# A New Proposal for the Interpretation of the Diagonal Compression Test on Masonry Wallettes: The Identification of Young’s Modulus, Poisson’s Ratio, and Modulus of Rigidity

## Abstract

**:**

## 1. Introduction

## 2. Problem Setting

## 3. Problem Solving in Parametric Form

#### 3.1. Stress State at Point $A$

**Theorem 1.**

**Theorem 2.**

#### 3.2. Strain State at Point $A$

**Theorem 3.**

- ${\epsilon}_{i}={u}_{i}$, where ${u}_{i}$ is the displacement component of point $P$ in the direction of the $i$-axis (normal component of the displacement, or normal displacement);
- $1/2{\gamma}_{ij}={u}_{j}$, where ${u}_{j}$ is the displacement component of point $P$ in the direction of the $j$-axis (tangential component of the displacement, or tangential displacement).

#### 3.3. Elastic Coefficients

#### 3.4. Limiting Values of the Parameter $k$

## 4. How to Identify the Coefficient $\mathit{k}$ and Obtain the Solution

## 5. The Elastic Coefficients Obtained for a Real Set of Experimental Data

- $\u2206H$ is the extension along the horizontal diagonal;
- $\u2206V$ is the shortening along the vertical diagonal (the compressed diagonal);
- $g$ is the gage length in the direction of both diagonals (the gage length for the identification of $\u2206V$ must be equal to the gage length for the identification of $\u2206H$ [9]).

## 6. Conclusions

- 40% of the ASTM tensile strength;
- 57% of the RILEM tensile strength.

## 7. Future Developments

## Supplementary Materials

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Meaning of the Direction Cosines in Equation (68)

- ${a}_{h}$, namely the cosine of the angle between the positive coordinate axis $s$ and the positive coordinate axis $h$:$${a}_{h}=\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta \right),$$
- ${a}_{v}$, namely the cosine of the angle between the positive coordinate axis $s$ and the positive coordinate axis $v$:$${a}_{v}=\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\pi}{2}-\beta \right)=\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta \right),$$
- ${a}_{z}$, namely the cosine of the angle between the positive coordinate axis $s$ and the positive coordinate axis $z$:$${a}_{z}=\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\pi}{2}\right)=0,$$
- ${b}_{h}$, namely the cosine of the angle between the positive coordinate axis $t$ and the positive coordinate axis $h$:$${b}_{h}=\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\pi}{2}+\beta \right)=-\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta \right),$$
- ${b}_{v}$, namely the cosine of the angle between the positive coordinate axis $t$ and the positive coordinate axis $v$:$${b}_{v}=\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta \right),$$
- ${b}_{z}$ namely the cosine of the angle between the positive coordinate axis $t$ and the positive coordinate axis $z$:$${b}_{z}=\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\pi}{2}\right)=0,$$
- ${c}_{h}$, namely the cosine of the angle between the positive coordinate axis $z$ and the positive coordinate axis $h$:$${c}_{h}=\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\pi}{2}\right)=0,$$
- ${c}_{v}$, namely the cosine of the angle between the positive coordinate axis $z$ and the positive coordinate axis $v$:$${c}_{v}=\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\pi}{2}\right)=0,$$
- ${c}_{z}$, namely the cosine of the angle between the positive coordinate axis $z$ and the positive coordinate axis $z$:$${c}_{z}=\mathrm{c}\mathrm{o}\mathrm{s}\left(0\right)=1.$$

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**Figure 1.**Stresses acting in the infinitesimal neighborhood of the center of gravity, $A$, of the masonry wallette.

**Figure 2.**Stress state at point $A$ in the modified Mohr plane (with normalized axes), according to the RILEM interpretation of the diagonal compression test.

**Figure 3.**Stress state for the infinitesimal neighborhood of point $A$ in: (

**a**) the reference frame of Figure 1; (

**b**) the modified Mohr plane of origin $B$, according to the RILEM interpretation of the diagonal compression test.

**Figure 5.**Rotation angle of the trace with positive pure shear stress, ${+\tau}_{0}$, in: (

**a**) the reference frame of Figure 4; (

**b**) the Mohr circle for the stress state.

**Figure 6.**Similarity between the right-angled triangle with vertices at the coordinate points $\left({\widehat{\sigma}}_{I},0\right)$, $\left(\mathrm{0,0}\right)$, and $\left(0,{\widehat{\tau}}_{0}\right)$ and the right-angled triangle with vertices at the coordinate points $\left(\mathrm{0,0}\right)$, $\left({\widehat{\sigma}}_{II},0\right)$, and $\left(0,{\widehat{\tau}}_{0}\right)$.

**Figure 7.**Stress state of an elementary cube with faces parallel and orthogonal to the trace with positive pure shear stress, in: (

**a**) the reference frame with origin in $A$; (

**b**) the modified Mohr plane.

**Figure 8.**Relationship between the rotation angles around: (

**a**) the origin $A$ of the reference frame in Figure 7a; (

**b**) the center $C$ of the Mohr circle for the strain state.

**Figure 9.**The Mohr circles in the ASTM, RILEM, and new interpretations of the diagonal compression test.

**Figure 10.**Plot of the normalized pure shear stress, ${\widehat{\tau}}_{0}$, as a function of the parameter $k$.

**Figure 11.**Plot of the two independent elastic moduli as a function of the parameter $k$: (

**a**) Young’s modulus, $E$; (

**b**) Poisson ratio, $\nu $.

**Figure 12.**Detail of the relationships between the elastic moduli and the parameter $k$, in the range $7\le k\le 12$: (

**a**) Young’s modulus, $E$, and modulus of rigidity, $G$; (

**b**) Poisson ratio, $\nu $.

**Figure 13.**Shear stress–shear strain curves for the ASTM, RILEM, and new interpretations of the diagonal compression test ($x/y$ plane).

**Figure 14.**Shear stress–shear strain curves for the new interpretations of the diagonal compression test, in the $x/y$ and $s/t$ planes.

**Figure 15.**Stress–strain relationships along the principal directions of stress and strain, for: (

**a**) the RILEM interpretation of the diagonal compression test; (

**b**) the new interpretation of the diagonal compression test.

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**MDPI and ACS Style**

Ferretti, E.
A New Proposal for the Interpretation of the Diagonal Compression Test on Masonry Wallettes: The Identification of Young’s Modulus, Poisson’s Ratio, and Modulus of Rigidity. *Buildings* **2024**, *14*, 104.
https://doi.org/10.3390/buildings14010104

**AMA Style**

Ferretti E.
A New Proposal for the Interpretation of the Diagonal Compression Test on Masonry Wallettes: The Identification of Young’s Modulus, Poisson’s Ratio, and Modulus of Rigidity. *Buildings*. 2024; 14(1):104.
https://doi.org/10.3390/buildings14010104

**Chicago/Turabian Style**

Ferretti, Elena.
2024. "A New Proposal for the Interpretation of the Diagonal Compression Test on Masonry Wallettes: The Identification of Young’s Modulus, Poisson’s Ratio, and Modulus of Rigidity" *Buildings* 14, no. 1: 104.
https://doi.org/10.3390/buildings14010104