Static reduction of shear or elastic moduli

[Mccoy]

Thread 'Static AND dynamic modulus degradation'

Oct 14, 2013

Dear colleauges,
this issue has being gnawing at the background of my mind for a while now, and it's about time to submit it to your valuable attention.
We know that elastic and shear moduli decrease their value with an increase in strain deformation. Tangent moduli, Eo and Go, may be 10 times the value of secant moduli at large deformations. The adjustment tangent-to-secant may be done in various ways, some of such are called 'degradation schemes', as per the articles by Fahey et al. 1994, advertised by Paul Mayne 1999 and herein attached (chapter 3). It is based upon the determination of...

• Mccoy
• Replies: 4
• Forum: Foundations by Terzaghi

It's interesting to revisit the same topic 25 years later. I wonder if there has been any evolution in a quarter of a century. Perhaps not so much. Is foundation engineering a stagnating field or is that just my impression, due to a lack of suitable personal updates on technical literature?

I recently did a search, facilitated by AIs, but not very much appeared. There is the linked paper by Fahey et al 2003, an interesting one because it refers to real-life raft foundations.

https://australiangeomechanics.org/admin/wp-content/uploads/2015/03/38_3_5.pdf

 

[Mccoy]

I submitter the issue of static reduction to Deepseek and Gemini. gemini provided the msot satisfactory conceptual illustration:

The concept of shear modulus reduction curves is primarily associated with dynamic (seismic) analysis in geotechnical engineering, where soil stiffness degrades with increasing shear strain due to cyclic loading. However, the underlying principle of soil stiffness being strain-dependent is also relevant to static conditions and can have indirect applications in the design of raft foundations.

Let's break down the concepts and their relevance:

Shear Modulus and Strain-Dependency in Soils​

The shear modulus (G) of soil represents its stiffness in resisting shear deformation. It's a crucial parameter for predicting settlement and assessing soil-structure interaction.

Soils exhibit non-linear stress-strain behavior, meaning their stiffness is not constant but depends on the level of strain they experience.

• Small-strain shear modulus (G0 or Gmax): This is the maximum shear modulus, measured at very low strains (typically less than 10−4%). It's often determined from geophysical tests like shear wave velocity (Vs) measurements. This represents the initial, undisturbed stiffness of the soil.
• Modulus Reduction Curve (G/G0 vs. γ): This curve illustrates how the shear modulus (G) reduces as the shear strain (γ) increases, normalized by the small-strain shear modulus (G0). While commonly used for dynamic analysis to account for seismic-induced strain, the concept of stiffness reduction with increasing strain also applies to static loading, albeit often at lower strain levels.

Static Conditions and Raft Foundations​

For raft foundations under static (non-seismic) loads, the primary concerns are:

1. Total Settlement: The overall vertical displacement of the foundation.
2. Differential Settlement: The uneven settlement across the raft, which can induce significant bending moments and shear forces in the raft structure.
3. Bearing Capacity: The ultimate load the soil can support without shear failure.

Traditional static design methods for raft foundations often simplify soil behavior as linearly elastic, using a single elastic modulus (Young's modulus, E, or shear modulus, G) or a modulus of subgrade reaction (ks). However, this simplification may not fully capture the real behavior of soil, especially for large rafts where significant stress bulbs and corresponding strains develop.

Applications of Shear Modulus Reduction (or Strain-Dependent Stiffness) in Static Raft Foundation Design​

While full-blown shear modulus reduction curves as used in dynamic analysis might not be directly applied in every static raft foundation design, the underlying principle of strain-dependent stiffness is relevant in the following ways:

1. More Accurate Settlement Prediction:
   
   • Non-linear Analysis: For more rigorous static analysis, especially with numerical methods like Finite Element Analysis (FEA), advanced constitutive models for soil can be employed. These models implicitly or explicitly incorporate the strain-dependency of soil stiffness. As the applied load increases and strains in the soil beneath the raft grow, the effective shear modulus (and Young's modulus) of the soil will reduce.
   • Impact on Settlement: A reduction in soil stiffness (represented by the shear modulus) directly leads to larger settlements. Therefore, accounting for this reduction, even under static conditions, can provide more realistic settlement predictions, which is critical for serviceability limit state design.
   • Differential Settlement: The non-linear behavior can also affect the distribution of stress and strain under the raft, influencing differential settlements, particularly for flexible rafts.
2. Improved Soil-Structure Interaction (SSI) Modeling:
   
   • For complex raft foundations, particularly combined piled-raft foundations (CPRF), understanding the load-sharing mechanisms requires accurate modeling of soil stiffness. If a simplified constant modulus is used, it might overpredict the stiffness of the soil where strains are higher, leading to an underestimation of settlement and potentially inaccurate internal forces in the raft.
   • Sophisticated SSI analyses can consider the varying stiffness of the soil with depth and stress level, which is essentially an application of the strain-dependent modulus concept, even if not explicitly plotted as a "reduction curve" in the traditional dynamic sense.
3. Back-Analysis and Calibration:
   
   • When back-analyzing the performance of existing raft foundations or conducting plate load tests, observed settlements can be used to infer the "operative" or "mobilized" soil stiffness. This operative stiffness will naturally be lower than the small-strain stiffness (G0) due to the strains induced by the applied loads. This implicitly reflects a point on a static shear modulus reduction (or stiffness degradation) curve.
4. Influence on Modulus of Subgrade Reaction (ks):
   
   • The modulus of subgrade reaction, often used in Winkler models for raft design, is not a fundamental soil property but an empirical value dependent on foundation size, shape, and soil properties. When considering non-linear soil behavior, the "effective" ks will decrease with increasing settlement and strain. More advanced methods for determining ks (e.g., from elastic theory) can implicitly account for the strain-dependent modulus.

Key Considerations for Static Conditions​

• Strain Levels: Static loads generally induce lower shear strains in the soil compared to dynamic (earthquake) loads. Therefore, the degree of shear modulus reduction will be less pronounced for static conditions.
• Laboratory vs. In-Situ: Static soil properties are often determined from laboratory tests (e.g., triaxial tests, oedometer tests), which capture stress-strain behavior. Dynamic properties (like G0 from Vs) are typically measured in-situ. Correlations exist to relate static and dynamic moduli, but care must be taken as these can be soil-specific.
• Empirical Correlations and Experience: In practice, for many routine raft foundation designs, empirical correlations and engineering judgment based on experience with similar soil types and loading conditions are often employed. While not explicitly using a modulus reduction curve, the chosen design parameters (e.g., Young's modulus) are often adjusted based on the anticipated stress and strain levels.

In summary, while "shear modulus reduction curves" are a cornerstone of dynamic analysis, the principle that soil stiffness is strain-dependent is equally valid under static conditions. For raft foundations, particularly for large or heavily loaded structures, considering this strain-dependency (even if not using a formal "reduction curve") through advanced numerical modeling or careful selection of design parameters, leads to more accurate and reliable predictions of settlement and better overall foundation performance.

 

[Mccoy]

INTERACTION OF STATIC REDUCTION AND DYNAMIC DEGRADATION.

The Gemini (2.5 flash) language model would tend to predict, according to theory, a summation of effects. Interesting. I also wonder, is that a simple summation or multiplication?

The interplay between cyclic degradation and static reduction in material properties, specifically focusing on shear modulus, is a critical concept in engineering, especially in geotechnical and structural applications. It highlights how a material's stiffness is affected by both sustained (static) loads and repetitive (cyclic) loads.

Let's break down the concepts first:

1. Shear Modulus (G)

• Shear modulus, also known as the modulus of rigidity, describes a material's resistance to shear deformation (i.e., its stiffness when subjected to forces that cause it to twist or shear). A higher shear modulus means a stiffer material.

2. Static Reduction of Shear Modulus (Gs)

• Static reduction refers to the decrease in shear modulus under a sustained, non-varying load or a single application of load. This reduction typically occurs due to:
  • Non-linear stress-strain behavior: Most materials, especially soils, exhibit non-linear behavior even under static loads. As the static shear strain (or stress) increases, the tangent or secant shear modulus decreases. This is often represented by a "backbone curve" (G vs. γ) where G reduces with increasing γ.
  • Plastic deformation/yielding: If the static load is large enough to cause plastic deformation or yielding, the material's internal structure changes, leading to a permanent reduction in stiffness, even after the load is removed.
  • Changes in effective stress: In soils, static loads can induce changes in pore water pressure, which in turn alters the effective stress. A reduction in effective stress (due to pore pressure build-up) can lead to a reduction in shear modulus.

3. Dynamic Reduction (Cyclic Degradation) of Shear Modulus (Gd)

• Dynamic reduction, or cyclic degradation, refers to the progressive decrease in shear modulus as a material is subjected to repeated (cyclic) loading, even if the individual load cycles are below the static yield strength. This phenomenon is particularly relevant in:
  • Soils: Earthquakes, wave loading on offshore structures, and traffic vibrations can cause cyclic loading. In saturated soils, cyclic loading often leads to the build-up of excess pore water pressure, which reduces the effective stress and, consequently, the shear modulus. This can lead to liquefaction in extreme cases.
  • Metals/Composites: Fatigue in materials can lead to a reduction in stiffness over many cycles, even if visible damage isn't immediately apparent. Microcracking and rearrangement of internal structures contribute to this degradation.
• Cyclic degradation is often characterized by a degradation index or a reduction in the secant shear modulus (Gsec) with the number of loading cycles (N) and cyclic shear strain amplitude (γc).

---

Illustrating the Relationship: Cyclic Degradation Imposed on a Static Reduction Condition​

Imagine a material, like a soil deposit, that is initially subjected to a static load, and then experiences cyclic loading.

Scenario: A soil layer under a building (static load) is then subjected to earthquake shaking (cyclic load).

Initial State (Static Condition):

• The soil is under an initial geostatic stress state. When the building is constructed, it imposes an additional static vertical load.
• Even before any dynamic loading, the shear modulus of the soil (Gs) will already be "reduced" from its maximum small-strain value (Gmax) due to the existing static shear strains and effective stresses induced by the building's weight. This is the static reduction. The relationship between shear modulus and static shear strain is typically nonlinear, as shown by a G-γ curve. As static strain increases, G decreases.

Imposing Cyclic Degradation:

• Now, an earthquake occurs, subjecting the soil to cyclic shear stresses.
• As the cyclic loading progresses, the soil undergoes cyclic degradation. This means that with each cycle of loading, the effective shear modulus (or the secant shear modulus measured during cyclic loading) further decreases.
• This cyclic degradation is imposed upon the already statically reduced shear modulus. It's not a separate process, but rather an additional reduction in stiffness that starts from the current statically-reduced state.

Visual Representation (Conceptual):

Imagine a graph with Shear Strain (γ) on the x-axis and Shear Modulus (G) on the y-axis.

1. Initial Stiffness (Gmax): At very small strains (almost theoretical), the material has its maximum shear modulus, Gmax.
2. Static Reduction Curve (G vs. γs): As static shear strain (γs) increases (due to static loads), the shear modulus (Gs) follows a non-linear degradation curve, starting from Gmax and decreasing. This is the static reduction. Let's say, under the building's weight, the soil is operating at a static shear modulus Gs1 corresponding to a static strain γs1.
   • Plot 1: A typical G/Gmax vs. γ curve for a given static effective stress.
     • The y-axis would be G/Gmax (normalized shear modulus).
     • The x-axis would be γ (shear strain).
     • This curve shows how G decreases as γ increases, even under static conditions.
     • So, for a specific static strain γs1 (due to the building's weight), the material has a static shear modulus Gs1=Gmax⋅(G/Gmax)γs1.
3. Cyclic Degradation (Superimposed): Now, the cyclic loading begins. The cyclic degradation acts on top of the existing static reduction.
   • The cyclic loading, especially in saturated soils, leads to pore pressure buildup, which reduces the effective stress. A reduced effective stress inherently leads to a lower shear modulus.
   • Additionally, repeated straining can cause rearrangement of particles and breakdown of bonds, further reducing stiffness.
   • So, as the number of cycles (N) increases, the shear modulus will progressively decrease from Gs1.
   • We can represent this as a series of degradation curves or points:
     • For N=1 cycle, the effective shear modulus might be Gd,N=1<Gs1.
     • For N=10 cycles, the effective shear modulus might be Gd,N=10<Gd,N=1.
     • And so on, until a certain level of degradation or even liquefaction (where G approaches zero).

Mathematical/Modeling Illustration:

A common approach in modeling is to consider the shear modulus at any given point in a cyclic loading scenario as a product of factors accounting for both static and dynamic effects:

G=Gmax⋅Rs(γs)⋅Rd(N,γc,…)

Where:

• Gmax is the maximum shear modulus at very small strains.
• Rs(γs) is the static reduction factor (a value between 0 and 1) that depends on the current static shear strain (γs) and effective stress. This factor captures the non-linear behavior under static loading. As γs increases, Rs decreases.
• Rd(N,γc,…) is the cyclic degradation factor (a value between 0 and 1) that depends on the number of loading cycles (N), cyclic shear strain amplitude (γc), stress ratio, drainage conditions, and other factors. As N increases (for a given γc), Rd decreases.

Example Application (Conceptual):

Imagine a soil element under a foundation:

1. Static Condition: The foundation imposes a certain static stress, leading to a static shear strain γs. From laboratory tests or empirical relationships, we determine that the shear modulus is Gs=Gmax⋅Rs(γs). For instance, if Gmax=100 MPa and Rs(γs)=0.7 for the current static conditions, then Gs=70 MPa. This is the starting point for dynamic analysis.
2. Cyclic Loading (Earthquake): The earthquake induces cyclic shear strains with an amplitude γc. As the shaking continues, the number of cycles (N) increases.
   • After, say, N=10 cycles, the material has degraded further. A degradation model might suggest that for these conditions, Rd(10,γc)=0.5.
   • The effective shear modulus during this cyclic loading would then be Geffective=Gs⋅Rd(N,γc)=(Gmax⋅Rs(γs))⋅Rd(N,γc).
   • Using our example values: Geffective=70 MPa⋅0.5=35 MPa.

This illustrates that the total reduction in shear modulus is a compounding effect. The cyclic degradation doesn't start from Gmax but from the already "statically reduced" shear modulus. This combined effect is crucial for accurately predicting material response under complex loading conditions, such as seismic events acting on existing infrastructure.

 

[Mccoy]

I also asked for specific literature citations, and at the end, there are some apparently valuable references; unfortunately, they may be behind a paywall.


Shear modulus degradation is a critical aspect in geotechnical engineering, particularly when analyzing soil behavior under various loading conditions, from static to dynamic (e.g., seismic). Here are citations focusing on shear modulus degradation, specifically addressing combinations of static and dynamic approaches or the relationship between them:

Key Papers and Reviews:

• Hardin, B. O., & Drnevich, V. P. (1972). Shear modulus and damping in soils: Design equations and curves. Journal of the Soil Mechanics and Foundation Division, ASCE, 98(6), 667-692.
  • Comment: This is a seminal work in dynamic soil properties, providing fundamental curves for shear modulus degradation and damping ratio as functions of shear strain. While primarily focused on dynamic properties, it lays the groundwork for understanding how modulus changes across various strain levels, which is relevant to both static and dynamic contexts.
• Vucetic, M., & Dobry, R. (1991). Effect of plasticity index on shear modulus reduction curves. Journal of Geotechnical Engineering, ASCE, 117(1), 89-107.
  • Comment: This paper highlights the significant role of the plasticity index in influencing shear modulus degradation, particularly under cyclic loading. It provides widely used normalized shear modulus degradation curves, which are essential for dynamic analyses.
• Ishibashi, I., & Zhang, X. (1993). Unified dynamic shear moduli and damping ratios of sand and clay. Soils and Foundations, 33(1), 182-191.
  • Comment: This research aimed to develop unified relationships for dynamic shear modulus and damping ratio for both sands and clays, providing a comprehensive framework for their estimation across different soil types.
• Kramer, S. L. (1996). Geotechnical Earthquake Engineering. Prentice Hall.
  • Comment: This textbook provides a comprehensive overview of geotechnical earthquake engineering, including detailed discussions on shear modulus degradation, its experimental determination, and its application in dynamic response analysis. It synthesizes many of the key findings from earlier research.

Papers Directly Addressing Static and Dynamic Relationships/Combinations:

• Benjamin, U. S., Tamunobereton-ari, I., Horsfall, I. O., & Mogaba, P. (2022). Comparative analysis of static shear modulus and dynamic shear modulus determined by geophysical and geotechnical investigation. Earth Sciences Malaysia (ESMY), 6(1), 01-10.
  • Comment: This paper directly compares static and dynamic shear moduli obtained from different investigation methods (geophysical for dynamic, geotechnical for static) and discusses the variations and correlations between them.
• Chae, Y. S.; Au, W. C.; and Chiang, Y. C. (1981). Determination of Dynamic Shear Modulus of Soils from Static Strength. International Conferences on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics. 7.
  • Comment: This older but relevant study investigates the correlation between dynamic shear modulus (from resonant column tests) and static strength (from undrained triaxial compression tests), suggesting a linear relationship in many cases. This explores the possibility of inferring dynamic properties from static tests.
• Senerakis, S., & Dritsos, S. (2018). Correlations and empirical relations between static and dynamic elastic ground parameters in shallow geotechnical site investigations. PURE Montanuniversität Leoben.
  • Comment: This paper presents a method for deriving empirical relations to estimate static moduli from dynamic moduli, aiming to improve accuracy compared to conventional linear approaches. This directly addresses the combination and correlation of static and dynamic properties.
• ResearchGate Discussion (2016). Young's modulus and shear modulus in static and dynamic analysis?
  • Comment: While not a formal publication, this discussion thread on ResearchGate includes valuable insights from multiple researchers regarding the differences and relationships between static and dynamic moduli, including when and how to use them in numerical simulations and the concept of modulus degradation.

Papers on Modulus Degradation Curves and Modeling:

• Oztoprak, S., & Bolton, M. D. (2013). Stiffness of sands through a laboratory test database. Géotechnique, 63(54), 70.
  • Comment: This research contributes to understanding the stiffness of sands, which includes aspects of shear modulus degradation across different strain levels.
• Ortiz-Pulido, M. P., Gaitán-Serrano, J. F., & Camacho-Tauta, J. (2014). Modelo unificado de curvas de degradación del módulo cortante en arenas del río Guayuriba. DYNA, 81(184), 77-84.
  • Comment: This paper presents a unified model for shear modulus degradation curves, which is relevant for both static (large strain) and dynamic (cyclic) loading conditions.
• MDPI (current). Evaluation of the Shear Modulus Degradation by a Modified Hyperbolic Function for Unconventional Geomaterials.
  • Comment: This relatively recent publication discusses a modified hyperbolic model for evaluating shear modulus degradation across various angular strains and effective confining stresses, providing a modeling approach for both initial (small-strain, dynamic-relevant) and degraded (larger strain, static-relevant) conditions.

For further in-depth study, consider looking into:

• Resonant Column Tests: These laboratory tests are crucial for determining the maximum (small-strain) shear modulus (G0) and initial portions of the shear modulus degradation curve, which are inherently dynamic properties.
• Torsional Shear Tests/Cyclic Triaxial Tests: These tests can provide shear modulus degradation behavior over a wider range of strains, bridging the gap between small (dynamic) and large (static) strain behavior.
• In-situ Seismic Tests (e.g., Downhole, Crosshole): These methods provide dynamic shear wave velocity, from which G0 can be calculated, offering insights into the undisturbed dynamic stiffness of the soil.
• Standard Penetration Test (SPT) and Cone Penetration Test (CPT) correlations: While providing static strength/stiffness indicators, many empirical correlations exist to estimate dynamic properties (like G0) from these tests.

When seeking information on "degradation combinations, static and dynamic," it's often about how the stiffness of the soil changes from its initial, very stiff, small-strain (dynamic) state to a less stiff, larger-strain (static or strongly dynamic) state, and how these different stiffness regimes relate to each other for design and analysis purposes.