T S (From table 7.7) 3 max me1 h d Pile section needs to be checked against this moment 10 Compute the soil reaction Pz = nh z yz(displacement of the pile at z) (refer to table 7.8 & Example 7.3) Compare this with the allowable Soil reaction : 1+ sinφ P = γ b z ( kg / cm 1 sinφ a ) = K σ b p v
Soil Dynamics week # 12 13/15
Soil Dynamics week # 12 14/15
15/15 11 Check on Design 1. Estimate the fixity of the pile head. In the absence of a realistic estimate, 50 percent fixity may be assumed. 2. Compute displacement. For 50 percent fixity, the displacement under seismic loading condition is 2.0 + 1.0 = 1.5cm. For any other fixity of the pile head, linear 2 interpolation may be made. 3. Compute the maximum bending moment as for displacements Mmax = 1213 t cm( ) 4. The soil reaction needs to be interpolated at each point and a new diagram obtained