Analysis of Composite Columns: Granular Material In-Fill Inside a Confining Annulus

Affordable housing is becoming a rapidly growing problem in all the major residential hubs worldwide. This problem has three major contributing factors: high population growth, limited supply of raw materials, and high transportation costs. This has led to housing becoming a luxury rather than a basic human need. Our study aims to analyse structural columns designed using recycled metals and readily available granular materials like sand. This will reduce the overall material cost and the cost associated with transporting bulky materials like steel and aggregate.

In order to understand the range of loads that these columns have to bear, we model the most basic of residential structures. As per the theory given in literature for solid mechanics and failure of soil, we derive the uniaxial strength equation for the proposed column designs. Now using the bearing load values, we try to validate whether these columns are practically feasible for construction purposes. Our study shows that if we use recycled metals or 3D printed materials like Poly Lactic Acid (PLA) for the confining annulus, these columns can indeed be used for civil construction. Finally, we focus on the practical considerations and geometric proportions of the columns for practical use on field.

Author: Utkarsh Gupta
Indian Institute of Technology Delhi
Mentors: Jose E. Andrade, Siavash Monfared
California Institute of Technology
Editor: Ann Zhu

1. Background

Concrete and steel have been used for civil construction since time immemorial. Al- though they are great for building structures from a strength-based perspective, they are bulky to transport, expensive, not readily accessible in remote locations, and not very sustainable.

A lot of ongoing research focuses on exploring materials that are environment friendly and cheap to replace concrete and steel in Reinforced Concrete (RCC) structures. Hector Archila et al. [1] explored the idea using bamboo as a replacement for reinforcing steel in RCC structures. But certain studies show that it is an ill-considered concept and not a sustainable alternative for steel [2] due to various strength, durability and economic reasons. P. Lalzarliana et al. [3] explored the idea of using waste plastic bottles filled with Recycled Concrete Aggregates, aggregates processed from construction and demolition waste, to replace bricks in masonry structures. These solutions may be sustainable, but do not eliminate the biggest challenge in the field of civil construction, which is transportation of bulky raw material.

With more and more innovation in the field of 3D printing and material science, it has now become possible to manufacture and realise complex, high-strength designs on the site of construction. This greatly reduces the need for any transportation of bulky materials from one place to another. In this study, we analyse columns made using granular material in-filled inside a confining annulus. For the in-fill, we assume a granular material like sand, which is not only cheap, but also readily available in remote locations. For the confining material, we propose to use recycled metals such as steel or aluminium, or a suitable 3D printed material. Given these properties, the composite columns should be considerably more cheaper and environmentally friendly than the current RCC columns used in concrete structures.

Using three separate theories: Jaky’s Theory [4], Theory of Solid Mechanics, and Bell’s Addition to Rankine’s Theory of Lateral Earth Pressure [5, 6], we find the ratio of applied vertical stress on the granular material to the lateral stress on the confining annulus. The yield strength of composite column, fycc is then derived using all three theories in the subsequent sections. A closer look at the results from these theories shows that there exists a correlation between them. This is where we introduce the dimensionless variable, Soil Characterisation Constant, S. After this, we move towards understanding the physical significance of each of the terms in the derived equation. Towards the end, we use the Buckingham π Theorem [7] to derive the π constants for the derived yield strength equation. In order to validate the columns for use in practical scenarios, we also consider three to four story residential structures and calculate the loads on their columns. Then, we check whether our designed columns can sustain those loads while staying inside the limits of reasonable geometric proportions.

2. Setup and Assumptions for Analysis

In order to analyse the composite columns, we assume a suitable design for the composite column that would help us carry out the numerical analysis. Figure 1 depicts a rough diagram of the same.

Figure 1: Setup assumed for the analysis of the composite column. The left figure demonstrates
the frontal view of the setup and the right shows the top view of the composite column.

Here, σapp is the uni-axial force that will be applied on the top cap and directly be transferred to the granular in-fill below. ’a’ and ’b’ are the inner and outer radius of the annulus respectively. h◦ is the height of the granular in-fill before any loading is applied on the setup.

In order to simplify the analysis procedure, we make certain assumptions based on the available literature on uni-axial compression of granular materials.

  1. During the loading range under consideration, we assume that the granular media behaves in a Linear Elastic manner [8].
  2. Under the considered loading range, we assume that there is no considerable crushing of the particles of the granular media [9].
  3. We ignore any friction between the particles of the granular media and the inner wall of the confining annulus (μ ≈ 0) [8].
  4. We ignore any hoop strains or radial strains that may be generated in the con- fining annulus (εθ ≈ 0) [8].
  5. Material properties for the granular media and confining material have been assumed as per the stress-strain plot shown in Figure 2.
Figure 2: Material Properties for the In-Fill Granular Media and the Material for the Confining Annulus (Egm = Young’s Modulus of the Granular Media, Ecm = Young’s Modulus of the Confining Material, fycm = Yield Strength of the Confining Material, fucm = Ultimate Strength of the Confining Material).

3. Vertical and Angular Stress in the Confining Annulus

The ultimate computational goal of this study is to derive an equation for the yield strength of the composite column. For this, we consider the boundary stress conditions where it is at its maximum, i.e the inner wall at the bottom of the container.

We know that mass of the in-fill granular material will remain constant throughout the compression cycle. We use this to find out the final density of the granular material as a function of the distance the top cap moves down into the annulus. Let that distance be ∆, as shown in Figure 3.

Figure 3: Displacement of the Upper Cap by ∆ when the applied uni-axial force on the top cap is increased from zero to a certain σapp.

Therefore we can say that:

M_{final} = M_{initial} = M \\             \rho_{final}V_{final} = \rho_{initial}V_{initial} = M \\             \rho_{final} = \frac{M}{\pi a^2(h_{\circ} - \Delta)} \label{rhoFinalEq} \hspace{10cm} (1) \linebreak


The vertical stress, σv, at a distance x below the top line can written as,

\sigma_{v}(x) = \sigma_{app} + \frac{Mg(x - \Delta)}{\pi a^2(h_{\circ} - \Delta)},\hspace{0.5cm}\forall x \in [\Delta,h_{\circ}] \hspace{8cm} (2) \linebreak


Where σapp is the uni-axial vertical stress that is applied on the top cap of the composite column. Substituting Equation (1) in (2),

\sigma_{v}(x) = \sigma_{app} + \frac{Mg(x - \Delta)}{\pi a^2(h_{\circ} - \Delta)},\hspace{0.5cm}\forall x \in [\Delta,h_{\circ}] \hspace{10cm} (3) \linebreak


According to the above equation, σv is directly proportional to x, i.e. the depth at which we are calculating the vertical stress. For maximum vertical stress, x = h. Therefore,

\label{eq:sigmaVMaxEq}             \sigma_{v,max} = \sigma_{v}(h_{\circ}) = \sigma_{app} + \frac{Mg}{\pi a^2} \hspace{10cm} (4) \linebreak


Equation (4) gives the maximum vertical stress at the bottom of the container as a function of the applied vertical stress on the top cap of the composite column. Using stress analysis, we can find out the angular stress in the wall of the annulus. Note that this stress will be tensile in nature.


Figure 4: Internal Pressure, pa and External Pressure, pb distribution on the Annulus Wall

Figure 4 shows the pressure distribution on the annulus wall. Using these notations, angular stress, σθ can we written as,

\sigma_{\theta}(r) = \frac{p_{a}a^2 - p_{b}b^2}{b^2-a^2} + \frac{a^2b^2}{r^2}\left(\frac{p_{a} - p_{b}}{b^2-a^2}\right) \label{angularStress}             \text{             } \hspace{10cm} (5) \linebreak


From Equation (5), we observe that σθ is inversely proportional to the radial distance from the center of the annulus. Therefore, the maximum angular stress will be on the inner wall of the annulus, i.e. at r = a.

\sigma_{\theta,max} = \frac{p_{a}(a^2 + b^2) - 2p_{b}b^2}{b^2-a^2} \label{eq:maxAngularStress} \hspace{10cm} (6) \linebreak


Using Equations (4) and (6), we derive the yield strength equation for the composite column as per the various lateral earth pressure theories mentioned in Section 1.

4. Yield Strength of Composite Column using Jaky’s Theorem, Theory of Solid Mechanics and Rankine’s Theory of Lateral Earth Pressure

We have already computed the vertical stress at the bottom of the annulus using the theory of mechanics. But in order to translate this vertical stress into lateral stress, we cannot employ these simple theories as they do not apply to granular materials like sand. Therefore, we use different lateral earth pressure theories given in the literature (Jaky’s Theorem, Rankine’s Earth Pressure Theory). To explore all avenues, we have also formulated the yield strength equation using the theory of solid mechanics. The theory and derivation using each of these theories has been discussed in the subsequent sections.

4.1 Jaky’s Theorem

Jaky’s Theorem states that the ratio of the lateral pressure to vertical pressure, K◦ for cohesionless soils in at-rest condition can be given by the following equation.

K_{\circ} = \frac{\sigma_{h}}{\sigma_{v}} = 1-sin(\phi) \hspace{10cm} (7) \linebreak


Using this we can say that

\sigma_{h} = \{1-sin(\phi)\}\sigma_{v} \hspace{10cm} (8) \linebreak

4.2 Theory of Solids Mechanics

Suppose we assume that the in-fill granular material is linearly elastic, homogeneous, and isotropic in nature. Then we can say that,

\varepsilon_{xx} = \frac{1}{E_{gm}}\left( \sigma_{xx} - \nu_{gm}(\sigma_{yy} + \sigma_{zz}) \right) \hspace{10cm} (9) \\                 \varepsilon_{yy} = \frac{1}{E_{gm}}\left( \sigma_{yy} - \nu_{gm}(\sigma_{zz} + \sigma_{xx}) \right) \hspace{10 cm} (10)  \\                 \varepsilon_{zz} = \frac{1}{E_{gm}}\left( \sigma_{zz} - \nu_{gm}(\sigma_{xx} + \sigma_{yy}) \right) \hspace{10 cm} (11)


Egm = Young’s Modulus of the In-Fill Granular Material, νgm = Poisson’s Ratio of the In-Fill Granular Material

As per the assumptions listed in Section 2, we can say that εxx ≈ 0 and εyy ≈ 0. Due to radial symmetry in loading, we can say that σxx = σyy = σh. Now using this in either of Equation (14) or (15), we can write

\frac{1}{E_{gm}}\left( \sigma_{h} - \nu_{gm}(\sigma_{h} + \sigma_{zz}) \right) = 0 \nonumber\\                 \sigma_{h} = \left\{\frac{\nu_{gm}}{1-\nu_{gm}}\right\}\sigma_{zz} = \left\{\frac{\nu_{gm}}{1-\nu_{gm}}\right\}\sigma_{v} \label{eq:lateralPressureTSM} \hspace{10cm} (12) \linebreak

4.3 Rankine’s Theory of Lateral Earth Pressure

Rankine developed the theory of active and passive earth pressure in 1857 for cohesionless soils [5]. Using this, he gave the coefficients of active and passive earth pressure.

K_{a} = tan^2(45^{\circ} - \phi^{'}/2) \hspace{10cm}(13)  \\ K_{p} = tan^2(45^{\circ} + \phi^{'}/2) \hspace{10cm}(14)


Ka = Coefficient of active earth pressure, Kp = Coefficient of passive earth pressure, φ’ = Effective angle of friction of the granular material

For soils with cohesion, Bell developed an analytical solution using the square roots of the pressure coefficient developed by Rankine, to predict the contribution of the cohesion part of the soil to the overall lateral pressure.

\sigma_{a} = K_{a}\sigma_{v} - 2c^{'}\sqrt{K_{a}} \hspace{10cm}(15) \\ \sigma_{p} = K_{p}\sigma_{v} + 2c^{'}\sqrt{K_{p}} \hspace{10cm} (16)


σv = Vertical stress, σa = Lateral stress in active case, σp = Lateral stress in passive case, c ‘= Effective cohesion of granular material

As per our assumed setup, we can say that the deformation in the annulus will be approximately as shown in Figure 5.


Figure 5: Approximate deformed shape of the composite columns on increasing the σapp. The red curved dotted lines represent the approximate deformed profile of the annulus as the applied load on the top cap of the container is increased.

We can clearly observe that our case is only concerned with the active earth pressure coefficient. So we can write

\sigma_{h} = K_{a}\sigma_{v} - 2c^{'}\sqrt{K_{a}} \hspace{10cm}(17) \linebreak

4.4 Derivation for Yield Strength of Composite Column

We have already derived the maximum vertical stress at the bottom of the composite column in Equation (4). Using Equations (8), (12), (17), we translate that vertical stress into lateral stress on the confining annulus. Then using Equation (6), we can compute the angular stress on the inner wall of the confining annulus. Now in order for the column to not fail, σθ,max ≤ fycm. This equation will have the σapp term. At the critical case, σapp can be written as σapp,max, and this σapp,max = fycc. Using this computation procedure, we find out that the yield strength of the composite column can be given as,

Although we have used three separate theories to formulate the yield strength equations, on close observation, it can be seen that these equations are not that different from each other. The fundamental part of the equation remains same in all three. In the subsequent sections, we combine these three into a single equation by introducing the Soil Characterisation Constant, S.

5. Towards a Combined Yield Strength Equation

5.1 Combined Yield Strength Equation Expression

Observing Equation (18) carefully, we see that in each of these equations, there is dimensionless term that characterises the in-fill granular material. Let us call that constant as the Soil Characterisation Constant, S. Using this constant, we can combine the three equations as,

f_{ycc} = \frac{f_{ycm}(b^2-a^2) + 2p_{b}b^2}{S_{\circ}(a^2 + b^2)} + \frac{2c^{'}}{\sqrt{S_{\circ}}} - \frac{Mg}{\pi a^2} \linebreak


Where

5.2 Understanding the Combined Yield Strength Equation

If we take a closer look at Equation (19), we observe that the equation can be split into four separate parts.

f_{ycc} = \underbrace{\frac{f_{ycm}(b^2-a^2)}{S_{\circ}(a^2 + b^2)}}_\text{Part 1} +  \underbrace{\frac{2p_{b}b^2}{S_{\circ}(a^2 + b^2)}}_\text{Part 2} + \underbrace{\frac{2c^{'}}{\sqrt{S_{\circ}}}}_\text{Part 3} - \underbrace{\frac{Mg}{\pi a^2}}_\text{Part 4}


Each component of the equation adds or subtracts to the yield strength of the composite column

(1)  Part 1: This is the strength that the column gains from the material properties of the confining material (fycm: Yield Strength of the Confining Material) and the geometry of the annulus (a: Inner Radius, b: Outer Radius), and the Soil Characterisation Constant, S◦.

(2)  Part 2: This is the strength that the column gains because of the outer confining pressure. We see that the higher the outer confining pressure is, the higher the strength of the column will be, which also makes sense intuitively.

(3)  Part 3: Here we see the role of cohesion of the in-fill granular media. In the simplest terms, cohesion is the tendency of the particles of a media to stick together. So, the more they stick together, the lower the lateral pressure that they apply on the confining annulus and hence, the higher the yield strength of the composite column will be. Thus, yield strength is directly proportional to the cohesion of the in-fill granular media.

(4)  Part 4: This term is due to the mass of the in-fill granular media. The greater the mass, the higher the applied lateral pressure on the confining annulus, and hence, the lower the strength of the composite column will be. Thus, this term has a negative sign as the higher its value, the weaker the composite column will be.

5.3 Understanding the Soil Characterisation Constant, S

As per Equation (19), we have three separate values of Soil Characterisation Constant, S according to the three theories that we have used for the calculating the yield strength of composite column.

In order to understand the variation of S, we take some arbitrary values of φ and νgm. As per the literature, we assume that the φ value can lie between 25◦ to 45◦ [10] and the νgm value can lie between 0.25 to 0.35 [11]. Using this range, we plot the S◦ values for the three different theories.

From Figure 6, we observe that the S would lie between 0.12 to 0.6 in all practical scenarios. So now we plot the variation of Yield Strength of Composite Column, fycc against the Soil Characterisation Constant, S. From Figure 7, we can say that the yield strength of composite column, fycc is somewhat inversely proportional to the Soil Characterisation Constant, S of the in-fill granular media. Therefore, in order to design composite columns with high yield strength, we require granular in-fill with high angle of friction, φ, and low Poisson’s Ratio, νgm (from Figure 7).


Figure 6: Soil Characterisation Constant, S◦ v/s Angle of Friction, φ and Poisson’s Ratio, νgm of the Granular Material for Jaky’s Theory, Theory of Solid Mechanics and Rankine’s Theory

Figure 7: Yield Strength of Composite Column, fycc v/s Soil Characterisation Constant, S for Steel (fycm =250MPa, a=190mm, M=460kg, h =2600mm, pb =0.101MPa, g=9.81m/s2) and PLA(fycm = 26.082MPa, a=75mm, M=7.6kg, h =270mm, pb =0.101MPa, g=9.81m/s2)

6. Comparing Yield Strength Equation Results from the Different Theories Used


Figure 8: Yield Strength of Composite Column, fycc vs Yield Strength of the Confining Material, fycm (a =75mm, t=10mm, pb =0.101MPa, M=7.6Kg, φ=30◦, ν=0.3, c′=0.020MPa)

Figure 8 shows the relation between fycc and fycm for the different theories that we have used to calculate the yield strength of the composite column. The plot clearly indicates that as we start using stronger materials for the confining material for the composite column, it becomes more and more critical which theory we choose to calculate the yield strength of composite column. The deviation between the yield strength given by different theories increases as the strength of the confining material increases.

One thing to note is that although the absolute difference in the yield strength may be increasing, the percentage increase remains almost the same, i.e. about 16% from Jaky’s Theory to Theory of Solid Mechanics and approximately 29% from Theory of Solid Mechanics to Rankine’s Theory. Figure 9 shows the ratio of the yield strength

Figure 9: Comparing the Yield Strength Results Calculated Using Jaky’s Theory, Theory of Solid Mechanics, and Rankine’s Theory of Lateral Earth Pressure

of the composite column calculated using the different theories under consideration. From the graph we can clearly observe that the ratio is almost a straight line under practical scenarios. Extrapolating the straight line on the y-axis, we see that the ratio between the yield strength using Rankine’s Theory and Theory of Solid Mechanics is approximately 1.28. The same is approximately 1.50 for Rankine’s Theory and Jaky’s Theory and 1.16 for Theory of Solid Mechanics and Jaky’s Theory.

7. Deriving π-constants for the Yield Strength Equation using Buckingham π Theorem

The Buckingham π Theorem provides a method for computing sets of dimensionless parameters from the given variables, even if the form of the equation is unknown. These “variables” are known as repetitive variables and should be chosen in such a manner that they do not form a non-dimensional pair among themselves. Using this theory, we derive the π constants for the combined yield strength equation derived in Section 5.1.

Since S is a dimensionless variable in all the three cases, we remove that from the equation for the Buckingham π analysis.

f_{ycc} = F(f_{ycm},b,a,p_{b},M,g,c^{'}) \linebreak


Now there are eight variables in the above equation: fycc, fycm, b, a, pb, M, g, c . Let us assume 3 variables as the repetitive variable, namely fycm, a, M. Using the Buckingham π Theorem, we get the π constants as:

\pi_{1} = \frac{f_{ycc}}{f_{ycm}}, \hspace{0.4cm} \pi_{2} = \frac{b}{a}, \hspace{0.4cm} \pi_{3} = \frac{p_{b}}{f_{ycm}}, \hspace{0.4cm} \pi_{4} = \frac{Mg}{f_{ycm}a^2}, \hspace{0.4cm} \pi_{5} = \frac{c^{'}}{f_{ycm}}  \hspace{8cm}(20)


If we take a look at π3, we know that in our study, pb = Atmospheric Pressure = 0.101 MPa. And the value of fycm is of the order of tens and maybe hundreds of MPa. Therefore the value of π3 is approximately zero, and hence can be ignored for the purpose of establishing structural similitude. Similarly for π5, we know that the cohesion for granular materials is around 0.02 MPa. Hence the value of π5 is also approximately zero and can also be ignored. Therefore, the primary π constants that will determine the structural similitude between the model and the prototype are:

\boldsymbol{\pi_{1} = \frac{f_{ycc}}{f_{ycm}}, \hspace{0.4cm} \pi_{2} = \frac{b}{a},\hspace{0.4cm} \pi_{4} =\frac{Mg}{f_{ycm}a^2}}

8. Conclusions

The current study focuses on deriving an equation to determine the yield strength of composite columns, fycc with a granular in-fill inside a confining annulus, under a uni- axial compressive load. In order to understand the range of loads for which we need to design the columns, we prepared analytical models of common residential structures which indicate that if our columns can sustain an axial load of 20 MPa or more, they can be a viable replacement for RCC columns.

We have used three separate theories to calculate the conversion ratio of the applied vertical stress on the granular material to the lateral stress on the confining material. Towards the end, we defined a new dimensionless number, Soil Characterisation Constant, S to identify the granular material that we have in-filled inside the annulus and combine the yield strength equations of different theories into one single equation. We have also derived the π constants for the yield strength equation using the Buckingham π theorem.

The theoretical calculations of this study suggest that it is possible to manufacture structural columns for low to moderately loaded buildings using the proposed design. If done successfully, these columns will not only be cheap, but also sustainable and environmentally friendly as they can be made entirely from recycled or natural materials.

9. Future Directions

In the course of this study, we have taken certain assumptions that may not be true on the field. Due to the theoretical nature of these assumptions, the natural next step would be to quantify the final Yield Strength of Composition Column empirically.

Using the π constants that we have derived in Section 5.4, an experimental study can be conducted between the model and the structural prototype of the composite columns to further validate their use in the field.

Once these columns have been validated for use in the field, it can completely transform the way we build low rise affordable housing structures. Given the focus on reducing the transportation needs associated with civil construction, these columns can also be used for the construction of extraterrestrial structures.

Acknowledgements

I would like to thank Prof. Jose E. Andrade for providing me this opportunity to work under his guidance as a Summer Undergraduate Research Fellowship at California Institute of Technology. This article and the research behind it would not have been possible without his constant guidance and support. I would also like to thank Siavash Monfared who constantly helped me in overcoming any challenges that I faced during the course of my research and showed me the path wherever I got stuck. Lastly, I would like to thank all the fellow graduate and undergraduate students who I met during this fellowship. Their advice and direction was also a contributing part of my research work.

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