Influence of self-weight on size effect of quasi-brittle materials: generalized analytical formulation and application to the failure of irregular masonry arches

Abstract

Up to the beginning of the twenty-first century, most of quasi-brittle structures, in particular the ones composed by concrete or masonry frames and walls, were designed and built according to codes that totally ignored fracture mechanics theory. The structural load capacity predicted by strength-based theories, such as plastic analysis and limit analysis, do not exhibit size-effect. Within the framework of fracture mechanics theory, this paper deals with the analysis of the effect of non proportional loadings on the strength reduction with the structural scaling. In particular, this study investigates the size-effect of quasi-brittle materials subjected to self-weight. Although omnipresent, gravity-load is often considered negligible in most studies in the field of fracture mechanics. This assumption is obviously not valid for large structures and in particular for geometries in which the dead load is a major driving force leading to fracture and structural failure. In this study, an analytical formulation expressing the relation between the strength-reduction and the structural scaling and accounting for self-weight, was derived for both notched and unnotched bodies. More specifically, a closed form expression for size and self-weight effects was first derived for notched specimens from equivalent linear elastic fracture mechanics. Next, equivalent linear elastic fracture mechanics theory being not applicable to unnotched bodies, a cohesive model formulation was considered. Particularly, the cohesive size effect curve and the generalized cohesive size effect curves, originally obtained via cohesive crack analysis for weightless bodies with sharp and blunt/unnotched notches, respectively, were equipped of an additional term to account for the effect of gravity. All the resulting formulas were compared with the predictions of numerical simulation resulting from the adoption of the Lattice Discrete Particle Model. The results point out that the analytical formulas match very well the results of the numerical model for both notched and unnotched samples. Furthermore, the analytical formulas predict a vertical asymptote for increasing size, in the typical double-logarithm strength versus structural size representation. The asymptote corresponds to a characteristic size at which the structure fails under its own weight. For large structural sizes approaching this characteristic size, the newly developed formulas deviate significantly from previously proposed size-effect formulas. The practical relevance of this finding was demonstrated by analyzing size and self-weight effect for several quasi-brittle materials such as concrete, wood, limestone and carbon composites. Most importantly, the proposed formulas were applied to the failure of semi-circular masonry arches under spreading supports with different slenderness ratios. Results show that analytical formulas well predict numerical simulations and, above all, that for vaulted structures it is mandatory accounting for the effect of self-weight.

Appendices

Appendix A: explicit analytical formulations for three-point bending tests

In this section, the expressions of SELF, CSELF, and GSELF are given explicitly for the case of simply supported prismatic three-point bending beams of height D, span S, width b, and notch length a. Using the notations presented earlier, one can define $c_{N1} = c_{N2} = (3/2)(S/D)$ from elasticity theory. The nominal stress and strength corresponding to the loads $P_{1}$ and $P_{u2}$, respectively, can be thus written as $\sigma _{N1}=(3P_{1}S)/(2bD^{2})$ and $\sigma _{Nu2}=(3P_{u2}S)/(2bD^{2})$. The beam weight is approximated by the point load $P_1$ acting at mid-span. Its magnitude can be calculated in way to produce the same central bending moment as the one generated by the dead-load. In the case of a simply supported beam, one has $P_1 = m_{beam}g/2$ where $m_{beam}$ is the mass of the beam. In terms of density $\rho$, $P_1 = \rho g bDS/2$. Therefore, the corresponding nominal stress can be written as $\sigma _{N1} = Df(\rho g)$ where $f(\rho g) = (S/D)^2(3\rho g/4)$. For a beam of span to depth ratio of 4, one obtains $\sigma _{N1}=12\rho gD$. The load-point displacements $u_1$ and $u_2$ associated to the loads $P_1$ and $P_2$, respectively, belong to the same nature since their corresponding geometry and boundary conditions are identical. Thus, the two dimensionless energy release functions are equal, i.e. $g_1 = g_2$. For three-point bending of span to depth ratio of 4 (Tada 1985): $g_1(\alpha ) = g_2(\alpha )=\alpha \left[ \Big (1.99-\alpha (1-\alpha )( 2.15-3.93\alpha +2.7\alpha ^2) \Big ) / \Big ( (1 \right.$$\left. +2\alpha )(1-\alpha )^{3/2} \Big ) \right] ^2$. In turns, the term related to gravity in Eqs. 7, 12, and 14 simplify. For a linear cohesive law, the characteristic length $\ell _{ch} = \ell _1$ and it is that notation that is adopted hereinafter. The SELF formula, valid for notched three point bending beams including the effect of gravity, assumes the following expression

$$\begin{aligned} \sigma _{Nu2}=-12\rho gD+\frac{\sqrt{\frac{G_fE}{c_fg'_{20}}}}{\sqrt{1+\frac{D}{D_{02}}}} \end{aligned}$$

(15)

The SEL is obtained by omitting the first term that carries the contribution of self-weight.

Fig. 15

a Irregular stone-aggregate distribution in a masonry arched structure; b Delaunay tetrahedralization procedure connecting the centers of the spherical particles and defining a lattice system; c two adjacent LDPM polyhedral cells enclosing the associated stone-aggregate pieces; d tetrahedron portion associated with a stone-aggregate and a triangular LDPM facet

The CSELF expression conjectured to be adequate for describing sharp notched beams including the effect of gravity writes as

$$\begin{aligned} \sigma _{Nu2}= -12D\rho g + f_t'\left( g_0\frac{D}{l_{1}}+ g'_0\frac{1+11\left( \frac{g_0D}{g'_0l_{1}} \right) ^n}{\beta _p^2g'_0+25\left( \frac{g_0D}{g'_0l_{1}} \right) ^n} \right) ^{-\frac{1}{2}}\nonumber \\ \end{aligned}$$

(16)

where $n=0.45$ (Di Luzio and Cusatis 2018). For the considered geometry, the elastic limit can be written as $\beta _e f_t' = (1-\alpha )^2(1 - 0.1773(D/S))^{-1} f_t'$ (Di Luzio and Cusatis 2018). The CSEC (Cusatis and Schauffert 2009) is recovered for weightless beams.

Last but not least, the GSELF formula, expected to capture the case of sharp, blunt and unnotched beams with weight simplifies as it follows:

$$\begin{aligned} \small \sigma _{Nu2}= & {} -12D\rho g + f_t'\left( g_{20}\frac{D}{l_{1}}{+} g'_{20}\frac{1{+}11\left( \frac{g_{20}D}{g'_{20}l_{1}} \right) ^n}{\beta _p^2g'_{20}{+}25\left( \frac{g_{20}D}{g'_{20}l_{1}} \right) ^n} \right) ^{-\frac{1}{2}}\nonumber \\{} & {} \quad \left( g_{20}\beta _e^2\frac{D}{l_{1}} +\frac{1}{\left( 1+ g_{20}\beta _e^2\frac{D}{l_{1}} \right) ^m}\right) ^{\frac{1}{2}} \end{aligned}$$

(17)

where $m=0.25$ (Di Luzio and Cusatis 2018). The plastic limit for a notched simply supported beam reads as $\beta _p f_t' = 3(1-\alpha )^2 f_t'$ (Di Luzio and Cusatis 2018). Similarly, the GCSEC (Di Luzio and Cusatis 2018) is obtained by omitting the weight of the beam.

Appendix B: the lattice discrete particle model

The Lattice Discrete Particle Model (LDPM) has been originally proposed by Cusatis et al. (2011a, 2011c) to simulate the mechanical interaction interactions among major material heterogeneities, i.e. coarse aggregates in concrete. Over the years, this model has been used to simulate other granular quasi-brittle materials such as mortar (Pathirage et al. 2019a; Han et al. 2020), fiber reinforced concrete and engineered cementitious composites (Schauffert and Cusatis 2011; Rezakhani et al. 2021; Feng et al. 2022), cycling (Zhu et al. 2022) or size-effect in concrete (Pathirage et al. 2023a). LDPM was also coupled to multi-physics models describing cement hydration from microscale simulations, heat transfer and moisture diffusion, alkali silica reaction, creep, aging (Alnaggar et al. 2013; Pathirage et al. 2019b; Yang et al. 2021, 2022; Pathirage et al. 2023b), or self-healing in concrete (Cibelli et al. 2022). More recently, LDPM was combined with machine learning for model feature selection and fast parameter identification (Lyu et al. 2023).

LDPM allows the characterization of irregular masonry as two-phase material, i.e. stone-aggregate and mortar. The potential failure is assumed to occur at the aggregate-mortar interface or within the mortar layer, which is consistent with typical experimental observations on irregular masonry.

The geometrical meso-structure of masonry is obtained through the following steps: (i) stone-aggregate pieces are assumed to be particles: they are randomly placed within the specimen volume through a trial and error procedure, from the largest to the smallest size. The particles follow a particle size distribution function which is defined from a set of mix-design parameters (cement content c, water-to-cement ratio w/c, maximum aggregate size $d_a$, minimum aggregate size $d_0$ and Fuller coefficient $n_f$). Figure 15a shows an example of particle placement inside the volume of a masonry arched structure; (ii) zero-radius particles are randomly placed on the external surface of the sample for the application of the boundary conditions; (iii) a Delaunay tetrahedralization procedure connects the centers of the spherical particles (or nodes), defining a lattice system (see Fig. 15b); (iv) a three-dimensional domain tessellation is then performed, resulting in a system of polyhedral cells, each of which encloses a particle (Fig. 15c). The polyhedral cells form a network of triangular facets that are assumed to be the potential material failure location (Fig. 15c, d). Three sets of equations are written to complete the discrete model framework: definition of strains at each triangular facet, constitutive equations which relate facet strain vector with facet stress vector, and particle equilibrium equations. The constitutive equations describe a softening behavior for pure tension and shear-tension and a plastic hardening behavior for pure compression and shear-compression.

If ${\textbf{x}}_{i}$ and ${\textbf{x}}_{j}$ are the positions of nodes i and j, adjacent to the facet k, the facet strains are defined as:

$$\begin{aligned} {\textbf{e}}_k=[e_{N}\ e_{M}\ e_{L}]^{T}=\Bigg [ \frac{{\textbf{n}}^{T}_{k}[\![ {\textbf{u}}_{k}]\!]}{l}\ \frac{{\textbf{m}}^{T}_{k}[\![ {\textbf{u}}_{k}]\!]}{l}\ \frac{{\textbf{l}}^{T}_{k}[\![ {\textbf{u}}_{k}]\!]}{l}\Bigg ] ^{T}\nonumber \\ \end{aligned}$$

(B1)

where $e_{N}$ is the normal strain component, and $e_{M}$, $e_{L}$ are the tangential strain components, $[\![ {\textbf{u}}_{k}]\!]={\textbf{u}}_{j}-{\textbf{u}}_{i}$ is the displacement jump at the centroid of the facet k, $l=\Vert {\textbf{x}}_{j}-{\textbf{x}}_{i}\Vert _{2}$ is the distance between the two nodes, ${\textbf{n}}_{k}=({\textbf{x}}_{j}-{\textbf{x}}_{i})/l$ and ${\textbf{m}}_{k}$, ${\textbf{l}}_{k}$ are two unit vectors mutually orthogonal in the facet plane projected orthogonally to the line connecting the adjacent nodes Fig. 1g. It was demonstrated (Cusatis and Schauffert 2010; Cusatis and Zhou 2013) that this definition of strains is completely consistent with classical strain definitions in continuum mechanics.

Similarly, one can define the traction vector as ${\textbf{t}}_{k}=[t_{N}\ t_{M}\ t_{L}]^{T}$, where $t_{N}$ is the normal component, $t_{M}$ and $t_{L}$ are the shear components. For the sake of readability, the subscript k that designates the facet is dropped in the following equations. In order to describe the behavior of the material, one needs to introduce the constitutive equations imposed at the centroid of each facet. The elastic behavior is defined through linear relations between the normal and shear stresses, and the corresponding strains as $t_{N}=E_N e_{N}$, $t_{M}=E_T e_{M}$ and $t_{L}=E_T e_{L}$, $E_N=E_0$ and $E_T=\alpha E_0$, $E_0 \approx E/(1-2 \nu )$ and $\alpha \approx (1-4\nu )/(1+\nu )$ are the effective normal modulus and the shear-normal coupling parameter, respectively, and E is the macroscopic Young’s modulus and $\nu$ is the macroscopic Poisson’s ratio of the masonry.

In order to describe the inelastic behavior, one needs to distinguish three sets of mechanisms.

The first mechanism is the fracturing and cohesive behavior under tension and tension/shear occurring for $e_{N}>0$. One can define the effective strain as $e=( e_{N}^{2}+\alpha (e_{M}^{2}+e_{L}^{2}))^{\frac{1}{2}}$, and the effective stress as $t=( t_{N}^2+(t_{M}^2+t_{L}^2)/\alpha )^{\frac{1}{2}}$ and write the relationship between stresses and strains through damage-type constitutive equations as $t_{N}=t e_{N}/e$, $t_{M}=\alpha t e_{M}/e$ and $t_{L}=\alpha t e_{L}/e$.

The effective stress t is defined incrementally as ${\dot{t}}=E_N {\dot{e}}$ and its magnitude is limited by a strain-dependent boundary $0\leqslant t\leqslant \sigma _{bt}(e,\omega )$ in which $\sigma _{bt}(e, \omega )~=~\sigma _0(\omega ) \exp \left[ -H_0(\omega ) \langle e_{\max }-e_0(\omega )\rangle /\sigma _0(\omega ) \right]$, $\langle x\rangle =\max (x,0)$, $\omega$ is a variable defining the degree of interaction between shear and normal loading defined as $\tan (\omega )=(e_{N})/(\sqrt{\alpha } e_{T})=(t_N \sqrt{\alpha })/(t_{T})$; $e_{T}$ is the total shear strain defined as $e_{T}=( e_{M}^{2} + e_{L}^{2})^{\frac{1}{2}}$, and $t_{T}$ is the total shear stress defined as $t_{T}=( t_{M}^{2} + t_{L}^{2})^{\frac{1}{2}}$.

The maximum effective strain is time dependent and is defined as $e_{\max }(\tau )=( e_{N,\max }^{2}(\tau )+\alpha e_{T,\max }^{2}(\tau ))^{\frac{1}{2}}$, where $\displaystyle e_{N,\max }(\tau )=\max _{\tau '<\tau }[e_{N}(\tau ')]$ and $\displaystyle e_{T,\max }(\tau )=\max _{\tau '<\tau }[e_{T}(\tau ')]$. The strength limit of the effective stress that defines the transition between pure tension and pure shear is

$$\begin{aligned} \sigma _{0}(\omega )=\sigma _{t}\frac{-\sin (\omega )+\sqrt{\sin ^2(\omega )+4 \alpha \cos ^2(\omega ) / r_{st}^2}}{2 \alpha \cos ^2(\omega ) / r_{st}^2}\nonumber \\ \end{aligned}$$

(B2)

where $r_{st}=\sigma _{s}/\sigma _{t}$ is the shear to tensile strength ratio, $\sigma _{s}$ is the shear strength and $\sigma _{t}$ is the tensile strength. The post-peak softening modulus is controlled by the effective softening modulus $H_0(\omega )= H_s/\alpha + (H_t-H_s/\alpha ) \left( 2 \omega /\pi \right) ^{n_t}$, in which $H_t= 2E_0/(l_t/l-1)$, $H_s=r_sE_0$ and $n_t$ is the softening exponent; $l_t$ is the tensile characteristic length defined as $l_t={2E_0G_t}/{\sigma _t^2}$, $G_t$ is the mesoscale fracture energy.

The second set of equations describes the mechanism behind pore collapse and material compaction $e_{N}<0$ under high confining pressures. The strain-hardening behavior in compression is simulated with the following strain-dependent boundary ${\dot{t}}_N~=~E_N{\dot{e}}_N$ and $-\sigma _{bc}(e_{D},e_{V})\leqslant ~t_N~\leqslant ~0$, where $\sigma _{bc} = \sigma _{c0} + H_c(-e_V -e_{c0})$ if $-e_V\leqslant e_{c1}$, otherwise $\sigma _{bc} = \sigma _{c1}$exp$[ (-e_V -e_{c1})H_c/\sigma _{c1}]$ and $H_c = H_{c1}+(H_{c0}-H_{c1})/(1+\kappa _{c2}(r_{DV}-\kappa _{c1}))$, $\sigma _{c1} = \sigma _{c0}+ H_c(e_{c1}-e_{c0})$, $e_{c1}=\kappa _{c0}e_{c0}$, $e_{c0}=\sigma _{c0}/E_0$, $e_{V}=(V-V_0)/V_0$ is the volumetric strain computed at the LDPM tetrahedral level, $e_{D}=e_{N}-e_{V}$, $r_{DV}=|{e_{D}}|/(e_{V0}-e_{V})$, for $e_D \leqslant 0$ and $r_{DV}=|{e_{D}}|/(e_{V0})$, for $e_D > 0$, $e_{V0}=0.1e_{c0}$, $\sigma _{c0}$ is the meso-scale yielding compressive stress, $H_{c0}$ is the initial hardening modulus, and $\kappa _{c0}$, $\kappa _{c1}$, $\kappa _{c2}$ are parameters governing the triaxial behavior at very high confinement.

The third failure type considered in LDPM describes the frictional behavior. In the presence of compressive stresses, the shear strength increases due to frictional effects. The frictional behavior is computed using a nonlinear Mohr–Coulomb model in which the internal friction coefficient varies from an initial value $\mu _0$ to zero with the following formulation:

$$\begin{aligned} \sigma _{bs}(t_N)=\sigma _s + \mu _0\sigma _{N0} - \mu _0\sigma _{N0} \text {exp}(t_{N}/\sigma _{N0}) \end{aligned}$$

(B3)

where $\sigma _{s}$ is the cohesion and $\sigma _{N0}$ is the so-called transitional stress.

Finally, the governing equations are completed by writing the equilibrium equations of each LDPM cell:

$$\begin{aligned} \sum _{k \in {\mathcal {F}}_I} A_k{\textbf{t}}_{k}+V_I{\textbf{b}}={\textbf{0}}, \ \ \ \sum _{k \in {\mathcal {F}}_I} A_k{\textbf{c}}_{k}\times {\textbf{t}}_{k}~=~{\textbf{0}} \end{aligned}$$

(B4)

where ${\mathcal {F}}_I$ is the set containing all the facets of a generic polyhedral cell I, $A_k$ is the area of the facet k, ${\textbf{c}}_{k}$ is the vector representing the distance between the center of the facet k and the center of the cell, $V_I$ is the cell volume and ${\textbf{b}}$ is the external body force applied to the cell.

In the simulation presented in the paper, the following LDPM mesoscale parameters are used (Cusatis et al. 2011b; Mercuri et al. 2020; Angiolilli et al. 2020):