Oscillations of retaining wall subject to Grob’s swelling pressure

The single-degree-of-freedom nonlinear problem describing the essential dynamics of an oscillating retaining wall based on non-quaking ground and subject to Grob’s swelling pressure is considered. The periodic solutions are derived using harmonic approximation. The amplitude-frequency relation is established by employing Lambert’s special function or alternatively using linearization of the nonlinear force. Analytical results are verified using numerical simulations.

Retaining walls and foundations in construction are often subject to the swelling pressure caused by expansive soils such as clay or soft rock. This pressure can result in significant vibrations of structures and hence lead to damage and economical loss. Predicting the effect of swelling pressure on structures is therefore an important problem in mechanical and civil engineering, see [1][2][3][4] . Vibration analysis of retaining walls is useful in many building and construction applications. These include building vibrations induced by high-speed trains moving on bridges 5 , seismic analysis in and around earthquake zones 6 , and vibrations incurred in construction sites 7,8 . This analysis is critical for retaining wall structures around power plants especially nuclear. The Fukushima nuclear disaster in 2011 was a result of the Tōhoku earthquake and tsunami. Vibrations of retaining walls caused by dynamic (seismic) loading were investigated analytically using linear approximation 9 . Experimental and numerical results for concrete retaining walls under low-frequency dynamic loading were reported in 10 . In this work, we present an analysis of the lumped mass model for retaining walls subject to the swelling pressure that obeys Grob's semi-logarithmic law for the volumetric stress 1 . The periodic solutions are derived using harmonic approximation. The amplitude-frequency relation is established by employing Lambert's special function or using linearization of the nonlinear force term. Analytical results are verified by numerical simulations. The approximations that we derive can be used to evaluate the frequency and amplitude of oscillations without time-consuming finite element calculations.
Mathematical model for retaining wall subject to swelling pressure Cantilever beam subject to Grob's swelling pressure. Swelled soil exerts pressure (stress) on the retaining wall. This pressure reaches its maximum at the original undeformed position of the wall which corresponds to the maximum of soil compression. Deflection of the wall from its original position results in expansion of soil and therefore a reduction of pressure that should asymptotically approach zero with any further increase of wall deflection (decrease of soil compression). A simple model satisfying this property that was confirmed by a series of experiments, including the combined swell-swell heave, the multi-stepped, and Hunder-Amberg swell tests 11,12 , is the exponential decay where u denotes the deflection of the retaining wall from its axial position, σ s is the present axial stress and σ 0 is the maximal stress for which swelling occurs (equilibrium stress with respect to swelling). The decay rate is given by where d is the thickness of swelled soil layer (compression is equal to u/d) and c is the experimentally fitted swelling parameter (see Fig. 1). Transverse deformations are not allowed in the abovementioned swelling tests. Eq. (1) is better known in the literature as Grob's swelling law describing the logarithmic dependence of deformation on pressure 1 This equation is well accepted in civil engineering areas and many applications can be found in the literature 2,3,13,14 .
Next, a small horizontal deflection of the retaining wall of length s can be described by the Euler-Bernoulli elastic beam equation subject to initial/boundary conditions  www.nature.com/scientificreports/ where x ∈ [0, L] is the axial position and L is the height of the retaining wall, E and ρ are the modulus of elasticity and the density of the wall material, respectively. A b = sH is the area of constant cross-section of rectangular cantilever and its moment of inertia is given by I = sH 3 12 . Notice that the boundary conditions correspond to the case of the cantilever beam whose base is clamped or fixed whereas the top end is free. The first equation in (Eq. 3) corresponds to the second Newton's law of motion. The retaining wall vibrates due to its restoring elastic force and the force resulting from the swelling and contracting clay.
Let us introduce the dimensionless variables and define where u c is some predefined characteristic deflection. Then, Eq. (3) can be written as where the tilde notation is omitted for brevity.

Mass lumped model. Let us assume that the initial/boundary value problem given by Eq. (6) has solution
of the form where Y j (x) are eigenmodes of the differential operator d 4 dx 4 subject to the boundary conditions listed. To study the essential dynamics of the wall subject to the swelling pressure, we employ a single-mode Galerkin approximation where Y (x) represents the first eigenmode of the cantilever beam whose base end is clamped, whereas the top end is free. Neglecting the higher frequency modes can yield simple yet accurate approximate solutions to many engineering problems 15 .
The scaled first eigenfunction (eigenmode) of the beam differential operator d 4 dx 4 subject to the boundary conditions is given by where denote Krylov's eigenfunction for the fourth order differential operator d 4 dx 4 subject to the boundary conditions for the cantilever beam 4,16 . The spectral parameter µ 1 = 1.8751040687... is the first positive root of the transcendental equation The scaled eigenfunction Y (x) in Eq. (8) has the following properties 4 Applying the trapezoidal rule to the integral on the right-hand side of Eq. (15) and using Eqs.(11)- (14) yields Substituting leads to the conservative single-degree-of-freedom oscillator equation in the dimensionless form where the tilde notation was dropped again for the sake of brevity, and dimensionless parameter p is defined as In this work, the non-linear oscillator equation (17) subject to zero-initial conditions is assumed Equations (17) and (19) constitute a zero-initial value problem for a conservative nonlinear oscillator. We expect a bounded periodic solution whose amplitude is a monotonically increasing function of the swelling parameter p. For higher values of p, the dynamics can be affected by the presence of higher-order harmonics.
Multiplying Eq. (17) by Ẋ (t) and integrating with respect to time, we get the conservation of energy where C = p due to the initial conditions (Eq. 19). Thus, We now show that all solutions to the initial value problem (17) and (19) are periodic.

Theorem 1
The initial value problem (17) with initial conditions (19) has a periodic non-negative solution for any positive lumped mass model parameters p.
Proof We establish the proof using simple phase plane analysis, cf., 4,[17][18][19] . The solution X(t) is periodic if and only if the phase diagram produces a closed curve. This holds true if the continuous function has two real roots s 1,2 , and g(s) > 0 for all s between s 1 and s 2 . Note that g(0) = 0 , and g(s) has only one local maximum due to the fact that g ′ (s) = 2pe −s − 2s = 0 has only one real root s * due to Rolle's theorem, and that g ′′ (s * ) = −2pe −s * − 2 < 0 holds true for the positive lumped mass model parameter p, i.e., g(s) is a concave function on the whole real line. Hence, the existence of the second root follows from the Intermediate Value Theorem. Now, if we set s * = X eq , the following condition is satisfied pe −X eq = X eq whence it follows that the solution to the lumped mass model is a constant X(t) = X eq representing the stable steady state of Eq. (17). To show that the periodic solution X(t) is non-negative, we rewrite Eq. (21) as follows Since the left-hand side of Eq. (23) is non-negative, it must hold true that X(t) ≥ 0. ∎ Integrating Eq. (23), we obtain for 0 ≤ t ≤ T/2 the exact solution in the implicit form

Remark Equation (27) can be rewritten as
The exact solution of the above transcendental equation can be expressed in terms of the generalized Lambert W function W(t 1 , t 2 ; z) 20

as follows
This special function is introduced in 20 as inverse to the mapping z → (z − t 1 )(z − t 2 )e z . Figure 2 presents the graph of the function W − √ 2p, √ 2p; z for different values of parameter p, evaluated by solving □ In the following theorem we estimate the period of oscillation.

Theorem 2 The solution to the initial value problem (17) with initial conditions (19) has a period
where A is the maximum deflection.
Proof By Eqs. (24) and (29), the period of the oscillation is given by The last integral in Eq. (30) can be approximated as follows since for all 0 < t < 1 and A > 0 . Therefore, where the angular frequency is given by

Harmonic approximations for nonlinear oscillator
In the following, we construct approximate periodic solutions to the non-linear oscillator equation (17) subject to zero-initial conditions in Eq. (19) by substituting the harmonic ansatz The equilibrium position is found by setting Ẍ = 0 in Eq. (17): (31) (36)  [20][21][22][23][24] which is defined as the inverse of the mapping z → ze z , and W(z) solves the equation Notice that the Lambert W function is uniquely defined for z ≥ 0 and increases monotonically. The Taylor expansion of the Lambert W function is given by The maximum deflection A can be found more precisely using the method of successive approximations. Substituting A i+1 = A i + A into Eq. (27) and neglecting terms that are of second order with respect to A results in Solving Eq. (43) with respect to A we obtain recursive relation The initial guess is based on the assumption that the center of the oscillations is close to the equilibrium position (which is a reasonable approximation if oscillations are nearly harmonic).
In particular, the maximum deflection after the first correction is given by Following the standard procedure, we approximate the frequency of oscillations by evaluating the derivative of restoring force f (X) = X − pe −X at the equilibrium 25 : The above expression for frequency (referred to as the Nayfeh frequency) coincides with frequency of the linear oscillator in Eq. (38) up to first order with respect to p while the approximations of the maximum deflection by Eqs. (46) and (36) are identical up to second order with respect to the parameter p. Also, the expression for the frequency given by Eq. (47) is consistent with Eq. (33) for the period of oscillations, e.g., for A ≈ 2W p as shown in Eq. (45).

Comparison of approximation methods
Numerical and approximate values of amplitude and frequency of oscillations vs. parameter p are presented in Fig. 3. Figure 3a shows the amplitude A, found by direct numerical solution of Eq. (17)  In Table 1 we list relative errors for the different amplitude approximations for two different values of p. Figure 3b shows the trend of the frequency ω vs. parameter p, found by direct numerical solution of Eq. (17) ( ω num -blue dashed line), linear approximation given by Eq. (38) (green dotted line), and the Nayfeh approximation given by Eq. (47) (red solid line). In Table 2 we list relative errors for different frequency approximations for two different values of parameter p.
(40) X c = W p , (44)        Fig. 4 we present numerical and approximate analytical solutions for the horizontal deflection of the tip of the cantilever wall u(t, L) vs. time t with the following set of parameters L = 2.54 m (Fig. 4a) and L = 5 m (Fig. 4b) Observe that while the approximation of amplitude A 2 can be found recursively and remains accurate even in the highly nonlinear regime shown in Fig. 4b, the Nayfeh approximation of frequency or period is not very precise because higher harmonics significantly affect dynamics in this regime.

Conclusions
This paper presented a simple approach to obtain approximate periodic solutions to the nonlinear oscillator describing the retaining wall dynamics subject to swelling pressure. The equation of the nonlinear conservative oscillator was established by using the single-mode Galerkin approach. It was shown that the zero initial value problem for the mass lumped model has only periodic solutions. Various approximations for frequency and amplitude of the periodic oscillations were verified against the numerical solution. In our forthcoming work, the relevance of transverse deformations of the retaining wall and its resulting oscillations under the influence of seismic vibrations will be investigated.