A Semi Analytical Method for Dynamic Analysis of Beam Resting on Inertial Soil

Abstract

Accurate dynamic analysis of beams supported on soil is essential for the safe and economical design of vibration-resistant and earthquake-resistant structures. Earlier contributions, such as the semi-analytical framework developed by Guenfoud, Bosakov, and Laefer (2009), provided reliable predictions for beams on elastic half-spaces with inertial properties. However, these formulations are mathematically intensive and often difficult to apply directly in engineering practice. Conventional soil models, including Winkler’s subgrade reaction method and continuum-based elastic models, either oversimplify soil behaviour or involve complex derivations, further limiting their practical utility. The present study introduces a modified version of the semi-analytical method for beams resting on an elastic half-space with inertial properties, reformulated to reduce mathematical complexity without compromising accuracy. The approach incorporates soil inertia effects while enabling more straightforward evaluation of displacements under vertical loading. To illustrate its effectiveness, the dynamic response of a beam measuring 10 m in length, 1 m in width, and 0.5 m in depth is investigated. The corresponding Eigenfrequencies and natural shapes of the beam are determined, and the results show close agreement with those obtained from established analytical methods. This refinement not only simplifies computational effort but also makes the analysis more accessible for engineering applications. The method thus provides a practical and reliable tool for studying beam–soil interaction dynamics, with direct relevance for the design of safer and more cost-effective structural systems.

Introduction

• A fundamental solution (or Green’s function) describes how a solid reacts to a static or dynamic point load at another location.

• These solutions are crucial for understanding soil–structure interaction (SSI), especially in the context of elastic and dynamic behavior of soil.

?? 2. Approaches to Modeling Beams on Elastic Soil

Two Main Methods:

1. Subgrade Reaction Model (Winkler model):

   • Soil is modeled as independent springs.

   • Deflection is local — no interaction between adjacent points.

   • Simplified but lacks realism.

2. Elastic Continuum Model:

   • Uses Boussinesq’s solution for a half-space.

   • Considers soil interaction across regions.

   • More accurate, but computationally complex.

???? 3. Categories of Dynamic Analysis Methods

1. Simplified Procedures

   • Use charts and empirical approximations.

   • Easy for practicing engineers.

2. Semi-analytical Methods

   • Discretize foundation surface.

   • Use Fast Fourier Transforms (FFT) and impedance functions.

3. Analytical Methods

   • Based on wave equations with boundary assumptions.

4. Dynamic Finite Element Models (FEM)

   • Include absorbing boundaries.

   • Suitable for layered or inhomogeneous soils.

5. Boundary Element Methods (BEM)

   • Reduce the problem to algebraic equations.

   • Handle wave propagation and far-field effects.

6. Experimental Methods

   • Field/lab tests.

   • Validate theoretical/numerical models.

???? 4. Method Used in the Study

• A semi-analytical method is adopted for modeling beam-soil interaction, focusing on:

  • Dynamic response

  • Natural frequencies

  • Mode shapes

• Based on the approach by Zhemochkin and Sinitsyn, and extended by Guenfoud et al.

????? 5. Problem Formulation

• Soil: Homogeneous, isotropic, elastic half-space (with inertia effects).

• Beam: Mass mmm, flexural rigidity EIEIEI, subjected to vertical excitation.

• Assumptions:

  • No damping, friction, or surface curvature.

  • Beam–soil interaction is discretized into rigid vertical connectors (not continuous).

Key Model Features:

• Beam divided into elements, each with a vertical rigid link at its midpoint.

• Discretization leads to an indeterminate system with unknowns:

  • Xi(t)X_i(t)Xi?(t): Liaison forces

  • Φ0(t)\Phi_0(t)Φ0?(t): Rotation angle

  • u0(t)u_0(t)u0?(t): Vertical displacement

???? 6. Governing Equations

• The core equations (Eq. 1) relate:

  • Green’s functions vijv_{ij}vij? (displacement response functions)

  • Inertial forces Jj(t)J_j(t)Jj?(t)

  • External loads ΔiP\Delta_i^PΔiP?

∑j=1n(vij+yij)Xj(t)−∑j=1nyijJj(t)+λiΦ0(t)+u0(t)+Δip=0\sum_{j=1}^{n} (v_{ij} + y_{ij}) X_j(t) - \sum_{j=1}^{n} y_{ij} J_j(t) + \lambda_i \Phi_0(t) + u_0(t) + \Delta_i^p = 0j=1∑n?(vij?+yij?)Xj?(t)−j=1∑n?yij?Jj?(t)+λi?Φ0?(t)+u0?(t)+Δip?=0 ∑j=1n(Xj(t)−Jj(t))λj=IyΦ¨0(t)\sum_{j=1}^{n} (X_j(t) - J_j(t)) \lambda_j = I_y \ddot{\Phi}_0(t)j=1∑n?(Xj?(t)−Jj?(t))λj?=Iy?Φ¨0?(t) ∑j=1n(Xj(t)−Jj(t))=mu¨0(t)\sum_{j=1}^{n} (X_j(t) - J_j(t)) = m \ddot{u}_0(t)j=1∑n?(Xj?(t)−Jj?(t))=mu¨0?(t)

? Summary of Key Takeaways

• The semi-analytical method provides a balance between accuracy and computational efficiency for modeling dynamic soil–beam systems.

• Green’s functions are central to capturing how soil reacts to forces.

• The system is modeled as discrete elements rather than a continuous contact surface.

• The approach enables evaluation of vibration modes, natural frequencies, and responses to loads, incorporating soil inertia but neglecting damping and friction.

Conclusion

This paper presents a semi-analytical approach for the dynamic analysis of beams resting on an elastic half-space with inertial properties. The method is developed to evaluate the Eigenfrequencies, mode shapes, and dynamic response of the beam subjected to external excitation. The analysis considers the interaction forces within the contact zone, which are essential for determining other physical parameters. The formulation is based on Green’s functions to represent the displacements in the contact region. Incorporating the inertial effects of the half-space, also known as Lamb’s problem, introduces mathematical challenges, particularly due to instabilities associated with hypergeometric functions. The present work overcomes these limitations and provides stable, reliable expressions for the solution. Results show that the computed mode shapes of the beam are irregular, and the proposed semi-analytical method is both computationally efficient and practical for engineering applications. Furthermore, the approach can serve as an approximation function in numerical models, enabling its application to more complex problems of wave propagation and dynamic soil–structure interaction.

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