An infinite Euler–Bernoulli beam on a Winkler elastic foundation describes how a long (theoretically infinite) beam behaves when supported continuously by an elastic medium, such as soil or bedding. The Winkler model assumes that the foundation reacts proportionally to local deflection, like a bed of independent springs. The governing differential equation EIyw(z)^(4) + kw(z) = q(x) balances bending stiffness EI and foundation stiffness k under load q(x) that represent in this case the local force. The key parameter is characteristic length L = (EI/k)1/4, defining how far deformations spread. For a concentrated load, deflection decays exponentially and oscillates slightly as it propagates along the beam. The solution enables the prediction of deflection, rotation, bending moment, and shear force, critical for designing foundations, pavements, rails, or pipelines resting on elastic supports.
Model assembly
03) Infinite beam on the elastic foundation
Solution for low-stiffness soils (LSS)
Low beam bending stiffness + Low soil stiffness
- Suitable for:
- Better energy dissipation.
- Moderate risk of punching failure.
- Be cautious:
- Excessive deformations.
- Sensitive to differential settlements.
04) Linear beam model, deformations, reactions, moments, shear forces
High beam bending stiffness + Low soil stiffness
- Suitable for:
- Improved global stiffness.
- Be cautious:
- Risk of cracking due to high bending stresses.
- Limited adaptability to uneven soil.
05) Linear beam model, deformations, reactions, moments, shear forces
Figure 06 illustrates the behavior for a relatively low-stiffness soil with a subgrade modulus of 16000 kN/m³ and varying heights of the footing strip.
06) Interaction of relatively low-stiffness soil with varying stiffness of the beam (closed-form solution)
Solution for high-stiffness soils (HSS)
Low beam bending stiffness + High soil stiffness
- Suitable for:
- Efficient stress transfer to the stiff soil
- Lower moment demand
- Be cautious:
- High local shear forces
- The most significant chance of punching shear failure
07) Linear beam model, deformations, reactions, moments, shear forces
High beam bending stiffness + High soil stiffness
- Suitable for:
- Stable system, minimal deflections
- Predictable linear response
- Be cautious:
- Higher construction cost
08) Linear beam model, deformations, reactions, moments, shear forces
09) Interaction of high-stiffness soil with varying stiffness of the beam (closed-form solution)
Response of a beam for low/high stiffness soils
10) Interaction of low and high-stiffness soil with varying stiffness of the beam