# Predictive model for the collapse load of masonry assemblage with two piers joined by a spandrel

## Abstract

The masonry assemblage composed of two piers connected by a spandrel can be considered a repetitive unit in large masonry walls with openings, occurring in masonry buildings. In this work, the collapse load of the above-mentioned masonry assemblage is predicted by solving a system of nonlinear equations, where the normal force in the spandrel is a root of an equilibrium equation of fourth degree. Piers and spandrel are assumed rigid and nonlinearity (crushing and no tensile strength) is concentrated at the pier-foundation and pier–spandrel interfaces. The model also takes into account the effect of a timber lintel supporting the spandrel and anchored into the two adjacent piers. This approach valid for assemblages with one spandrel can be extended for the evaluation of the collapse load of structures composed of *N* piers connected by *N* − 1 spandrels. The system of nonlinear equations is easily solved with an iterative method and the collapse load provided by the solution agrees well with the experimental result.

## Keywords

Masonry Pier Spandrel Collapse load Crushing## List of symbols

*B*_{i}Width of the

*i*th pier*H*_{i}Height of the

*i*th pier- \(B_{\text{s}}\)
Width of the spandrel connecting the piers

- \(H_{\text{s}}\)
Height of the spandrel connecting the piers

*t*Thickness of piers, spandrel and lintel

- \(H_{\text{lin}}\)
Height of the timber lintel supporting the spandrel

*a*Length of the anchorage of the timber lintel

- \(\gamma_{\text{m}}\)
Specific weight of the masonry material

*G*_{i}Weight of the

*i*th pier- \(G_{{{\text{t}}i}}\)
Permanent load on the top of the

*i*th pier- \(G_{\text{s}}\)
Weight of the spandrel connecting the piers

*F*Horizontal force representing the live load

*λ*Multiplier of the force

*F*- \(\lambda_{\text{c}}\)
Collapse multiplier of the predictive model

- \(H_{\text{F}}\)
Height of the application point of the variable load

*λF*- ℓ
_{i} Side of the closed interface between the

*i*th pier and the foundation- \(\ell_{\text{L}} , { }\ell_{\text{R}}\)
Sides of the closed interfaces between the spandrel and the left and right piers, respectively

- \(f_{{{\text{c}} \bot }}\)
Compressive strength of the masonry when the compression force is perpendicular to the mortar bed joints

- \(f_{{{\text{c}}\parallel }}\)
Compressive strength of the masonry when the compression force is parallel to the mortar bed joints

- \(\sigma_{{{\text{c}} \bot }} , { }\sigma_{{{\text{c}}\parallel }} ,\)
Average stresses in the contact area between elements (\(\sigma_{{{\text{c}} \bot }} = 0.85f_{{{\text{c}} \bot }}\), \(\sigma_{{{\text{c}}\parallel }} = 0.85f_{{{\text{c}}\parallel }}\))

*N*_{i}Compressive normal force at the base of the

*i*th pier- \(N_{\text{L}} , { }N_{\text{R}}\)
Compressive normal forces at the left and right ends, respectively, of the spandrel

*T*_{i}Shear force at the base of the

*i*th pier- \(T_{\text{L}} , \, T_{\text{R}}\)
Shear forces at the ends of the spandrel

- \(\sigma_{\text{lin}} , { }\tau_{\text{lin}}\)
Normal and tangential stresses at the horizontal contact surfaces between the lintel and the adjacent left pier

*c*,*μ*Cohesion and the friction coefficient, respectively, at the contact surfaces between the lintel and the adjacent left pier

- \(T_{\text{b}} , \, T_{\text{t}}\)
Tangential forces at the contact surfaces \(a_{\text{b}}\) and \(a_{\text{t}}\), respectively

- \(\sigma_{\text{lin,b}} , \, \sigma_{\text{lin,t}}\)
Average normal stresses at the contact surfaces \(a_{\text{b}}\) and \(a_{\text{t}}\), respectively

- \(N_{\text{b}} , \, N_{\text{t}}\)
Normal forces in the transverse sections \(s_{\text{b}}\) and \(s_{\text{t}}\), respectively

- \(M_{\text{b}} , \, M_{\text{t}}\)
Bending moments in the transverse sections \(s_{\text{b}}\) and \(s_{\text{t}}\), respectively

- \(H_{\text{I}}\)
Height of the intrados of the timber lintel

- \(F_{\text{exp,max}}\)
Experimental collapse force

- \(f_{\text{t}}\)
Tensile strength of the masonry

## Notes

## Compliance with ethical standards

## Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Lagomarsino S (2015) Seismic assessment of rocking masonry structures. Bull Earthq Eng 13:97–128CrossRefGoogle Scholar
- 2.Giordano A, De Luca A, Mele E, Romano A (2007) A simple formula for predicting the horizontal capacity of masonry portal frames. Eng Struct 29:2109–2123CrossRefGoogle Scholar
- 3.Milani G, Beyer K, Dazio A (2009) Upper bound limit analysis of meso-mechanical spandrel models for the pushover analysis of 2D masonry frames. Eng Struct 31:2696–2710CrossRefGoogle Scholar
- 4.Roca P, Cervera M, Gariup G, Pela L (2010) Structural analysis of masonry historical constructions. Classical and advanced approaches. Arch Comput Methods Eng 17:299–325CrossRefMATHGoogle Scholar
- 5.Caporale A, Feo L, Luciano R, Penna R (2013) Numerical collapse load of multi-span masonry arch structures with FRP reinforcement. Compos B Eng 54:71–84CrossRefGoogle Scholar
- 6.Mendes N, Lourenço PB (2014) Sensitivity analysis of the seismic performance of existing masonry buildings. Eng Struct 80:137–146CrossRefGoogle Scholar
- 7.Portioli F, Casapulla C, Cascini L (2015) An efficient solution procedure for crushing failure in 3D limit analysis of masonry block structures with non-associative frictional joints. Int J Solids Struct 69–70:252–266CrossRefGoogle Scholar
- 8.Addessi D, Sacco E (2016) Enriched plane state formulation for nonlinear homogenization of in-plane masonry wall. Meccanica 51:2891–2907MathSciNetCrossRefGoogle Scholar
- 9.Augenti N, Parisi F, Prota A, Manfredi G (2011) In-plane lateral response of a full-scale masonry subassemblage with and without an inorganic matrix-grid strengthening system. J Compos Constr 15:578–590CrossRefGoogle Scholar
- 10.Brandonisio G, Mele E, De Luca A (2015) Closed form solution for predicting the horizontal capacity of masonry portal frames through limit analysis and comparison with experimental test results. Eng Fail Anal 55:246–270CrossRefGoogle Scholar
- 11.Parisi F, Lignola GP, Augenti N, Prota A, Manfredi G (2011) Nonlinear behavior of a masonry subassemblage before and after strengthening with inorganic matrix-grid composites. J Compos Constr 15:821–832CrossRefGoogle Scholar
- 12.Saloustros S, Pelà L, Cervera M, Roca P (2017) Finite element modelling of internal and multiple localized cracks. Comput Mech 59:299–316CrossRefMATHGoogle Scholar
- 13.Beyer K (2012) Peak and residual strengths of brick masonry spandrels. Eng Struct 41:533–547CrossRefGoogle Scholar
- 14.Vanin A, Foraboschi P (2012) In-plane behavior of perforated brick masonry walls. Mater Struct Mater Constr 45:1019–1034CrossRefGoogle Scholar
- 15.Italian Building Code - In Italian: Norme tecniche per le costruzioni (NTC2008) - D.M. 14 Gennaio 2008Google Scholar
- 16.Caporale A, Luciano R (2012) Limit analysis of masonry arches with finite compressive strength and externally bonded reinforcement. Compos B Eng 43:3131–3145CrossRefGoogle Scholar
- 17.Augenti N, Parisi F (2011) Constitutive modelling of tuff masonry in direct shear. Constr Build Mater 25:1612–1620CrossRefGoogle Scholar