A novel graphical assessment approach for compressed curved structures under vertical loading

Abstract

Geometry forms the basis for the study of equilibrium of masonry structures. The dependence of the stability of masonry structures on geometry rather than on material strength was well known to the ancient builders who, over time, empirically worked out rules of proportion for good structural building design. We propose a new graphical approach to assess, the equilibrium in compression for planar and non-planer arches and curves according to the Safe Theorem of Limit Analysis. The approach extends an existing two-dimensional funicular polygon strategy into three dimensions for the first time. The method finds robust applications in the study of masonry flying staircases made of monolithic blocks, giving an alternative for equilibrium to the pure torsional solution. The advantage of the presented approach is that it can be solved solely graphically. The approach is demonstrated by assessing the equilibrium of the main flying or open-well staircase in the Palazzo delle Poste (Trapani, 1924, designed and constructed by Francesco La Grassa), an expressive yet poorly understood structure.

Keywords

Structural mechanics equilibrium analysis curved structures no-tension material limit analysis safe theorem graphic statics vault masonry stairs

Introduction

For masonry structures, geometry is key to their equilibrium. Ancient master builders¹ developed masonry design rules that relied on geometry, not on material strength.

This reliance on geometry is due to the unreliability of the tensile strength of traditional masonry materials. In building codes, masonry structures are designed in such a way to resist all external loading through compressive internal stresses.

For an historical approach to the structural analysis of masonry constructions, the reader is referred to Huerta.^2–4 The question of what specific unilateral model should be adopted to model masonry is still a matter of debate. Its definition and development have taken centuries to emerge from the first insights of 17th and 18th century geometers (Hooke⁵; de La Hire⁶; Couplet⁷; Danyzy⁸; Frézier⁹; Mascheroni¹⁰), to the systemization of the theory of masonry and its placement into the framework of Limit Analysis in the 1960s by Heyman.¹¹

Heyman^12,13 made a significant contribution to the development of Limit Analysis for ductile materials such as steel. Steel has a limited strength capacity and can resist a limited range of internal forces. When its fracture stress is reached, steel does not fail suddenly and is able to maintain “indefinitely” these limit values. Technical regulations and standards (e.g. Ministero delle Infrastrutture e Trasporti¹⁴) encourage the use of ductile materials in construction. Ductility in masonry refers to its ability to adjust its geometry in response to changes in boundary conditions, such as settlements, or to dynamic pulse, by utilizing its infinite compression strength and neglecting its tensile strength. This enables the masonry structure to withstand a certain range of internal forces.¹⁵ Assuming that the masonry construction prevents slippage between the brick or block elements, researchers^16–26 have shown that for such a material, two Limit Analysis Theorems are valid to establish the stability of a structure. In other words, these theorems can be used to assess whether a structure can be maintained in equilibrium close to its design geometry. The inability to maintain equilibrium in the vicinity of the design geometry results in collapse. The values of external loads that produce the transition from a state of equilibrium to a state of non-equilibrium, are called collapse loads. In particular, the First Theorem (Safe Theorem) asserts that if there is a possible equilibrium solution compatible with the limited strength of the material (for Heyman’s unilateral model, an equilibrium state of pure compression), then collapse is not possible, and the structure is stable. This raises the question what the level of safety is for a defined structural geometry and a set of the external loads. Different strategies to measure the degree of stability of a unilateral structure have been proposed such as the geometrical safety factor criterion.²⁷

In this paper, a novel graphical approach is presented to analyze the stability of spatial masonry curved structural systems with the Safe Theorem. The method is applied and demonstrated to a flying or open-well masonry staircase (also called Roman-style staircase,²⁸ or, with the incorrect term, cantilevered stairs). The example of open-well staircases is very relevant today because they are emblematic of the complex relationship between contemporary structural engineering on the one hand, and structural restoration and rehabilitation of historic masonry buildings on the other.^29,30

Several challenges arise when assessing the structural stability of such staircases today.³¹ First, most contemporary analysis methods (such as finite element analysis) were developed for contemporary structural systems (such as concrete or steel beam column systems), whose behavior is very different from that of traditional masonry structures. Adapting this methodology to traditional masonry constructions requires a very complex input data set in terms of 3d geometry and boundary conditions. These data are challenging to define properly because of the uncertainties about the real restraint conditions at the surrounding wall.³² Second, technical Standards and Building Codes address masonry structures composed or recti-linear and planar elements such as columns and walls. They do not cover spatially curved systems like the flying staircase. Third, contemporary civil engineering education does not include the transmission of knowledge of ancient building techniques and prefers analytical and abstract methods of approach based on calculus and symbolic analysis, not graphical approaches such as Graphic Statics that better fit with the modern structural materials such as reinforced concrete, steel, and timber. The remainder of this paper is arranged as follows. In section 2, the main constructive and geometrical typologies of flying staircases structures are briefly described, and a particular case study which will serve to demonstrate the graphical method of analysis is presented in detail. In section 3, the ideas behind the new-old strategy to assess the equilibrium of masonry structures are briefly presented, while in section 4, the methodology of the proposed approach and its application to the case study of the open-well staircase in the Palazzo delle Poste (Trapani, 1924) are shown.

Flying or open-well stair structures

Flying or open-well stairs can be categorized in four different types from a structural perspective (see Figure 1). Types A and B refer to stairs composed of monolithic steps formed by one or two pieces, respectively. Type C, a typical so-called Roman stair, is an open-well stair made of vaults. Type D is a tile vault stair. The approach presented in this paper can be used to evaluate these four different types of flying stairs.

Figure 1.

Main types of open-well stairs: (a) one-piece monolithic steps (spiral stair of the convent of San Domingos de Bonaval) representing Type A; (b) two-piece monolithic steps (early 20th century house in the City Center of Cagliari, courtesy of Matteo Lai) representing Type B; (c) masonry vaults with bricks arranged in a stretcher bond (19th century house in Naples) representing Type C; (d) monolithic steps arranged in a flat bond (or “by quarter”) Guastavino helicoidal tile stair, First Church of Christ Scientist, New York City, New York, ca. 1900, representing Type D.

According to Heyman’s model,¹¹ the elements forming the masonry interact through unilateral contact. In types A and B, the elements are large monolithic pieces that form the steps. In types C and D, the elements are the bricks making up the masonry construction. To overcome the challenge of modeling the interaction between such a large number of elements that compose the structure, we adopt a continuous model. The analysis presented is applied to open-well stairs of Type A. In the Conclusions section, we discuss how the method can be adapted for type B, C, and D flying staircases.

Open-well staircases of Type A are a structural type that has evolved extensively over the centuries. Their geometry in plan and elevation and the step shape can vary widely.

For an in-depth study of the construction technologies of the original type of these staircases, the reader is referred to the manuals of Rondelet³³ and Breymann.³⁴

Many characteristic examples of this kind of stairs can be found in Southern Italy. In particular, the case study of the main staircase in the Palazzo delle Poste (Trapani, 1924, designed and constructed by Francesco La Grassa) will be analyzed in detail to illustrate our graphical approach.

The case study

The Palazzo delle Poste flying staircase has steps made of solid stone (or more specifically pietra palazzo, a compact limestone, also known as pietra rosone or pietra misca) and is surrounded by masonry walls made of tuffaceous ashlars from Favignana (Figure 2).

Figure 2.

Pictures of the main staircase of the Palazzo delle Poste (Trapani, 1924): (a and b) down-top view of the stair; (c) close-up showing the stereotomy and technological details of the steps. (courtesy of Scalvedi and La Grassa³⁵).

The main staircase serves three storeys with three ramps between two storeys and is set into a rectangular wall box (see Figure 3(a)). The two parallel ramps have an overhang of 1.70 m, and reach two intermediate landings, made from the juxtaposition of two large stone slabs with rounded corners, which include the smaller transverse ramp, having also an overhang of 1.70 m. The single step, although having a serial shape, has a curvilinear outline apt to contain the rebate of the step above, with a thickness of the contact area that reaches 6 cm, thus considerably greater than the torus visible along the lateral edge (see Figure 2(c)).

Figure 3.

Main staircase of the Palazzo delle Poste (Trapani, 1924): (a) plan section and (b) view of the i-th step with the local reference system.

This example, similar to the one considered by Heyman,²⁷ contains all the elements necessary to demonstrate our graphical equilibrium analysis and can be adapted for more complex case studies. Each of the steps shown (Figure 3(a)) is made of a monolithic block of dimensions $L x b x h$ (Figure 3(b)); at one end—the outer end—each step is embedded into the wall, while at the other end—the inner end—is free. The riser of the previous step rests on the back edge of the lower tread; and so on for the subsequent steps. The step at the lower end of the ramp (Figure 2(c)) rests on the ground. In practice, tread connections can be established through the process of notching. This involves cutting the undersurface (soffit) of each tread to create a smooth surface and reduce the overall weight of the staircase. Notching is a critical aspect of staircase design and construction, as it eliminates the potential for infinite stresses that could be generated by linear contact between treads. The notched joint also provides a safety factor against defects in the staircase, and its orientation effectively mitigates the risk of tread slippage.

Figure 4.

Axonometric view of one of the ramps of the staircase of Figure 3.

An old-new strategy to assess the equilibrium of masonry structures

There is a way of looking at the equilibrium of structures that can be traced back to Hooke’s well-known anagram on chains and the equilibrium of arches—Ut pendet continuum flexile, sic stabit contiguum rigidum inversum.⁵ In more general terms, this way can be thought of as Strut-and-Tie models.³⁶ An exemplification of the Strut-and-Tie method in the context of masonry equilibrium is the Thrust Network Analysis (TNA).^37–41 The basic idea of the method is to devise within the body of the structure a truss made of 1D elements capable of transmitting the applied external forces (usually represented by the weights of the structure and superstructure) to an appropriate foundation capable of supporting them.

According to this interpretation, the thrust polygon, which can be constructed as a possible funicular polygon of equilibrium of the interaction forces between the stair elements, can be interpreted as a 1D structure on which the external forces act, and stresses are concentrated. In the case of masonry, which is assumed to be unable to transmit tensile forces between its elements, such a structure, must be purely compressed by the loads. In the case of a masonry arch consisting of elementary ashlars, such an interpretation of the thrust line as a structure within the structure is due to Rankine,⁴² who called it the “Linear Arch.”

The equilibrium of open-well masonry staircases has already been analyzed using Linear Arch Static Analysis (LASA).^43–46 In this method a strut and tie type of approach is adopted to seek the equilibrium of the structure by inserting within the masonry one or more Linear Arches represented by generally non-planar smooth curves on which the vertical loads acting on the structure, are lumped as loads per unit length. These Linear Arches transmit the loads to the supports through compressive normal stresses, thus proving the stability of the structure based on the Safe Theorem of Limit Analysis. One of the key ideas arising from this Linear Arch static analysis approach is that the equilibrium of a spatial curve can be decomposed into two plane analyses obtained by projecting the Linear Arch onto the “planform” (the horizontal plane) and onto a vertical plane. In the case where the curve is parameterized with the arc length along the projection of the curve onto the planform, a more convenient choice for the vertical plane is represented by the development into a plane of the vertical cylinder containing the spatial curve.

The idea of working with forces projected onto the planform is similar to that proposed by Pucher for membrane analysis. Pucher’s intuition,⁴⁷ leads to an effective formulation of membrane equilibrium. Indeed, the choice of the projected stresses as the primary variables makes the surface equilibrium problem of the structure independent of the surface geometry and decoupled from the transverse equilibrium problem which it depends upon. This idea is also at the basis for the Membrane Equilibrium Analysis (MEA), formulated for masonry structures in Refs.,^48–51 and applied to the design of new structures,⁵² and masonry spiral stairs.^53–55

In the case of plane arch equilibrium, the projected stress is defined by the thrust force, and the counterpart of the continuum approach associated with the construction of the pressure curve (i.e. thrust line) is represented by the discrete approach defined by the graphical construction of the funicular polygon of the successive resultants of the internal contact forces.

Using a discrete approach, we construct a statically admissible stress state based on the identification within the structure of a purely compressed one-dimensional space polygonal structure capable of transferring the vertical forces applied to it to the supports (in the case of the stairs: the perimeter walls and the ground). The forces acting on the flying stair structure, which can be determined either analytically or graphically, are obtained here using a graphical method.

A graphical method for the assessment of equilibrium in 3D

Geometry, stresses, and loads

From a structural point of view, the step, considered as a 1D element of length $L$ whose stress characteristics are concentrated on its middle-line, is a beam-type element, capable of transmitting, within some limits, all possible tensile and compressive stresses. Here $L$ is defined as the distance between the free end and the wall (Figure 3(b)). Usually, to represent the stresses into the step, as shown in Figure 3(b), a local orthogonal Cartesian reference is adopted with origin $O^{'}$ placed at the wall surface, the $z^{'}$ axis laying along the middle-line, and the $x^{'}$ and $y^{'}$ axes arranged horizontally and vertically (downward).

The stress characteristics are the axial internal contact force $N$ directed along the z' axis, the shear forces $T_{x}$ and $T_{y}$ directed along the $x^{'}$ and $y^{'}$ axes, the bending moments $M_{x}$ and $M_{y}$ and the torque $M_{z}$ . These generalized stresses are produced by the external forces represented by the self-weight, the live load and by the reactions transmitted to the end section of the step at the wall. In addition, those determined by the interactions between the steps along the contact lines between one step and another and between the end steps and the landings have to be considered. As for the external loads acting on the step, that is, the self-weight and the live load (here assumed uniformly distributed), denoted cumulatively $Q$ , they are applied for simplicity at the centroid G of the step itself (Figure 5(a)).

Figure 5.

(a) forces transmitted to the steps at their free ends and (b) forces nomenclature and possible equilibrium regime.

The reactions transmitted from the wall to the step at the end section, which are denoted with notation $N °$ , $T_{x} °$ , $T_{y} °$ , $M_{x} °$ , $M_{y} °$ , $M_{z} °$ , depend on the actual constraint present at the end of the step and on the way the socket in the wall is made.

Usually, since the depth of the socket is a small fraction of the length $L$ (typically 1/4 of the width b³³), the constraint provided by the wall, although capable of supplying sufficient reactions of the type $N °$ , $T_{x} °$ , $T_{y} °$ , can only provide negligible values of the bending reaction $M_{x} °$ . Therefore, in the first instance, the step can be considered as simply supported at the wall end. As for $M_{y} °$ , for the same reason, it can only have negligible values, while the kind of embedment realized at the built-in end at the wall, effectively limiting the rotation about the z-axis, can supply substantial values of torque $M_{z} °$ .

It is also reasonable to assume that the contact between the step and the walls of the socket be unilateral, thus restraining any possible force $N °$ to be one of pure compression ( $N ° \geq 0)$ , while on $T_{x} °$ and $T_{y} °$ there is no limitation. However, a final check of the main stresses is advisable.⁵⁶

The action that a unilateral constraint of this type can transmit to the step, is represented by a torque $M_{z} °$ and a force of local components { $N °$ , $T_{x} °$ , $T_{y} °$ } with $N ° \geq 0$ , applied at an internal point $C$ of the step end section that can be generally different from $O^{'}$ . The translation of the force { $N °$ , $T_{x} °$ , $T_{y} °$ } from the origin to a different center of pressure $C$ can be achieved by exploiting small values of the bending moments $M_{x} °$ , $M_{y} °$ , which however are limited by the unilateral nature of the constraint.

The other reaction forces come from the contact of the step with other steps (or with the landings) along the contact line between one step and another. In some cases, this contact is punctual and occurs only at the free end. In others, it may occur at one or more points, or it may be distributed along the entire line. In the equilibrium analysis that follows, we assume that the contact occurs only at a discrete number of points located along the contact line. The mutual reaction force transmitted to the contact points can occur in all directions subject to restrictions due to the one-sided nature of the contact, the actual size and orientation of the joint, and the friction coefficient between the steps and between the steps and the wall.

Possible equilibrium scenarios for open-well stairs

We introduce and illustrate a simple graphical technique which is a geometrical version of the method put forward in Angelillo et al.⁴⁴ and Olivieri et al.⁴⁶ described in Section 3. The main idea is to identify a 1D space polygon inside the structures capable of transferring the vertical forces applied to it to the perimeter walls and to the ground. This can be achieved by introducing and using a new graphical approach.

The equilibrium problem under the action of the specified vertical loads is reduced to two plane graphical analyses obtained by projecting the one-dimensional polygonal structure onto a vertical plane, called π, and onto the horizontal plane (the planform), denoted $γ$ .

As mentioned in Section 3, open-well staircases, also called cantilevered staircases usually do not cantilever from the walls; their equilibrium depends on the interaction forces between one step and the next, which grow from top to bottom and ultimately transfer the load to the ground. Collapses of open-well staircases in which the lower ramp or only a few lower steps had been removed in the course of restoration or renovation, have been reported in Price and Rogers.⁵⁷

The approach to guarantee the equilibrium of this type of structure has been described in Refs.^27,58,59 This mechanism of stress transmission, which we refer to as Mode 1, is concisely described next.

The analysis refers to the particular case of Figure 3, and a ramp extracted from it and depicted in Figure 4.

We consider the following dimensions: $L = 1.7 m$ , $b = 0.3 m$ , $h = 0.17 m$ , to which there corresponds a horizontal length $B = 10 (0.3) = 3.0 m$ and a vertical height $H = 10 (0.17) = 1.7 m$ . Then, the slope of the staircase with respect to the horizontal is $H / B = 0.56$ while the slope with respect to the vertical is $B / H = 1.76$ , corresponding to a slope angle $β = 60.46 °$ (see Figure 5(b)).

Mode 1 We assume that the step is simply supported at the two ends to which it transfers two equal vertical forces of value $Q / 2$ by bending to the points depicted in Figure 5(b).

The equilibrium of the vertical forces transmitted to the free ends of the steps of the ramp of Figure 4 taken from the stair depicted in Figure 3, is analyzed in the vertical plane $x_{2}$ - $x_{3}$ , with reference to the diagram shown in Figure 5(b). Figure 5(a) shows graphically the forces applied to the plane structure depicted in the same figure.

The 10 triangles of Figure 5(b), representing the cross section of the steps, appear to be connected to one another by internal hinges (the dots in the same figure), the lowest step by an external hinge at the bottom edge (n° 10). In contrast, the top step is considered free of constraints at the upper right vertex (n° 0) and subject to a known (possibly zero) purely vertical force $q (0) = q °$ coming from the remaining part of the structure, as shown in Figure 5(a).

Neglecting any other type of constraint present in the step except the restriction that each step rotates about a plane orthogonal to the wall, and that restricts the rotation of the triangles of Figure 5(b) into their plane, we add 10 constraints on rotation about the axis $z^{'}$ (for this ramp parallel to the global axis x₂ of each step.

The 20 reaction components ${R_{x} (i), R_{z} (i)}$ in the hinges and the 10 torque reactions $C (i)$ , can be determined by solving the $3 x 10 = 30$ equilibrium equations in the plane of the 10 triangles. The solution can be graphically obtained as shown schematically for a generic step in Figure 5(a), by solving sequentially the equilibrium step-by-step starting from the top step.

Referring to Figure 5(b), defined $q (0) = q °$ the load present at the upper right corner of the free end of step 1 (node 0), and having set for all the others the applied loads: $q (i) = Q / 2$ , according to the scheme of Figure 5(a), at the $i$ -th node of the $i$ -th step, we obtain:

R_{x} = 0, R_{y} = \sum_{j = 1}^{i} q (j), C (i) = \sum_{j = 1}^{i} q (j) b, i = 1, 2 \dots, n

(1)

$n$ being the total number of steps that make up the ramp ( $n = 10$ in the example of Figure 5(b)). Based on this equilibrium solution, the support $n$ receives a vertical load

F = \sum_{j = 1}^{n} q (j),

(2)

while the step n sustains a net vertical load equal to $Q / 2$ and a torque

M_{t} = \sum_{j = 1}^{n} q (j) b .

(3)

Based on this equilibrium solution, we can evaluate the torque sustained by each step. A computation related to a staircase of tread $b = 0.3 m$ , and riser $h = 0.17 m$ , and a load on the step $Q = (1 + 4) kN = 5 kN$ (i.e. obtained considering the self-weight and the prescribed overload for stairs), gives the following torque for a staircase consisting of 20 steps

M_{t} = (n - 1) \frac{Q}{2} b = 19 (2.5 k N) (0.30 m) = 14.25 kN m .

(4)

Considering that for the ratio $b / h = 1.875$ , the value of the torsion constant is $α = 4.13$ , we estimate a value of τ elastic, equal to

τ = α \frac{M_{t}}{b h^{2}} = 4.13 \frac{14.25}{(0.3) {0.16}^{2}} = 7.66 MPa .

(5)

This value of $τ$ turns out to be critical for any kind of stone. This means that another way of verifying balance must be found for staircases of more than one floor in which the weight of the steps is not borne by the landings but channeled to the ground. This also shows that the torque sustained by the step increases linearly with the number of steps and becomes unbearable for stairs with a development of more than 20 steps.

A different mechanism for transferring forces to the wall and ground that does not involve the linear growth of torque with height produced by Mode 1, is Mode 2, described below.

Mode 2 This alternative way of transferring the load acting on the stair to the ground, which is essentially based on the activation of axial contact forces into the steps, requires only a minor contribution from torque reactions. This approach stems from the idea of inserting a space polygonal Linear Arch within the staircase, which carries the vertical forces transmitted to it from the steps. The Linear Arch is supported laterally by compressed horizontal connecting ideal rods (here referred to as the ideal struts) also contained within the stair.

The polygonal Linear Arch represents a kind of spatial funicular polygon whose stability is ensured by some additional compressed elements (the struts). In the conventional construction of the funicular polygon for an arch, the loads and their lines of action are specified, while the shape of the polygon, that is, the ordinates of the polygon nodes together with the thrust force, are the unknowns. In contrast, in the approach we propose here, the shape of the spatial funicular polygon representing the path of the internal forces transmitted through the steps is assigned together with the vertical forces transmitted to it by the steps, while the direction of the compressed connecting struts and the forces transmitted by them are considered as unknowns together with the thrust forces. The thrust forces, denoted $S (i)$ , $i = 1, 2, .., n$ , represent the projections onto the planform (the plane $γ$ ) of the axial generalized stresses $N (i)$ , $i = 1, 2, .., n$ , exerted along the sides (denoted $n (i)$ in Figure 7(a)) of the space polygon. The nodes of the space polygon, which are considered to be located along the interfaces between the steps (not necessarily at the internal end of the steps) are $n + 1$ in number and the sides of the polygon are $n$ in number (Figure 5(a)).

If the thrust in the first side of the polygon is given, the unknowns are then the $(n - 1)$ thrust forces $N (i)$ , $i = 2, .., n$ and the $2 (n - 1)$ reaction components $M (i)$ of the horizontal truss elements (denoted $m (i)$ in Figure 7(a)). These $3 (n - 1)$ unknowns can be determined by solving the $3 (n - 1)$ equilibrium equations at the nodes $1, 2, \dots, n - 1$ .

The construction of the Linear Arch, shown in Figure 6(a), can proceed by starting from the assignment of an appropriate polygon, called polygon 1, in the planform. This choice is arbitrary, and we can alter it (possibly through optimization) to obtain different results.

Figure 6.

(a) projection of the Linear Arch (polygon 1) and of the nodes onto the planform; (b) 3d view of the Linear Arch, nodes and its projection (polygon 1, dashed).

Specifically, the position of the nodes shown in Figure 6(a)) and the corresponding polygon has been obtained by assuming a distribution along a parabolic arc of the nodes from 0 to 10 and considering a rise of the arc in the horizontal plane $γ$ of value $4 / 5 L$ .

The specific shape of the spatial funicular polygon representing the Linear Arch that is considered here, is then generated by placing the vertices of the polygon (the nodes of the structure) at the interface between the steps. The resulting shape is shown in Figure 6(b). The sides of the Linear Arch are denoted $n (i)$ and the axial forces transmitted by them are $N (i)$ . The thrust forces, that is, the projections onto the planform $γ$ of the axial forces $N (i)$ are denoted $S (i)$ . Finally, the projections of the axial forces $N (i)$ and thrust forces $S (i)$ onto the plane π defined by the axes $x_{2}$ , $x_{3}$ are denoted by $N_{π} (i)$ and $S_{π} (i)$ . A representation of the elements $n (i - 1)$ , $m (i - 1)$ , $n (i)$ , $m (i)$ , $n (i + 1)$ , $m (i + 1)$ and of the nodes $i - 1, i, i + 1,$ relative to the $i$ -th step, is shown in Figure 7(a).

Figure 7.

The Linear Arch as a spatial funicular polygon. (a) 3d view of the detail of a step; in continuous line the “tie ro,” relative to step i, dashed the “truss element,” relative to the adjacent steps; n(i) denotes the sides of the Linear Arch, m(i) the horizontal connecting “truss element.” (b) projection of the Linear Arch into the vertical plane π (polygon 2) with the indication of the forces (of length proportional to their values) due to the weight transmitted by the step.

The vertical loads $q (i)$ applied to the nodes are assigned. Specifically, the value of the vertical load applied to node zero is:

q (0) = q ° .

(6)

As for the loads $q (i)$ for $i = 1, 2, \dots, n$ , considering an inversely proportional distribution to the distances $g (i)$ of the nodes from the wall, we assume

q (i) = \frac{L}{2 g (i)} Q .

(7)

The determination of the solution for the case at hand will be obtained graphically by referring to the values of the forces $q (i)$ represented pictorially in Figure 7(a). In the first analysis, we take $q ° = 0$ , since we assume initially that there is no load acting at the top of the stair.

The graphical solution proceeds by preliminarily addressing the equilibrium in the vertical plane, that is, on a polygon that is the projection of the Linear Arch into the plane π (parallel to the $x_{2}$ - $x_{3}$ plane) as shown in Figure 7(b). This projection, called polygon 2, is a special funicular polygon since it is formed by $n = 10$ parallel sides (Figure 7(b)). Equilibrium on polygon 2 can only be maintained by considering the presence at the nodes of horizontal forces. In this case, these forces $p (i)$ are the projection onto π of the horizontal axial contact forces $M (i)$ exerted by the truss elements (called $m (i)$ in Figure 7(a)). We define such components $M_{π} (i)$ , we can write: $p (i) = M_{π} (i)$ . These force components can be determined through nodal equilibrium via the force diagram which is a slight variation of the typical approach³⁶ used for the construction of the classical plane funicular polygon as drawn in Figure 8.

Figure 8.

Diagram of forces corresponding to polygon 2.

The approach proceeds as follows:

The forces are ordered by taking them from the funicular polygon 2 of Figure 7(b) from left to right and arranging them sequentially from top to bottom. In Figure 8 it has been assumed: $q (0) = q ° = 0$ , whilst in Figure 11 $q (0) = q ° \neq 0$ .

The last force $S_{π} (0) = q ° \tan β$ is horizontal and represents the projection on the $x_{2}$ axis of the axial contact force acting on the side $0 - 1$ of the funicular polygon: $N_{π} (0) = q ° / \cos β$ , which is assumed as prescribed at the 0 end of the Linear Arch.

The horizontal forces $p (i)$ are then obtained by drawing lines parallel to the sides of the funicular polygon 2 through the end points of the given vertical forces, and intersecting these lines with the horizontal line drawn through the bottom vertices of the given vertical forces, which is used as the starting point in the force diagram (Figure 8). The projected thrust forces (Figure 8) are

S_{π} (i) = S_{π} (0) + \sum_{j = 1}^{i} p (j) .

(8)

Once the projected thrusts $S_{π} (i)$ (and the forces $p (i)$ ) that satisfy equilibrium with the vertical forces have been determined, we can proceed to determine the forces necessary to sustain equilibrium in the horizontal plane $γ$ , by using the other plane funicular polygon representing the projection of the Linear Arch onto the planform (polygon 1, see Figure 6(a)). Indeed, once the projected thrusts $S_{π} (i)$ are known, the thrust forces $S (i)$ , representing the projections of the internal axial forces $N (i)$ exerted by the line elements $n (i)$ onto the plane $γ$ , are obtained from the known directions of the sides of the polygon drawn on the planform, as shown in Figure 9.

The so-obtained normal forces $M (i)$ acting horizontally on the nodes and transmitted by the rods $m (i)$ of Figure 6(a) to the nodes of the space Linear Arch, are taken from the force diagram of Figure 9 and graphically represented as forces applied at the nodes of Polygon 1 (by adopting a bigger scale) in Figure 10.

Figure 9.

Force diagram of the thrust forces $S (i)$ relative to the Linear Arch rods n(i) obtained by intersecting the vertical lines drawn from the projections $S_{π} (i)$ obtained earlier with concentric lines parallel to the sides of the funicular polygon 1 and passing through an arbitrary pole $P$ , located along the first vertical line on the left. The horizontal nodal forces $M (i)$ transmitted by the rods $m (i)$ to be applied to the nodes $i$ are obtained by imposing the equilibrium, that is by connecting the end vertices of the forces $S (i)$ (red arrows).

Figure 10.

Forces transmitted from rods $m (i)$ to nodes $i$ , required to sustain equilibrium with applied forces and normal stresses in rods $n (i)$ . Here, the forces transmitted by the $m (i)$ rods in the case q° = 0 are represented.

From Figure 10, one can see that, in this case, the horizontal forces transmitted by the rods $m (i)$ , do not intersect the attachment section of the step, but rather the interface between the step $i$ and the one immediately preceding it ( $i - 1$ ). The effect of these forces, which are eventually transmitted to the walls or to the upper landing, is to produce a torque into the step. Indeed, they have a line of action which is applied at the lower edge of the step and to reach the interface of the preceding step they have to be lifted by a lever arm which is the size of the rise $h$ of the step. These forces can be resolved into two components, one in the direction of the interface and the other orthogonal to it. The one parallel to the interface can run all the way to the wall. It is observed that, in this case study, this is a small traction in the upper two steps of the stair, meaning that the last part of these two steps is submitted to a tensile force. The components of the forces orthogonal to the interface, $p (i)$ , must be lifted and produce a torque (i.e. a moment of type $M_{z}$ ) of value $C (i) = p (i) h$ . Notice that this moment is positive as opposed to the sign of the torque produced by the forces in Mode 1, which is negative. All the moments transmitted to the steps must be balanced by the moment reaction $M_{z} °$ , that is $M_{z} ° = \sum_{j = i}^{i + p} C (j)$ .

In the present case the maximum moment is transmitted to the first step:

M_{z} ° (1) = C (1) + C (2) + C (3) + C (4) + C (5) = 3.08 kN m

Such a value, which is one order of magnitude less than the one produced in Mode 1, is reduced if the effect of the load coming from the landing or produced by the weight of the upper ramps is considered. This is studied next.

We repeat the analysis on the same staircase ramp (the third one) by assuming the same arch shape in projection. Here, we assume $q ° = 19 kN$ , since we consider the load coming from the upper part of the stair transmitted by the other Linear Arches, which take the load of the upper ramps, and part of the load coming from the two landings. The results summarized in Figure 11 shows how the torsional effect is reduced:

M_{z} ° (1) = C (1) + C (2) + C (3) = 2.01 kN m

Figure 11.

Third part of the stair where q° = 19 kN: (a) Funicular polygon and (b) diagram of corresponding forces, (c) projected normal forces $N_{π} (i)$ relative to the Linear Arch rods n(i), and (d) forces transmitted by the $m (i)$ rods.

The flowchart in Figure 12 provides a clear overview of the steps involved in the Mode 2 procedure. It displays a well-organized sequence of events, making the process easy to follow and understand, contributing to the transparency and reproducibility of the research.

Figure 12.

Flowchart of the Mode 2 procedure.

Conclusion

We have presented a new graphical approach for assessing the stability of open-well staircases and have applied it to the main staircase of the Palazzo delle Poste (Trapani, 1924, designed and constructed by Francesco La Grassa). This stone staircase has a rectangular plan and is composed of monolithic stone steps built into the wall.

The presented approach extends the existing 2D funicular polygon approach for assessing arch equilibrium into 3D and represents the graphical counterpart of Linear Arch Static Analysis (LASA). This graphical approach allows to graphically compute equilibrium in compression for arches and curves, which can be non-planar. The method finds robust application in the case study of masonry staircases made of monolithic blocks, giving an alternative for equilibrium to the pure torsional solution. This alternative equilibrium regime guarantees to keep the tangential stresses into the stone material composing the steps within reasonable limits when the length of the stair exceeds one coil.

A graphical approach in solving structural problems, even in today’s age of advanced numerical tools, allows the engineering practitioner, researcher or student to visually understand the effect of varying parameters in the definition of form and its effect on the membrane behavior in a shell without resorting to complex numerical modeling. This method also allows for evaluation using hand calculations without relying on computer software. This might be useful in the context of preliminary evaluation or evaluation on-site. The graphical approach offers a simple and effective way to understand the forces at play, making the structural assessment intuitive and straightforward.

The main result of the equilibrium analysis conducted is that the linear growth of the torsional moments with height, predicted by Mode 1, is eliminated using Mode 2, in which the torsional moment does not increase with height, while the internal axial forces in the connecting rods $m (i)$ and in the rods $n (i)$ of the Linear Arch increase with height.

Compression is the most favorable stress regime for masonry, but, of course, also an excess of compression must be kept under control. To this end, combining Mode 1 and Mode 2 could be of interest.

In particular, in the case study we considered, we see that for a staircase of 10 steps, there are $10 - 1 = 9$ hyperstatic reactions corresponding to 9 possible arbitrary combinations of Mode The approach can be applied to a broader range of case studies, such as elliptical, spiral or circular staircases made of monolithic blocks, and can be extended to deal with flying staircases of type B, C and D. For Type B the contribution of the railing needs to be accounted for. The Linear Arch (or a discrete number of them) can be assumed external to the steps and running through the lines of axis of the two parts. For types C and D, considering the setting and the size of the elements, several Linear Arches must be considered, and the curvature of polygon 2 can also be exploited.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research was partially funded by the ROBELARCH, a Princeton University Global Collaborative Network.

ORCID iD

Carlo Olivieri

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