simple expression of the dynamic stiffness of grouped piles

24SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES⎛α⎞ ⎛⎞2R0R0k ≅ ⎜ +⎟ =⎜ +⎟s2πξkα1µLa0.4 µ La2πξkα20. 8πξkα1(2.19)⎝ α1La⎠ ⎝ La⎠For the dynamic case (ω →∞), non-dimensional frequency a 0converges on:ω R0a0= i = ia(2.20)vT pEquation (2.16) is thus approximated by:k ≅ L i ⋅ 2πξα ⋅ a + 0. πξ α(2.21)s( )µa k 18k1From Equations (2.19) and (2.21), soil stiffness will presumably be approximated as:⎧⎛R⎫0⎞k ⎨⎜⎟s≅ µ La2πξkα2+ 0.8πξkα1+ i ⋅ 2πξkα1⋅ a⎬(2.22)⎩⎝La⎠⎭Even without the soil above the active pile length, the pile group exhibits its ownstiffness, k (Figure 2.7b), which is described as:kg4EIp µ L0α3gα3= α3 3= µ L3 3 0LaLaα0122≅ (2.23)⎛ d ψ ( ζ )where, ∫ ⎟ ⎞α3=⎜ dζ20 ⎝ dζ⎠Both ksand k gsustain the massamong the piles. This massmgLa02 20ψ z)mgof the embedded pile group with soil caughtmgis approximated by:20≅ ∫ ρ πR( dz = ρ πRL α(2.24)ssaTherefore the overall stiffness k xxof the pile cap for sway motion is given as:kxx = ks + kg −mgω 2 (2.25)From Equations (2.22), (2.23) and (2.24), Equation (2.25) is rewritten as:⎡⎪⎧R ⎛⎞⎪⎫⎤0α⎢⎨⎜32k⎟xx≅ µ La2πξkα2+⎬ + ⋅ ⋅ − ⋅ ⎥⎢0.8πξkα1+3i 2πξkα1a πα1a (2.26)⎣⎪⎩La⎝ α0 ⎠⎪⎭⎥⎦Substituting Equation (2.3) into Equation (2.26), one obtains:⎡⎪⎧R ⎛⎞⎪⎫⎤0α⎢⎨⎜32k⎟xx≅ µ L0 2πξkα2+⎬ + ⋅ ⋅ − ⋅ ⎥⎢0.8πξkα0α1+2i 2πξkα0α1a πα0α1a (2.27)⎣⎪⎩L0⎝α0 ⎠⎪⎭⎥⎦It is now obvious that k xxin Equation (2.27) has the following simple form withfrequency-independent stiffness k 0, and damping and mass parameters c 0and m 0respectively:2≅ k + i ⋅ c ⋅ a − m ⋅(2.28)where,k xx0 00a1