, Volume 53, Issue 7, pp 1803–1817 | Cite as

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

New Trends in Mechanics of Masonry


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.


Masonry Pier Spandrel Collapse load Crushing 

List of symbols


Width of the ith pier


Height of the ith pier


Width of the spandrel connecting the piers


Height of the spandrel connecting the piers


Thickness of piers, spandrel and lintel


Height of the timber lintel supporting the spandrel


Length of the anchorage of the timber lintel


Specific weight of the masonry material


Weight of the ith pier


Permanent load on the top of the ith pier


Weight of the spandrel connecting the piers


Horizontal force representing the live load


Multiplier of the force F


Collapse multiplier of the predictive model


Height of the application point of the variable load λF


Side of the closed interface between the ith 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 (\(\sigma_{{{\text{c}} \bot }} = 0.85f_{{{\text{c}} \bot }}\), \(\sigma_{{{\text{c}}\parallel }} = 0.85f_{{{\text{c}}\parallel }}\))


Compressive normal force at the base of the ith pier

\(N_{\text{L}} , { }N_{\text{R}}\)

Compressive normal forces at the left and right ends, respectively, of the spandrel


Shear force at the base of the ith 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


Height of the intrados of the timber lintel


Experimental collapse force


Tensile strength of the masonry


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Civil and Mechanical EngineeringUniversity of Cassino and Southern LazioCassinoItaly

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